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COHPUTER-AIDED DESIGN
OF EXTRUSION DIES FOR THERMOPLASTICS
A thesis
submitted in fulfilment of
the requirements for the Degree
of
Master of Engineering
at the
Department of Mechanical Engineering,
University of Canterbury,
Christchurch, New Zealand.
by
JULIUS YAO BADU
B.E. (Hons)
University of Canterbury
March 1975 - June 1976
TO MY PARENTS
"NUSIANU MIATENU AW'J LA, MI ~t\f'JE,
GAKE NUSI M~E MATENU AW'J
HADE 0 LA, MILE WOGE."
YULIUS KAISAR.
"ANYTHING WHICH IS POSSIBLE HAS
BEEN DONE, ANYTHING IMPOSSIBLE
WILL BE DONE."
GAlUS JULIUS CAESAR.
i
ACKNOWLEDGEMENTS
I wish to express my deepest gratitude to
Doctors K. Whybrew and R.J. Astley, my supervisors,
for their valuable suggestions and assistance throughout
this work and for showing great interest in the research.
I am very much indebted to the Staff of the
Computer Terminus (University of Canterbury) for their
help during the course of this work.
An inestimable debt is owed to several colleagues
who, by personal kindness, have cheerfully shared with me
their wisdom. Thanks are also due to my friends who
helped me in several ways and who inspired me to complete
this work.
The author wishes to express gratitude to
Helen Oteng for her motherly care and constant
encouragement given to me during the course of study.
My sincere appreciation goes also to Mrs A.J. Dellow for
her expert typing of the manuscript.
June 1976.
J. BADU
B.E. (Hons).
ii
iii
SUMMARY
This report presents a survey of the literature on
the design of extrusion dies for thermoplastics. The need
for scientific die design and the major problem areas are
discussed.
In the literature survey, the major problems
isolated were:
1. Material Properties
2. Die Flow Analysis
3. Flow Instability and Melt Fracture
4. Melt Elasticity.
Attention was then confined to Die Flow Analysis.
The finite element method was proposed for solving two
dimensional flow problems in complex geometrical
configurations commonly encountered in polymer extrusions.
The two finite element methods adopted used the
variational principle in the formulation of the problem.
Also, different variational functionals and nodal variables
were used.
The results of the two methods were successfully
compared with finite difference, analytical and existing
numerical solutions.
The flexibility of the finite element methods makes
them very suitable for problems involving complex boundary
geometries. It is shown that the finite element method has
great potential for use in flow problems, and represents a
powerful new tool for the analysis of viscous flows.
QUOTATION
ACKNOWLEDGEMENTS
SUMMARY
TABLE OF CONTENTS
LIST OF FIGURES
LIST OF TABLES
NOMENCLATURE
ABBREVIATIONS
INTRODUCTION
CHAPTER
TABLE OF CONTENTS
SECTION 1 (LITERATURE SURVEY)
1 INTRODUCTION
iv
Page
i
ii
iii
iv
vii
ix
xi
xiii
xiv
3
(a) Description of the extrusion process 3
2
3
4
5
(b) Regular practice in die design 4
PROBLEMS OF EXTRUSION DIE DESIGN FOR THERMOPLASTICS
MATERIAL PROPERTIES
Introduction
(a) Non-Newtonian fluids
(b) Flow behaviour of thermoplastics
DIE FLOW ANALYSIS
Introduction
(a) Entrance effects
(b) Flow channel design
(c) The effect of die design on the quality of products
FLOW INSTABILITY AND MELT FRACTURE
(a) Die entry effect
(b) The concept of wall slippage
(c) Die exit effect
6
8
8
9
1 1
15
15
16
17
22
23
23
25
26
CHAPTER
6
7
8
9
1 0
1 1
12
v
Page
(d) Behaviour of linear polyethylene (L.P.) and branched polyethylene (B.P.) 28
(e) What is to be done? 29
(f) What can be done? 29
MELT ELASTICITY 31
(a) Cause of die swell 31
(b) Various studies carried out on die swell 32
(c) What can be done to reduce die swell? 34
CONCLUSION AND RECOMMENDATIONS
SECTION 2 (ANALYSIS OF FLUID FLOW)
BASIC THEORY
(a) Introduction
(b) Equations governing flow
STREAM FUNCTION ANALYSIS
(a) Introduction
(b) Finite element formulation of the problem
(c) Problems in this analysis
ANALYSIS USING VELOCITIES AND PRESSURE
(a) Introduction
(b) Finite element formulation of the problem
(c) Advantages and disadvantages of this analysis
RESULTS AND DISCUSSION
(a) List of results
(b) Discussion
GENERAL DISCUSSION
38
45
45
45
49
49
49
54
59
59
59
65
66
67
86
96
(a) Constitutive ~quations 96
(b) Problems encountered in the solution of the "stream function analysis" 98
(c) Problems encountered in the solution of the "velocities and pressure analysis" 101
(d) Comparison of both analyses 106
CHAPTER
1 3
REFERENCES
APPENDICES
A1
A2
A3
A4
AS
A6
B1
c
D
P1
P2
CONCLUSION AND RECOMMENDATIONS
(a) Conclusion
(b) Recommendations
Dimensional analysis
Finite element analysis of fluid flow within a converging channel using w and ¢
Dimensionless boundary conditions (W and ¢)
Evaluation of u (the x-component velocity) of an element
Pressure drop (6p) across the channel
Theoretical velocities and viscosities
Finite element analysis of fluid flow using u, v and p
Part of the initial computer programme
Estimation of costs involved in the research work
Resulting computer programme from the first analysis
Resulting computer programme from the second analysis
vi
Page
108
108
1 1 2
1 1 5
120
123
129
130
132
135
1 38
145
153
154
184
vii
LIST OF FIGURES
FIGURE Page
1 Elements of an extruder 2
2 General flow curve for a pseudoplastic fluid 10
3 Obstruction introduced by a spider leg 18
4 A sketch of recoverable shear strain versus shear stress for "Styron" 24
Sa Shear stress versus apparent strain rate for L.P. 27
Sb Shear stress versus apparent strain rate for B.P. 27
6a The shape of the required die by the compression method 35
6b Die profile 37
7 Converging channel flow 44
8 Triangular element showing nodal points 48
9 Mesh used in the analysis 50
10 Initial mesh 57
11 Typical finite element 61
12 Velocity profile for Newtonian flow in a parallel channel 88
13 Velocity profile for non-Newtonian (n = 0. 5) flow in a parallel channel 89
14 Dimensionless pressure drop versus taper angle 92
15 Initial layout of the system matrix equation 97
16 A portion of the mesh used 99
17 Unfavourable way of labelling the nodal variables 102
18 Favourable way of labelling the nodal variables 103
FIGURE
1 9
20
21
22
Channel boundary division for the forward difference formula
Channel boundary division for the central difference formula
Channel boundary division for the backward difference formula
Flow region showing the nodal points along the bottom boundary
viii
Page
131
131
131
133
TABLE
1
2
3
4
5
6
7
8
9
1 0
11
12
LIST OF TABLES
Number of cycles of iteration for non-Newtonian (n = 0. 5) parallel channel flow using a (9X9) mesh
Dimensionless flowrate for parallel channel flow for a Newtonian fluid
Dimensionless flowrate for parallel channel flow for non-Newtonian fluid (n = 0.5)
Dimensionless flowrate for converging channel flow at x = 0. 5 for a Newtonian fluid
Dimensionless flowrate for converging channel flow at x = 0. 5 for a non-Newtonian (n = 0. 5) fluid
Dimensionless velocities for parallel channel flow (first analysis) for a Newtonian fluid
Dimensionless velocities for parallel channel flow (second analysis) for a Newtonian fluid
Dimensionless velocities for parallel channel flow (first analysis) for a non-Newtonian (n = 0. 5) fluid
Dimensionless velocities for parallel channel flow (second analysis) for a non-Newtonian (n = 0. 5) fluid
Dimensionless pressure drops for converging channel flows (first analysis) for a Newtonian fluid
Dimensionless pressure drops for converging channel flows (first analysis) for a non-Newtonian (n = 0. 5) fluid
Dimensionless pressure values for flow in parallel channel (second analysis) for a Newtonian fluid
ix
Page
68
70
70
71
71
73
74
75
76
78
78
79
TABLE
1 3
1 4
15
16
Dimensionless pressure values for flow in parallel channel (second analysis) for a non-Newtonian fluid
Dimensionless nodal viscosities for parallel channel flow using a (9x9) mesh for a non-Newtonian (n = 0. 5) fluid
Dimensionless nodal viscosities for parallel channel flow at x = 0. 5 (first analysis) for a non-Newtonian fluid (n = 0.5)
Cost and storage requirements for the computer programmes resulting from the analyses
X
Page
80
82
83
85
xi
NOMENCLATURE
Symbol Definition
a, b local nodal point coordinates.
A area.
c1 to c 6 constants in stream function distribution.
F
H
H1
I2
i, j '
[K]
[Ke]
L
m
n
k
viscous dissipation functional.
depth at any section along the axial length.
channel depth at inlet.
second invariant of rate of deformation.
subscripts referencing nodal points.
system viscous stiffness matrix.
elemental viscous stiffness matrix.
length of converging channel.
subscript referencing element number.
flow behaviour index of a polymer; power-law index.
N shape functions. with subscripts
P pressure.
6P overall pressure drop.
Q volumetric flowrate.
u velocity in the x direction
u' dimensionless value of u.
v velocity in the y direction; volume of an element.
v' dimensionless value of v.
vc characteristic velocity.
w width of flow normal to x-y plane.
x, y Cartesian coordinates.
X', Y' dimensionless Cartesian coordinates.
Greek letters.
]1
Tw1' th b · t su scr1p s
cp'
X
l)J'
channel taper angle.
constants in the velocity distribution.
reference shear rate.
characteristic shear rate.
elemental area.
viscosity.
effective viscosity at y . 0
cha~acteristic viscosity.
viscous stress.
characteristic viscous stress.
sum of velocity components u + v.
dimensionless ¢ •
integral of functional F.
stream function.
dimensionless stream function.
overrelaxation factor.
optimum overrelaxation factor.
dimensionless pressure drop
dimensionless flowrate.
xii
xiii
ABBREVIATIONS
The following abbreviations were used in this work.
FD Finite Difference
FE Finite Element
FEM Finite Element Method
LA Lubrication Approximation
LP Linear Polyethylene
BP Branched Polyethylene
xiv
INTRODUCTION
Many flow problems in polymer processing are regarded
as difficult to solve without gross over-simplifications or
excessive labour. The two main sources of difficulty are
the intricate geometrical configurations which are
encountered and the complex rheological properties of
polymer melts. Consequently one finds that theoretical
treatments are often restricted to simple, sometimes
unrealistic, geometries and Newtonian fluids.
Industrial processes involving viscous fluids are
invariably associated with geometries of a complex nature;
for example, mixing vessels, screw extruders and extrusion
heads. Herein lies the potential of the finite element
method as a design procedure, because it is not limited to
geometries associated with the major coordinate systems
which, in practical terms, tends to be the case with other
methods. Furthermore, because of the way modifications to
the geometry can be effected the potential also exists for
"Computer-aided Design" in its entirety.
The present study has been restricted to the analysis
of two-dimensional, creeping, non-Newtonian flows using the
finite element method.
Finite element method approximates the flow by
dividing the flow region into small subregions or elements
and analysing the flow in terms of, say, velocities at the
corners of these elements.
XV
The FE methods are well established in the fields of
structural and solid mechanics (Zienckiewicz (60)), but
have not been widely used in solving fluid mechaniQS and
heat transfer problems. General examples include Martin
(25) and Oden and Somogyi (13). A general FE formulation
for fluid flow problems has been developed by Oden (35).
Zienkiewicz (60) has also described a FE formulation of
two-dimensional flow problems in terms of u, v and p.
Various authors (Oden (35), Martin (25) and Card (12)) used
the variational principle and demonstrated the FE analysis
of two-dimensional fluid flow. Atkinson et al. (1,3) applied FE
methods to two-dimensional Newtonian flow problems. Palit
and Fenner (37) also applied the FE method to two-dimensional
slow non-Newtonian flows.
The object of this work is to compare the type of
analysis used by Palit and Fenner (37) with the results
using a new type of analysis; a different variational
functional and nodal variables.
The present work contains two major sections:
Section 1 contains the literature survey. It
critically reviews the published literature on extrusion
die design for thermoplastics and lists the major problem
areas.
Section 2 presents the analysis of the flow of a
Newtonian and non-Newtonian fluid within a two-dimensional
channel. Two FE methods were adopted and the differences
would be pointed out in the course of this work.
1
SECTION 1
LITERATURE SURVEY
2
Fig. 1 Elements of an extruder.
CHAPTER 1
INTRODUCTION
Little has been written about the detailed design
of extrusion dies and it is unlikely that this state will
change appreciably for some years to come. The die maker
is a craftsman who depends on personal experience to an
extent which is uncommon in modern industry. The mystique
which appears to surround the craft is increased by the
reluctance of companies to talk about their success - the
solution to the problems of a difficult die may give a
distinct advantage over a competitor.
(a) DESCRIPTION OF THE EXTRUSION PROCESS
Fundamentally, the process of extrusion consists
of converting a suitable raw material into a product of
specific cross-section by forcing the material through
an orifice or die under controlled conditions. This
definition, as applied to the extrusion of thermoplastic
materials, covers two general processes - screw extrusion
and ram extrusion.
Thermoplastics are extruded predominantly through
screw extruders; more specifically, through single-screw
extruders. The elements of a typical single-screw extruder
are shown in Fig. 1.
3
4
The plastic material is fed from a hopper through the
feed throat into the channel of the screw. The screw
rotates in a barrel which has a hardened liner. The screw
is driven by a motor through a gear reducer, and the
rearward thrust of the screw is absorbed by a thrust bearing.
Heat is applied to the barrel from external heaters, and the
temperature is measured by thermocouples. As the plastic
granules are conveyed along the screw channel, they are
melted. The melt is forced through a breaker plate which,
in some cases, supports a screenpack. The melt then flows
through the adapter and through the die, where it is given
the required form.
When the thermoplastic material leaves the extrusion
die it is usually in the form of a melt or a very soft mass.
It has often little decisive form at this stage and even
this would rapidly be lost unless it were handled in a
suitable manner as it leaves the die. The product from the
extrusion die therefore must, in a large number of cases,
be looked upon as a semi-finished raw material which may be
given its correct form and dimensions by subsequent
processing whilst it is still mouldable. The methods of
doing this naturally depend on the thermoplastic being used
and on the desired product. This subject will not be
discussed here. The reader is referred to a suitable work
on this topic.
(b) REGULAR PRACTICE IN DIE DESIGN
In most extrusion shops, it is still a regular
practice to shape the die cross-section as judged
appropriate from previous experience and to proceed from
there by trial and error. The die is then installed, and
5
a run is begun. Samples of the chilled extrudate are taken,
and the dimensions are checked. Where the thickness is
excessive, the corresponding points along the die lips are
peened with hammer and punch while the extruder is running,
so as to close them slightly and reduce the section
thickness. It is hoped that some day plastic flow would
be understood well enough so that the crude method of
peening extrusion dies would be eschewed by die makers
forever.
The work presented here reviews the present state of
scientific die design and investigates methods of analysing
flow in extrusion dies.
CHAPTER 2
PROBLEMS OF EXTRUSION DIE DESIGN FOR THERMOPLASTICS
INTRODUCTION
The rheologist makes measurements of viscosity and
elasticity under carefully defined steady flow conditions;
the practical processor commonly works in less ideal
circumstances. An example of this contrast is extrusion,
where the rheologist measures the flow through a lon~
cylindrical die, but the practical processor operates with
relatively short, tapered dies, often of slit or other
profile.
6
A typical tubular extrusion die of 150mm diameter
costs about a thousand dollars to make. At this price the
processor has a problem which is expensive to solve by trial
and error, and he legitimately turns to the rheologist for
assistance. The processor needs to know what will be the
swell ratio as the melt leaves the die and what will be the
pressure drop through this die. He also needs to know at
what throughput rate non-laminar flow will occur. The ideal
die will maximise output rate of smooth extrudate and
minimise pressure drop and swell ratio, and such
optimisation commonly requires an accurate choice of taper
for the converging flow regions in the die.
A survey of all available literature on die design
was carried out to ascertain the various problems. The
following major problem areas were isolated.
These are:
(1) Material Properties.
(2) Die Flow Analysis.
(3) Flow Instability and Melt Fracture.
(4) Die Swell.
The above areas are further discussed in the next
four chapters.
7
CHAPTER 3
MATERIAL PROPERTIES
INTRODUCTION
Ideally, during extrusion, different screws are
required for each material because of the change in
extrusion viscosity values from one material to another.
The same requirement applies to die design, although
the necessity is, perhaps, not so stringent. A change
in material viscosity brings about a corresponding
change in flow properties, with the result that a die
arrangement suitable for a polyamide, for example, would
be completely inadequate for an unplasticised vinyl
material. In general, the higher the extrusion viscosity
of a material the greater the necessity for streamlining
the interior of the die.
The dependence of die design on material
characteristics need not be overemphasised. This part
of the work therefore briefly looks at non-Newtonian
fluids, flow behaviour and properties of thermoplastics.
The rheological equation describing thermoplastics and
its limitations are also discussed.
8
(a) NON-NEv'iTTONIAN FLUIDS
Classification
The Newtonian viscosity, ~, depends only on
temperature and pressure and is independent of the rate
of shear. The diagram relating shear stress and shear
rate for Newtonian fluids is the so-called "flow curve",
and is therefore a straight line of slope~.
Non-Newtonian fluids are those for which the flow
curve is not linear. The viscosity is not constant at
a given temperature and pressure, but depends on other
factors like shear rate, the apparatus in which the
fluid is contained, or even on the previous history of
the fluid.
These fluids can be classified into three broad
types:
(1) Fluids for which the shear rate is a function of
the shear stress only.
(2) Those for which the relation between shear rate
and shear stress depends on the time the fluid has
been sheared or on its previous history.
(3) Systems which have the characteristics of both
solids and liquids and show partial elastic recovery
after deformation - viscoelastic fluids.
Viscoelasticity results in stresses normal to
the dir~ction of shear stress. When viscoelastic jets
emerge from capillaries, the normal stresses result in
an expansion of the jet as opposed to the usual vena
contracta in Newtonian fluids. This is the Barus effect.
9
Shear stress
Fig. 2
pseudoplastic •n T = ky
k = constant
n = flow behaviour index< 1.
Strain rate
General flow curve for a pseudoplastic fluid.
___,.
0
11
Rheological Equation and Limitations
In this work, attention is confined to pseudoplastic
fluids which are described by the equation
·n T = ky
where k is a constant and n (less than one) is the flow
behaviour index.
A typical flow curve for this fluid is as shown
in Fig. 2. It can be seen that n is not constant over
the whole range of the shear rate. This is not a serious
drawback because all that is required is the value of n
which describes the flow over the particular range
encountered in a particular problem. The chief limitation
of this equation is its inability to portray correctly
flow behaviour at shear rates in which the fluid is
approaching Newtonian behaviour. Thus, extrapolating of
data taken over a modest range of shear rates from the
non-Newtonian into Newtonian region may result in
appreciable errors.
Unfortunately there is no ready solution to this
problem and hence one must obtain rheological data at shear
rates corresponding to those used in process.
(b) FLmv BEHAVIOUR OF THEID10PLASTICS
Introduction
Flow through dies, tubes and viscometers all
represent steady-state processes. Thus steady-state
behaviour represents a convenient starting point from which
to analyse flow behaviour. Then time-dependent effects may
be superimposed in the form of simple additions.
The non-Newtonian behaviour of polymer melts in
steady flow is attributed to two characteristics of the
molecular structure of these materials.
1 2
(1) The asymmetric shape (great length as compared to the
radial dimensions) of the molecules themselves results in
an orientation of the particles when a velocity gradient is
imposed on the polymer molecules.
(2) The size of the flowing elements - if they are in
groups of molecules rather than single particles - is
decreased as the shear rate is increased. The restoring
tendencies are the intermolecular forces.
Solid or fluid-like behaviour
Polymeric materials may exhibit either solid-like
or fluid-like behaviour. The kind of response is determined
by the Deborah number.
Deborah number is defined as the ratio of natural
time to duration time of a process (or the residence time
of the material in the process). Natural time is the time
required for the relaxation moduli of the material to decay.
Due to the large forces applied during processing of
polymers the residence times are very short. Analyses of
flow of polymers neglect these short residence times and
consider the deforming polymer to be a purely viscous fluid.
Obviously this is not a generally valid approach.
It has been shown by Netzner et al. (29, 30, 42) that
the behaviour of polymer materials under short residence
times is not at all well proximated by fluid-like behaviour
and that solid-like behaviour may instead be exhibited under
13
conditions of sudden acceleration of the fluid.
The importance of memory of a fluid element for its
previous deformation history and the possibility of either
a solid-like or fluid-like behaviour of a given material
may be illustrated clearly through consideration of the
behaviour of polymeric melts in short dies.
Deborah no. as we recall is defined as:
duration of fluid memory duration of deforming process
Thus, large resident times and small Deborah nos. imply
fluid-like behaviour, while small residence times and large
Deborah nos. imply solid-like behaviour.
If the span of the fluid memory exceeds the residence
time in the die, i.e. large Deborah no., the emerging
extrudate is able to recall its state prior to entry.
Otherwise the extrudate is able to recall only the flow
conditions in the die. The behaviour of the fluid on
emergence from the die should reflect such differences in
memory.
Swelling of extrudates is known to decrease
approximately exponentially to a constant value as the die
length is increased. In a sense therefore, the tendency of
the extrudate to return to a state representing its original
configuration before entering the die increases as the
Deborah no. increases.
The entry to a die is subjected to rapidly changing
deformation rates and the fluid should exhibit solid-like
behaviour because the Deborah no. is high. Metzner et aZ.
(31) reported that the instability which occurs is due to
the high stresses imposed being relieved.
Any changes which decrease the Deborah no. in
the entry region, such as the use of conical rather than
sharp-edged entry, should therefore result in postponement
of the instability to higher flowrates.
Other characteristics
1 4
One of the major problems encountered in extrusion
die design and the study of plast flow is the specification
of the flow properties of commonly extruded thermoplastics.
For generality, simplifications are made in the
representation of the flow properties.
Two chief, readily observable characteristics of
thermoplastics are:
(1) A decrease in apparent viscosity with increasing
shear rate, and
(2) A capacity for elastic recovery when the stress is
suddenly removed.
These two are strongly dependent on the material, the
temperature and in some cases on the previous shear history
of the melt.
Due to the very high coefficients of bulk elasticity
of polymers, incompressibility can also be assumed in the
study and the analyses of their flows.
15
CHAPTER 4
DIE FLOW ANALYSIS
INTRODUCTION
Published work so far on die flow analysis presents
results as either algebraic formulae or numerical solutions
obtained from finite difference computations.
The recently developed finite element method, an
alternative numerical technique, has been applied to
non-Newtonian flow and die design analysis.
This method approximates the flow behaviour by
dividing the flow region into small (triangular) subregions
or elements and analysing flow in terms of, say, velocities
at the corners of these elements.
The principal advantage is the ease of application
to complicated geometries. Analysis would enable us to
predict pressure drop, distribution of flow in dies, etc.
In wire-coating dies, analysis can be done to compute the
relationship between pressure drop and flowrate. Useful
predictions can be made for the wire tension and shear
stresses which are important when the strength of wire
and likelihood of melt fracture are considered.
The following discusses briefly, entrance effects,
flow channel design, dead spots, flow past obstructions
and how die design affects the quality of a product.
1 6
(a) ENTRANCE EFFECTS
When a polymer enters a capillary, a drop in the
available pressure occurs. This drop is referred to as the
entrance loss and results in there being less pressure
available to drive the polymer through the capillary. The
same effect is observed in the flow within a die.
The original available pressure, p, is given by:
p =
where 6p is the entrance drop and 6p is the actual e cap
pressure loss within the capillary.
The entrance loss is most usually explained in terms
of the energy needed to suddenly change the stream from a
fat slow stream to a thin fast stream with stretched
molecules. Calculations have shown that some of this loss
is used up in forcing the melt to flow through the wide
chamber upstream of the orifice itself.
Various suggestions have been put forward as to how
to determine the entrance drop. These are given in
references (45, 46 and 57). Bagley (4) has shown a method
of determining the entrance drop which is applicable to all
thermoplastics. He proposed that the entrance loss be
expressed as an effective length in orifice diameters.
This was found to be between 2 and 3 diameters. To correct
for entrance loss, this is added to the actual tube length
to give a fictitious effective length. Bagley (4, 45) also
gave a method based on end correction curves which may be
used to apply an approximate correction to flow curves for
which no entrance corrections have been made.
(b) FLOW CHANNEL DESIGN
Given the dimensions and required output rate of an
extruded product, one of the tasks of the die designer is
to calculate the dimensions of the die at the exit and
define an appropriate flow channel in the body of the die
17
in order to achieve an acceptable pressure drop and minimise
the likelihood of flow defects.
The internal shape of the die adaptor (the part of
the die body which precedes the land) is designed to allow
the material to flow easily towards the smaller cross
sectional area of the die.
Dead space and consequences
When a melt encounters an abrupt change in section
during its flow the material conforms to a natural angle of
streamline flow in the region in front of the die entry.
This pattern of flow results in the presence of dead spots.
Materials which stagnate in the die are subjected
to a change in viscosity. This alters the flow properties
of that part of the polymer, and polymer degradation is
liable to occur if the temperature is high enough. There
is therefore a variation in the quality of the extrudate.
This could, however, be avoided by reducing the half
angle of entry to the die from 90° to the natural half angle
given by:
tan a = 12n/A quoting Powell (44)
where n = the apparent viscosity corresponding to the shear
rate (or shear stress) at the die entry; and A = tensile
viscosity corresponding to the tensile rate at the die entry,
I .,. Flow line
around web
Fig. 3
V///) ........... "-..!' .~
spiqer leg
> • 0 • • 0 + +
.r I . I weld line
highly sheared polymer
Obstruction introduced by a spider leg.
• • • • shearing gradually
disappearing
1-' co
which in approximate calculations may be taken at the
critical tensile rate or tensile stress.
The convergent angle for high viscosity polymer,
e.g. UPVC, should not exceed 60° and for a low viscosity
material like nylon, the angle can be relatively obtuse.
High viscosity materials tend to stagnate at the walls
while flow takes place through the centre.
Flow past obstructions
19
With some materials of high melt viscosity it is
difficult to achieve proper homogeneity after the melt has
been broken by an obstruction like a spider leg.
Obstructions by spider legs and breaker plates often result
in weld line formation.
Weld lines are liable to occur whenever two flow
fronts meet. They are formed when the flow during extrusion
becomes divided due to an obstruction and rejoins again.
A digrammatic representation is as shown in Fig. 3.
The flow lines part around the webs and there is a
region of high shear adjacent to them. Because polymer
which has flowed around the web is sheared to a greater
extent than the polymer on either side of it the die is
presented with a non-uniform polymer melt. In film or
extruded pipe this lack of uniformity shows up as visible
lines known as weld lines or memory lines. These lines
cause local extrudate weakness.
Weld lines arise from the fact that a molten polymer
takes some time to relax to a normal, unoriented state after
being sheared. In view of this, rigid materials require
greater compaction to erase weld lines.
The following ways have been suggested as to how to
eliminate the non-uniformity introduced.
(1) The web should be placed some way from the die so as
to give the polymer a long time in which to relax
after being parted.
20
(2) Some degree of mixing of the melt should be encouraged
after it had rejoined on the die side of the web.
(3) An adequate hydrostatic pressure should be maintained.
(4) For the tube dies another suggestion is to utilise an
off-set die in which the melt flows round the cone
from a side entry.
The first suggestion is proven in practice as one
way of achieving the second by which mixing is encouraged
by designing spiral or overlapping flow channels in the
die body. A theoretical solution to the second suggestion
is to place a constriction in the flow channel between the
web and the die itself so as to create a region of mildly
distorted or turbulent flow.
The problem that arises in practice is that of
estimating the effect of obstructions placed in the channel
of flow, or of asymmetries at the entrances or the exits.
Pearson (39) reported that where the index of power
law dependence is n, the effect of the obstruction is n
times as strong as for a Newtonian fluid (n = 1). In other
words, the entry or exit length required to smooth out the
effects of obstructions in a channel of uniform depth is
proportional to n.
Using a slender-body approximation, Pearson (39)
designed a spider leg. This resulted in a thin obstruction
21
aligned in the flow direction without producing any
disturbance to the flow.
Channel Design
The re~istance to flow in the shaping zone is higher
than in the feed channels. Nevertheless the feed channels
may have considerable influence on the flow. Their lengths
may be different for the separate sections of the orifice,
and accordingly the cross-sections along the lines of flow
have to be wide where the path is shorter, so that the
resistance to flow in them shall be identical.
The correct choice of flow passages from extruder to
die-lips is the essential prerequisite of a successful die.
Clearly the general topological configuration of these
passages will be governed by the basic type of die required,
but their detailed shape must be determined by reference to
the detailed flow behaviour of the materials to be extruded.
Pearson (40) developed an analytical process to
predict channel geometry required to give a specified output.
The choice of die channel length is determined only by the
duration of the relaxation process, which in turn depends
on the rate of deformation (the output from the extruder)
and the temperature of the melt in the die.
Blymental et al. ( 11 b) reported that the channel
length, L, is given by;
L h
0 = .1_ K1/V
6
where K and V are constants dependent on the temperature
and Tp is the relaxation time of stress.
.
The same reference reported
T p = •-v Ky
where y is the rate of shear.
"T II
p as being given by;
(c) THE EFFECT OF DIE DESIGN ON THE QUALITY OF PRODUCTS
It would in many ways be convenient if during
extrusion the polymer remained essentially unaltered,
but this is not so. It is well known that polymers may
be modified ,through being "worked" (i.e., subjected to
high shear stresses in the melt) and that, furthermore,
the effect of exposure to high temperatures during
fabrication processes may be to modify yet further the
22
structure of the polymer. For a given extruder, the degree
of working achieved on the screw is affected by the die
design, as this largely controls the back-pressure.
Alterations to die design may thus affect the quality of
the extrudate in two ways:
(1) By providing a different flow path for the polymer
and indirectly,
(2} By modifying the polymer itself through alterations
in the degree of shear experienced on the screw.
Quality of a product may be defined, among others,
in terms of uniformity of flow. In order to achieve a
uniform rate of melt flow to all points of a die, lip
analysis of the complex flow behaviour is required.
Pearson (39) showed that it is possible to obtain uniform
output by using a narrow-channel approximation. Pearson
(40) also developed a numerical technique of designing
cross-head dies to achieve uniform output.
CHAPTER 5
FLOW INSTABILITY AND t-1ELT FRACTURE
One of the primary criteria of extrudate quality
is the presence or absence of extrudate roughness or melt
fracture.
23
It is widely observed that during extrusion a
breakdown from uniform to irregular flow takes place above
a certain critical flowrate. Below this rate the extrudate
is smooth, but above it the extruded polymer is rough,
twisted and distorted. This phenomena sets the upper limit
for polymer processing and is known as melt fracture.
No unanimous opinion has yet been reached to account
for the onset of melt fracture. Some authors (Tordella
et aZ. (53, 5, 7, 14)) favour a die entry effect; Spencer
·et aZ. (50) are in favour of die exit effect, while others
support the concept of slip at the walls.
(a) DIE ENTRY EFFECT
It has been acknowledged widely by Tordella et aZ.
(53, 7, 47) that melt fracture is a result of a tearing
mechanism occurring at the die entry. However, evidence
exists which favours the fact that under certain conditions
melt fracture may be induced in the die land. A study by
Han et aZ. (20) gave a clear indication that extrudate
Recoverable shear strain
Fig. 4
..,_____ . 1 point · ~t1ca
shear stress
A sketch of recoverable shear strain versus shear stress for "Styron".
N ~
25
distortion is attributable to irregularities in the flowrate
at the tube wall.
Change of Properties
A sudden change of elastic properties was reported at
the critical point by Han et al. (20). A sketch of such a
change for 'Styron' is as shown in Fig. 4.
Above the critical point, the storage capacity
(recoverable shear strain) suddenly increased with shear
stress. This increase in the amount of stored energy is
released at the tube exit and may well go to increase
the severity of the extrudate distortions.
The "entry region" theory was also supported by
Howells and Benbow (21) who reported the cause to be due
to the failure of the polymer melt to sustain high tensile
stresses arising in the entry region. The breakdown of the
network which occurs causes the noticeable backflow or
recoil near the entry region.
Schulken and Boy (48) pointed out that shear rate
and shear acceleration at the inlet may be the cause of melt
fracture. Elastic oscillations, as a result of a sudden
change in velocity at the entry region, are no longer damped
by the viscosity of the medium when the critical point has
been passed.
(b) THE CONCEPT OF WALL SLIPPAGE
Another explanation so far given for flow instability
is the concept of slip within the capillary walls.
Benbow and Lamb (8) showed that local slipping causes
bulk flow effects throughout the whole volume of the melt.
Slipping at the wall, as reported by Tordella et aZ. (54, 8)
is suggested to be the initiator of flow instability.
Lupton (24) and Maxwell and Galt (26) suggested that
boundary slip is a relevant and perhaps a determining
factor in the breakdown phenomenon.
26
Several authors (Yun et al. (58, 18, 9)) have indicated
that wall slippage occurs when high-density polyethylene
melts are extruded. There are some serious grounds to
consider local slip of polymer systems relative to the
capillary wall as one of the causes of melt fracture at
high shear stresses. Nevertheless, melt fracture is
sometimes detected for branched polyethylene, propylene
and polystyrene in the absence of wall slippage.
Although slip flow is the dominant mode of
transport during melt fracture, several authors favour
the idea that slip is an effect rather than a cause of
melt fracture.
(c) DIE EXIT EFFECTS
There is not much evidence in the literature
in favour of flow instability being associated with
die exit effects. Spencer (50) explains, however, that
due to the high velocity gradient at the walls, there is
a very large amount of elastic energy stored and a core
of a relatively small amount of elastic energy. The
system is unstable and instability takes place at the
exit.
27
Shear strain
hysteresis loop
Apparent strain rate
Fig. Sa: Shear stress versus apparent strain r.ate for L.P.
Shear stress
melt fracture
Apparent strain rate
Fig. Sb: Shear stress versus apparent strain rate for B.P.
(d) BEHAVIOUR OF LINEAR POLYETHYLENE (L.P.)
AND BRANCHED POLYETHYLENE (B.P.)
It was observed and reported by Bagley et aZ. (6)
that discontinuity occurred in the flow curve for linear
polyethylene (L.P.). With reference to Fig. Sa, a sudden
increase in the flowrate without any increase in
differential pressure was noted at a critical pressure
value. On decreasing the shear stress from an unstable
flow condition, discontinuity occurred at a lower shear
stress than on increasing it from a stable condition.
Consequently a hysteresis loop is formed.
28
Branched polyethylene (B.P.) exhibited melt fracture
without any discontinuity in the flow curve as evident in
Fig. 5b.
Flow visualisation studies on L.P. and B.P. have
shown that, at the die entry, B.P. shows very large
circulatory dead space while for L.P. the dead space is
extremely small.
Other studies also show that flow disturbance
occurred at the inlet region for B.P., and that for L.P.
the breakdown of the flow occurred within the capillary.
General conclusions of other workers can be
summarised as below:
(1) Melt elasticity is responsible for flow instability.
(2) Extrudate distortion is primarily due to uneven,
elastic strain recovery, and
(3) Instability initiates at the capillary inlet for B.P.
and within the capillary at the wall for L.P.
29
An important result established for L.P. is that melt slip
at or near the wall is the cause of flow curve discontinuity
occurring at the onset of unstable flow.
(e) WHAT IS TO BE DONE?
In order to understand the phenomenon of slip(
it is necessary to resolve whether it occurs at the
melt-die wall interface or between layers of melt. Quite
a large number of major problems regarding capillary flow
instability are still left unresolved. In particular, the
rheological differences between linear and branched
polyethylene which result in the different ~nstability
sites and flow patterns within the reservoir feeding the
capillary are unexplained. Additionally, the precise
nature of the flow breakdown must be elucidated, although
the slip process which occurs within the capillary for L.P.
lends credence to the fracture mechanism proposed by
Tordella (53).
(f) WHAT CAN BE DONE?
It has been fairly well established that melt
fracture occurs only at and above a certain critical shear
rate. By changing certain variables a quality product can
still be obtained by extruding even above this value.
One such variable is viscosity which can be lowered
by increasing the temperature or using a resin of higher
melt index. Higher temperatures lessen the resistance to
flow and result in high shear rate without melt fracture
30
occurring. Since resistance to flow is lessened, the
pressure is lowered and the shear stress is increased.
The use of a resin that is less viscous or a higher melt
index has the same effect as raising the temperature.
Reducing the angle of entry and increasing the die parallel
length can also result in extrudates of excellent quality
at high output rates.
CHAPTER 6
MELT ELASTICITY
(a) CAUSE OF DIE SWELL
Die swelling is the most obvious measure of the
elasticity of a polymer. It is caused by reversible
elastic deformations developing in a polymer melt while
it is flowing in the die.
31
An extrudate, on issuing from a die, is noted to
have a cross-section much larger than the hole from which
it emerges. The main cause of which is the viscoelasticity
of the resin.
Stresses developed in the melt by the screw extruder
tend to relax on its flow through the die. At the die exit
the residual stresses cause the transverse widening of the
stream. Also, during flow in the die, the transverse
velocity gradient leads to orientation of the molecules
along the stream. It is reported by Clegg et aZ. ( 14, 49, 59)
that at the die exit disorientation of the molecules occurs
and leads to the deformation of the extrudate. A certain
increase in the cross-section has also been reported to be
due to the equalisation of the velocity profile at the exit.
Observations
For one-dimensional flow, the swelling is in the
32
direction of the velocity gradient, e.g. a round rod swells
to a diameter larger than that of the round hole. For
two-dimensional flow, however, e.g. in a square-section
hole, the shear rate (or shear stress) is not uniform in
all directions. It is highest in directions parallel to
the sides of square and least along the diagonals. Uneven
swelling occurs and the stream swells more at the sides
and less at the corners of the square. The corners are all
rounded and the final shape is between a square and a
circle.
The situation is messier for non-symmetrical sections.
Other effects of melt elasticity
The effects of melt elasticity are not confined to
die swell alone. A polymer of high swelling ratio
corresponds to one having a long elastic memory (long
relaxation time) and hence possesses poor drawdown
characteristics. Because of the relatively high melt
tension generated during drawdown, such a polymer extruded
as a tubular film would have good bubble stability. In the
case of wire covering, however, the polymer would yield a
covering showing a greater degree of retraction than one
made from a polymer with a low swelling ratio.
(b) VARIOUS STUDIES CARRIED OUT ON DIE SWELL
Investigations were carried out by many authors on
the problem of die swell. Their findings are briefly
presented here.
Metzner and associates (27) measured the swelling of
round strands of polyethylene and polypropylene at various
shear rate and L/D (length to diameter) ratios. They
observed that for a small L/D ratio there is less time for
the stresses set up to relax; therefore, more swelling
results.
33
An experimental investigation by Kaplun et al. ( 2 3)
revealed an exponential dependency of swelling ratio on the
relative length of the channel. An attempt was made at
obtaining the swelling ratio theoretically. A comparison
of the calculated and the measured values showed a relative
error of more than 10% only with a conventional shear rate
of more than 300 g/cm 3 •
Several other authors (Poller et al. (43, 23)) confirmed
that die swell decreases with increase in L/D ratio. The
maximum value of L/D for which swelling reaches a minimum
and remains virtually constant is very important. Kaplun
et al. ( 2 3) find it to be 15-2 0 for high pressure
polyethylene.
Another important aspect of die swell is the effect
of shear rate on it. Several authors have tried to
correlate die swell ratio with critical shear rate.
Clegg (13) observed that swelling increases with
shear rate up to a maximum (critical point) and decreases
thereafter. Kaplun et al. (23) reported the increase in
swelling with increase in shear rate; but reaches a limit
(probably the critical point) beyond which swelling is
constant.
From these investigations,we can say that die swell
is constant beyond a particular shear rate. Thus, when the
34
cross-section of an extrudate is distorted (as a consequence
of uneven swelling on sections with non-uniform shear rate,
e.g. a square section), by using the above relationship it
is possible to select the optimal rate of extrusion.
Kaplun et al. (2 3) studied "swelling" under conditions
which excluded the effects of contraction during cooling and
of elongation under the weight of the extrudate itself.
The following were established:
(1) The optimal extrusion conditions.
(2) The most suitable length of the moulding channel,
and how to predict
(3) The increase in thickness of extrudate resulting from
the elastic after effect only at the given ext~usion
rate.
(c) WHAT CAN BE DONE TO REDUCE DIE SWELL?
It has already been established that the amount of
swelling is not uniform for most extrudates. There is more
drag at certain points and the resulting shape of the
extrudate is different from the die orifice. The "swelling"
effect has become very important for "free" extrusions -
where the extrudate is not subjected to any mechanical
treatment and therefore viscoelastic effect and thermal
contraction are at a maximum.
Is it possible therefore to compensate for the
effect of swell in the initial die design? A suggestion
has been made that this can be done by cutting a "hole" of
the appropriate shape in a mass of compound and compressing
the mass under pressure, thus finding the shape of the
"void" which represents the shape of the required die.
35
Piston
Chamber
Square void
Material
Direction of applied pressure
!
resulting shape of square void
Fig. 6a: The shape of the required die by the compression method.
This is as evident in Fig. 6a.
However, while this method gives some idea of a die
for square, oblong and straight sided solid sections, the
method gives quite a reverse effect when applied to more
complicated sections. The illustrations in Fig. 6b show
the type of die required to produce a square rod and also
show the suggested die by the "compression" method.
Other suggestions as to how to compensate for die
swell are:
(1) To reduce the die opening.
(2) To extrude at the optimal extrusion rate where the
swelling ratio is constant, and
(3) For "drawdown" extrusion, the extrudate can be pulled
out at a velocity greater than the average velocity
in the die.
36
37
Profile of die required to produce the square object shown.
Profile of die as suggested by the cor:1pression method.
Square object.
Fig. 6b: Die profile.
38
CHAPTER 7
CONCLUSION AND RECOMMENDATIONS
The previous four chapters discussed the various
problems in the design of extrusion dies for thermoplastics.
The following conclusions were drawn and suggestions were
made under the four main headings.
MATERIAL PROPERTIES
A brief survey was made into the flow behaviour and
flow properties of polymers. It was established that
polymeric materials could exhibit either solid-like or
fluid-like behaviour.
It would represent a tremendous advance if the type
of behaviour to be exhibited could be predicted under any
set of conditions. This would probably require the setting
up of a criteria under which the polymer could be described
as either behaving like a fluid or a solid. An investigation
of such nature will therefore be suggested.
The class of non-Newtonian behaviour which is most
important industrially is pseudoplastic. Thus, Newtonian
together with pseudoplastic behaviour in steady flow are of
primary importance in the processing of thermoplastics.
FLOW ANALYSIS
Obstructions like spider legs, etc. are known to
cause weld lines and thus non-uniformity in products.
Obviously the extent of non-uniformity introduced by
obstructions will differ for various polymers because of
the differences in their flow behaviour. Thus the degree
of obstructive effect on different sets of polymers could
be looked at.
Pressure drop occurs at the entry region of every
die. Various suggestions have been put forward (discussed
in Chapter 4) as to how to determine it. The type of flow
pattern displayed by a polymer at the die entry could be
related to the entry pressure drop. Branched and linear
polyethylene have been shown (Chapter 5) to exhibit
different flow patterns at the die entry. Perhaps,
the rheological differences between L.P. and B.P. could
give an insight into the mechanism of the pressure loss.
39
In order to make die design automatic and scientific,
it is required that the flow of polymers be really
understood. Analysis of the flow of polymers within
various channels would enable us to predict their flow
behaviour in industrial processes.
The FE method briefly discussed in Chapter 4 could
be used to obtain the flow characteristics in terms of
dimensionless parameters. From these, the physical
parameters may be obtained using the geometric dimensions
of the problem region.
FLOW INSTABILITY AND MELT FRACTURE
Flow instability has been thoroughly discussed
but the exact nature of flow breakdown is still not clear.
Suggestions have already been put forward as to how to
minimise the occurrence of melt fracture.
The site of flow instability is still in dispute.
In my opinion, evidence seems to be more in favour of
either "entry region" or "within the capillary" than at
the "exit region 11•
A study on branched polyethylene supported the
"entry region" theory and linear polyethylene went in
favour of the "slip theory within the capillary".
It would add tremendous knowledge to melt fracture
if the rheological differences between linear and branched
polyethylene that led to the different instability sites
could be stated.
40
Linear polyethylene is known to exhibit melt fracture
together with a discontinuity in the flow curve while
branched polyethylene shows no such discontinuity. The
"hysteresis loop" effect displayed by linear polyethylene
is still not understood. These two polymers are also known
to show different flow patterns at die entries.
Slip has been established to be the cause of flow
curve discontinuity and it occurs between polymer to polymer
layers very near to the die walls.
Extrudates are known to distort severely beyond the
critical point. This is due to the sudden increase in
stored energy which is released at the die exit. The reason
for the sudden increase in elastic recoverable properties
4'1
for polymers might give an explanation for flow instability
and could be investigated.
Investigations into various aspects of flow defects
suggested in this work, if approved, could be carried out
with silicone rubber at room temperature because it exhibits
the same flow defects as polymers at high temperatures.
MELT ELASTICITY
Die swell was thoroughly discussed but its
distribution among various flow layers is still not very
clear. Several authors have reported the maximum die swell
to occur at the outer layers because shear rate (and shear
stress) is maximum there, while others favour the idea of
the maximum occurring along the axis of flow. This is
because the fastest flow is along the axis and would have
had the least time for its stresses to be relaxed.
Most of the work on die swell so far included the
effect of contraction during cooling and the weight of the
extrudate itself. More realistic results about elastic
after-effects of polymers would be obtained if die swell
is measured under conditions which exclude the effects of
contraction and elongation.
It has been established that die swell is constant
beyond a particular shear rate. Non-uniform shear rate (or
shear stress) distribution is known to exist for sections
like square dies. This results in non-even swelling. If
the extrusion can, however, be carried out at the optimum
extrusion rate for the polymer in question, "swelling" will
be uniform for such a profile.
Internal stresses are known to exist in extrudates
after processing, partly due to the redistribution of flow
at the die exit. The effects of these residual stresses
on the mechanical properties of polymers have so far not
been covered. Assessment of the quality of extrudates
42
could also be suggested by the determination of the internal
stresses.
The stress distribution determination could lead to
birefringence studies and would also be recommended.
43
SECTION 2
ANALYSIS OF FLUID FLOW
Y axis
L
H1
...._.._ ~~
Direction of fluid flow
1 boundary /
I I
H2
.p
.p
Direction of fluid flow
----------------------~~ ~
" I''>.>''>;.> ; >; ,>;; 11 i , > ; ; ; ; ; ; J > ; / > / .~ 1 / / / 1 ~ / ~·;; / / / ; / / 1 ; / / 1 1 7 1 / 7t ;> X-axis
Fig. 7: Converging channel flow.
CHAPTER 8
BASIC THEORY
(a) INTRODUCTION
The equations of continuity, momentum, and energy
are mathematical formulations of fundamental physical
principles and are independent of the nature of the fluid.
One or more of them, together with the rheological
equation of the fluid, are always involved in the
mathematical formulation of flow problems.
45
In general, a flow problem cannot be formulated
completely by using only the equations of continuity,
momentum and energy. A rheological equation will be
required to relate the stress components to the rate of
strain of the fluid, and an equation of state will be
required for all problems where the density is not constant.
In addition, equations giving the temperature and pressure
dependence of the other fluid properties may be required.
(b) EQUATIONS GOVERNING FLOW
Fig. 7 shows the geometry of the problem region
in Cartesian coordinates.
In this analysis, the flow is assumed to be
incompressible, isothermal and creeping (thus the inertia
forces are negligible compared with the viscous pressure
46
forces).
According to the principle of conservation of mass,
u and v must satisfy the equation of continuity. That is;
au + av 0 ax ay = ( 8. 1 )
The equations of momentum can be written as;
ap OTXX +
a1:xy = ax ax ay ( 8. 2)
ap = oTxy
+ aTYY
ay ax ---ay ( 8. 3)
Neglecting the cross-viscosity term in the general
constitutive equation the viscous stress components become;
Txx =
Tyy =
and Txy =
au 211 ax
211 av ay
au 11(ay
( 8. 4)
+ av) ax
Assuming also that viscosity is a function of the second
invariant of the rate of deformation tensor, the viscosity
can be written as;
= ( 8. 5)
where
4I 2 = 2 (au) 2 + 2 (av)2 + (au + av)2 ax ay ay ax
In general, the melt viscosity depends on the pressure
temperature and the strain rates. The effect of the
pressure and temperature is ignored for the present purpose.
Attention is confined to the power-law form variation
which represents more closely the stress-strain rate
relationship of a wide variety of polymer melts.
47
Thus the equation of state becomes;
= (4 I)n-1/2 2
( 8. 6)
The solution of the above equations completes the analysis
of the flow of the fluid within the specified geometry.
The results were generalised by obtaining the
required variables in dimensionless form. Further details
on this are presented in Appendix A1.
·s~u~od TEpou ou~Moqs ~U8W818 JETnnuE~Jili :s ·o~~
sn
CHAPTER 9
STREAM FUNCTION ANALYSIS
(a} INTRODUCTION
This FE analysis adopts an app~oach similar to that
of Palit and Fenner (37).
The flow region was divided into subregions and the
relevant variables required were in terms of the values at
the discrete nodal points in the mesh. An appropriate
(polynomial) function of position is assumed for each
49
variable within each element. A variational formulation is
then used to find the solution for the nodal point variables
by minimising the functional.
(b) FINITE ELEMENT FORMULATION OF THE PROBLEM
This sub-heading is further divided into four main
groups. These are:
1. Elements and choice of mesh.
2. Nodal variables and shape functions.
3. Variational function.
4. Solution procedure.
1. Elements and choice of mesh
A mesh of triangular elements with nodal points at
the corners (Fig. 8) was selected as being the simplest and
OS
51
easiest to use.
The principal advantage is the ease of application
to complicated flow geometries. For non-Newtonian flow
problems, there is the further advantage that the viscosity
is constant over each element, simplifying the analysis.
The flow region was divided up into a mesh of
triangular elements as shown in Fig. 9.
2. Nodal variables and shape functions
The nodal points chosen are at the vertices of the
triangular elements. In this work ~ and ¢, the stream
function and linear sum of velocity components, were used
as the nodal variables.
A quadratic distribution of the stream function was
assumed within each element.
Thus;
( 9. 1)
where the C's are constants, expressible in terms of the
nodal point variables.
It is not always easy to ensure that the chosen
function will satisfy the requirement of continuity between
adjacent elements. Thus the compatibility condition on such
lines may be violated (though within each element it is
obviously satisfied) .
3. Variational function
The nodal point variables are required to satisfy the
condition that the total viscous dissipation over the flow
region should be a minimum. This is equivalent to the
52
equilibrium condition expressed by equations (8.2) and (8.3).
The functional used was obtained by integrating the
local viscous dissipation of 4~I 2 over the region.
That is;
( 9. 2)
where ~ is the viscosity and r 2 is the second invariant rate
of deformation of an element.
The functional x was differentiated with respect to
each nodal variable and equated to zero for each element.
The condition to be satisfied for every nodal point variable
such as \jJ. is; l
ax a\jJ.
1
= 0
This leads directly to the elemental stiffness equation
which takes the form:
[ Ke 1 [ l/Je ' <Pe 1 =
( 9 • 3)
( 9. 4)
where [Ke1 is the elemental stiffness matrix, and [l/Je,¢~1 is
a matrix of the required elemental nodal variables.
For details on the formation of matrices [Ke1 and
[lj!: , ¢·' 1 refer to Appendix A2. e e
The contribution from all the various elements
results in the system matrix equation of similar form, i.e.
[K1 [l/J,¢1 = [01 ( 9. 5)
where [K1 is the system stiffness matrix and [l/J,¢1 is a
column matrix containing the nodal values of l/J and <P.
4. Solution procedure
Before an attempt is made at solving the equation
(9.5), some basic information necessary for the FE
formulation must be obtained. These are: nodal point
coordinates, elements and their associated nodal numbers,
degree of freedom numbers associated with each node, and
the boundary conditions.
Nodal and elemental information
53
These constitute a considerable amount of data which
was very time consuming and required a large amount of
manual effort. The nodal variables were numbered in such
a way as to minimise the bandwidth of the system stiffness
matrix and thereby reduce the amount of storage space
required. Refer to Appendix P1. These numbers also refer
to positions in the solution vector-matrix where the
variables are stored. Within this matrix, the nodal
variables were numbered in this fashion; the unknown values
first, followed by the known non-zero values and then the
zero nodal values.
Boundary conditions
The type of velocity distribution used for the
boundary conditions depended on the physical situation of
the problem. Following the work of Palit (36), fully
developed flow boundary conditions were used, both at the
inlet and outlet. The boundary conditions in dimensionless
form are as shown in Appendix A3.
Solution of equations
After the introduction of the boundary conditions,
the equations were solved for ~ and ¢ with the aid of a
digital computer. The solution subroutine used is Symsol.
Refer to Appendix P1.
The viscosity of all elements for the Newtonian flow
54
was known in advance, but not for non-Newtonian flow. It
was therefore necessary after every cycle of iteration to
update the elemental viscosities for the non-Newtonian case.
Convergence and solution accuracy
In FE analysis, the following types of convergency
must be considered:
(a) As the element sizes are reduced, the results must
tend to the true solution of the physical problem.
(b) A decrease in~he convergent criterion results in an
increase in the number of iterations and the nodal
point variables should tend to constant values.
Convergence test was applied after every cycle. The
relative change of each nodal point variable was computed
and the magnitude of these changes summed up. The criterion -5
used was that the above sum should be less than 10
A check on the FE accuracy can be obtained by
comparing the FE and LA solutions. For parallel channel
flow, the solutions should be identical. The flowrates at
every section of the channel could also be used as a further
check on the FE accuracy. Any variation in the flowrates
from section to section is a measure of the solution
accuracy. This is particularly useful for tapered channels.
(c) PROBLEMS IN THIS ANALYSIS
The various problems encountered in using ~ and ¢
as the nodal variables can be divided into the following
headings:
1. Boundary conditions.
2. Compatibility~
55
3. Extracting pressure and velocity values.
4. Geometric problems.
Boundary conditions
The boundary conditions often used for FD formulations
are stream function and vorticity. Vorticity is often very
difficult to prescribe at the boundaries. In this work,
~ and ¢ were used as the boundary conditions. Even though
they are relatively straightforward to specify, compared
with the FD boundary conditions, they did introduce some
problems in the analysis.
First of all, the boundary conditions depended on the
flow behaviour index of the polymer whose flow was being
analysed. A different set of boundary conditions was
therefore required for the Newtonian and non-Newtonian
flows. Thus, if the analysis for another polymer (whose
flow behaviour is known) is required, a different set of
boundary conditions has to be written. For further
details on the boundary conditions refer to Appendix A3.
The boundary conditions are also dependent on the
inclination of the channel.
The preparation of the boundary conditions is thus
laborious and time-consuming.
A solution to the above problem would be to find a
set of boundary conditions independent of both the taper
of the channel and the type of polymer in use.
Compatibility
This analysis leads to continuous velocities over
each element. There is compatibility across element
interfaces in ~ and ¢, the sum of the velocity components,
but not in the individual velocities. This is therefore a
great drawback to this type of analysis since there is
discontinuity in velocities between adjacent elements.
Extracting pressure and velocity values
56
The nature of this analysis requires the pressure
and the velocities to be obtained from the nodal variables.
The solution of the matrix equation (9.5) yields
~ and ¢ (at a nodal point).
But
¢ = u + v . (9.6)
Therefore, if either u or v is found, the other is obtained
directly from the above equation.
The derivation of the velocity u of an element is as
shown in Appendix A4. Once the velocity of an element is
found, the nodal point velocities can be found using an
averaging process over all the elements involving that point.
The pressure drop across the channel was obtained by
integrating equation (8.2) along the bottom boundary of the
channel. The stress gradients were obtained in terms of the
nodal viscosities and velocities. Finite difference
approximations were used to evaluate the derivatives in
equation (8.4). Further detail on the estimation of the
pressure drop is evident in Appendix A5.
Geometric problems
From the look at the coefficients of the shape
functions in Appendix A2, it is found that some of the
coefficients become infinite when E = 0.
57
Fig. 10: Initial mesh.
58
i.e., when
a. + b. = ak + bk J J
or when
a. +b. = 0 or ak + bk = 0 J J
With regards to the mesh (Fig. 10) isosceles right angled
triangles with their hypotenuse making an angle of 135°
with the x-axis satisfy this condition. The initial mesh
(Fig. 10) used, contained elements as described above and
the problem was realised.
Palit (36) also reported a similar problem in his
study. Similar difficulties were also experienced by
Card (12). A modified area coordinate system as described
by Card (12) and Zienkiewicz (60) can be used to get over
these difficulties.
Another solution to this problem as adopted by the
present author is to divide the flow region into a mesh
as shown in Fig. 9.
For the converging channel, the mesh had to be
modified to suit the actual boundary. The y coordinates
of all the nodal points were changed to y 1 , given by
y' = y ( 1 _ ta~ a) (9.7)
where a is the angle of taper of the channel and L is the
length of the channel.
CHAPTER 10
ANALYSIS USING VELOCITIES AND PRESSURE
(a) INTRODUCTION
When the literature on computational methods in
fluid mechanics is examined, the overwhelming tendency is
to use stream function for the incompressible plane flow
class of problems. Generally speaking, the sequence of
calculations is to find the stream function, obtain the
velocities and then the pressure drop.
This, however, seems to be a lengthy approach.
The FE analysis following solves the same flow problem
as was discussed in Chapter 9, but obtains the pressures
and velocities directly.
In this approach also, a variational principle
formulation of the problem was adopted and a different set
of nodal variables used. This work also serves as a check
on the first analysis of Chapter 9.
59
Further details of this analysis are presented under
the subsequent headings.
(b) FINITE ELEMENT FORMULATION OF THE PROBLEM
The first step towards the FE formulation is to
represent the problem region with a finite number of
60
elements interconnected at a discrete number of points -
nodal points. Any arbitrary shape of the element can be
chosen. The parameters relevant to the physical problem
are then selected. The values of these parameters
(e.g., velocity, pressure, temperature, etc.) at the nodal
points are called the nodal variables. Nodeless variables;
for example, element stress, could also be used. Element
functions for the variables are then expressed in the form
of polynomials of the coordinates. The choice of the
element shape and the form of the element shape function
affects the degree of accuracy of the solution, the
numerical analysis and the computing time.
1. Elements and choice of mesh
In general, the elements can be of any shape, but
triangular elements were chosen in this work to represent
the continuum. They are particularly suitable for
approximating arbitrary geometries. A typical element is
as shown in Fig. 8. The mesh used is similar to that of
the first analysis (refer to Fig. 9).
2. Nodal variables And shape functions
. \ The s1mplest triangular elements (i.e. elements
with three nodal points at the vertices) were used in order
to make the analysis simple and suitable for a wide range
of applications. The velocities (u and v) and pressure (p)
were used as the nodal variables.
A linear variation pressure and velocities were
assumed over each element. This is a better approximation
than the first analysis where the pressure was assumed to
t-J:j
1-'·
\.Q
....... .....
8 ~ 1-'·
0 Pl
1-'
H1
1-'·
::::s
1-'·
rt
(!)
(!) 1-'
(!) E:! (!) ::::s
rt .
1-'·
0
t<l I
... 0
~Pl
~--------------------------
p. .....
.... :X: I Pl
:X:
1-'· rn
L.J
.
Pl
u . ... tJ
' u
.
rn
~9
62
be constant within each finite element. Fig. 11
describes the geometry of a typical triangular element m
with the nodal points i,j,k. The three components of
element velocities are;
u. l
[u] = u. J
uk
The velocity distribution within the element is thus given
by the linear polynomial
(10.1)
where the constants a 1 , a 2 and a3
can be evaluated in terms
of the element dimensions and the nodal viscosities.
In general, the nodal variables u, v and p are given
by; u = N.u. + N.u. + Nkuk l l J J
(10.2)
v = N.v. + N.v. + Nkvk l l J J (10.3)
and
p = N.p. + N.p. + Nkpk l l J J (10.4)
where the N's are the shape functions given by;
Nr 1
[ar + b + cr y] = X 26 r (10.5)
For further details see Appendix B1.
3. Variational function
The formulation of the problem requires the
satisfaction of equations (8.1) through to (8.6).
A formulation technique using the concepts of variational
calculus was developed.
63
The variational method requires in general the
minimisation of a functional x over the continuum. X is
an integral quantity obtained mathematically. Although
it is apparently treated as a mathematical quantity, it has
a physical meaning associated with the energy of the system.
For the equilibrium of the system, the functional
chosen must be minimised, with respect to the nodal
variables. The functional x chosen for the present problem
is given by;
X = (10.6)
Since the differential of X, with respect to the nodal
variables, is to be zero, it is only necessary to consider
the local viscous dissipation term as being proportional to
the equivalent term for the pressure forces.
For a typical element the differential
respect to the nodal variables is given by;
where
axe u. du-:- l
l
axe av-:- v.
l l
= [Ke]
[K ] is the elemental stiffness matrix. e
of X with
(10.7)
An outline of general variational procedures and
the evaluation of [K] is given in Appendix B1. e
The final equations are obtained by adding the
contributions of each element and equating to zero, thus
minimising the total integral.
The resulting equation is of the form;
64
[K] -: l ~ [O] (10.8)
where [K] is the system stiffness matrix and
column matrix containing the nodal variables
of u, v and p.
4. Solution procedure
is a
The basic information required in the solution of
equation (10.8) is similar to those listed under Chapter 9.
The only differences are in the degree of freedom numbers
and the boundary conditions.
There were three degree of freedom numbers per node
denoting the three nodal variables. The coordinates and
the degree of freedom numbers were generated with the aid
of a digital computer (refer to Subroutine Genode, Appendix
P2). This considerably reduced the effort in preparing the
data manually.
A set of boundary conditions (a great deal simpler
than those of the previous analysis) was employed here.
A dimensionless pressure of unity and zero was prescribed at
the inlet and outlet, respectively, of the channel.
A solution procedure, similar to the previous one,
was also adopted here (Reference: Symbol Subroutine
Appendix P2).
(c) ADVANTAGES AND DISADVANTAGES OF THIS ANALYSIS
This analysis was found to have great advantages,
over the previous one. Most of the problems encountered
previously were eliminated here.
65
No geometric problem was encountered in the analysis.
The only geometric problem which may have occurred was
already solved in the first analysis.
The boundary conditions used were independent of
the flow behaviour index of the polymer. Thus, the same
set of boundary conditions could be used to analyse the
flow of a Newtonian as well as a non-Newtonian fluid.
The pressure and the velocities being the nodal
variables needed not be extracted after the solution was
obtained. The required pressure drop encountered during
the flow could be obtained without any further numerical
integration. The pressure and velocities were found to be
continuous over each element and compatible between adjacent
elements. Hence the condition of compatibility is satisfied
over the whole domain; a great advantage indeed.
CHAPTER 11
RESULTS AND DISCUSSION
Some results are presented for parallel and
converging channel flows. The main objective is to
compare the results of the two analyses and to
demonstrate the power of the FE method.
Part (a) shows the list of results; which are
discussed under Part (b) .
The FE results are compared with the analytical
and FD solutions in terms of dimensionless flowrate,
velocities, pressure gradients and viscosities.
The results of the two analyses are presented
together for (9x9) and (17x17) meshes; and at various
0 0 0 0 taper angles of the channel of 8 = 0 , 10 , 20 , 30
and 40°.
66
67
PART (a} : LIST OF RESULTS
NUMBER OF CYCLES OF ITERATION FOR NON-NEWTONIAN (n - 0. 5)
PARALLEL CHANNEL FLmi\T USING A 9 x 9 MESH.
Table 1
CONVERGENCE NO. OF CYCLES CRITERION USED OF ITE.RATION
0.001 6
0.0001 8
0.00001 12
68
69
FLOWRATE RESULTS FOR FIRST ANALYSIS.
70
DIMENSIONLESS FLm'VRATE FOR PARALLEL CHANNEL FLOW
Table 2
NEWTONIAN FLUID
POSITION X MESH SIZE ALONG CHANNEL ( 9 x9) (17X17) EXACT
0 0.98042 0.99465 1.0
0.5 0.97761 0.99420 1 . 0
1 . 0 0. 9804.2 0.99465 1 . 0
MAXIMUM % ERROR FROM THEORETICAL 2.239 0.580 0 VALUE
Table 3
NON-NEWTONIAN FLUID (n = 0. 5)
POSITION X MESH SIZE ALONG CHANNEL ( 9x 9) (17X17) EXACT
0 0.98536 0.99629 1 . 0
0.5 0.98409 0.99607 1 . 0
1 . 0 0.98536 0.99629 1 . 0
MAXIMUM % ERROR FROM THEORETICAL
I 1 . 5 91 0.393 ·o
VALUE I
71
DIMENSIONLESS FLOWRATE FOR CONVERGING CHANNEL FLOW AT X = 0.5
THEORETICAL FLOWRATE = 1.0
Table 4
NEWTONIAN FLUID
INCLINATION MESH % ERROR FROM OF CHANNEL (17X17) THEORETICAL VALUE
00 0.99420 0.580
1 0° 0.99408 0.592
2 0° 0.99340 0.660
30° 0.99196 0.804
40° 0.98890 1 • 11 0
Table 5
NON-NEWTONIAN FLUID (n = 0. 5)
INCLINATION MESH % ERROR FROM OF CHANNEL (17X17) THEORETICAL VALUE
00 0.99607 0.393
1 0° 0.99515 0.485
20° 0.99341 0.659
30° 0.99056 0.944
4 0° 0.98591 1 . 4 09
72
VELOCITY RESULTS FOR BOTH ANALYSES
DIMENSIONLESS VELOCITIES FOR PARALLEL CHANNEL FLOW
(FIRST ANALYSIS)
Table 6
NEWTONIAN FLUID
POSITION POSITION ALONG CHANNEL
73
ACROSS THE THE OR- X = 0.25 X = 0.50 X= 0.75
CHANNEL ETICAL MESH MESH MESH MESH MESH MESH (y) VALUE (9X9) (17X17) (9X9} (:t7X17) (9X9} (17X17)
0.00 o .. ooooo 0.01910 0.00552 0.02232 0.00580 0.02384 0.00591
0.125 0.65625 0.64640 0.65325 0.65708 0.65651 0.66239 0.65775
0.250 1.12500 1.10824 1.1202 7 1.11319 1.12188 1.11943 1.12345
0.375 1.40625 1.38941 1. 40175 1. 38862 1. 40159 1. 39445 1. 40314
0.500 1.50000 1.48496 1.49606 1. 48076 1.49492 1.48496 1.49606
0.625 1.40625 1.39445 1. 40314 1. 38862 1.40159 1.38941 1.40175
0.750 1.12500 1.11943 1.12345 1.11319 1.12188 1.10824 1.12027
0.875 0.65625 0.66239 0.65775 0.65708 0.65651 0.64640 0.65325
1.000 0.00000 0.02384 0.00591 0.02232 0.00580 0.01910 0.00552
MAXIMUM % ERROR 1.501 0.457 1. 283 0.339 1.501 0.457
DIMENSIONLESS VELOCITIES FOR PARALLEL CHANNEL FLOW
(SECOND ANALYSIS)
Table 7
NEWTONIAN FLUID
POSITION POSITION ALONG CHANNEL
74
ACROSS THE THE OR- X - 0.25 X = 0.50 X = 0.75
CHANNEL ETICAL MESH MESH MESH MESH MESH MESH (y) VALUE (9X9) (17X17) (9X9) (17X17) (9X9) (17X17)
0.0000 0.00000 0.00000 0.00000 0. 00000 0.00000 0.00000 0.00000
0.125 0.65625 0.66667 0.65882 0.66667 0.65882 0. 66667 0.65882
0.250 1.12500 1.14286 1.12941 1.14286 1.12941 1.14286 1.12941
0.375 l. 40625 l. 42857 1. 41176 1. 4285 7 l. 41176 1. 42857 l. 41176
0.500 1.50000 l. 52381 l. 50588 1.52381 1.50588 1.52381 1.50588
0.625 1.40625 l. 42857 1. 41176 l. 4285 7 1. 41176 l. 42857 1.41176
0. 750 1.12500 1.14286 1.12941 1.14286 1.12941 1.14286 1.12941
1.875 0.65625 0.66667 0.65882 0.66667 0.65882 0.66667 0.65882
1.000 0.00000 0.00000 0.0000 0.00000 0.00000 0.00000 0.00000
MAXIMUM % ERROR 1.588 0.392 1.588 0.392 1.588 0.392 I
75
DIMENSIONLESS VELOCITIES FOR PARALLEL CHANNEL FLOW
(FIRST ANALYSIS)
Table 8
NON-NEWTONIAN FLUID (n = 0. 5)
POSITION POSITION ALONG CHANNEL
ACROSS THE THE OR- X = 0.25 X = 0.50 X = 0. 75
CHANNEL ETICAL MESH MESH MESH MESH MESH MESH (y) VALUE (9X9) ( 17X17) {9X9) (17X17) {9X9) (17X17)
0.000 0.00000 0.02531 0. 00720 0.02672 0.00727 0. 02729 0. 00729
0.125 0.77083 0.77535 0.77210 0. 77968 o. 77299 0.78126 0.77322
0.250 1.16667 1.16784 1.16712 1. 16 796 1.16704 1.16933 1.16725
0.375 1.31250 1.31098 1. 31245 1. 31054 1. 31225 1. 31234 1. 31252
0.500 1. 33333 1.33094 1. 33272 1. 32969 1. 33246 1. 33094 1. 33272
0.625 1. 31250 1. 31234 1. 31252 1. 31054 1. 31225 1. 31098 1. 31245
0.750 1.16667 1.16933 1.16725 1.16796 1.16704 1.16784 1.16712
0.875 0.77083 0.78126 0.77322 0. 77968 0.77299 0. 77535 0. 77210
1.000 0.00000 0.02729 0.00729 0.02672 0.00727 0.02531 .0.00720
MAXIMUM % ERROR 1.353 0. 310 1.148 0.280 1. 353 0.165
DIMENSIONLESS VELOCITIES FOR PARALLEL CHANNEL FLOW
(SECOND ANALYSIS)
Table 9
NON-NEWTONIAN FLUID (n = 0. 5)
POSITION POSITION ALONG CHANNEL
76
ACROSS THE THE OR- X= 0.25 X = 0.50 X = 0.75
CHANNEL ETICAL MESH MESH MESH MESH MESH MESH (y) VALUE (9x9) (17xl7) (9x9) (17x17) (9x9) (17x17)
0.000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
0.125 0. 77083 0.79027 0. 77554 0.79027 0.77554 0.79027 0.77554
0.250 1.16667 1.19354 1.17322 1.19354 1.17322 1.19354 1.17322
0.375 1.31250 1. 33875 1. 31894 1.33875 1. 31894 1. 33875 1.31894
0.500 1.33333 1.35489 1. 33864 1. 35489 1. 33864 1. 35489 1.33864
0.625 1.31250 1. 33875 1.31894 1.33875 1. 31894 1. 33875 1.31894
0.750 1.16667 1.19354 1.17322 1.19354 1.17322 1.19354 1.17322
0.875 0. 77083 0.79027 0.77554 0.79027 0.77554 0.79027 0.77554
1.000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
MAXIMUM % ERROR 2.522 0.611 2.522 0.611 2.522 0.611
77
P·RESSURE RESULTS FOR BOTH ANALYSES
78
DIMENSIONLESS PRESSURE DROPS FOR CONVERGING CHANNEL FL01ii1S
(FIRST ANALYSIS)
Table 10
NEWTONIAN FLUID
ANGLE OF MESH MESH LUBRICATION TAPER OF SIZE SIZE APPROXIMATION CHANNEL ( 9x 9) (17x17) VALUES PAL IT
00 11.64146 11.91259 12.00 12.43
10° 16.54130 16.13 17.36
20° 27.36746 24.27 29.05
30° 63.33640 47.78 69.41
40° 382.60895 269.05 546.70
Palit did not state the mesh size for his results.
Table 11
NON-NEWTONIAN FLUID (n = 0. 5)
ANGLE OF MESH MESH TAPER OF SIZE SIZE L.A. CHANNEL ( 9X 9) (17X17) VALUES
00 6.00431 5.49810 5.65685
10° 6.63719 6.86762
20° 9.01633 8.89443
30° 14.63545 13.38584
40° 38.62187 35.15758
79
DIMENSIONLESS PRESSURE VALUES FOR FLOW IN PARALLEL CHANNEL
(SECOND ANALYSIS)
Table 12
NEWTONIAN FLUID
POSITION POSITION ACROSS THE CHANNEL
ALONG THEOR- y= 0.0 Y= 0.25 y = 0.5 CHANNEL ETICAL MESH MESH MESH MESH MESH MESH
on VALUE (9X9) (17X17) (9X9) (17X17) (9X9) (17X17)
0.000 12.0000 12.19048 12.04706 12.19048 12.04706 12.19048 12.04706
0.125 10.5000 10.66667 10.54118 10.66667 10.54118 10.66667 10.54118
0.250 9.0000 9.14286 9.03529 9.14286 9.03529 9.14286 9.03529
0.375 7.5000 7.61905 7.52941 7. 61905 7.52941 7.61905 7.52941
0.500 6.0000 6.09524 6.02353 6.09524 6.02353 6.09524 6.02353
0.625 4.5005 4.57143 4.51765 4.57143 4.51765 4.57143 4. 51765
0.750 3.0000 3.04762 3.01176 3.04762 3. 01176 3.04762 3.01176
0.875 l. 5000 l. 5 2381 1.50588 l. 52381 1.50588 1.52381 l. 50588
1.000 0.0000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000
MAXIMUM % ERROR 1.587 0.392 1.587 0.392 1.587 0.392
80
DIMENSIONLESS PRESSURE VALUES FOR FLOW IN PARALLEL CHANNEL
(SECOND ANALYSIS)
Table 13
NON-NE1r\ITONIAN FLUID (n = 0. 5)
POSITION POSITION ACROSS THE CHANNEL
ALONG THE OR- y = 0.0 y = 0. 25 y = 0.5 CHANNEL ETICAL MESH MESH MESH MESH MESH MESH
(X) VALUE (9x9) (17X17) (9X9) (17x17) (9X9) (17X17)
0.000 5.65685 5.74602 5.67777 5.74602 5.67777 5.74602 5.67777
0.125 4.94975 5.02777 4.96805 5.02777 4. 96805 5.02777 4.96805
0.250 4.24264 4.30952 4.25833 4.30952 4.25833 4.30952 4.25833
0.375 3. 53553 3.59126 3.54860 3.59126 3.54860 3.59126 3.54860
0.500 2.82843 2.87301 2.83888 2.87301 2.83888 2.87301 2.83888
0.625 2.12131 2.15476 2.12916 2.15476 2.12916 2.15476 2.12916
0. 725 1.41421 1. 43651 1. 41944 1. 43651 1. 41944 1. 43651 1.41944
0.875 o. 70711 0.71825 0. 70972 0. 71825 0.70972 0. 71825 0. 70972
1.000 0.00000 0.00000 0.00000 0.00000 o.ooooo 0.00000 0.00000
MAXIMUM % ERROR 1.577 0.370 1.577 0.370 1. 577 0.370
81
VISCOSITY RESULTS OF FIRST ANALYSIS.
82
DIMENSIONLESS NODAL VISCOSITIES FOR PARALLEL CHANNEL FLOW
USING A ( 9X9) MESH
Table 14
NON-NEWTONIAN FLUID (n = 0. 5}
POSITION THE OR- AVERAGING DIRECT ACROSS THE ETICAL PROCEDURE % EVALUATION %
CHANNEL VALUES VALUES ERRORS VALUES ERRORS
0.000 0.35355 0.40496 14.40 0.36567 3.40
0. 1 25 0.47141 0.48024 1.87 0.46827 0.67
0.250 0.70711 0.71983 1. 80 0.68850 2.63
POSITION
DIMENSIONLESS NODAL VISCOSITIES
FOR PARALLEL CHANNEL FLOW AT X = 0.5
(FIRST ANALYSIS)
Table 15
NON-NEWTONIAN FLUID (n = 0. 5)
FINITE ELEMENT ESTIMATION ACROSS THE THEORETICAL MESH SIZE MESH SIZE
CHANNEL VALUES ( 9X 9) (17X17)
0.000 0.35355 0.36567 0.35659
0. 125 0.47141 0.46827 0.47118
0.250 0.70711 0.68850 0.70523
0.375 1.41421 1.25519 1.39177
0.500 - - -
0.625 1.41421 1.25519 1.39177
0.750 0.70711 0.68850 0.70523
0.875 0.47141 0.46827 0.47118
1 . 00 0 0.35355 0.36567 0.35659
MAXIMUM % ERROR 11.244 1 . 58 7
83
84
COST OF THE RESULTING PROGRAMMES.
COST AND STORAGE REQUIREMENTS FOR THE COMPUTER PROGRAMMES
RESULTING FROM THE ANALYSES
Let Programme 1 result from the stream function analysis and Programme 2 from the velocity and pressure analysis.
Table 16
PROGRAMME 1
MESH RUNNING COST OF STORAGE STORAGE COST AS A SIZE TIME PROGRAMHE CAPACITY % OF THE TOTAL COST
gx 9 53 sees $2.10 7 K 1
1 7X 1 7 12.5 mins $20.00 20 K 25
PROGRAMME 2
MESH RUNNING COST OF STORAGE STORAGE COST AS A SIZE TIME PROGRAMi'1E CAPACITY % OF THE TOTAL COST
9X9 3 mins $5.25 1 0 K 14.5
17x17 36 mins $70.00 41 I<; 47
85
86
PART (b) : DISCUSS]ON
87
PRELIMINARY RESULTS
An initial investigation was performed to determine
the most economical convergence criterion to be used in the
analysis. This was done for a non-Newtonian fluid (n= 0.5)
in a parallel channel. The results are as shown in Table 1.
It is seen that a convergent criterion of 10- 3
resulted in 6 cycles of iteration, and for a convergent
criterion of 10- 5 , 12 cycles of iteration were required.
The latter value was therefore chosen and used for
all subsequent analysis.
Flowrate
For two-dimensional plane flows, the accuracy of
the FE solution is investigated by analysing the flowrate
across a parallel channel.
In Table 2, the dimensionless flowrates at various
positions along the channel are compared with the analytical
solutions for the Newtonian case.
For steady, isothermal, incompressible flow the
dimensionless flowrate at each section along the channel
is, by definition, equal to unity.
It is thus seen that the FE results approach the
true solution as the element size is decreased.
Table 3 shows the comparison between the flowrates
for n = 0.5 using the two meshes. It is found that, similar
to the Newtonian case, the results converge to the true
values as the mesh is refined.
Using the finer mesh alone, the flowrates at
position x = 0.5 were obtained for various angles of taper
Position y across the
channel 1 • 0
.875
.750
.625
.500
.375
.250
. 125
0
Fig.
.2
12:
Dimensionless
• 4 . 6 . 8 1.0 1 • 2 1 • 4 1.6 1.8 velocity
Velocity profile for Ne1.vtonian flow in a parallel channel.
00 OJ
Position y across the channel
1 • 0
• 1 25
.2 • 4 • 6 . 8 1.0 1 • 2 1 • 4 1 • 6 1 . 8
Fig. 13: Velocity profile for non-Newtonian Crt~ 0.5) flow in a parallel channel.
co 1..0
90
of the channel as presented in Tables 4 and 5. The accuracy
of the FE method is thus seen to decrease as the inclination
of the channel is increased.
Velocities
The velocity values across the channel were found
for parallel flow at positions X= 0.25, 0.5 and 1 .0. The
results for both analyses are as shown in Tables 6 through
to 9.
The velocity plots (Figs 12 and 13) of the first
analysis are given to demonstrate the nature of the profile.
For the Newtonian flow, the profile is parabolic while the
parabola flattens for the non-Newtonian case.
All the results illustrate the accuracy of the FE
method and also the fact that the finer mesh values tend
towards the analyticql solution. Along any streamline, the
results from the first analysis reveal that the velocities
vary from position to position. This is probably due to the
compatibility condition which is not satisfied in this
analysis. With the second analysis, where compatibility is
satisfied over the whole domain, it is noted that velocity
is constant for any chosen streamline.
The velocities obtained from the first analysis are,
on the average, lower than those from the second analysis.
This could be explained by the fact that the dimensionless
flowrate was unity for the second analysis while slightly
lower flowrate values were obtained with the first analysis.
Both analyses showed almost the same errors. The
largest error was obtained with the non-Newtonian flow using
the second analysis as evident in Table 9. In this case,
however, there was a considerable improvement in the
results, as a finer mesh was used.
Pressure
The first analysis compares the FE results of the
pressure drops for converging channels of taper ranging
from 0° to 40° with LA solutions and the solution by Palit
(36). (Refer to Tables 10 and 11). The results of the
second analysis, which only dealt with parallel channel
flow, are presented in Tables 12 and 13.
91
For parallel channel flow, Palit (36) reported an
error of the order of 4% in the pressure drop for Newtonian
flow with a (9X9) mesh. The first analysis of this work
showed an error of 3% for the same mesh and 0.73% for a
(17X17) mesh. The results of the second analysis are seen
to be the best of the three. Errors in the order of 1.59%
and 0.39% for the coarse and fine mesh respectively were
obtained in this analysis.
For the non-Newtonian case, however, the first
analysis yielded errors in the range of 2.8 to 6.2%. One
of the major sources of error in the evaluation of the
pressure drop is the differentiation process used. Error
is also due to the fact that the viscosity is assumed to be
constant for each element. This results from the assumption
that the stream function varies quadratically over each
element. This assumption has the advantage of ensuring
correct convergence as tpe element size is decreased. The
price to be paid for these simplifications is a loss of
accuracy at the nodal points.
Dimensionless pressure drop
60
50
40
30
20
10
0
:::;..-
---- --~ -
n = 1
t/ / /
"/ /
y
/
I
• I
I I
I
/ n = 1 (LA)
'/ n=op / n = 0. 5 (LA)
1.0 l'V
Angle of ~-----------------L------------------~-----------------L------------------._------------~inclination
1 o0 20° 30° 40° of channel
Fig. 14: Dimensionless pressure drop versus taper angle.
93
Fig. 14 shows the plot of the dimensionless pressure
drop against the taper angle. The dotted ones show the
IA solution; while the solid ones represent the FE results
of the first analysis.
Fig. 14 reveals that the FE and LA solutions diverge
as the angle of inclination of the channel is increased.
The difference between the two solutions is more noticeable
from about the 15° angle onwards. A breakdown of the
0 LA solution beyond 10 was also reported by Benis (10).
The FE solution is also seen to be higher than the
LA solution because of the two-dimensional nature of the
flow.
Table 10 shows differences in the author's results
and those of Palit (36). (It was not obvious from Palit's
report what mesh size he used.) The author's approach to
the evaluation of 6p was different from that of Palit. In
both cases, the 6p was evaluated along the bottom boundary
of the channel. However, 6p is known to be very much
dependent on the viscosity values. Palit used an averaging
procedure to obtain the nodal viscosities along the boundary
while the approach adopted here evaluates the viscosity
directly by the use of equation (8.6). The value of the
theoretical viscosity at the boundary of the channel is
0.354. The values obtained from the averaging procedure
and the direct evaluation methods are 0.405 and 0.366
respectively. The corresponding errors, as compared with
the theoretical value, are 14.4% and 3.4%. Table 14 shows
the values and errors for other positions using the two
methods.
It can be seen that the errors in both cases are
almost the same except at the boundary, where the error in
the averaging procedure is very high. It is then obvious
why the estimated values of the pressure drops by Palit
are unsatisfactory.
94
Further explanation as to why the viscosity values,
using the two methods, are almost the same and only differ
significantly at the boundaries is given under Chapter 12.
Also refer to the Visco Subroutine of Appendix P1 for
further details on the direct evaluation of the viscosities.
Viscosities
Finite element nodal viscosity predictions using the
"direct evaluation'' method are presented in Table 15. Also
included in this table is a set of results for the
analytical viscosities. The analysis for the theoretical
viscosities is as shown in Appendix A6.
In both cases it is seen that the row corresponding
to the y = 0.5 position is left blank. This is because the
denominator used du~ing the evaluation tended to be
extremely small at that position. It resulted in an
infinite value for the viscosity which is physically
impossible.
Improvements in the results are still observed as
the mesh is refined.
Storage
All the computing for this work was done on a
Burrough 6718 at the University of Canterbury Computer
Terminus.
95
The results of the cost and the storage requirements
of the programmes are presented in Table 16.
It was found that the storage requirement was largely
dependent on the FE mesh. The computing time was also found
to be largely dependent on the number of nodal variables
required, the degree of non-Newtonian behaviour and to a
lesser extent on the inclination of the channel.
With reference to the table, we can see that
programme 2 is the more e~pensive of the two and it demanded
a lot of storage capacity.
It is obvious from the results that there is
inefficiency in the second progra~me so far as the (17x17)
mesh is concerned. It is therefore essential for this
programme to be improved if it is to be used for further
melt flow analysis in future. In Chapter 12, some
suggestions will be put forward as to how to improve upon
this programme.
CHAPTER 12
GENERAL DISCUSSION
In the earlier chapters, melt flows in parallel
and converging channels were analysed using the FE method.
Two approaches were adopted; the first one was similar to
the type of analysis used by Palit and Fenner (37) where ~
and ¢ were used as the nodal variables. In the second
analysis a different variational functional was used and
nodal variables u, v and p were required.
The results of the two methods were presented and
discussed under Chapter 11.
This chapter discusses features associated with the
two methods and the general problems encountered with the
solutions.
GENERAL DISCUSSION
(a) Constitutive Equations
96
Polymeric fluids are viscoelastic. Therefore any
constitutive equations describing them should include both
viscous and elastic effects of the polymer melts. Viscosity
may be very large compared with melt elasticity in some
flows, whereas in others the elastic effects may dominate.
Not much is known about melt elasticity. Pickup (41),
Metzner et al. (28) and Cogswell and Lamb (15) have shown
NK
...
NK X 0 D =
K
NDF N 011 II>' -<1 ~
NDF 1 I K1 I K2 I I I D1
I INDF NK = Initial no. of D.O.F.
t T ~
lN NDF = Unkno\'m no. of D.O.F.
D2 I N = No .• of non-zero D.O.F.
Fig. 15: Initial layout of the system matrix equation.
\0 -.1
9B
that the stresses caused by elongation are greater than the
shear stresses in certain flow situations, for example,
converging flow of viscoelastic materials. Pearson (39)
discussed constitutive equations which include the elastic
effects, but they are very complex for practical use and
the required material parameters are difficult to measure.
Thoughout this work, the melt was treated as a
Stokesian fluid. Viscosity relations which are functions
of pressure, temperature and strain rates can be used.
However, in this investigation, the temperature and pressure
dependence were neglected. And also the cross viscosity
term was ignored. The flow was also assumed to be steady,
incompressible, laminar, viscous and slow moving. For slow
non-Newtonian flows it was further assumed that the inertia
forces were negligible, compared with the viscous and
pressure forces.
In extrusion processes, elastic effects are not
predominant, since melts are subjected to large rates of
deformation for relatively long periods. For isothermal
flow, therefore, a power-law fluid model was good enough to
represent the pseudoplastic nature of polymer melts.
(b) Problems encountered in the solution of
the "stream function analysis"
The initial problem encountered in this solution was
a storage one.
When the system matrix equation;
[K] [1/J,¢] = [0]
was formed, it was put in the form as shown in Fig. 15.
Increasing y
direction
3 y=Q.2sr-----------~--~----~-----------
G E
Y = .12s~ A j
A
B D
y = 0 1 t
Fig. 16: A portion of the mesh used.
Increasing viscosity direction
Boundary of mesh
\.0 \.0
1 00
The original idea was to divide the equation into matrices
K1, K2, D1, D2, etc. The sizes of the matrices are also
shown in the figure.
Attention was confined to matrices K1, K2 and part
of the column vector [¢,~] which is D1 and D2.
The loading vector is then equal to [K2] x [D2] ;
and the final matrix equation is
[K1] X [D1] = - [K2] X [D2] (12.1)
This procedure is evident from Appendix C.
The above idea worked well for a (9x9) mesh which
had the size of [K] being (162x162). For a (17x17) mesh,
however, the size of [K] was (578x578). Storage problems
were therefore encountered because the B6718 computer in use
had an array limit of 65535 elements; thus approximately a
size of (256x256).
To overcome this problem, advantage was taken of the
symmetric nature of the stiffness matrix, and its banded
form. The loading vector ([K2] X[D2]) was also formed as
each finite element was brought in for processing. This
procedure is as outlined in the ASSEMB Subroutine of
Appendix P 1 .
It was mentioned in the previous chapter that Palit's
estimation of the pressure drop during flow was unsatisfactory.
This was because of the incorrect values of the viscosities
obtained along the boundaries of the channel. The following
explanation might give an insight into why Palit's
averaging procedure of viscosity evaluation led to
unsatisfactory results.
With reference to Fig. 16 and using an averaging
1 01
procedure, nodal point 2 has contribution towards its
viscosity from 6 elements (A, B, C, E, F and G). The last
three elements are in the high viscosity region and the
first three are in the low viscosity region. The effect of
the low and high viscosity regions is that a fairly accurate
value for ~2 of 0.48024 is obtained. The theoretical
viscosity value for nodal point 2 is 0.47141. But since
nodal point 1 is along the boundary, it has contribution
from only 3 elements (B, C and D), all in the high viscosity
region. Because there are no elements below nodal point 1,
the value of its viscosity and, for that matter, all points
along the boundary is higher than expected. This method
gives the viscosity of nodal point 1 as 0.40496 while the
analytical value is 0.35355.
A solution to this problem was to obtain the boundary
viscosity values by a "direct evaluation method" as
already discussed.
(c) Problems encountered in the solution of the "velocities
and pressure analysis"
A problem of "division by zero" often occurred in the
SYMBOL subroutine of this part of the work. This subroutine
uses the same idea as the Gaussian Elimination process.
Division by zero could be caused by either of the following:
(1) The stiffness matrix is singular (i.e. no inverse
exists) leading to a "no solution'' (i.e. not a
physical problem) .
OR
(2) An unfavourable choice of degree of freedom numbers
was employed.
30
6 4( ,4- 24, 25 1 26
s.--.,-----,~ /~21,22,23
IZ11S,19,20 /' 15,16,17
2~---, A 12,13,14
8th row ~
'X
8th column ~
0 1\l
l
~--------,_--~string _of zeros
Fig. 17: Unfavourable way of labelling the nodal variables.
8
,. 9,10,11
2¥ ./~ 12,13,14
15,16,17
¢' A18,19,20
30
Fig. 18: Favourable way of labelling the nodal variables.
0 w
104
This is further explained below.
In the normal Gaussian Elimination procedure,
division by zero is avoided by rearranging the equations.
With the symmetric banded matrix, however, division by zero
can occur even if the matrix is not singular. This occurs
when there is no direct connection between the previously
labelled degree of freedom numbers and the present ones.
It can best be explained by means of a diagram.
A portion of the nodal variables are labelled as
shown in Fig. 17. This results in a string of zeros in
the eighth column of the stiffness matrix as shown in the
figure. A "division by zero" message is therefore obtained
from the computer during the reduction process of the matrix.
A way to overcome this problem is to label the nodal
variables in a fashion such that they are all directly
connected, as evident in Fig. 18.
A more serious problem encountered was that this
analysis gave very good results for the pressure drop in
parallel channel flow, but failed to yield any satisfactory
results for other inclinations of the channel.
An explanation of this can be given with reference
to one of the equations of flow, say equation (8.2), i.e.
= chxx + aTxy
ax ay
Assuming a constant viscosity, we can write this equation
as;
ap ax = (12.2)
Comparing both sides of the differential equation
we can say that the pressure distribution should be a degree
1 05
lower than the velocity distribution. In this \vork,
however, linear distribution was assumed for both the
velocities and the pressure, and therefore led into problems.
While this research was in progress, a similar work by
Tanner (51) reported a solution to the problem by assuming
a quadratic distribution for the velocities and linear
distribution for the pressure over each element.
The problem of running cost of the programme was
also faced in this analysis. Due to the large number of
nodal variables required and the extremely large arrays
encountered, a considerable amount of money was spent
as far as the finer mesh is concerned. Typical arrays
stored in this work are:
Element (512,5), Anode (289,5), Delta (867),
ABC(512,9) and K(735,53). Refer to Appendix P2 for details
on these arrays.
For parallel channel flow, all the elements are
right-angled triangles. There are also two types of
elements present; one with hypotenuse at 45° and the other
with hypotenuse at -135° with the x-axis.
For parallel channel flow therefore, the area of
the elements needs to be evaluated once. This programme
evaluates all entries of the array ABC (512,9), i.e. 4608
of them. For parallel channel flow this is not necessary
because there are only two types of elements present. Thus
only ABC (1,9) and ABC (2,9) are required, i.e. only
18 elements instead of 4608. This would lead to a lot of
savings in computer time.
For converging flow, however, where the elements are
all different in area, it would be required to evaluate all
the 4608 entries.
whether:
It becomes necessary to investigate
(1) It would be cheaper to store all the 4608 entries
during the processing time, which could run into
more than 30 minutes as mentioned before.
OR
(2) Have no storage facility for array ABC and
then evaluate every entry of the array as it is
required in the programme.
Improving the rate of convergence of the procedure by
106
using an overrelaxation factor, as suggested by Palit (36),
could result in a shorter computing time. Section 2.8 of
Palit (36) gives an idea of how to determine the optimum
overrelaxation factor for a given mesh.
(d) Comparison of both analyses
The second analysis is found to be the more general
of the two. The same set of boundary conditions is used for
the analyses of both the Newtonian and non-Newtonian flows.·
Thus, for the second programme, the elemental viscosities
were assumed to be constant (~ = 1) during the first
iteration and the resulting solution was for Newtonian flow.
Subsequent iterations required the viscosities to be updated
until convergence was finally achieved. Hence, the solution
for the non-Newtonian flow problem. This programme can
therefore be used to analyse any type of polymer melt flow
so long as the flow behaviour index (n) is known.
With the resulting programme from the first analysis,
however, different boundary conditions were used for both
the Newtonian and the non-Newtonian flow problems. The
programme had to be run separately to obtain the two
107
solutions. If the analysis for another polymer is required,
a different set of boundary conditions has to be written in
order to run the programme. This is rather a slow and
time consuming approach. The programme would, however, be
extremely powerful if a set of boundary conditions could be
found independent of the flow behaviour index of the fluid.
The second analysis which uses the pressure as one
of the nodal variables gives better results for the pressure
drop in parallel channel flow. As regards cost, the second
programme was found to be more expensive and demanded a lot
of storage capacity.
Both analyses used a mesh of triangular elements with
nodal points at the corners. This cut down the algebra
involved in generating the stiffness matrix. However, higher
order polynomial functions are worth looking into. The use
of such polynomials means more nodal points per element.
The stiffness matrix formation would involve more algebra
and numerical integration. The total number of elements
required, however, to achieve a certain accuracy will be
less than that for three point triangular elements.
Oden and Somogyi (35) reported shorter computing
times for higher degree polynomial function and, correspond
ingly, more nodal points per element.
Further work on this topic should therefore consider
parabolic and cubic functions.
CHAPTER 13
CONCLUSION AND RECOMMENDATIONS
(a) CONCLUSION
The main idea behind this work was to develop a
FE technique suitable for analysing more general flow
situations. Two methods were adopted and discussed in
Chapters 9 and 10.
The FE results of the first analysis were success
fully compared with theoretical, LA and other numerical
solutions where they exist. The results of the second
analysis compared favourably with those of the first.
The second method gave better estimates of pressure drops
in parallel channels, but failed to give any reasonable
result for converging channel flows.
108
Throughout the work, themelt was treated as Stokesian.
Elastic effects are noted not to be predominant in extrusion
processes and hence, for isothermal flow, a power-law fluid
model is a good enough representation of the pseudoplastic
nature of polymer melts.
The second analysis was found to be the more general
of the two. The boundary conditions were independent of the
fluid. Different boundary conditions were required for each
fluid flow when the first programme was used. The boundary
conditions, in this case, depend very much on the flow
109
behaviour index of the fluid in question.
The use of three point triangular elements led to
considerable cut down of the algebra involved in generating
the stiffness matrix. The use of higher order polynomials
(more nodal points per element) involves more algebra and
integration in obtaining the stiffness matrix. To achieve
the same accuracy, however, higher order polynomials require
less elements. They are also noted to lead to shorter
computing times.
The input data for flow problems like the type
discussed consists of geometric dimensions of the flow
region, rheological equations, boundary conditions and the
FE representation in terms of nodal points and element
interconnections. For the problems considered, the number
of nodal points and elements were in the order of several
hundreds. It is thus desirable to use an internal mesh
generating technique. This considerably reduces the effort
involved in the preparation of the mesh data manually.
The running time of each programme is very much
dependent on the convergent criterion used. 10-5 wasfound
to be the most economical convergent criterion and was
therefore used throughout the analysis.
The velocity plots from the first analysis showed the
normal expected trends. The Newtonian velocity profile was
parabolic, while for the non-Newtonian flow the parabola
flattened. The velocity results from the first analysis
were lower, on the average, than the analytical values.
The characteristic velocity was chosen so that the flowrate
became unity. For the second analysis, the FE predictions
11 0
for the velocities were slightly higher than the analytical
values.
The results presented in this work demonstrate the
type of accuracy achieved by the FE method. Increasing mesh
refinement results in convergency to the analytical solution.
The accuracy of the FE method was observed to decrease as
the taper of the channel was increased.
Pressure drop predictions by the author are better
than those of Palit (36) due to a more direct method of
evaluating the viscosities. The evaluation of the viscosity
by an averaging procedure is known to be unsatisfactory. In
the second analysis, linear variation of pressure over each
element was assumed. This led to even better results for
the pressure drop. FE predictions could be improved further
if the elastic nature of polymer melts is taken into account
in the analysis.
In general, it is observed that the pressure drop
decreased with increase in non-Newtonian effect.
The computer in use, Burrough 6718, placed a
restriction on the size of the system stiffness matrix.
The array limit of the computer is 65535 elements; thus
approximately a size of (256 x 256). The original method
(Appendix C) of obtaining the system stiffness matrix and
the loading vector was therefore abandoned. The new method
adopted is shown in the ASSEMB Subroutine of Appendices P1
and P.2.
The reduction process of the system stiffness matrix
can yield a "division by zero" message, even if the matrix
is not singular. This occurs if the degree of freedom
numbers are unfavourably arranged as explained in Chapter 12.
1 11
The second programme was the more expensive.
The cost was mainly in the storage capacity requirements.
It is therefore essential to reduce the running time and
the storage capacity of the second programme. Several
suggestions as to how this can be achieved have been put
forward in Chapter 12(c) of this work.
The rate of convergence of the programmes can be
increased by the introduction of an overrelaxation factor
(w) as suggested by Palit (36). For a given mesh the value
of w at which the total number of cycles of iteration is
minimum, is termed the optimum overrelaxation factor w t op .
It is desirable to determine w t for the meshes in order op
to economise on the computing times.
The present work shows that the use of the FE method
for 2 D flow problems is elegant; and results of acceptable
accuracy can be produced with reasonable expenditure of
co~puting effort. The accuracy attained in a particular
problem is clearly a function of the number of elements
used and their distribution, which in turn governs the
time for solution.
To predict polymer melt flow behaviour in industrial
processes, the first step is to generate a suitable mesh
for the flow geometry. With the rheological equation and
the boundary conditions known, the two FE methods discussed
in Chapters 9 and 10 may be applied to obtain the flow
characteristics in terms of dimensionless parameters such
as pressure, flowrate, velocity, etc. From these, the
physical parameters may be obtained using the geometric
dimensions of the problem region. Alternatively, practical
problems can be formulated and solved in dimensionless
1 1 2
variables.
We have used the two programmes discussed to solve
a complex flow problem. They give us some insight into the
type of problems that might be encountered in the study of
non-Newtonian flow in extrusion dies.
(b) RECOMMENDATIONS
Early attempts to analyse the extrusion process
considered only the assumed Newtonian flow of a polymeric
material after melting. With the advent of readily
accessible, high-speed digital computers, it was possible
to include the non-Newtonian behaviour.
Polymeric fluids are viscoelastic. Thus, more
realistic predictions by the FE method would be obtained
if the constitutive equation includes both viscous and
elastic effects. Other parameters like temperature
sensitivity behaviour of polymer melt viscosity could also
be included.
In this work a mesh of triangular elements with nodal
points at the corners was selected as being the simplest and
easiest to use. For the second analysis, a linear
polynomial was also used to define uniquely the variables
within each element. The use of higher order polynomial
functions and improved elements to investigate the accuracy
and computing times of the present solutions could be
suggested. An efficient matrix assembly and solution
procedures should be developed to deal with these functions
and elements. The technique of matrix handling and solving
the algebraic equations for the present formulations may
1 1 3
also be improved, thereby economising on computer storage
and time. Palit (36) pointed out that three point elements
whose shapes are roughly equilateral give well-behaved
stiffness matrices for more general flow problems. It may,
therefore, be useful to develop a technique of generating a
mesh of equilateral triangles. It might also be desirable
to generate all the necessary data within the computer to
get rid of the manual effort and the time involved.
With both programmes discussed, the first pass
computes the constant viscosity solution, and iteration
proceeds from this. We then recompute the viscosity at
each point and use the new values in the next cycle of
computation. The process was noted to be very slow. It
is therefore recommended that a modified value of the
viscosity is used each time, and also that an overrelaxation
factor be introduced in order to increase the rate of
convergence of the solution.
The second programme gave very good results for
parallel channel flow. It is, however, known to be
inefficient and it is essential to improve upon it and
extend it to the analysis of converging channel flow.
Attention has so far been confined to isothermal,
two-dimensional flows. The FE method, however, can be
applied to more general problems. The Cartesian FE
formulation can easily be extended to axi-symmetric
problems, such as melt flow in converging cylindrical and
annular dies, and wire coating dies.
The formulations presented here are applicable to
solid stress analysis and lubrication problems. For
1 1 4
example, the flow in the narrow clearance between journal
and bearing which is identical to couette flow with
pressure gradient. These formulations are also applicable
to free surface flows (for example, open channel flows) and
there are possibilities of extension to unsteady fluid flow
and heat transfer problems.
Different element properties (e.g. viscosity,
conductivity, elasticity, etc.) can easily be introduced
in the FE formulations. The methods presented, therefore,
may be applied to two-phase flows which are very common in
the compressing and metering sections of screw extruders
and other inhomogeneity problems.
The use of solid elements like tetrahedrons may
make the methods extendible to three-dimensional flow
problems.
REFERENCES
1. Atkinson, B. (1971) Trans. Inst. Chern. Engrs.,
48, T276, "Application of the finite element
method to creeping flow problems."
2. Atkinson, B., Brocklebank, M.P., Card, C.C.H.
115
and Smith, J. M. ( 1 9 6 9) A. I. Ch. E. J. , ·.1.2 ( 4) ,
pp.548-553, "Low Reynolds Number Developing Flows."
3. Atkinson, B., Card, C.C. and Irons, B.M. (1970)
Trans. Inst. Chern. Engrs., ~ (7-10), pp.276-284,
"Application of the finite element method to
creeping flow problems."
4. Bagley, E.B. J. Appl. Phys., 28, 5, pp.624-627,
(1957), "End corrections in the capillary flow
of polyethylene."
5. Bagley, E.B. and Birks, A.M. J. Appl. Phys.,
31, 556, (1960).
6. Bagley, E.B., Cabott, I.M. and West, D.C. J. Appl.
Phys., 29, 1, pp.109, (Jan. 1958).
7. Bagley, E.B. and H.P. Schrieber, Trans. Soc. Rheol.,
5, 341 (1961).
8. Benbow, J.J. and Lamb, P. SPE, Trans. 3, 7, (1963).
9. Bartos, O.J. Poly. Sci., B3, 1025, (1965).
10. Benis, A.M. (1967) Chern. Engng. Sci., ~'
pp.805-822, "Laminar flow of power-law fluids
through narrow three-dimensional channels of
varying gap."
11. Bird, R.B. SPE. J. 11, 35, Sept. 1955, "Viscous
heat effects in extrusion of molten plastics."
11b Blyumental, M.G., Lapshin, V.V. and Akutin, M.S.
"The flow of polymer melts in extrusion dies."
12. Card, C.C.H. 1968). Ph.D. Thesis, Univ. of Wales,
Swansea, "A study of polymer extrusion."
1 1 6
13. Clegg, P.L. Brit. Plastics, 39, pp.96-103, Feb. 1966.
14. Clegg, P.L. "Rheology of Elastomers," p.174,
Pergamon Press (1958).
15. Cogswell, F.N. and Lamb, P. (1969) Conf. Practical
Rheol. Polymer Processing, London, paper 2, "Material
Properties Relevant in Melt Processing."
16. Conte, S.D. (1965) Elementary Numerical Analysis,
McGraw-Hill.
17. Eirich, F.
1 96 9.
"Rheology," vol. 5, Ch. 2, Academic Press,
18. Galt, J.C. and Maxwell, B. Mod. Plast. 42, 4115,
1964.
19. Gee, R.E. and Lyon, J.B. Ind. Eng. Chern. 49, 956,
1957, "Non-isothermal flow of viscous non-Newtonian
fluids."
20. Han, C.D. and Lamonte, R.R. Polym. Eng. and Sci.,
11, 5, pp.385-394, Sept. 1971. "A study of polymer
melt flow instabilities in extrusion."
21. Howells, E.R. and Benbow, J.J. Trans. Plast. Inst.,
30, 240, 1962.
22. Hutton, J.F. Proc. Royal Soc., 287, 222, 1965.
23. Kaplun, Ya. B. and Levin, A.N. Soviet Plastics, 2,
p.48-53, Feb. 1966.
2 4. Lupton, J.M.
of Poly.
A. I. Ch. E / S.P.E. Symp. on Fund.
Proc., Phil., Pa. 1963.
25. Martin, H.C. (1968). Proc. 2nd Conf. on Matrix
Meth. inStruct. Mech., pp.517-535, "Finite Element
Analysis of Fluid Flows."
26. Maxwell, B. and Galt, J.C. J. Poly. Sci., 62, 550,
1962.
27. Metzner, A.B., Carley, E.L. and Park, I.K.
Mod . P 1 as tics , 3 7 , 1 1 , pp . 1 3 3 , 1 9 6 0 .
28. Metzner, A.B., Uebler, E.A. and Chan Man Fong, C.F.
( 1 9 6 9) ' A. I . Ch. E. J. ' 1 5 ' ( 5) I pp. 7 50-7 58 ,
"Converging flows of viscoelastic materials."
29. Metzner, A.B. and White, J.L. A. I. Ch. E. J.,
11, p.989, 1965.
117
30. Metzner, A.B., White, J.L. and Denn, M.M. A. I. Ch. E.
J., 12, 5, p.863, Sept. 1966.
31. Metzner, A.B., White, J.L. and Denn, M.M. Chemical
Engineering Progress, 62, 12, p.81-92. "Behaviour
of viscoelastic materials in short-time processes."
32. Middleman, S. and Gavis, J. Phys. Fluids, 4, pp.355,
1 9 61 .
33. Oden, J.T . (1969) Int. J. Num. Meth. Engng.,
..:!_, (3), p.247-259.
34. Oden, J.T. (1970) Proc. A.S.C.E., 2_§_, EM4, p.529-534,
"Finite element analogue of Navier-Stokes equations."
35. Oden, J.T. and Somogyi, D. (1969) Proc. A.S.C.E.,
~' EM3, p.821-826, "Finite element applications in
fluid dynamics."
36. Palit, K. Ph.D. Thesis, London Univ., (1972).
37. Palit, K. and Fenner, R.T. A.I. Ch. E. J., 18, 6,
p.1163-1170, Nov. 1972.
38. Pearson, J.R.A. (1966) Mechanical Principles of
39.
40.
41 .
Polymer Melt Processing, Pergamon Press.
Pearson, J.R.A. Trans J. Plastics Inst., 30, 230,
1 962.
Pearson, J.R.A. Trans. J. Plastics Inst., 31, 125,
1963.
Pickup, T.J.F. (1970) Ph.D. Thesis, Univ. of
Cambridge, "Converging flow of viscoelastic fluids."
42. Pipkin, A.C. Technical Report No. 9, Army Research
0 f f ice ( 1 9 6 5 ) .
11 8
43. Poller, D. and Reedy, O.L. Mod. Plastics, 41, p.133-
138, March 1964, "Designing dies for monofilaments."
44. Powell, P.C. Polymer Eng. and Science, 14, 4,
p.298-307, April 1974, "Design of extruder dies
using thermoplastic melt properties data."
45. Ryder, L.B. SPE Journal, p.1305-1309, Dec. 1961,
"End correction implications in die design and
polymer processing."
46. Ryder, L.B. 15th Annual Tech. Con£. SPE, New York
City, Jan. 1959. "The significant flow properties
of thermoplastics."
47. Schrieber, H.P., Bagley, E.B. and Birks, A.M. J.
Appl. Polymer Sci., 4, 362, 1960.
48. Schulken, R.M. and Boy, R.E. SPE J., 16, 423, 1960.
49. Spencer, R.S. and Dillon, R.E. J. Coll. Sci.,
3, p.163, 1948.
50. Spencer, R.S. and Dillon, R.E. J. Coll. Sci.,
4, 241, 1949.
51. Tanner, R.I. Dept of Mechanical Engineering, Univ.
of Sydney, Sydney, N.S.W. 2006, "Finite Element
Methods for the solution of some non-Newtonian
fluid mechanics problems."
52. Toor, H.L. Ind. Eng. Chern. 48, 922, 1956, "The
energy equation for viscous flow."
53. Tordella, J.p. ' J. Appl. Phys., 27, p.474, 1956.
54. Tordella, J.p. ' J. Appl. Poly. Sci., 7, 215, 196 3.
55. Vinogradov, G. V., Malkin, A. Ya. and Leonev, A.L.
Kolloid z' 1 91 ' 25, 1963.
56. Westover, R.F. Poly. Eng. Sci., 6, 83, 1966.
57. Westover, R.F. and Maxwell, B. 13th Annual Tech. Con£.
SPE, Jan. 1959, "Flow behaviour and turbulence in
polyethylene."
58. Yun Ui. SPE Trans., 4, 295, 1964.
59. Ziabicki, A. and Kendzierska, K. Koll. Z.,
171, No. 2, 1960.
1 1 9
60. Zienkiewicz, O.C. (1971) The finite element method
in engineering science, McGraw-Hill.
120
APPENDIX A1
Dimensional Analysis
Consider the various equations governing the flow of
the fluid as below:
ap aT XX +
aTxy ax = ax -ay- ( 1 )
au + av 0 ax ay = (2)
11 = 110 r~r-1 ( 3)
Let us define a characteristic velocity and height being
vc and H respectively. We then have the following dimension-
less quantities (variables with dashes):
u' u x' X = - = v H c
v' v Y' y
= = v H c
n-1 v
110 (H~CJ Yc c and (4) = 11c = H
where y = the reference shear rate and 11 0 is the effective ' 0
viscosity at a reference shear rate of y • 0
i.e. llo
v c H
121
i.e. vo [v; r 1 -n-1 Yo
( 5)
Also TXX 'v' = __ll_ T = --
]Jc XX Tc
and P' = _E_
T (6)
c
from ( 1) ,
aTXX = +
(lx
Substituting dimensionless quantities in the above equation
we obtain
=
This implies that
aP' ax' =
aTyx' -a-·
y
which is similar to equation (1).
(7)
( 8)
Similarly, the continuity equation in dimensionless
form becomes
Now for
412
we have
412 =
au' ax'
the
=
v 2 c
H 2
+ av' ay'
= 0
equation of state:
2 au 2 + 2 av 2
(au + av] 2
+ ax ay ay ax
~ au' 2 av' 2 (au' dv'J] + 2 + -,
ay' + a I ax a I y X
(9)
Now define the dimensionless 4I 2 thus: 4I 21 as
the terms in the bracket on the previous page.
= v 2
c
H 2
4I I 2
From equation (6),
].1' = L J.lc
i.e. [~ r1 _1 [HY 0
] ].10 Yo J.lo Yc
i.e. [/4I2 H)(or-1
'ito vc
i.e. [lli12 _ll_r1 vc
From equation (10),
v /4I 2
c v'4L""' = H 2
].1' [vc /4L' H 2
n-1
VHJ n-1
i.e. ].1' = ( ) n-1 /4I 2 '
122
( 1 0)
which is also in the same form as the original equation of
state. Hence, variables without dashes will be used
to represent dimensionless quantities.
123
APPENDIX A2
Finite element analysis of fluid flow
within a converging channel using ~ and ¢.
Consider the geometry of the problem region as shown
in Fig. 7.
The equilibrium conditions to be satisfied are:
ap aT aT XX + yx
ax = ax 3Y ( 1 )
ap aT aT = ____E_ + yy
ay ax ay ( 2)
Variational approach
The total potential energy of the system is given by
The integral of this energy must be minimised in terms of
the nodal values. The above expresses the equilibrium
condition as in (1) and (2).
To satisfy the continuity equation:
au ax + av
ay = 0,
we introduce the concept of a stream function (~).
i.e. u = -~ ay and v = ~ ax
Now for u and v to vary linearly over the element, the
stream function over an element must be given by
( 3)
( 4)
124
tjJ = c1 + c 2x + c 3y + c x 2 + c y2 + c6
xy ( 5) 4 5
where the c's are constants expressible in terms of nodal
point variables. Now
412 2 au 2 + 2
av 2 + [au+ "vj' ( 6) = ax ay ay ax
u = a¢ and v = a1jl - ay ax
Thus a1jl u = = - c3 - 2 c y - c
6x - ay 5
au 2 c5 ( 7)
ay =
And
v = a1jl = c2 + 2 c 4x + c6y ax
av 2 c 4 ( 8)
ax =
Also
au ( 9) ax = - c
6
and av ( 1 0) ay = - c
6
Substitution in (6) gives
= ( 11 )
Consider an element (m) as shown in Fig. 11.
The nodal variables are:
u.,v.,1jl., l l l
u., v., 1/J., J J J
corresponding to the points i, j, k respectively.
BUt 1/J = c 1 + C 2 X + C 3 y + C 4 X 2 + C 5 y 2 + C 6 xy .
Thus, six constants to be determined with the nine
nodal variables.
We therefore define a new variable
= u + v =
The nodal variables are therefore reduced to
cj> . ' ljJ . ' 1 1 cj> . ' ljJ . ' J J
Substitution in the equation for ljJ gives
From ( 12) ,
Thus cj> = A + Bx + Dy
Substitution in (16} gives
( 1 7 ' ( 1 8)
cj>. = A 1
cj>. = A + J
Ba. + J
cj>k = A + Bak +
and ( 1 9) give
B+D -2- = c4 - c5
Db. J
Dbk
where 6 is the area of the element given by
125
( 1 2)
( 1 6)
( 1 7)
( 1 8)
( 1 9)
c!>· 1
( 2 0)
Further elimination gives
where E =
J =
K =
a1 =
and
= 1 E
26JK(J-K)
a. + b. J J
ak + bk
T
46 (J -K) l)J. l
46K
-46J
- [46ajK-46akJ + (bj -bk) (aj 2 K-ak 2 J)
126
( 21 )
( 2 2)
Thus r 2 can be found by substituting (20) and (22)
in to ( 11 ) •
Now the viscous dissipation functional is
from (3).
The integral required to be minimised in terms of
the nodal variables is therefore
X = f f f F dv ( 2 3)
= Wr r 211 I 2dA J J
127
where W is the width of flow normal to the x-y plane.
Substituting for r 2 gives
X = ( 2 4)
The condition to be satisfied for every nodal point variable
such as ¢· is l
= 0 .
This leads directly to the fluid element stiffness equation,
which takes the form:
[a e axe J [Ke] [</Je,¢e] = ft' (25) Cl¢e
Differentiation of ( 2 4) with respect to the nodal variables
gives equation ( 2 5) as
y 2 1 y1y2 y1y3 y1y4 Y1Ys y1y6 </1.
1
y1y2 y 2 2 y2y3 y2y4 y2y5 y2y6 ¢.
1
y1y3 y2y3 y 2 Y3Y4 Y3Ys y3y6 </J. 4]J6W 3 J (26)
y1y4 y2y4 y3y4 s 2+. 2 1 y4 S1S2+Y4Y5 S1S2+y4y6 ¢.
J
Y1Ys Y2Ys Y3Ys S1S2+y4ys s 2+y 2 2 5 S2S3+YsY6 </Jk
y1y6 y2y6 y3y6 s1s3+y4y6 S2S3+YsY6 s 2+y 2 3 6 ¢k
where y1 = 46 (J-K) E
y2 46K = E
Y3 -46J = E
a· y = 1
4 E
y5 Cl.2
= E
128
a3 y6 = E
81 = (ak - a. + b. - bk)/46 J J
82 bk- ak
= 46
and a. -b.
83 = J J 46
APPENDIX A3
Dimensionless boundary conditions ·(lp and ¢).
The boundary conditions prescribed in this work
are the stream function 1)! and the sum of the velocity
components ¢ .
Conditions at the top of the channel are:
1)! = 0 and ¢ = 0.
At the bottom we have
1)! = - 1 and ¢ = 0.
In general,
1 = 2(1+n) ·[2YH1 l [2YH1 l --1 -n --1 H H
and
= (~) 1+n ~. [2YH1 ]~ + 1J 1 - -- 1
H
2YH1 --1
H
129
where H is the depth of the channel at the point of interest.
y = is the dimensionless coordinate
and H1 is the height at the inlet of the channel.
130
APPENDIX A4
Evaluation of u (the x-component velocity) of an element.
It was assumed in the formulation of the problem
(Appendix A2) that the stream function distribution over an
element is given by
( 1 )
where the c's are constants expressible in terms of nodal
point variables. But
u = ( 2)
i.e. ( 3)
All that is required here is to determine c 3 , c 5 and c 6 for
an element. Now
¢ = u + v
i.e.
It is assumed that~·, ¢., ~., ¢., ~k and ¢k are known at l l J J
the three vertices of the element.
c 6 has already been determined in Appendix A2. So
also is (c 4 - c 5 ) in the same appendix.
By the use of equations (1) and (5) and the nodal
values, c 3 and c 4 can be found.
Thus enabling u[equation(3)] to be obtained.
( 4)
( 5)
a b ~ a
n n+1 n+2
Fig. 19: Channel boundary division for the forward 'difference formula.
a b ·-a
n-1 n n+1
Fig. 20:- Channel boundary division for the central difference formula. ·
b - a a . ~ ~
ri-2. n-1 n
Fig. 21: Channel boundary division for the backward difference formula. ·
"'
131
APPENDIX A5
Pressure drop (6p) across the channel.
From the basic equations of flow we have
ap ax =
aT XX
ax + aT xy ay
The integral of the above gives
pressure drop.
i.e. r dp ·- dx =
J ax
where T XX
au = 2 f1 a X
I: a ( 2 au) dx + ~ ax flax ay
and Txy =f.!(~~+~~)
the required
(au ay
+ av)dx ax
The f1 here refers to the node at which the integral is
evaluated.
Now the partial derivatives can be approximated by
finite differences.
The development of Taylor's series for f(x + 6x)
about (x,y) gives
Ax~ (6x) 2 a 2f f(x+6x,y) = f(x,y) + Ll ax (x,y) + 2! (x,y) .
ax 2
With this, the finite difference formulae are derived.
Forward Difference Formulae
Consider an interval as shown in Fig. 1 9.
f f + f' + a2
f" = a 2 n+1 n n n
f f + b f' + b2
f" = n+2 n n 2 n
1 32
( 1 )
( 2)
( 3)
( 4)
( 5)
( 6)
P + op
1 2 3 4 5 6 7 8
Fig. 22: Flow region showing the nodal points along the bottom boundary.
p
~
9
_. w w
Eliminating f" from (5) and ( 6) we obtain n
b 2 f - a 2 f - (b 2 -a 2 )f f' = n+1 n+2 . n
n ab(b-a)
and hence
2[bfn+ 1 - af - (b-a)f ] f" n+2 n = n ab(a-b)
Central Difference Formula
With reference to Fig. 20, we obtain similar
expression for the central difference formula as
and
f' n =
f II = n
2 2 2 2 a fn+ 1 - (b-a) fn_ 1 - (a -(b-a) )fn
ab (b-a)
2[afn+ 1 + (b-a)fn_ 1 - b fn]
ab(b-a)
Backward Difference Formula
to Fig.
and
The backward difference formula with
21 is
f' = n
f II = n
2 b
2f (b
2-a 2 )f a f 2
- + n- n-1
ab(b-a)
2[afn_ 2 - bfn_ 1 - (a-b)fn]
ab(b-a)
n
reference
134
For a channel as shown in Fig. 22, the forward and backward
difference formulae were applied at nodes 1 and 9 respective-
ly. For the intermediate points, the central difference
formula was used.
APPENDIX A6
Theoretical velocities and viscosities
The second invariant of the rate of deformation
tensor is given by:
= Clu 2 Clv 2 Clx + 2 Cly 2 ( 0 u 0 v) 2 + - +Cly Clx
For fully developed flow,
au = o, av o, au 0 and = = ax ay ay
4I 2 = au 2
ay
Now 11 = (/4I)n-1 2
i.e. 11 = (~~Jn-1 a'(
Also ()p = ____l5:X ax ()y
'( au xy = 11 ay But
ap = a (11 au
ax ay ay
i.e. ap = a (au)n- 1au ax ()y ay ay
ap = _Q__ (~~Jn· ax ay
Integrating (9) we obtain
av = 0 • ax
1 35
( 1 )
( 2)
( 3)
( 4)
(5)
( 6)
( 7)
( 8)
( 9)
1 36
r~~t = dp dx y + A. ( 1 0)
For non-Newtonian flow, n = 0. 5, we have
du rdp r , dy = dx y (11)
Integrating ( 11 ) we have
r~~r 3
u = y + B 3 ( 1 2)
at y = 2 ' u = 0
B 1 r~~ J' = 24
r~~ J' [f 1 l .. u = +24 ~ ( 1 3)
Let dp = 'IT (the dimensionless pressure drop) . dx p
'IT 2
+ .l) u = ___£ (y3 3 8
( 1 4)
The dimensionless flowrate 'ITQ is given by
= f u dy ( 15)
f-~ 'IT 2
.J
_E (y 3 + .l) 2 dy 0
3 8 i.e. ( 1 6)
'IT 2
p 32" i.e. =
But 'ITQ = - 1
'IT 2 = 32 p
.. 'IT p = 132
= 4ff .
From ( 1 4) ' we have
32 (y 3
+ 1 ) . u = 3 8
As a check, at y = 0, u = 4/3 and at y = .:!,. -z, u = 0.
Now the dimensionless viscosity ~ as from (4) is given by
~ =
i.e. [~~] n-1
Substitution for u = 3{ (y 3 + i) and n = 0.5 gives
1 at y = 2 ~
= 1
412Y
= 0.354
Similar procedure is adopted for Newtonian flow .
.With n = 1 in equation (10), we obtain
u =
and 1T = p
2 1 6[y --4
1 2
137
138
APPENDIX B1
Finite element analysis of fluid flow using u, v and p.
Consider an element as shown in Fig. 11. Let the
velocity variation within this element be linear, as shown
below.
where a 1 , a 2 and a 3 are constants to be determined.
Applying the known nodal values we obtain
u. l
=
u. = J
=
elimination gives
=
where 6 is the area of an element. 26 is given by:
1
26 = 1
1
x. l
x. J
y. l
y. J
Further elimination gives a 1 and a 3 as
=
( 1 )
( 2)
( 3)
( 4)
(5)
( 6)
(7)
139
and
1 Gk -xj' xj-xJ ~~l ( 8) a3 = xi-xk, . 26
These can be written as
1 u~ a1 = ri'
a.' a~ u. ( 9) 26 J J
uk
1 Gi' bJ [i a2 = 26 b.' u. ( 1 0)
J J uk
and
1 Gi' cJ [i a3 = 26 c. ' u. ( 11 )
J J uk
where
a. = xjyk - xkyi l
a. = xkyi - xiyk J
ak = x.y. - x.y. l J J l
b. = y. yk l J
b. = yk - y. J l
bk = y. - y. l J
c. = xk - X, l J
c. = x. - X J l k
ck = x. X. J l
Substitution of (9), (10) and (11) into (1) yields
i.e. u =
where theN's are the shape functions given by
=
Similarly,
v =
and
p =
Now
dU. 2 av 2 = 2 + 2 -- + ax ay
au = 216 [ bi ui ax
av 1 = 26 ~.v. ay l l
au 2
ay
+ b.u. J J
+ c.v. J J
au 1 ~.u. + = c.u. ay 26 l l J J
+ av ax
2
+ bkuk]
+ ckvk]
+ ckuk]
av 1 ~.v. + bjvj + bkvk] = ax 26 l l
+ 2~. av ay ax
i.e. ~~ 4I2 = (u. ,u, ,uk) _1_ 2 [B) :~ +(v. ,v. ,vk)-\ [c) ~ j 1 +(u. ,u, ,uk)-
2 l J 2 6 J l J 26 J l J M ~ vk
1
Gj + (u,,u,,uk) _1_ [D) ~~ + (v.,v.,vk) -[B) l J 46 2 l J 26 2
140
( 1 2)
( 1 3)
( 1 4)
( 15)
( 1 6)
( 1 7)
( 1 8)
( 1 9)
( 2 0)
( 21 )
[c) ~~ (22)
where
and
i.e.
_1_ X 462
Now
[B) =
[C] =
[D) =
4I2 =
2 b. ' l
b .b. 1 J l
2 c.
l '
c. c. ' J l
ckci'
c. b.' l l
c .b.' J l
ckbi'
b. b.' l J
c. c.' l J
2 c. ' J
ckcj,
c. b.' l J
c. b.' J J
ckbj'
( u . , u . , uk , v . , v . , v k) l J l J
cick
cjck
ck 2
cibk
cjbk
ckbk
X
2b. 2+e. 2 ,2b.b.+e.e.,2b,b +e.ek,e.b,,e.b.,e.bk l l l J l J l k l l l l J l
2b.b.+e.e. ,2b. 2 +e. 2 ,2b.bk+e.ek,e,b.,e.b.,e.bk J l J l J J J J J l J J J
2bkb. +eke . , 2bkb . +eke . , 2bk+e 2 , ekb. , ekb . , e b
l l J J k l J k k
2 2 e.b.,e.b, ,ekb,,2c. +b. ,2c,c.+b.b,,2c.ck+b,b l l J l l l l l J l J l l k
2 2 c.b.,c.b,,ckb.,2c.c.+b.b. ,2c. +b. ,2c.ck+b,bk
l J J J J J l J l J J J J
X = f (4~I 2 + 2~·Vp)dA A
1 41
u. l
U, J
uk
V, ( 2 3)
l
V, J
vk
( 2 4)
Let
and
4I 2 has
where
and
x1 = t 4ll I 2dA
x2 = fA
2~ • \lpdA
x1 = f 4ll I 2 dA
x1 = 4I 2 ll~
already been obtained as in equation ( 2 3) •
x2 = fA
2~·\lpdA
~·\lp = ap +
ap u- v-ax ay
Using equations (13), (15) and (16) we obtain
[AB] =
[AC] =
p. l
p. J
pk
a. b., l l
a .b., J l
akbi'
a. c., l l
a. c. , J l
akci,
a. b., l J
a .b., J J
akbj,
a. c., l J
a. c., J J
akcj,
aibk
ajbk
akbk
aick
ajck
akck
p. l
p. J
pk
1 42
( 2 5)
(26)
( 2 7)
( 2 8)
(29)
For an element, the partial differential of X with respect
·to the nodal variables (u. ,v. ,p.), (u. ,v. ,p.) and (uk,vk,pk) l l l J J J
is given by
143
ax e u.
au. 1 1
~ v. av.
[Ke] 1 (30)
1 =
where [Ke] equals.
r-2a.b. 2a.b. 2aibk 2 2 2b . b . +c . c . , 2bkbi+ckci, c. b. c .b., cibk ,
l. J l. J 2b. +c. , , , , 11 I I u. l. l. l.J 1.] l. l. l. J fl fl l.
2a.b. 2a.b. 2ajbk 2 2 2bjbk +cj ck , c.b. c.b. cjbk ,
J l. J J 2b.b.+c.c., 2b. +c. , , , fl
, fl
, u. l.J l.J J J J l. J J fl J
2bkb.+ckc., 2b.b +c.ck, 2 2 ckbi ckbj ckbk ,
2akbi 2akbj 2akbk 2bk +ck , , , , , I I u l. l. ]k J fl fl fl k
2a.c. 2a.c. 2aick c.b. c.bJ ckbi
2 2 2c.c.+b.b,, 2ckc.+bkb.,
l. l. l. J , , , 2c. +b. , , , I I V, l. l. J l. l. l. l.J l.J l. l. fl fl fl l.
xj 2a.c. 2a.c. 2a.ck fl c.b. c.b. ckbj 2c. c . +b. b . , 2c. 2+b. 2 , 2cjck+bjbk, J l. J J J v. I ( 3 0) 4/:,. , , , , , l. J J J 1.] l.J J J fl fl fl J
2~ci 2~c. 2akck cibk cjbk ckbk 2ckc.+b b., 2c.c +b.b , 2ck2+bk2 J ,
' , ,
' , I I v l. k l. J k J k fl fl fl I k
2a.b. 2a.b. 2akbi 2a.c. 2a.c. 2akci l. l. J l. l. l. J l. 0 0 0 , , f , , , , , p, fl fl fl fl fl fl l.
2a.b. 2a.b. 2akb. 2a.c. 2a.c. 2~cj ~ J J J l. J J J 0 0 0
I I pj
' , ,
' , , , ,
fl fl fl fl fl fl
2a.b 2ajbk 2akbk 2aick 2ajck 2akck l. k 0 0 0 I Pk .
, , , , '
, , ' I fl fl fl fl fl fl --'
+=o +=o
145
APPENDIX C
Part of the initial computer programme
146
~·•IWTATIOIIS ~·: 'i
1( ... ·:·,': ~\· ..... ~· .. ..,,_ • .., ... ·.': ., ... .. ·: 'i'.' •,': ...... ., ......... ·.'t .... .......... ~· ........ .. ·: .. • .... ,a ........ ~·:·,':;',· .......... ~ ..................... .., ... ;': .,., i\· ...... ·:.· .. ·: ;': -.·: ... ·: ·:: ·:~ ".1t ., .............. ·: • ..... ~·; i': ·,',• i\' ;': ;': ·,':;': ..... -:: ..... ;': ., ...
~·: !\ ~·· .. ,~, ..................... ..
AJ,f1J,AK,81\::U1CAL COOfW!II:\TF.S OF i\11 ELEI'F.IIT 1/IT:l IU-:::rccr Tn THE !1 llrJ'lE. ;\!1 ELE'IEIIT IS LA8r:LLEO !1, !?., \3 Ill 1\11 /\!ITICU1C!\1/IS[ FASHIOII,
C(IIDF.-l)=STDRF.S THE PRODUCT OF f\2(11DF,tl) .t\IID OELT?.(H).
DELTA (til\) =N~RA.Y OF Kll\J\111 AiiD lJI1KtiOI-Itl NODAL \1 AF: !ABLES.
DELTl (IIDF)::fU".QUIRED llO[);~L V/'IR!J\BLES.
DELT2(1l)::l\IIOI·IIl I,IODAL VALUES( PART OF),
~·: K ;':
K(NK, IIK)=SYSTE11 STI FFtiESS !lATRIX.
KE(6,1.)):::ELE11EllTAL STTFFIIESS IIATRIX.
Kl(!!OF,I!DF)=A SU8·11ATR!X OF THE SYSTEtl STJFFIIESS H-'\TRIX.
K2(11DF,Il)=l~ SLJEl-1·1/\TRIX OF TfiE SYSTEII STIFFIIESS !lATRIX TO flE IIULT!Pl.IEO BY OF.LT2(1l)
K3(1JDF, !f31/)=THE: lJPPF.R TrUAIIGULAR CLEIEI!TS ·oF KI(I!DF,N) ARE STCJREO Ill THIS f{ECTI\IIGULAR iii\TfUX. TillS FOF\IIS THE RESULTI fiG STI FFIIESS 1\L\TRIX TO GE USED Ill FIIIDIIIG THE IIOOAL VARIABLES.
HDELTI\(G)=ARRAY COIITI\!IJIIIG Tf!E D,O.F. lllJ~13ERS OF A HOD F..
SUGROUT f1iE :=ti.LCK(KF 1 ELCIIET 1 1\IIODE, l ,liE, 1~1<, Tllii 1 1-1,H)
c c ;': ....... ,., •,': ..,.,.., ... -.'.· ..... ;\• ..,., ·,': .... ·:.·-,':;': ·.': ·.': ·.': ·:: ·.': ·.':·::·:: ;':·:: ...... ·:: ~:-: '.'t ..,., ..... ·.': ·,': ·,': ·.·:-.': ••• ....... ..,., ......
C C1\Lt.:UL!\TF.S THE ELF.I!":IlT.i\t. STIFFiiESS 1 11\Tf~!X c ... ·.~ ...... ., ............... .., ... ';\' .., ............... ·::·.': ..... ;': ;\• ............ ;': ..... ·.': ;'; ~·· .':·,': .., .............................. ~· .. ~· ... · .. ·:: ',':;·'· ·.· . ..,· .. ;\·-.,· ..... · .. c
REAL K!::((J,6) 0 Ir IE i IS I 0! I f:L E11E T (liE 1 I{) , 1\l JOD E ( IHI, If)
c C EL:::rtEIJT>\1. Gt OIIETRI C CIJIIST/\IJTS RCQU IRED. c
c
CI\LL Gi\ 1 1:1J~T( 1)1 '(!?I G'·l I r,I, 'GS. (;() '[11 '8?., T ~)) ) ~~~F. i\' 1\.11 fl J ' ;'\!\, fll~, J fl 'I I 1(1\ y I F, TE L E 11 ET I /\! IOlJ E, I I F /\R I I u , F 1\f"{ I liE ' IlK' IH I I \I J H)
F 1 ::F,\RI\W:G I F2 ..:F t\RI \W:G2 F 3= F /\R l·ttJ:': G3
C C/\LCUU\TF: KE II()..J. c
1<. E ( 1 , 1 ) =F 1·:: c; 1 I( E ( I I 2 ) :: F 1 :': Glf I( E ( 1 I 3 ) :: F p G2 KE(1 ,lf)=FJ:':G5 KE( 1 ,5)=FJ:':G3 i\ E ( 1 , 6) =F I:': G6 K E < 2 , 2) = F 1\ R' 1 u-:, c < s1·::·:,?.) + (r;lf .,.,.,., 2) ) I(E ( 213) =F?.:':Glr 1< E < 2 I 1 f) = F 1\ R r 1 u·:: ( f3 p f3 2 + G 1 f ·:: G 5 ) KE( 2,5 )::FJ·::c;lr I( E ( 2 , ()) =F 1\R r IW: ( 81 :': f3 3 + Glp': G6) K E ( 3 , 3 ) ::: F2 :'.· G2 I\[ (.3 I If) ::F2·.'•G5 I~E (3, 5) =F?.:':GJ l<f:: (3 ,G) =-=F2:'•G6
. 1<. E ( If , l f ) ::FA P r I'J:': ( ( f3 2 :'::': 2 ) + ( G 5 ~·d: 2 ) ) l:E·(4, 5) =F3:':G5 !<. E ( lr , 6 ) =F /\R 1\U:': ( f32 ,., l3 3 + li5''' G6) 1\ E ( 5 , 5 ) = F 3 ·:: G 3 I<E ( 5, ti) =F J~':G6 K E ( 6 , 6 ) ::: F '~ R! 1 lJ:': ( ( 8 3 ~·: :': 2 ) + ( G6 ~·: ·::?. ) ) DO ?.6 12=1 ,5 DO 26 J:: 12+1 ,6 I<E(J, I2)=i~E( I2,J)
26 COI!TI rJUE RETURII OlD
147
c
:~llflnJlUT r 'IF. r r1s r:~ T( 1\, F:IJ' lET 1 J\llflfJ!~, r, T' I[ I !·!!':I : !i I, \I I II)
c **************************************** C TillS l'l~;r:rns Tli[ EU~I':~'IT/\l. ST!FF 11ESS C 1\:\'Trn:< 1:1r1 TilE SYSTE:I ~;TlFF'IC:SS !11\Tf(f:(, c **************************************** c
REi\L l:(llfZ,IIK),I<E(6,G) OIIIEIIS!'l!l ELE:iET(IIF.,h),,\!IODE('Iii,lf):
T! !') ELT /1 ( 6) c C I 1001\L I llli\!JF:J; S, c
c
l I =EI.UIFT( I I 1) 12:: ~: L ~~· 1 C f ( I, ? ) l3=EL.E!IET( !,3)
C D,O,F. tllJIWERS, c
ii[)ELT.'\ ( 1) =-;\! lfJ[) E( ! 1 , 3) :In El. Til (2) ::,A, 110[) r:( I 1 ,If) lin EL Tt, (3) =/\ '!OD 1:( !?. , 3) II fJ r:: l.T II ( If ) :: 1\ I I 0 D E ( I ?. 1 Lf ) :IDELTA(5)::;\I!O!lF.( !3,3) liD ELTII ( (i) =II llnDE( 13 ,If)
C 0!3TI\TII LOC/\L CfJOROIIJ;.'\Tt.S \liTH C RESPECT Til THE !1 110f1C OF [f\CH ELEIIE~IT(!\S ORIGIII)
,~.J=MIDrJ[( I?., 1 )··MJODF:( II, I) rlJ=LJ.IIODE( I2,2)-AII[)DE( fl ,2) /IX=i\l'IODE( I3, 1) -J\IJDDF.( !1 I I) FlK=AIIODF.( I~ ,n-AIIfJDE( II ,2)
c C KE CAl-l 1101/ BE CALCULATED, c
c
Ct\LL CALCI<(I<E, El.FJ1ET I IIIlO[)[, I I liE, lll~,IHI, HI,H)
on 25 I1 :;J ,6 DIJ 25.1=1 ,6
C I !·ISERT K E I! ITO K I WI·/, c
I< ( i I 0 EL T A ( I 1 ) 1 110 E L T A ( .J) ) :: K ( i I D E L T A ( I 1 ) 1 I 10 EL T A ( ,J ) ) +I< E ( I 1 , ,J ) 25 cmn r tiUE
P.ETURII EI,ID
148
c
SUflHOUT I liE ~>lJil( 1<, ELDIET 1 AIIODE, HE 1 tiK, HIN,I/ 1 11)
c ****************************** C THE SYSTEit ST!FFI!ESS !1!\TfU:< IS. C ASSErtrli.ED c ****************************** c
c
RF.AL 1\(i!IZ,I!K) [)II tEllS; flll El. :-:1\F.T( I!E 'If) I !\I lODE( !Ill, II) DO ?. 3 I ::c 1 , I !K 0 0 2 3 .J::: 1 , Nl<
23 K(I,J)=o.o DO 21+ IC::: 1 I liE
C ELE'IEIITJ\1 STIFFr!f:s~; 1\4TRIX IS 1\[(!tl!RED TCl C BE IIISERTED !liTO THE SYSTEfl ST!FFIIESS tV\TRTX. c
CALl. I !~S ERT( K, EUTE T, AI lODE, I C, liE, I!!< 1 r!tl, \1, H) ?..L1 CO!!T !I·IUE
RETIJRII 010
149
sun JHll IT I 11r 1 s n U\ T ( f~ 1 1 f~ 2 1 n EL T 1 , o E L T/. , K ; T [) [ I.T.I\ I I 10 F I 11 I Ill()
c c ************************************ C THIS IS(]li\TI:~; l(l,f::>,~·JF.LTl,OEU?,FfWI\ C SYSr~il 11/\TIUX F.OUAT!OII c ****************************~~****** c
c
REAL K ( ll!(ll!l() II( I ( l!llF ,IIDF) I
TK 2 (II[) F I I!) lJiiiEIIS!Oil DELT/1(111<) 1 0F.LT1 (tiDF),lJELT:?.(II)
C I< 1 ISOLATED, c
c
DO 28 !=1 ,ffDF DO 2 8 J:: 1 ! 10 F f( 1 ( 1 , J ) = rd 1 , .J )
2 8 C :J liT I lllJf:
C K2 ISOLATED. c
c
Dfl 29 I=1 ,IIDF o:; 2:) .J=IIDF+ 1, 'IOF+fl f( 2 ( I , .J., I 
SUBROUTINE KAY3 (K3,K1 ,NDF,Imv,K,DELTA, TK2,DELT1,DELT2,NE,NK,NN,N)
c c *************************************** C SUBROUTINE TO TAKE THE UPPER TRIANGULAR C ELEMENTS OF K1 AND PUT THEM INTO K3. c *************************************** c
REAL K(NK,NK) ,K1 (NDF,NDF), TK2(NDF,N) ,K3(NDF,IBW)
DIMENSION DELTA(NK) ,DELT1 (NDF), TDELT2(N)
DO 27 I=1,NDF DO 27 J=1,IBW K3(I,J)=O.O
27 CONTINUE c C K1 REQUIRED. c
c
CALL ISOLAT(K1,K2,DELT1,DELT2,K,DELTA, TNDF,N,NK)
DO 127 J=1,IBW DO 127 I=1,NDF+1-J
C K1 INTO K3. c
K 3 ( I , J ) = K 1 ( I , I +J- 1 ) 127 CONTINUE
RETURN END
1 5 1
c Slltir;:wn 11::: f'Wnur( c, :~:2,:F.I.T2, ·:~)F, :1, II)
c *********************************** C T!!IS i!'.ll.Tlf':.TFS T\f'l rL'\T!:rt:F') !~? ;\'I'J C Dr::LT? ;\~1'1 :)w~n:s 1'1
APPENDIX D
Estimation of costs involved in the research work
The various costs encountered during the course of
computation are as detailed below.
B6718 CPU Time
I/0 Processor Time
Memory Integral
Slow IO Operations
File Open Time
Total
$210.58
67.90
111.82
170.65
1 8. 2 8
$578.23
153
154
APPENDIX P1
Resulting computer programme from the first analysis.
155
,',II rJT 1\ T I rJ i IS .,., *******"********~***~*********************************************
..... -.':·.':;': ... ·:
l\J, ll.J, ~1\, iH~::LOCr'\t.. COCli\D I ilt\'i"ES OF /\II ELEI~ liT \/IT II i \ E S I' E (: T T ,) r:1 E l 1 : i d D l; • I\ t I ELE! lUll I·; l. .\_)I·:U ::: l [ 1, [', '.· ':1 :1·1 'diT I CLOCI\\IISE FtiSfl I Oil.
A:lOl.H:(;III,l;)::J\r(RI\Y Ct'JiiTl\IIIliiG I I·IFOI~!li'ITIOI! ABDUT ;\LL TilE llllDES. Tilt: FiRST H/0 PGSITIOIIS F C~ T I IE(;<) ,·\: !D ( Y) CO:JR U I: l/1 TE S ( GLOGi\1.) J\1 iD TilE THIRD 1\110 FOURTI~ Fa\ TilE 0. 0. F. I· I OS.
ARE A= ARE 1\ OF /\t'l E LEI \E NT
~·, .... , .... , ..,•: .... , 8 1 S=BETAS(GE011ETRIC CotiSTAIITS OF EACII ELEt1EIH)
CELTA(NDF)=ARRAY II~ I·IHICH PREVIOUS SOLUTIDI'I FRm1 DELTA IS STORED
C(i-.JDF)=ARRAY It~ 1/HICH SOLUTION FOR TilE UIIKNO\·HIS IS STORED. LATER TRAI'-ISFEREO I liTO DELTA
C45=C4-C5
~·: 0 -;':
D' S=UNKI-10\m tWDAL VALUES OF AN ELEHUlT.
DELTA(NK)=ARRAY OF REQUIRED NODAL VALUES. THESE ARE STREAM FUilCT I Otl( PSI I) AllD SU~I OF VELOCITIES(PHII)
DH=DIVISOR TO DIMENSIONALISE ALL COORDINATES
D I THO=V !\LUE OF I2
DL=OIE LOIGl'H
DMUX( )=(MRAY) DERIVATIVE OF (IV) I.J.R.T .X
Dt·1UY( )=(MRAY) DERIVATIVE OF(I1U) \·I.R.T.Y
DP=DIFFERENC~ IN PRESSURE BETHEEN INLET AND OUTLET
DUX( )::;(AHRAY) DERIVATIVE OF (U) H.R.T.X
DUY( )::(MRAY) DERIVATIVE OF (U) \·J.H.T,.Y
DVX( )::(MRAY) DEidVATIVE OF (V) \I.R.LX
OVY( )::(ARRAY) OF.R:VATIVE OF (V) \I.R .. T.,Y
DXOU~( )::(AHRAY) SECOtiD DERIVATIVE \JF(LJ)\-/,H,T.X
DXDVY( )::(AHRAY) DERIVATIVE OF (OVY) \.J •. R."f.X
OYDUY( )=(MRAY) SECOND DERIVATIVE OF (U)H.R.T.Y
ELEtlE T (II~, 4) = /l.RRAY C otHA Irll tH1 I IIFORI·iA TI OH ABOUT ALL THE ELH1E IHS. THE FIRST THREE POSITIOtlS FOR TilE ~JOOAL liDS. i\110 THE FOURTH FOR THE EL Et1EIIT VISCOSITY.
FBI::FLO\/ BEHAVIOUR lt~DEX OF THE POLn1ER
FAR=FOlJR TI~1ES AI~ ELEit I'ITAL AREA
~·: G ~·:
G'S=GAI·l~IAS(GEOI·IETRlC CO~lSTAIHS IJF EliCH ELHIHIT)
H=HE IGHT UF CHANNEL AT INLET
"i'( I ..,·: ;': '".\• ;': .,, 'i'(
PS=COUHTER; DEGREE OF FREEOOI·1 1Hlt1BERS; tlOOE tlUt·lBERS
I B\/=B.l\ NOH I D TH
;': i'n': ;': 'i':-
JfS:=COUtiTER, DEGREE OF FREEDOI·l Nut·IBERS; HODE llU1·1BERS.
KE(6,6)::ELE11EIHAL STIFFI·JESS t1ATRTX.
156
K(l~OF, IO~/)=SYSTE11 STTFFIIESS 11ATRIX HI f1AIWED AIIO RECTMIGULAA FORt·t
-.,'( t·l ';'( ..,., ~·:;': ';'('i'(
r~tU=ELH\EIJT VISCOSITY.
fllJIHJDE(I~II)=MRAY COflTAIIIII·IG THE VISCOSITY OF EACH NODE,
f~=!Wtl ZERO flO OF 0, Od,
NDELTA(6):::fiRRAY COI'ITAINII~G THE o.o.F .. IWSd DF AH ELUIE Iff
NDF=UIIKI'Ir.Mtl ~lO. OF 0. b. F •
NE:::fW OF ELHIEHTS
IIK=HIITIAL 110. OF D.O.F(tiO. OF f•IODES ~~2)
NtJ:::t-10 OF NODES
;': Q ";'( ·::-,':·.':'i'n':
Ql ,Q2,Q3=FLO\.JRATE AT HlLET,f·HDHAY AND OUTLET OF THE DIE,
i'( ;': ;': ;': ')'(
T=TANGEiJT OF l\1~ AHGLE
..,•: ;': i': ..,., ;':
U(NE,J)=X-VELOCITY ARRAY OF ALL THE ELEMEHTS. THE FIRST EIHRY REFERS TO THE 11 POSITIDtl OF THE ELEIIEtlT,THE SECmiD TO THE I2 POSJ"I'Iotl AND THE THIRD TO THE I3 POSITION~
UNODEO!tn :::X-VELOCITY ARRAY OF ALL THE NODES
;': v ·:: ..,., ;'('";1( ";'('","'!
V(tiE,3)=Y-VELOCiTY ARRAY OF ALL THE ELEIIENTS THe FIRST EIHRY REI-ERS TO IHE Il POSITIOtl OF THF. ELEI-1EI~T, THE SECmlD TO THE 12 POSITion AI~D THI!. THIRD TO TH~ I3 POSITION
VNdDE(Nti):::Y-VELOCITY ARRAY OF ~\LL ·rHE NODES
'vl~\·1 t D TH DF [} t g
157
S U BR OU TI fJE I NP U T ( ~~ K, NE , NN, NO F, N, I 8\.f, D L, H, \.f, TE LU1lT, AI WOE, DELTA, OH, ALPHA)
c ~*******~~*****~***************** C THIS INPUfS ALL THE AVAILABE DATA c *********~***************~******* c
D II·1E NS ION EL Ef1E T ( NE, L+) , ANODE( W~ ,4) , DELTA( NK) c
158
C DATA /\BOUT f~LL THE ELEI-IEI'ITS. THE FIRST THREE POSITIONS· IS FOR THE C NODAL rHJt1BERS AI~D THE FOURTH FOR THE VISCOSITY. c
c
viR IT E ( 6 , 8 )
8 F 0Rt1A T (31 X, 50H EL Et1E NT I l·lF ffit!A TI ON•'n':;':;'n'n':;':;'n'o':;': EL Et-lE NT I f.IF ORI'IA T I ON, I TJ.It Tj1X,10H El.Et,IENT ,10H tlODE 110 01 101: NODE t·JO., IOH tlODE ~10,,10H V r I seas ITY)
DO 9 I= 1 , NE READ(S,/) P,I2,I3
E L E f1ET ( I , lf) :: 1 • 0 \.JRITE(6,10) I,I1 ,I2,I3,ELHIET(I,4)
10 FORt'lA'f(31X, !5,4X, IS,SX, IS,SX, IS,S:<,F10.5) El. Et1ET ( I, 1 ) :: I 1 + 0. 1 E L D 1 ET ( I , 2 ) = I 2 +0 • 1 E l. E t1E T ( I, 3) = I 3 +0. 1
9 GotH I NUE
C INFORI1ATION ABOUT ALL THE ~lODES IS READ IN c
c
\.JR IT E ( 6 , 1 1 ) 11 F0Rt1AT ( 1 HO, 30X, 1 NODAL I i~FOR~IAT I ON•':.'n'n':.':·.'n>:,·n•n•:tWDAL I NF OR~1A T I Otl 1 II
T31X, 1 tlODE ~lO. 1 , 1 :<COORD 1 , 1 Y COORD 11
1 DOF NO. 1
T 1 x, I DOF NO. I/)
DO 12 I=l,t,ltl R E A 0 ( 5 , /) A t W D E ( I , 1 ) , AN 00 E ( I , 2 ) , J 1 , J 2 ANODE( I, 1 )=MIODE( I, 1) /DH ANODE( l,2)::AIWDE(I,2)IDH \·IR IT E ( 6 , 1 3 ) I , AN 0 D E ( I , 1 ) , AN 0 D E ( I , 2 ) , J 1 , ,J 2
13 F0Rf1AT(31X, I5,4X,2F10.5, I5,5X, 15) ANODE( I,3)=J1+0.1 ANODE( I ,4)=J2+0,1
12 CONTINUE
C READ IN INITIAL NODAL VALUES c
~/R I TE ( 6 , Jlf ) 14 FORt·lA T( IHO ,30X, 1 BOUtlD ARY CDi'lD IT IONS 1 I
T31X, 1 **********************'II T31X,IOH DOF flO, ,10H PSII , TIX,!OH DOF NO, ,10H PSII ,/)
DO i 5 I=I.JDF+ 1 ,NK READ(S,i) DCLTA( I)
15 COIHINUE DO 16 I=tiDF+l ,NK,2 \·IR IT E ( 6 , 1 7) I , 0 E L T A ( I) , I+ 1 , D E L T A ( I + 1 }
117 FOW1AT(31:<, IS,I-IX,F9.6,2:<, I5,4X,F8.6) 6 COIH I tJUE
RETURtl END
SUl3ROUTI !,IE COORD( I, MJODE,Nli,DL, ALPHI\) c c *******~****~********************* C THE GLLJQJ\l CJOi\DII!i\TES GIVCII Iii C T II E I , II' U T S lJ lJ f( ::-1 U T I II E i\:\ E F OR 1\ C Uil! FURil r:ECTAIIGUL/\;\ i;CSII, i\i.JD!FIElJ C 3ELU'./ TJ ,\CC:Ju;;y F:ii~ liiE Tfd't:i~Ii!G C OF TilE DIE. TilE X COOI\D!IIATES C FEIIAill TilE SAllE. c ***~****************************** c
0 U\Eil~ lOll /\[lODE( llil,4) T=T.I\11\ 1'\LPI-Ii\J R=T/ DL MIOD E ( I , 2) = MIOD E ( I , 2) ~._. ( 1 • 0-A 1100 E ( I , 1 ) ~·:R) Jl ::AI~ODE( I ,3) J2=AiWDE( 1,1+) RETURH E HO
159
SUBROUTIIJE UPDATE( ELEI-IET,AIWDE,DCLTA, I, HIE,HK,IHI, ITEI~)
c c ********************************** C THIS SUBROUTINE UPDATES THC VALUES C OF VISCOSITY FOR EACH E.LEt1ElH c *********~~*~*********************
c
o II IE r~s I otl ELEI-IET (tiE, Lf), ArJODE(tltJ, 4), TDELTA( 1,1~:)
REAL l·lU
C NODAL lWllBERS c
c
I 1 :: E L E t1E T ( I , 1 ) I2=ELEI-1ET( I, 2) I 3:: E L E t1E T\ I ,3 )
C D.O.F. HU11BERS c
c
J 1 :: /'\ N 00 E ( I 1 , 3 ) J2=AiJODE( I 1 ,4) J3=ANODE( !2 ,3) Jlf::ANODE( !2,4) JS=MIOOE( I3,3) J6=ANOOE( 13 ,L:)
C VALUES OF THE UNKNOWNS c
c c c
c c c
c
01 ==DELTA( J1)
ar;BttfM j) ~ D4=DELTA( J4) DS=DELTA(JS) D6=DELH\(J6)
VALUES OF ELE11ENTAL CONSTAIJTS REQUIRED
CALL GAI1!3ET(G1 ,G2,G3,GLf,G5,G6,B1 ,82,63, TARE A, AJ, IJ J, AK, BK, JAY, KAY, F, E L E 11£ T, AllOD F., T I, FAR~lU, FAR, NE, NK, N~J,\1, H)
CALCULATE I2 ( ITWO) NOW
C6=G1*D1+G2*D3+G3*05+G4*D2+G5*D4+G6*06 C45=B1*02+B2*D4+B3*06 0 IT\.JO::( C6~':C6)+(CLf5~':CLf5) R=SQRT ( 0 I T\.JO)
C CALCULATE THE VISCOSITY ( t1U) IF THE FLOW C BEHAVIOUf\ INDEX(FBJ) OF THE POLY11ER IS KIJO..nl c
FBI=1.0 t1 U = ( 2 • ~h': ( F 8 I - 1 • 0 ) ) ~: ( R ·:d: ( F 8 I • 1 • 0 ) ) E L El-\£ T ( I , 4) :: t1U RETURN EHD
160
c c c g c c c c
c c c
70
71
SLJ8ROUT IIJE 1\SSEI-18( K, KE, Et.EtiET, AtWDE, DF.LTI\ 1 C, ~lE, NK, TtlN, \-/, H, NDF, 18~/)
THIS CALCULATFS TilE STI FFIJESS tiATfU X OF EACH ELEt-lEtH /\flO IIISERTS IT IlrfO THE SYSTEil STIFFIJF.SS I'~ATHIX. TilE ltATRIX IS 81\I~DED AIJD TilE LOAD!t~G VECfOR OBTAHIED IN THE PROCESS ************************~****************
D ItiUIS Iotl El.EI1ET ( l·lE ,l}) 1 Af,IODE( Nt:, 4), DELTA( NK), TC(tWF, 1 ),llOELTA(6)
RE.i\L K(IJDF, !Oii),i<.E(6,6) D 0 7 0 I~~ 1 , NO F DO 70 J:::1, IB\4 K ( I, J) =0. 0 C(l~1>=o.o CONTI f·lUE DU 50 1=1 ,NE DO 71 lC=1,6 DO 71 J::: 1 , 6 . KE( IC,J):::O.O
CONTI HUE
ELEMENTAL GEotiETRIC COi'lSTANTS REQUIRED
CALL GM1BET(G1 ,G2,G3,GI~,G5,G6,81 ,82, T83,:\REA,AJ~ BJ,AK, BK,.JAY ,KAY, F, TELEI-H:. T' MIOOE, I I FARIIU I FAR I fJE I l·lK, Nil I \1' H)
c C CALCULATE ELHIENTAL STIFFNESS f1ATRIX N0\4 c
F 1 =FARI1U>':GJ F2= FARIIU>':G2 F 3= FARI·IU~• G3 K E ( 1 , 1 ) :::F P G 1 K E ( 1 , 2 ) ::: F P G4 K E ( 1 , 3 ) ::: F 1 ~' G2
~Hl:~l~~n~a~ K E ( 1 I 6) =F 1'' G6 KE(2,2)=FARtlU>':( ( 81>':81 )+(G4>':Gll)) KE(2,3)=F2>':G4 K E ( 2 , L~)::: F ARI'IIJ>': ( 81 ,., 82 + G4 -:: G5) K E ( 2 , 5 ) :::F 3 >': G4 KE(2,6):::F/\RMU*(B1*83+G4*G6) KE ( 3, 3) =F2>':G2 KE ( 3, Lf) ::F2>':G5 KE ( 3 I 5) =F2>'1 G3 K E ( 3 , 6 ) :: F 2-:: G6 KE(4,4):::FARMU*((B2*B2)+(G5*G5)) K E ( 4 Is)::: F 3 >': G5 KE ( 4, 6) =FARHW ( B2~1 83+G5>':G6) K E ( 5, 5) = F 3 ,., G3 K E ( 5 , 6) = F 3 >': G6 KE(6,6)=FARMU*((B3*B3)+(G6*GG)) 00 26 12=1 ,5 DO 26 ,J:::I2+1 ;6 K E ( ,J , !2 ) ~: I(E ( I 2 , J)
26 CotHI NUE
1 61
c C ELE~·IENTAL STIFFNESS t·1J\TRIX IS AVI\!LABLF.; NO\-/ c C 08Tf.\IN TilE o.o.F. NOS. c
c
II =ELEIIET( I, 1 ~ I 2 = E L E I lET ( I I 2 ) I 3 = E L El 1 E T ( I I 3 ) IIDELTA( I )::AI~ODE( 11 ,.3) N 0 E L T A ( 2 ) ::A I WD E ( I 1 , If ) IWELTA(3)=MJODE( 12,3) NDELTA(lf)=I\IWOE( I2,L+) I JOEL TA( 5 ):::AIJODE( 13,3) NDELTA(6)=ANODE( 13,4)
C CHtCK ALOI-IG TilE RO\-/ OF tl/\TRIX. C IF IT IS BEYOND THE RAI~GE OF (K) IIEGLECT C IT. IF IT IS \IITHII~ THF RAI1GE, GO C ALONG THE COLUMN. c
DO 2 KI=I,6 c C IF BFYOND K1 S RANGE, TRY NEXT ROW c
c c c
c c c c
IF(NDELTA(KI).GT.NOF) GO TO 2 I~K!=NDELTA(KI)
GO COLUI·11~/ISE
DO 3 KJ=l ,6 NKJ=NOEL TA( KJ)
IF BEYOND K 1 S RANGE, OBTAIN THE LOADING VECTOR
IF(NKJ.GT.NDF) GO TO 4
C HITHIN K' S RI\NGE. NO'v/ IN BAI~DED FORt1 c
NKJ1 =I~KJ-I·!Kr+ 1 IF(NKJl.LT.I) GO TO 3 K ( NKI, NKJ 1 )=K( NKI, HKJ I )+KE( KI, KJ) GO TO 3
4 C ( HKI, 1 )::C(NKI, 1) -KE(KI ,KJ)~':OELTA( I~KJ) 3 com I NUE 2 CmlT I NUE 50 CONTINUE
RETURN EI~D
162
c c c c c
c
c
163
SUBROUTINE SYMSOL(NN,M1.NDIM,KKK,A,B)
BANDED SYI111::.TRIC t·\ATRIX EQUATIOI~ SOLVER CROlJT t·1ETHOD . ****************************************************************** D Ir1E NS I Ol ~ A ( 1 ) , B ( 1 )
LOC( I, J)=I+ (J-1 )~':IWH1 GO TO (1000~2000),KKK
C REDUCE 1-tl\TRIX c
c E
c c c
c
1000 DO 280 N=1,NN ~I i :: LO C ( l·l, 1 } DO 260 L= 2, t-M NL=LOC( N, L) C=A(NL)/A(I~1) I=N+Lw1 I F ( r·l~ I • L T • I) G 0 T 0 2 6 0 J=O DO 250 K=L,l't1 J=.J+1 I J=LOC( I, J} NK=LOC(N,K)
250 A( I.J)=A( IJ)-C1':A(NK) 260 A(NL)=C 280 GOtH I NUE
2000
285 290
300
400
GO TO 500
REDUCE VECTOR
DO 29CJ N= 1? I~N N 1 =l.OC( II, 1 J DO 28 5 L= 2 t-1M tJL::: LOG (II, L) I=ll+ L-1 IF(l~l·l.LT. I) GO TO 290 8 ( I } = B ( I ) - A ( I H.) ,·: B ( N) B( t~) =B( II) /A( 1~1)
8,\CK SUBST I TUT I Oil
II=Nt·l !1=11"1 IF(Il.EQ.O) GO TO 500 DO 400 1(::2,11'1 IJl(::J.OC( II,K) L=II+K-1 I F ( ll; 1. L T • L) G 0 T 0 1 f 00 D ( i:) ::2 { II) -~~ ( t li() ~·: B ( L) c.; i I T I IJlJ [ GCl TO 300
500 RETURII Ell()
SUEllUJUT I II[ COIIV[(i( lJEl.TI\, C, IIDF, KODE, iH<) c c ************************ C Tl~[ TEST FIJH COIIVERGEIICY c ************************ c
Dli\OISIOIJ fJEl.TA(I·IK) ,C(IlDF) I<ODE=l lJll Jlt I= 1, llOF E..:i\tJS ( C( I) -D t~LTi\ (I)) -0. OJOO 1 ~~ 1\bS ( C( I))
3L1 IF(t:,GT.O.O) I~OOE=l(ODE+I ;;E"!URII c:::D
164
c
SlH3ROlJT lllE VEL(U,V, I, ELIJtET ,OELTA 1 1\ilODE 1 I!E, Ill~, T : : ll , \·1 I ll )
c ******************~************************** C Cf\LCUL!\TIUII OF tlOOi\l. POIIH VE!.CJCITIES, U H~.<
C TliE X Clli\f'OilE:1T l\IIU 1_1 FtJf{ THE Y CDI\f'l)!'IEilT FOr\ C ti\CII EL.EillilT. c ***"******************************~~********* c
o r; tE : 1s r o:1 u < : <E 1 3 ) I v < :a:: , 3 ) I r:. t. E: :c T c :It: , i: ) I o E u id : 1: ~) , 1\ll oo E < 1 1:1, h ) c C ilODAL IIUitDERS c
c
Il=ELEIIET(I,l) l2=CLEi!ET( I ,2) J3:.:t:t.Eii:T( 11 3)
C D.o.F. IIUi\BERS c
c
Jl =1\:lOiJE( !1,3) .J2c.:Afi0DE( 11,4) j 3:: ~~:! OD = ( I 2 I 3 ) J';:.:x::x:( l2 ,4) J.)::I\:!DLJE( !3 13) J6=;'\ilOOE( 13 ,4)
C VALUES OF THE Ulli<iiOUiiS c
c
01 =DELTI\ ( Jl) U2=DELTI\( ,J2) D 3:: DE LT .c\ ( J 3 ) D I~ ::DELTA ( JL~ ) DS=DELTI\(JS) D6=DELTI\( J6)
C VALUES OF ELE11EIJTAL COIISTAIHS REQUIRED c
c
CALL GAIIBET(Gl ,G2,G3,GI~ 1 G5,G6,B1 ,G2 1
T 8 3 tARE A I t'\J' 8 j I AI<' B K, JAy I KAy' F I TEL EI'IE T 1 AIWD E 1 I 1 F ARIIU, FAR, ~IE, IlK 1 I~N 1 H 1 H)
C OBTAIH C6,CS,Cl.:·S,Cl~ AND C3 c
C6=GPD1 +G2~':QJ + G3~'•05 + G4~':02+ TG5;'•04+ G6;':06
C 5 =0. 5;': C6 + ( W: ( 02 ~·.· ( AJ ~ AK) ) + ( 04 ;': AK) T-(06;'•AJ)) /FAR C45=81*02+B2*04+B3*06 CLt=C45+C5 C 3:: ( ( 1-J;'' ( D .3 -0 1 ) ) - ( 0 2 ;':A J ) - ( Clp': ( A J;': A J)
T+C5*(BJ*BJ)+C6*AJ*BJ)/H)/JAY c C OBTAIN THE X 3, Y VELOCITIES tiDY/ c
c
Ui =" C3 U2::-(C3+2.0*C5*BJ+C6*AJ) U3=-(C3+2.0*C5*BK+C6*AK)
C STORE THE X-VELOCITY VALUES IN THE (U) ARRAY C MID THE Y"VELDCITY VALUES IH TilE (V) ARRAY c
U(I,l)=Ul V( I, 1) ;:O?.-Ul U ( I, 2) =U2 V ( I , 2):: DL> -U2 U( I,3):::UJ V ( I ~ 3 ) :: 06 ~ U3 f{ETURtl END
165
c
SUBROUT I I~E VELfJC(UNODE, VIIOOE,U, V, ELEI·IET, DELTA, T A IJOO E, HE , liK, N~l, \-1, H)
c ********************************************** C THIS rALCULATES THE X ArlO THEY VELOCITY C COHPOilEIHS nf: EACH tWOE. THE COtHRIBUTIONS C OF VARIOUS ELEI·\S NTS TO THAT I lODE IS TAI~E tl rrno C ACCOUtlT. c ********************************************** c
0 !liENS I 0! I Ul l 00 E ( HN) , V~l 00 E ( l·l~l) > U ( HE , 3 ) , V ( I·IE ,3) , TEL El IE T ( I JE , 4) , 0 E LT A ( t1K) , A IWD E ( liN, 4) , ~~~~ 00 E ( 81 )
DO 7 3 IH:: 1 I HN UI,IODE( N1) =0.0 VI·IODE( Nt·I)=O.O ~JI~ODE( HI) =0
7 3 COIHI NUE c C NODAL NU~IBERS c
c
DO 772 J::J,NE I 1 :: E L E t ·IE l( I I 1 ) 12=t::LEt·IET( 1,2) l3=ELEt1E T( I ,3)
C NODAL VELOCITIES c
UNODE( 11 )::LJtiOOE( 11 )+U( I, 1) V~lODE( 11 )=Vi·IOOE( 11 )+V( I, 1) NNODE( 11 )::tltiODE( Il )+1 UNODE( l2)=UIWOE( 12)+U( I,2) VI~OOE( l2)=VtWDE( 12)+V(I,2) NtlODE( 12)=Nti0DE( 12)-1·1 UIIOOE( I3)=UIIODE( I3)+U( 1,3) Vt·JODq 13)=Vt-IODE( 13)+V( I,3) t~I·IODEt I3)=1HIOOE( 13)+1
772 CONTINUE cc C NODAL VELOC rTY DBTA HIED I~OH. c
DO 773 1=1 ,NN UI~OOE( I) =UtiOOE( I) /NNODE( I) VI•IODE( I) =VIIODE( I) /NI!ODE( I)
773 COil TI I~UE RETURN END
166
c SUBHOUT I !lE VISCO( l·llltlODE, ELU1ET, DELTA,
TANOD E, UllOOE, VllOD E, NE, HK, tHl,\-/, H)
c *****~******~********* C VISCOSITY OF EACH NODE c ********************** c
0 IHEI'lS I uti ELE: lET ( NE, If), DELTA( l·lK), AIJODE( W·l, If), TUII ODE ( HI~) , V rJGD F. ( 1~11) r DUX ( 8 1 ) , DVX ( 8 l ) , DUY ( 8 1 ) , OVY ( 8 1 )
REAL t·\UJJODE( tHn c C I~E\HOHIAII FLO\~ THEFO~E VISCOSITY EQUALS 1.0 c
c
GO TO 560 DO 4 9 t I I :: 1 , l·li'l 1\UNODE( IH )=0.0
lf9 COt·ITI NUE
C VISCOSITY 1\T EACH fiOOE IS EV.£\LUATED C fiY 08 TA I tH I~G THE VALUE OF If! 2 AT C EACH PO I llT c c C FINITE DIFFERENCE METHOD IS USED TO C OBTAI!~ THE DIFFEREHTIALS c c C OBTAIN DERIVATIVES W.R.T.X c
DO 50 I:: 1 , 73 , 9 A=AI~OD£(1+1,1)-Af.IODE( !,1) B=AIIODE( 1+2, 1 )-AIIOOE( I, 1) DUX ( I ) :: ( IJ~': s~·:u t·J ODE ( I+ 1 ) - M: A~': W·l ODE ( I+ 2) ~ ( s~·, B- N: A)
T~··u rwo E < I ) ) 1 ( A~·· s~·, ( s-A) ) DVX ( I):: ( B~'•B~'•VtlODE( I+ 1)- A~'•M•Vf.JODE( 1+2)- ( s~·:s-N: A)
T~'•VIWDE( I)) I( A~':B~':( B-A)) 50 CONTI HUE
DO 51 1=9,81,9 A 1 =ANODE ( I, 1 ) -ANODE ( I-1 , 1 ) Gi =ANODE( I,1 )-ANODE( I-2, 1) DUX (I)::( APAPUt·lODE( l-2) -BP8J:':UHODE( I-1)
T+ ( ( 8 1 ~·, B 1 ) - ( A 1 ~·,A 1 ) ) ~·:u II ODE ( I ) ) I (A 1 ~·, B 1 ~·, ( B i -A 1 ) ) DVX( I)=(A1~'•At~'•VNODE( I-2)-B1~•81~':VNOOE( I~t)
T + ( ( 8 P B 1 ) - (A 1 ~·,A 1 ) ) ~·, VN 00 E ( I ) ) I (A P 8 P ( B 1 -A 1 ) ) st cmnr NUE
DO 52 ,1=2,8 DO 53 IL=0,8 I=J+ I U:9 A2=ANODE( I, I )-ANODE( 1~1, I) B2=AIWDE( I+ 1, I) -AtJODE( 1··1, 1) DUX ( I) = ( A2 ~: A2 ~·:u f.l ODE ( I+ 1 ) - ( 82- A2) ,., ( 82- A2 ) ·::u r JOD E ( I -1 )
T - ( ( A2 -::A 2 ) - ( ( 82 - A2 ) ~·: ( B 2 - A2 ) ) ) ~·, W J 0 D E ( l ) ) I ( A2 ~·, R 2 ~·: ( B 2 - A2 ) ) D V X ( I ) :: ( A2 ·:: A 2 ,., V N D D E ( i + 1 ) - ( B 2 -A 2 ) ~~ ( !3 2 - l\2 ) ~~ V r I D D E ( I - 1 )
T- ( ( A2>': A2)- ( ( 132-A2) ~·:( 82 -A2)) PVtlODE( I)) I ( A2>':B2>': ( 82 -A2)) 53 CDI'JTINUE 52 COHTINUE
167
c c c
c
tlO\.J 0 B T A 1 N 0 EIU VAT 1 V E S H. R. T • Y •
DO 54 J:: 1 , 9 AY=ANOOE( i+9 2)-AIWDE( 1,2) BY=AIHJOE( l+ JG,2)-i\IIOOE( !,2) DUY( I)::( uy·::£3'(qJNOOE( I+ 9)nAY·.':AYl':LJtiOOE( 1+18)
T- ( ( B'(l': BY)- ( AY•< A Y) ) ·::LJ t lODE ( I) ) / ( AY·.'dJ'(l': ( BY" AY) ) DVY ( I) :: ( 8Y•': 8 y·:: VllOD E ( I+ 9)- AY•': AY•'<V II ODE ( I+ 18) - (( B Y•': [l Y)
T - ( A Y ,·,·A Y ) ) ,·,v l·lO D E ( I ) ) I ( A Y ,., 8 Y '~ ( 8 Y ~ A Y )) · 54 Cat-IT! NUE
DO 55 I=-'73,H1 AY=MIODE{ 1,2)·ANOOE( l-9 2) BY=AIJODEt I ,2) -ANODE( I-JS,2) DUY( I)::((AY•':f\Y•':LJtiODE( 1-IB)wBY•'•BY·.':UtJODE( 1-9)
T + ( ( BY,., BY) - ( A Y•': AY) ) ·::u tl ODE ( I) ) ) I (A y,·, B Y•': ( IW- AY)) DVY( I)=(AY•':AY•':VNODE( !-18)-BY·::sy·::VtWDE( 1-9)
Tr ( ( G y,·, BY) - ( A Y•': A Y) ):': V NOD E ( I) )/ ( A Y•': By,·, ( C '(-A Y} ) 55 corn I HUE
DO 56 ,J:::J0,64,9 DO 57 I L=0,8 I=J+IL-'•1 A Y = 1\ N 0 D E ( I , 2 ) -AN OD E ( I -9 , 2 ) BY=ANOD~:( I+9,2)-Ail0DE( I-9,2) DUY( 1):: (AY:':Ay·::UfiODE( I+9)-(BY-AY) .'•(BY-/W)
P U llO D E ( I - 9 ) - ( ( A Y \., A Y ) - ( ( BY - A Y) ,., ( 8 Y -A Y ) ) ) ,., U ~lO 0 E ( I ) ) V (AY*BY*(BY-AY))
DVY( I)= (AY\':AYl':VtlODE( 1+9)-(BY-AY)•':(BY-AY) PVIIODE( I-9) -( ( AYl'<AY) -( ( BY~AY) •':( BY-AY)) )•':VtiOOE( I)) V (AY*BY*(BY-AY))
57 CmlTI NUE 56 CONTINUE
C VALUE OF 412 c
c c c c
DO 58 1= 1 , NH FOURI2=2.0•':DU~~( !)":DUX( I)+2.0•':D'JY( I)
PDVY( I)+(DUY( i)+DVX( I) )•':(DUY( I )+DVX( I)) R=SQRT(FOURI2)
VISCOSITY OF EACH HODE; FOF< LARGE VALUES OF THE VISCOSITY, IT IS SET :: 1000
t'1UIIOD E( I) =1 000.0 IF(R.GE.0.000001) 11UIIODE( 1)=1.0ISQRT(R)
58 CuiHI l iUE 5 60 DO 561 IH:.: 1 , Nil
I IU: l OD E ( II I ) :.: 1 • 0 561 CJIITI IIUE
RETURI~
END
168
su:m:JUT!Il:~ DELT.4P(DP,CLF.IET,OELT1\,AWH1f:,U,V,IIE, T: II~ I IIi I I I I ' II )
169
c c c c c c
TillS C/\LCULJ\Ti::S Tilt Clli\IIGE !II PHF.SSUf\E UET\/[[11 !liLET ~'\110 OUTLET t\LlWG THE X"I~XI S
c
D lt1E !JS I 0! I E L E! lET ( liE , 4) 1 DElTA ( til~) , AtWU E ( t,H I~~.) , T u t 1 oo E < 3 1 ) , u < 1 !E , 3 ) , v 11 o o d n 1 ) , v < r 1 E , 3 ) , oux ( 9) , TGXDUX(9) ,D!IUX(9) ,OUY(9) 1 0YDUY(9) ,DVY(9) 1
TOXD'.'Y ( 9) I DVX(9) IF( 9), Dl!UY ( 9) REAL l·IUtlOOE( B I)
C VALUE OF ALL NODAL VELOCITIES REQUIRED c
CALL VELOC(U~WDE,VllODE,U,V,ELEI·'f:T,DEL.T/\,ANOOE, HIE, NK, Nil,\·/, H)
c C VALUE OF ALL tlODAL VISCOSITIES REQUIRED c
CALL VI SC 0( t1Ut!OD E1 EL E11E T 1 OELT A 1 A!~OD E, U tWO E, VIIODE 1 i~E, NK, Ntl, H, H) c C FINITE D I FFE NCE t·IETHOD IS USED HEr-E C THE FORi/ARD Arm BACI<I.JARD D IFf-EREtiCE Fo:\l·IULA C IS USED FOR THE Et!D POI tHS C CE lHHAL DIFFERENCE F CRt1lJLA FOR OTHERS c c C DBTA HI FIRST Atm SECOt!D D IFI:"EREIHIAL OF C U 1·1. R. T • X c
A=ANDDE(2,1)-ANODE(1,1) B=AtWDE(3 I 1 )-AI~DDE( 1 I 1) . DUX ( 1 ) = ( 8 •': B•':Ut·l ODE ( 2)- A•': f-\·:: U~IOD E (3 ) - ( 8•': 8- A•': A) •':UN 00 E ( 1 ) )
T/ ( A•':B":( B-A)) D XDUX ( 1 ) :: ( 2. 0•': ( B•': Uti ODE ( 2) - A•': UI·J ODE (3 ) ~ ( 8- A) •':LJ tlOO E ( 1 ) ) )
T/ (A•':B•':( A-B)) A 1 =A IWO E ( 9, 1 ) - Al-100 E ( 8 1 1 ) B 1 =·AIIOD E(9 1 1) -A!-IDDE(7 i) DUX(9)=(A1~AI*UNODE(7~-8i*B1*UNODE(8)+((B1*R1)-(Al*A1))
PUtJOOE(9)) /( APBJ:':( Bl-Al)) D XOUX ( 9):: ( 2. Q;': ( A PU I·WD E (7 ) -B 1 1':Ut JOD E ( 8) - (A 1 - B 1 ) •':
TU~JODE(9))) /( APBP( B1-f\1)) DO 50 1::2,8 A2=AI~ODE( !,1 )-At~OOE( 1-1, I) B2=AI~ODE( I+ 111 )-At-lODE( INl '1) DUX( I)=(A2•':A2•'1UNODE( 1+1 )-(B2-A2)•':(B2-A2);':U!IODE( I-1)
T- ( ( A2 1'1 A2 ) •· ( ( f3 2- A2 ) ;': ( B 2 - A2 ) ) ) •':LJ t I 0 0 E ( I ) ) / ( A 2 ,., B 2 •': ( B 2 - A2 ) ) 0 XDUX ( I):: ( 2 • 0>': ( A2 ~:LJ tl DOE ( I+ 1 ) + ( 82 -A2) ;':U H 00 E ( I~ 1 ) -82 1'1
TUrWDE( I)))/( A2•':B2•':( 82-1\2)) 50 CLHH I i~UE
~;-
c DOTAI_N FIRST AND SECOND DIFFERENTIAL OF U · C AND V W.R.T. X AND Y. c
c
DO 51 1=1 ,9 1\Y::At~ODE( 1+9,2)-AIWDE( I,2) 13Y=AilGDE( I+ 18,2)-,L\IWDt::( 1,2) OUY( I j::: ( 13Y>': BY>':lJJ 100 E ( I+ 9)- t\Y·:: t\Y''IUtlOD E ( I+ 18)- ( ( BY''I BY)
T- ( AY>': A Y) ) ·:.-u IWO:: ( I) ) I ( AY·:.- 8Y>': ( BY- AY) ) DYOUY( I)=(2,0>':(BY,':lJIIODE( 1+9)~AY,':UIWDE( 1+18)~(BY-AY)
PUIWOE( I))) I(AY>':BP( AY-8Y)) DVY ( I ) :: ( 8 Y>'1 B Y>'1 Vt lOO E ( I +9)- I\Y''1 A Y>':V I I 00 E ( I+ 18) - ( ( BY'''
TBY) -( AY''1 AY)) >'1VIIODE( I)) I ( 1\Y>'I(Jy,·~ (BY-A'()) 51 CDNT:I~UE
DXDVY ( 1)::: ( 8''18>'10VY ( 2) -A''IJ\''10\fY {3)- ( ( D>'1B) -( (\>'1 A) )'':DVY ( 1)) Tl ( A>':B>':( 8-A}) DXDVY(9)=(AI*A1*0VY(7)-B1*B1*DVY(8)+((R1*131)-
T ( A P A 1 ) ) ·:.-ov Y ( 9) ) I ( A l ,.,. B P ( B 1 -A 1 ) ) Oo 52 I= 2, 8 A2=Ai~ODE( I, 1) -MIOOE( l-1, 1) B2=AtWDE( I+ 1,1) -AI lODE( I-1, l) DXOVY (I):::( A2''1A2>':DVY ( !+ 1) -(( 82-1\2) ''1 ( 82-t\2) )>'•DVY (I -1 ~
T- ( ( /\2 ,.,1\2 ) - ( ( f32- A2 ) ·:: ( l32 - /\2 ) ) ) ,., D V Y ( l ) ) I ( i\2 ,.,. 82 ·:: ( 132- A2 J )
52 COI:T I NUE
C OBTA HI THE D l FFERE t~T IAL OF U MID V c
c
. 0 VX ( 1 ) = ( 8>': 8''1 Vt JOD E ( 2) ~A,., A;'1V I I 00 E (3) - ( [3'.'1 B- N· A) PVIIODE( 1)) I(A'''8'''(8-A))
DV X ( 9)::: (A 1 ,.,. A PVI·J ODE ( 7 ) - B P R I ,.,.VI I 00 E { 8) + ( ( 8 1 ,.,. B 1 ) T - ( A 1 ,., A 1 ) ) ,·,.v I JO D E ( 9 ) ) I ( A 1 ·:: 8 1 ,.1 ( B 1 - A I ) )
DO 54 1=2,8 -A2=ANODE( I, 1 )·ANODE( I-1, 1) B2=AHODE( I+ 1,1) -ANOOE( 1··1, 1) DVX ( I)::: ( 1\2 ,.,. A2'':Vtl 00 E ( I+ 1 ) - ( 82- A2) ·:: ( 82 ~A?.) ·::Vtl GO E ( I -1 )
T- ( ( A?.,.,. A2 ) - ( ( 82- A2 ) ,., ( 82- A2 ) ) ) ,.,. V N 00 E ( I ) ) I ( A2 ,.,.112 , ... T ( B2 -A2))
SL• COI·ITI HUE
C OBTAIN THE DIFFERENTIAL OF VISCOSITY. c
00531=1,9 AY:::AJ-IODE( !+9,2)-AI-IODF.( 1,2) BY=ANODE( I+18,2)-AtJODE( I,2) 0 1'1UY ( I):: ( [3Y~·.- BY,':fv\UII 00 E ( I+ 9)- AY''' AY>"t1UIIOD F. ( I+ 18)
T -((BY~'.-8Y)-(iW'''AY))":1·1UIIODE(I))I(AY'''8Y•'•(BY~AY)) 53 COJITII~UE
01-\UX ( 1 ) :: (8''' 8'''11UJ.IOO E ( 2)- A,., A'':J1UIIOD E ( 3) -( ( g,·, f3)- ( N: A)) •':f1UHODE( 1)) T I ( A\'lfl''' ( B-A))
01·\UX (9)::: (A 1·::AJ ~·:t1U~IODE (7) -81 ,.,.8 1 ,·:t-1UHODE( 8) + ( ( 81 '':BJ)-T ( A1 ,., A 1 ) ) ,·,.1-\UilOD E ( 9) ) I ( AI ~·.- B !''• ( B 1- A 1 ) )
DO 58 1=2~8 A2=AtWDE( 1,1 )-At-lODE( I-1, 1) B2~AI~ODE( I+ I, 1 )-ANODE( I-1 1) 01·\UX( I)=(A2'''A2'':t-1UtiODE( 1+1 ~-(B2-.A2)•':(82-A2)·.':f1U~10DE( I-1)
T- ( ( A2'''A2)- ( ( f32-A2) ~·.- ( 82 -A2))) ~·.-t·\UIWOE( I)) I ( A2''.-U2~'.-( B2-A2)) S8 COIHINUE
170
C THE I tHEGIV\110 OF CIIAIIGE I 1·1 PHESSURE C (DELTI\-P) IS OIHI\IHF.D FOH EI\Cil O~IE OF THE C PO lilTS ALONG TilE fJOUrJOARY • c
C·
DO 55 J:::l ,9 F ( I):: 2. 0•': ( 1\UtiOD E ( I) ·::o XDUX ( i) +0 t·\UX ( [) •'rDlJX ( li)
T +1\UIWD E ( l) ·:: ( DYDUY ( I) +D XDVY ( I) ) +D 1"1UY ( 1) ,·: ( OUY ( I) T+DI/X(I))
55 COIH I NUE SUI 1:::0.0 DO 56 I::: 2,8 SUII=SUH+F( I)
56 COI·IT I NUE
C IIHERGRATE llSHIG TRAPEZOIDAL RULE TO OBTAI!~ THE C CHAIIGE It~ PRESSURE c
D P:::- ( F ( 1 ) + 2. 0•': S Ul·1+ F ( 9 ) ) I 1 6 • 0 RET URI~ END
171
c
SUOROUT It IE FLORET ( Q I ,Q2 1QJ , ELEI·E T, ToE LT A , AtHJD E, u , LJI'l no E, 1 j[ , I~K, t Hl, 11, H)
c **********~~***************************** C TillS CALCULATES THE FLOWRATE AT D IFFERF.IIT C SECT! OtiS OF THE CHAIU•IEL C Ql FOH TilE ii-ILET C Q2 FOR TilE IIIDDLE SECTION C Q3 FOR TilE OUTLET c **********************************~****** c
c
0 !fiE tiS I Otl ELEt·l£ T( HE, 4), DELTA ( NK), AIIOD E( I~N, 4), TU ( I IE , 3 ) 1 UtI ODE (lit I)
Sut\1 =o. o SUt\2=0 .0 SUI\3=0.0 DO 7 1::2,8 SUI 1 I = SUI\1 +U IWD E ( ( I~·: 9) -8) SUt12=SUI12+UIIODE( ( 1":9)-4) SUi13::SUI13+l!IIODE( 1'':9)
7 COIITI NUE
C TO OBTA!IJ THE FLCJ\./KATE, THE PRODUCT C (U~':DY) \·lAS HITERGRATE ACROSS THE CHAi~NEL c
Ql ::( Ul!OOE( 1) +2.Q·.':SUI11 +UIIOOE(7 3)) It 6.0 Q2:: ( U 1'1 00 E ( 5 ) + 2. o~·: SUf\2 +UtFJD E ( 77) ) I 1 6. 0 Q3 = ( u ttoo E ( 9 > + 2. o~·: sur 13 +U t·l oo E ( 8 1 ) ) 1 1 6. o RETURN END
172
c c c c c c c
c c c
c
SUOROUT! 1/E Gi\IHJET( G 1, G2, (13 'GLf I G5 I G6 I f.ll, 82 I 133' TAR EA, .t\J, L3 J, AK 1 BK, .J/\ Y, KAY t F, E lEI \E T, ANODE 1
T I ' FAR llU ' F 1\f{ ' HE ' t~K' I Ill)\/ I H}
THIS EVALUJ\TES VARIOUS ELEI\Et!TAL COIISTAIITS THUS TilE GJ\ti:IAS(G'S) TilE 1JlT1\S(3 1 S), /~liD THE 1\RE A lJF ,\I I t: L El IE trr *********************************~*******~**
Dii\EiiS !llil ELEI\ET( IIE,If) ,· i\i:ODE( fl:t,Ir) REAL I ·,u , J/i Y, 1(;\ Y
IIOuAL liU:It3ERS AI!D VISCOSITY
I 1 = EL E:\E T ( I, 1 ) 12=ELEI\ET( 1,2) 13=ELE:'iET( I,;::) IIU::[L EI·IE T (I, 4)
C OBTAIII TilE LOCAL COORDHIATES 110\·/ \IITII RESPECT· C TO THE 11 NODE. c
c
AJ:::MIODE( 12, t )-MIODE( It, 1) BJ::AIWDE( I2,2)-AI~ODE( It ,2) AK=L\NODE( 13, t )-MIODE( It, 1) BI(=ANOD[( i3,2)-AIIODE( It ,2)
C OTHER GEm\ETRIC COHSTAI-HS. c
c
AREA=ABS(O.S*((AJ*BK)-(AK*BJ))) JAY=AJ·I-BJ KAY=AK+BK FAR::Lf .O'':ARE/\ F=2. 0''' AREJ\":KA Y~•JAY''' (JAY- KAY) \·J;: 1 • 0 H:..: t. 0 FARI·\U= F AF~,··t1U,'•\1
C GAMMAS AND BETAS c
G 1:: f FAR~: ( JAY-KAY)~: ( H'':H) ) /F G2= ~ ( FAR'':KAY) ,·:( W:H)) /F G3=(-1 .O*(FAR*JAY)*(H*H))/F . ALPHA I =-KAy,·, JAY'': ( BJ,': J\J+BK'': AK) -AK'':BJ,':
T (KAy,·, ( -2. 0~• AJ- B J) +JAy,·: AK)- AJ,': BK'': (KAy,·, TAJ+JAY*(-2.0*AK-BK))
ALPHA2 =KAy,·, ( JAy,·: AK~• BK- ( BK,': ( AJ~: AJ) + AK,;• ( BJ~: B J) ) ) ALPH A3= JA y,•: ( KA y,•: B J'': AJ- ( B J'': ( AK'': 1\K) ) - ( 1\J,': ( B K'': B K) ) ) G4=ALPHAI/F GS=ALPH/\2/F G6=ALPIM3/F Bl=(((AK+BJ)-(AJ+BK))*H)/FAR B2=((BK-AK)*H)/FAR B3=((AJ-BJ)*H)/FAR RETURtl J::HD
173
174
c *******************~*********************~*~********************~* C ,., F I i l I T E E L U t: liT 1\ i ~~\ L Y S 1 S ,., C :': : lA Ill fJI\lH11,111 I .. C ,., IIESH= (~lX:J J ·:: c *****~****************~************·'****************************** c c c C LH IE I:;, l ~JI I ALL NU{i\ Y S
,... I..
Dlili:ilS!OI! /\IInDE (81,1f), C(9il,1), Dl::l.T~\(162); TEL E I IE I ( 1 20 , Lf) I lJ ( I ;~::;I 3 ) ' ! !(! [ L T A ( 6 ) , u TJD E ( 8 I ) , TV:JODEt81), V(123,3)1CEl.iA(98)
f<EAL K(~l~J,JG) ,I<.E( 6,6) 1 lllJIIODE(81) \li< I TE ( 6, 998)
99 8 F OR.i1A T ( 1 flO I 3 OX' I :'::';;';:'::';:'n'::',·:'::'<:'::'::'<:'::'::'n'::'"'::'::'::':;';:'::':\'n':>n':·.'::'::'::'::'::'n'.-:'::'n':·.'::':;';;'n'::'r:'n'::'::'d: T:';·.': I ,
T'*********************'l T31X, ,,., 1\llf\l.YSIS FOR llE\/Totiii\11 FLQI./; FLO\-/ ElEHAVIOUR T I II D E;\ = 1 ;': I I T31:<,':': THROUGH 1\ DIE OF 1\il t\IIGL.E SIID\/!1 1
T ' 13 E L mt ,·:~ TI31X 1 '***********************************************************' T, T'**u***********'ll) .
C ,·..,·:·::·::·:: C Oi I TiW L I i lTE R (1ER S •':·:.-:'::'::': A llD 0 T H El\ I i lF OF;;~!\ T I Oil C READ I i I ,., ,.,., . ..,.,,., c
READ ( 5' /)Ill~ I I IE I tnl, liD F' I l, I f3\ I' D L' HI \I I Oil \ /R I T E ( 6 , 7 ) i lf(, liE , II! I , I W F 1 I i, IG\·1, D l , H , \I, 0 H
7 FOR11AT(.31X,l;I}H COIIHWL IIITEGERS , .. ,.,.,.,,.,.,.,,.,., . ..,.,,.,,.,CDIITRDL Ii'ITEGERS,I,I, T31:<,281i IIHTIAL tiD, OF D.D.F. tlK=,I5,1, T31Xt23H tiD. OF ELEI\EHTS IIE=,I5,1, T31X,28H 110, OF I'IODES 1111=,15,1, T31X,2811 REDUCED HO. Of o.o.F. IIDF=,I5,1, T31X,28H tlO. OF KNO\·HI D.D.F. II=, 15 1 1, T31X,28H l3AIID\·/IDTH IB\.1=, 15,1 T31X,28fl DIE LEI'IGTH DL=,Fio.4,1, T31X,28H HEIGHT OF DIE AT It!LET fi::,F10,4·,1, T3 1 X, 2 811 \II D Tf I OF 0 IE I.J::, F 1 0. Lf , I, T3 1 X' 2 8 H II 01,1- D HIE ,,IS I OI·IAL I z ER DH=, F 1 0. 4 I I' I, I)
TEST=O. 00001 AL.=O. ALPHA= (3. 11~ 159 265 5:': AL) I 18.0 \·/RITE ( 6 999) ALPHA
999 FDRt·IAT ( j HO, 30X, ,,,.,.,, . ..,.,.,.,.,...,.,,.,, . ..,., ... ,,.n':•'::':;\·:'::'n>::'::'n':;':-:n·,,·,,·,.,·,,·,-: .. ,·..,·::·::·:,·::';·::;'::'n'(•'::'::';·::,'<:'(;'::':-:: T, . ..,.,, ,
T,*************************'l T31X 1
1 ''' THIS OlE iS TAPERED AT All i\IIGLE= 1 ,E12,Lf, T' RADIAliS,.,,
TI31X, 1 ***********************************************************' T, , 1 ***********~******'II)
c ('
c c c c c c c c c c c c
c c c
c c c c
33
39
L10
OBTAI :1 1\LL. TilE IJATA I~UlU l ftEO.
CALL I : i I' UT ( lli\, : :c , i 1:1 , ! llJ F, i!, I fl\ I~ D l., ! I,\ I, E L E: 1ET : 1\: l ;)[) E, LH: LT i\ , Tll!l, ,\LPHI\)
LIS'fiiiG Til[ SUlli\OUTI:IES :\:ill PIUilTl\IG F.LEIIET, A110DE, f\!lD TilE llDU:I:Hv~Y co:llJ Ill CJ:1S :J:iLY.
DO 33 1=1 1 1111
:\D;)IFY TilE C:!Cl.';.JJ::.\TES OF tf\Cil iiDDE lst~Cf\USE TilE DIF. IS L~f'ERED i\T J\!1 AfiGLE ALPIIA.
CALL COORD ( I, ;\liDO E 1 11:1 ~ D L, ilL Pill\) C Jl 1 TI r,IUE I TER::Q IF(ITER.EQ.O)GO TO 1+1 DO 1+0 I:: 1 ,I~E
IJEI·/ VALUES OF VISCOSITY REQUIRED.
C.ii.LL UPDATE(ELE!'1ET,AilODE,DELTA, I,IIE,IIK,IItl, ITER) CotHI HUE
OBTA HI TilE STJ FFtiESS t1ATRIX AI~D THE LOADII~G VECTOR.
41 CALL ASSEI'IB(K,KE,ELEtiET,AI,IODE,DELTA,C,NE, HIK, Nl~ ,I·/, II, NDF, I BH)
c C SOLVE THE 11ATRIX EQUATiml TO OBTA II~ THF. C U~IKIW~/IIS. c
c
CALL SYt\SOL(NDF, !8\-/,IIDF,l ,K,C) CALL SYt1SOL( tJOF, I [3\·/, I~DF, 2, K, C) ITER:::ITER+l I F ( ITER·· 1 ) l13 , 4 3 , 5 5
C FIRST TitlE ROUI•ID c c C INSERT VALUES OF U~IKNO .. ItiS IIHO DELTA. c
43 DO 41+ I:: 1 , tiDF DELTA ( I) ::C ( I , 1 )
41+ corn I IIUE 1·/R I TE ( 6 , L1 5 ) ITER
175
45 FORI1AT(lli0r30X,39H ITERATIOH NU1\BER·::~1 ''1~1 ITERI\TIOII NUI\OER ::,IS.,/) I·IR I TE ( 6 , l16 J '
1+6 FORIIAT(31X,10H DOF 110. ,IOH PSII Tl:<,lOil DOF llO, ,lOll Pili! ,/)
DO L17I:::J IWF,2 \ 11\ I T E ( 6 , ~ d ) I , D E L T A ( I ) , I+ 1 , 0 E LT 1\ ( I + 1 )
1+8 F Ofz:\.L\T (3 1 X 1 I 5 , 5 X, F 1 0. 5 ; 15 , 5X, F 1 0. S) 4 7 C u i lT I i ;u E
176
C CALCULATE TilE ilcJOi\L VELOCITIES DF EACH C~.u:;:IIT c
c
0 J Lt ') l -~ 1 , I it: C i\ L L V El ( U , V , I , E LEI r T , D [ L T A , i~: I 00 E 1 IE , I If~, lilt ,
T\.',H) 49 COIITI IIUC
C U\l..CULATE THE VELOCITY uF EACH !lODE c
c
CALL VELOC(UIIODE,VIJODl:,U,V,El.EI'ET,DF.LTI\,MIOOE, HIE , t I K , I -HI , i I , H )
C I·IOOAI. VISCOSITY
c c c
c c c
c
C A L L V I S C 0 ( ; \U II OD E , E L E 11 ET , 0 E lT A , A: I GD F. , lJ i I 0 D E , '.!i I U D E , 1· I E , II K , Ill l , 1-1, H ) 1/iziTE(G,If:JJ) ITER
Lf 50 F 01~1 \AT ( 1 HO' 3 O:<, I ITER A TI 01~·.'.:'; 110. -:I I IS , /) \.'RITE( ),I,:J1)
1+51 F;J,;: i/-1 T (31 X 1 1 : llJ:J [;'n':·:::IIJ. 1
1 5X 1 1 X- VE LOCI TY 1
1 SX, T1 Y·· 1.'ELGCITY1 ,5X, 1VISCOSITY 1 )
DO If 52 I:: I , i 1: I 1·/f\ I TE ( 0,G53) I ,UIIDDE( I), VI lODE( I), t\U:IODE( I)
453 FOI~IIAT(31>:, IS, 10X,F10.5,5X,F10.5,5X,F10.5) 452· CONTI liUE
CALCULATE THE CHAIJGE I II PRESSURE .
CALL DEL TAP ( DP, ELF. 1\E T, DELTA, I\ I JDlJ F, U, V, I IE 1 ilK, I HI, \I, H) \liUTE(6,53) ITEI(,lJP
53 FD~91i\T(1i10 1 fq0,1?H I_J~~~\·!:Iml ~0._ ;,15,/l/, T3L"!.JH Cll,-1.ll..1c. I,l Pk.:;,JlJI,E DP-,flO.S,/,J
CALCULt\TE TilE FLO\/RATE
CALL FLORET ( Q 1 , Q 2 , Q3 , E LEI \E T, DC L T A, TfdiOOF.,U,U:IOD.E, IJE, IlK, 1111,\/,H)
\·/R I T E ( 6 I 5 Lf ) I T E R ' Q 1 , Q 2 , 0.3 SLf FDRi\AT(31:<,16H ITERAT!Oii 1!0. =,IS,/,/,
T31X,22H IIILET FLOIIRATE U.l=,FlO.S/,/, T31:<,2211 ii!DI/:\Y FLD\~R/\Ti: q2~;,FJO.S,/ 1 / 1 T.::l1 X, 2 ?.II nUTLET F L 0\ 1R J\T E Q3 =, F I 0 • .S , /)
GCJ TO 39
C TEST FOR COiiVERGE I ICY. c
55 CALL COIIVEG( DELTA, C, IJDF ,KODE, I~K) \/RITE(6,J)ITER,KODE-1
3 FOi\iif~T( 11l0,30)( 1 1 ITER!\TIOII~'::'n':II0.= 1
1 I5// T31X, 1 110. OF U:IIZ:IO\J:lS i!DT COIIVtr;G:::D YET=', IS)
DO 59I=1 ,i!DF CELTA( !)=DELTA( I) Df::I.TA( I)=C( I, 1)
59 COi !TI iiUE !F(I(DDE.IIE.l) GO TO 39 \/RlTE(6,(>6) TEST
66 FORI~T(11ID,30X, '************************************************** T-::·::t, ~ ********************'/ T31X, 1 ;': VALUE OF COIIVERGEIH CRITERlOtl IS',F10.5, T 1 X, I ;':I
T/31X, '***********************************************************' T, T'*************'//)
177
I·IR I TE ( 6, ~; 6) ITER 56 F ORr I/\ T ( I 110, 3 o:<, 5 91l ITEI~ATI at I NUt H3E R •'n'n'n': ITERATION 11Utt8ER ,.,.,,.,.'! TEHA T I
c
Tml I~U:\BER =,15,/,/, TI;2X,J9H PRESE 11T PREV!OUS,11X,I9H PRESEtiT PREV!OUS,/ 1
T31X,57H DOF IW. PSI! PSI! OOF fiu. PHil PHII) 00 571=1 ,IIOF,2 1/R I TE ( 6, 58) I 1 C ( I , 1 ) , CELT A ( I ) , I+ 1 , C ( I+ 1 1 1 ) , C E L TA ( I+ 1 )
58 F ORI·1A T ( 3 1 X, 15 , 5 X , 2 F 1 0. 5, I 5 , 5 X, 2F 1 0. 5) 57 COII'fl tiUE
C CALCULATE TilE ~IOOAL VELOCITIES OF EACH C ELEt·1EtH c
c
DO 60 1=1, tiE CALL VEL ( U , V, l , E L E t1E T 1 0 E L T A; AI·! ODE., I~E , IH~, N~l,
HI, H) 60 CmiTI NUE
C CALCULATE THE VELOCITY OF EACH NODE c
c
CALL VELOC(UI'JOOE,VNDOE,U,V,ELEf'[T,DELTA,AI•IOOE, TilE, IlK, l~tl, \·/,H)
C NODAL VISCOSITY c
c
CALL VISCO( 11UtlODE, ELEI'£ T, DELTA, AI lODE, UN ODE, V~IODE, HE, ~IK, Nil, 11, H) WRITE(6,6SO) ITE~
650 FOR1'1AT( 1 HO,JOX, 1 ITERAT!Oti""''i·iO.=i, 15, /) 1·/R ITE ( 6, 6 5 1 )
651 FORI·1AT(31X, 1 1mDE•'n':•':N0 1 ,SX, 'X-VELOCITY' ,SX, T' Y~VELOCITY' ,SX, 'VISCOSITY')
DO 6 52 I= 1 , fit! \·/RJTE(6,Lf53) I,UtiODE( I),VIIDOE( I),I·1UiWDE( I)
6 53 F ORI·IA T (3 1 X , 15 , 1 0 X, F I 0. 5 , 5 X, F 1 0. 5 , 5 X, F 10. 5 ) 652 CmlTI NUE
C CALCULATE THE CHANGE HI PRESSURE c
c CALL DELTAP(DP,ELEI·\ET,DELTA,ANOOE,U,V,NE,f.lK,IIti,\-/,H) \/R ITE ( 6, 53) ITER, D P
C CALCULATE THE FLOWRATE c
100
CALL FLORET(Q 1 ,Q2 ,Q3, ELEI''ET, DELTA, TANOOE,U,UNOOE,NE,NK,NN,W,H)
\.JR I TE ( 6 I Sl+ ) IT E R , Q 1 , Q 2 ' 03 IF(KOOE.EQ.1)GO TO 100
GD TO 39 \·IR I TE \ 6 , Lf 5) ITER HR!TE{6,4f)) DO 631=1 ,IWF+N,2 1-IR I T E ( 6 , 6 If ) I , 0 E LT A ( I) , I + 1 , 0 E L T A ( I + 1 )
6L: F OR~1A T (3 1 X, I 5 , 5 X, F 1 0. 5, I 5 , 5 X, F 1 0. 5) 63 COI'ITI NUE
101 103
\.JR ITE ( 6, 1 0 1) KOD E FOR1'1AT(31X 1 'KODE =', Il , 1THEFORE COI·IVERGEIICE TOOK PLACE') STOP END
,·: J\tlA L Y S I S ·:: THROUGH
FOR tiC\HmiiJ\tl FLO\-/; FLO\-/ BEHAVIOUR A DIE OF All U.llGLE SH0\-111 BEL0\1
C OH TRO L I liTE GER S ·::,·n·:·:n·:,'n':,·:,'n': C OIIT RO I. I tHE GE r- S
II~ITIAL tiO, Of D.O.F. 110, OF ELEI\E IHS 110, OF NODES REDUCED 110. OF O,O.F. 110, OF KII0~/~1 D. U. F. BAIJD~/!OTH DIE LEI~GTH HF!GHT OF OlE .L\T INI.ET II IDTH OF 0 IE 1101'1-D It\EI~S IOHt\L IZER
NK:: NE:: Nil=
NO F= tl=
I BH= DL=
H:: \·/=
DH=
162 128
81 98 Lf6 13
1. 0000 1 .oooo 1. 0000 o.oooo
178
I NDEX=1 1':
***************************************************************************** ,·:THIS DIE IS TAPERED. /\T AN AIIGl.E= 0, RADIAilS ·:: *****************************************************************************
ELEt-\E NT I NF 0Rt1A T I mt,•:,'n'n'n>n':;•:,•:,·:·::,': EL EME HT I NF OR 1·1A T 1 OH
ELH1E NT t·IOOE ~JD. HODE HO. ilODE t~O. VISCOSITY 1 1 11 10 1 .ooooo 2 1 2 11 1. 00000 3 2 12 11 1 .ooooo l} 2 3 12 1. 00000 5 3 13 12 1.00000 6 3 4 13 1. 00000 7 4 14 1 3 1.00000 8 4 5 llf 1. 00000 9 5 15 1Lf 1 .ooooo
10 5 6 15 1 • 00000 11 6 16 15 1. 00000 1 2 6 7 16 I. 00000 13 7 17 16 1 .ooooo 14 7 8 17 1. 00000 15 8 18 17 1. 00000 16 8 9 18 1.00000 17 10 20 19 1. 00000 18 10 11 20 1 .ooooo 19 11 21 20 1 • 00000 20 11 1 2 21 1. 00000 21 12 22 21 1. 00000 22 12 13 22 1 .ooooo 23 13 43 22 1.00000 zl.f 13 1lf 23 1 .ooooo 7.5 14 24 23 1 .00000 26 1l• 15 24 1. 00000 27 15 25 2l• 1 ,00000 28 15 16 25 I. 00000 29 16 26 25 1. 00000
179
30 16 17 26 1. 00000 31 17 27 26 1. ooooc 32 17 18 27 1 .ooooo 33 19 29 28 1. 00000 3L+ 19 20 29 1. 00000 35 20 30 29 1. 00000 36 20 21 30 1. 00000 37 21 31 30 I .ooooo 38 21 22 31 1. 00000 39 22 32 31 1 .ooooo l~O 22 23 32 1. 00000 41 23 33 32 1 • oonoo 42 ~l 2lf 33 1 .ooooo 43 31+ 33 I .ooooo 1•4 21+ 25 31+ 1. 00000 45 25 35 ]I+ 1,00000 46 "'l 26 35 1. 00000 47 26 36 35 I. 00000 L1-8 26 27 36 1 .ooooo LJ.9 28 38 37 1 .ooooo 50 28 29 38 1. 00000 51 29 39 38 1 .ooooo 52 29 30 39 1. 00000 53 30 40 39 1,00000 51+ 30 31 40 1. 00000 55 31 41 40 1,00000 56 31 32 41 1. 00000 57 32 42- LJ. 1 1,00000 58 . 32 33 42 1. 00000 59 33 LJ.3 '• 2 1. 00000 60 33 34 43 1 ,00000 61 3L+ 44 ~~3 1. 00000 62 Jl• 35 44 1 .ooooo 63 35 45 44 1. 00000 64 35 36 I+ 5 1. 00000 65 37 lj. 7 46 1.00000 66 H 38 47 1. 00000 67 48 I+ 7 1 .ooooo 68 38 l§ 48 1 .ooooo 69 39 48 1. 00000 70 39 40 LJ.9 1 .ooooo 71 40 50 49 1. 00000 72 1+0 41 so 1.00000 73 41 51 so 1 ,00000 74 Lf 1 42 51 1. 00000 75 42 52 51 1 ,00000 76 42 Lf3 52 1.00000 77 43 53 52 1 .ooooo 78 43 44 53 1. 00000 79 Lf4 54 53 1. 00000 80 Lflf 45 5L+ 1.00000 81 1-•6 56 55 1. 00000 82 46 47 56 1 .ooooo 83 47 57 56 1.00000 84 47 48 57 1 .ooooo 85 !J.8 58 57 1 .ooooo 86 1•8 49 58 1. 00000 87 49 59 58 1 ,00000 88 49 50 59 1,00000 89 50 60 59 1. 00000 90 50 51 GO 1. 00000 91 51 61 60 1 .ooooo
180
92 51 52 61 l. 00000 93 52 62 61 1 • 00000 91• 52 53 62 i. 00000 95 53 63 62 1 .ooooo 96 53 51+ ze 1 • (1()()00 9'1 55 65 1 .ooooo 98 55 56 65 1. 00000 99 56 Go 65 1. 00000
iOO 56 57 66 1. 00000 101 57 67 66 1 .00000 102 57 58 67 1 .00000 103 58 68 67 1 .ooooo 104 58 59 613 1. 00000 105 59 69 6.8 1.00000 106 59 60 69 1.00000 107 60 70 69 1 .ooooo 108 60 61 70 1 .ooooo 109 61 71 70 1. 00000 110 61 62 71 1. 00000 111 62 72 71 1 .ooooo 11 2 62 63 72 1.00000 113 61• 74 73 1 .ooooo 114 64 65 71~ 1. 00000 115 65 75 71• 1 .ooooo 116 65 66 75 1 .ooooo 117 66 76 75 1. 00000 118 66 67 76 1,,00000 119 67 77 76 1 .ooooo 120 67 68 77 1.00000 121 68 78 77 1. 00000 122 68 6·9 78 1. 00000 123 69 l9 78 1 .ooooo 12L• 69 70 79 1. 00000 125 70 so 79 1. 00000 126 70 71 80 1 .ooooo 127 71 81 80 1 .ooooo 128 71 72 81 1. 00000
1 81
N 00 f\L l t~F ORI-1.1\ T I ON>'n'o'n'o'n'n>n'r>'n'r NOD /\L l NF OHttf\ T I ON
NODE NO. X COORD y COORD D OF NO, DOf NO.
I o. 00 000 o.ooooo 145 1116 2 0. I 2 5 CJ() o.ooooo ll• 7 14fl 3 0. ?.5000 o.ooooo 149 150 /. 0.37500 o.ooooo 151 15?. 5 0. 50000 0 .ooooo 153 151} 6 o. h 25 00 o.ooooo 155 156 7 o. 75000 o.ooooo 15 7 158 l3 0.87500 0. 00000 159 160 9 1,00000 0. 00000 161 162
10 o.ooooo 0. 1 2500 11} 1 l'f2 11 0,12501) 0.12500 1 2 12 0. 25000 0. I 2500 3 I+
13 0.37500 o. 12500 5 6 Jlf o.;:;oooo o. 1 2 ~;no 7 8 15 0,()2500 0. 1 2500 9 10 16 0.75000 0.12500 11 1 'I ,_ 17 0.87500 o. 12500 13 )L} 1fl 1,00000 0. 1 2500 11}3 ]lf4 19 o.ooooo 0. 2 5000 IJ7 1311 20 0. I 25 00 0.25000 15 1 (, 21 (I. 25000 0.25000 17 1!3 22 0 • .37500 0. 2 5 000 19 20 23 0. ;10000 0.25000 21 22 21f O.G25UO U.25000 23 Zlf 25 0 ~ /5 CJ UCJ CJ.25000 25 ,,,..
.-.... 0 2(, o. :l? 500 o. 25000 27 2fl 27 1 .oo (Jl)Q 0. 25000 139 JlfO .-,n , ..... ) u. 000 GO 0.37500 133 J3lf 29 0.1251)0 0 • .37500 29 ··lO 30 0.2500(.) 0.37500 31 J2 31 0. 3 75 00 0. :J! 500 33 Jlf 32 0. ~iOOOO 0,3/500 35 y;
§4 0,62500 0.37500 37 3il 0. 7 sooo U,37500 39 I10
35 o. ,:;7 ::;oo 0.37500 41 [~ 2 36 1. 00000 0.37500 130 136 37 o.ooooo 0. 50000 129 130 38 o. 12500 0,50000 Lf3 If I+ 39 0. 25000 o.soooo :+5 lf() 40 o.:nsoo 0. 50 000 lf7 Lf8 ~~ 1 0. 50000 0. 5000 0 Lf9 50 l~ 2 o.r.25oo o.soooo 51 52 lj.J 0.75000 0.50000 53 5/.f IJ.Lf 0. 87 500 0.50000 55 5h 45 1. 00000 0. 50000 131 132 116 0. 00000 o.t>zsoo 125 126 lf7 o. 1 2500 0.62500 t;7 58 lf8 0.?.5000 0.62500 )9 60 49 u. 3 7500 0 J>2500 G1 6?. so 0 'soooo 0.62500 63 fill 51 o.G2500 ,O.E12500 65 (i6 52 0. 7 5000 0.62500 67 fiB 53 u.fl7500 0. 6 2 500 69 70 Slf 1. 00 UCl U () ,()2 sou I 27 1 ')·:> ... ,} 55 o.uuuoo 0.7:)000 121 122 56 0,1:2~)i)) il. ?:> ) J ) ·,~ l "I•)
,' ... r:-, ..J, 0.?.5000 0.75C>:_l0 73 71, 5G f). 17 r, 00 n _ 7 r~ n n,·l 7C:: ., (..
182 ...
5:) O,.~iOOOO o.; ~;ooo 77 /8 tiO Ll. I> :!S OJ u.; ~)ouu /:) ;)\)
61 u. ;1 ;iuoo o.;:i(J,,iu nt i3?. 62 (). :i; )00 l).7)000 f}J ()/+
(i3 1,0l100() 0. 7 s 00 () 123 1 ?.Lf (ilf 0 • 00 000 o.r>;soo 11 7 11 13 65 o. 1 ;:soo o.:)750o nr
U,) B6 fi6 0.?.5000 O.fl7SOO C\7 80 67 0.37500 o.:nsoo 8;J 90 68 o. :;oooo o. G7 500 91 n 69 0,62500 0.87500 93 9Lf 70 0. 7 ~)000 0.37500 95 96 71 0.117500 0. 87 500 97 98 72 1. 00000 0.87500 119 120 73 o.ooooo I. 00000 J9 100 7 Lf 0. 1 2 500 1. 00000 101 102 75 0.?.5000 1. 00000 103 1 OLf 76 0 • .37500 1.00000 105 106 77 0' s 0000 1 .ooooo 107 108 7i3 0.02500 1. 00000 109 110 79 o. 75000 1.00000 111 112 80 0 .fl7 500 1 .ooooo 11.3 I Jlf 81 1 .ooooo 1. 00000 115 116
183
OOll!IDMY C OtiD lT I OtiS **********************
DDF tw. PSI I DOF NO. PSII
99 ··1.oonooo 100 0,000000 101 -1 • 000000 10 2 o.oooocJ 103 -1 • 000000 1 ()If o.ooooc::J 105 ~ 1. 000000 106 o. 000000 107 -1.000000 lOB o. 000000 109 ~ 1 .oonooo 110 0. 000000 111 -1.000000 11 2 0,000000 11 3 -1 .onoooo Jjl} 0 .oooo 00 11 5 -1 ,000000 116 o.oooooo 117 -0.957030 118 0.656250 119 -0.957030 120 0,656250 121 ~0.8Lt3750 122 1.125000 123 -0,81t3750 124 1. 125(100 125 -0.683590 126 1. If 062 50 1 2 "1 -0.683590 128 1 .~~o6:~5o 129 -o. 5ooooo 130 1. 500000 131 -0.500000 i32 1. 500000 133 ··0.316410 1Jif 1,Lf06250 135 -0.311)1}10 136 1. 406?.50 137 -0.1511250 138 1.1 ;~sooo 139 ~0.156250 140 1.125000 141 -O.Oif2970 Jlf2 0.6 56?.50 1Lf3 -0. 0lf2970 1 Lflf 0. 6%250 llf 5 0,000000 Jlf6 0,000000 Jl;7 0,000000 . Jlf 13 o.oooooo ~ lf9 o.oooooo 150 o.oooooo 151 0. OiJ 000 0 15 2 0,000000 153 0,000000 i 51, o.oooooo 15 5 0,000000 i56 o.oooooo 15 7 0. 0 C()00C1 153 0,000000 1 ,. " ;J..- (), 000000 1 ()0 o.oooooo 161 0. 00000 0 1()2 0. 000000
1 84
APPENDIX P2
Resulting computer programme from the second analysis.
****************************************************************** :~NOT AT I OtiS ******************************************************************
:'r A :'r
A(3)=COIITAitiS THE FIRST THREE GEOIIETRIC COtiSTAIITS OF At~ ELEIIF:HT,
ABC(NE,9)=/\RRAY OF ELEI1EI·ITAL GEot1ETRIC COIISTIHHS.
IHIODE(1~11,5)=ARF\AY Cm!TAIU!I1G ll·lFORIIATint~ Al30UT 1\LL THE ~lODES, THE FIHST H/0 POSITIOIIS FOR THE (X) 1\tlD (Y) COORDII~ATES(GLOGAL) MID THE LAST THREE ~OR THE ll,O,F, I·IOS, I.E. X-VELOCITY,Y-VELOCITY,AI~D THE PRESSURE RESPECTIVELY,
ALPHA=AilGLE OF TAPER OF CHAt·II~EL,
B(3)=COI'IT,'\INS THE SECOI10 THREE GEOtiETRTC CotiSTMITS OF 1\H ELEI1ENT,
'",'( c 'i't
C(3)=COtiTA!NS THE THIRD THREt GEOf-1ETRIC CotiSTAtiTS OF AN ELEfviEiH.
CE(I~K,l)=ARRAY H! 1/HlCH THE SOLUTiotl FOR THE UtiKIIO\ItiS IS STORED.
CELTAUIK)=PREV!OUS VALUE OF THE SDLUTIOtl VECTOR(CE) IS STORED II~.
:': D :':
DELTA(IIK)=ARR/\Y OF REQUIRED HODAL V/\LUES, THESE ARE THE (X) MID (Y) VELOCIT!i=:S, MID THE PRESSURE.
Dl=DIE LEI·IGTH,
".'( ·:: ..,·: ..,•: ;'r
:·: E :': 'i'('i':..,·:-.'(".'(
ELEiiET{!lC,S):::Af=~RAY CotiTA!llltiG TIIFORt1/\Tir:tl AROUT ALL THE ELE11Et1TS, THE FII~ST THREE POSITIO!:S FOR THE tiUDAL f·IOS, A liD THE LAST HJO FOR TilE IJ I SCOS ITY !1110 THE AREA RESPECTIVELY,
1 85
~·: F ~·t
-;'r .,.r..,•r..,·:~'t
FBl=FL0\·1 BEHAVIOUR li·IDEX OF THE POLYIIF.R.
F OIJ R i 2 ::VAL lJ E 0 F ld 2
H=HEIGHT OF CHM!NEL AT ItiLF.T.
I 1 S=COUtiTER; DEGREE OF FREEDot1 t1Ut1BERS; NODE NUI·IBERS
I B\I=BA l~D\-1 I 0 TH
ITEP=COUtlTER lJF THE 110. OF ITERAT!Oi'IS OONE.
~·( J >'t
J' S=COlHHER 1 DEGREE OF FREEDot1 NUt1BERS; ~lODE 11Ut1BERS.
K(I~DF, 18\·I)=SYSTEII STIFnlESS tiATRlX HI RECTi\NGULM A NO 8 A 1~0 E 0 F OR t·\,
KE(9,9)=ELE11EtHAL STIFFtiESS nATRIX.
KDDE=KEEPS CDUHT OF THE liD. OF UI~K\IOI·IS ~lOT. COtiVERGED.
K1(6,6l=A SUB-MATRIX OF 412.
K2(6,3)=A SUB-MATRIX OF KE(9,9)
LCOUNT(I·Iti)=KEEPS COUIH OF THE t·IUI1BER OF ELEI1EIITS COtlTRIBUTHJG TO A t!ODE,
11U=V!SCOSTTY OF Atl ELEI\EIH
1·\U~IODE(tlll):::ARRAY OF THE VISCrJS ITY OF Ei\CH tlODE.
186
N=tiO, OF IJOil ZERO O,O,F.
NDELTA(6):::.f\HR~\Y CClllTAIIHIJG THf O,O,F, tto,; OF E1'.CH !·lODE,
HDELT(9)=ARRI\Y CCIIITAirlltK1 THE O,O,F, tiOS OF 1\N El.Ht::NT.
NDF=UIH<rW\-/H 110, OF 0 ,0, F,
NDOF=COUTER FOR THE O,O,F. NUMBERS
NE:::tlO, fJF ELEI\OlTS,
NK=INITIAL NO, OF D,O,F,
NKI,NKJ=D,O,F, NO, OF AN ELEMENT.
NN::NC, OF IIDOES.
NNODE=IWDAL liUt\BER •
..,., ..,·: .. ,., ;': -,': ·:: Q ~':
Q(l)=FLO\/RATE AT VARlOUS SECTIONS OF THE CHAllllEl.,
;': T ..,·: ~~ l': "i'( "i': ~·,·
T=TANGENT OF AN ANGLE,
~·: "\~ ~·: ..,.,..,.( ",'( .. ,.( .., ...
\./=\.flDTH OF DIE
X= X- COORD I I~ATE (GLOBAL) OF A tlODE, XM=X-COORDII,IATE OF THE CHITROIO OF Ml F.LH1ENT.
XI ,X2,X3=THE X LOCAL COORDHIATES OF THE VERTICES Of At~ ELEt1EIH HITH RESPECT TO ITS CENTROID,
..,·:..,·:;': "i':-,':
Y=Y-COORDJt.lATE(GLOBAL) OF A NODE.
nl=Y-COORDWATE OF THE CENTROID OF All F.LH\ENT,
Yl, Y2 Y3=THE Y LOCAL COORD! IIAHS OF THE VERTICES OF AN ELEtldiT \liTH Rf::SPECT ro ITS CEIHRtJID,
187
SUBROUTINE CONTRO(NK,NE,NN,NDF,N,IBW,DL,H,W,ALPHA) c c ********************************* C THIS READS IN CONTROL INFORMATION c ********************************* c
READ(5,/)NK,NE,NN,NDF,N,IBW,DL,H,W
c C OBTAIN ANGLE OF TAPER OF THE DIE c
AL=O. 0 ALPHA=(3.141592655*AL)/18.0 WRITE(6,1)NK,NE,NN,NDF,N,IBW,DL,H,W,ALPHA
1 FORMAT(/31X,' CONTROL INFORMATION*******CONTROL INFORMATION'//
T31X, 'INITIAL NO. OF D.O.F.' ,6X,'NK=' ,I5/ T31X,'NO. OF ELEMENTS' ,12X,'NE=' ,I5/ T31X, 'NO OF NODES' ,16X,'NN=' ,I5/ T31X, 'UNKNOWN NO. OF D.O.F. I ,5X, 'NDF=' ,t5/ T31X, 'NON ZERO NO. OF D.O.F.' ,6X,'N=' ,I5/ T31X, 'BANDWIDTH' ,17X,'IBW=' ,I5/ T31X,'DIE LENGTH' ,17X,'DL=' ,F10.5/ T31X, 'HEIGHT OF DIE AT INLET' ,6X,'H=' ,F10.5/ T31X, 'WIDTH OF DIE' ,16X,'W=' ,F10.5/
188
T31X, 'ANGLE OF TAPER OF DIE ALPHA=' ,F10.5,3X,'RADIAN'///) RETURN END
1 8 9
SUBROUTINE SOLVE(ELEMET,ANODE,DELTA,ABC,NK,NE,NN,NDF,N,IBW,DL,H, TW,ALPHA,K1,K2,K,KE,CE)
c c
c
DIMENSION ELEMET(NE,S) ,ANODE(NN,S) ,DELTA(NK), TABC(NE,9) ,CE(NK,1),Q(9),CELTA(243)
REAL K1 (6,6) ,K2 (6,3) ,K(NDF,IBW) ,KE(9,9)
C ELEMENTAL AND NODAL INFORMATION REQUIRED c
CALL INPUT(ELEMET,ANODE,DELTA,ABC,NK,NE,NN,NDF,N,IBW,DL,H,W,ALPHA) c C JUST LISTING ANODE AND DELTA ONLY. c
c
GO TO 650 ITER=O KODE=O
200 ITER=ITER+1
C SYSTEM STIFFNESS MATRIX (K) AND LOADING VECTOR (CE) REQUIRED c
c
CALL ASSEMB(K,KE,ELEMET,ANODE,DELTA,CE,ABC,K1,K2,NK,NE,NN,NDF,N, TIBW,DL,H,W,ALPHA,ITER)
C MATRIX EQUATION SOLVER REQUIRED. c
c
CALL SYMSOL(NDF,IBW,NDF,1,K,CE) CALL SYMSOL(NDF,IBW,NDF,2,K,CE) IF(ITER.GT.1) GO TO 400
C NEWTONIAN SOLUTION IS NOW AVAILABLE C TO BE PRINTED. c
c
CALL OUTPUT(ITER,DELTA,ANODE,ELEMET,Q,CE,NK, TNE,NN,NDF,N)
GO TO 200
C TEST FOR CONVERGENCY. c
c
·100 CALL CONVEG(DELTA,CE,NDF,KODE,NK) WRITE(6,3)ITER,KODE-1
3 FORMAT(1HO, 'ITERATION***NO.=',IS// T1X,'NO. OF UNKNOWNS NOT CONVERGED YET=' ,IS)
DO 450 I=1, NDF
C PREVIOUS VALUE OF THE SOLUTION IS STORED C IN (CELTA) ,AND THE PRESENT VALUE IN (DELTA). c
c
CELTA(I)=DELTA(I) 450 DELTA(I)=CE(I,1)
IF (KODE.NE.1) GO TO 200
C NON-NEWTONIAN SOLUTION IS NOW AVAILABLE TO BE C PRINTED. c
CALL FINALE(CELTA,DELTA,ANODE,ELEMET,NK,NN,NE,NDF,N,Q,ITER) 550 WRITE (6,600) KODE 600 FORMAT(1X,'KODE=' ,I1,2X,'THEREFORE CONVERGENCE TOOK PLACE') 650 CONTINUE
RETURN END
~; LHll( OUT I liE I liP LJT ( l:l. U 1 F. T, t\ IIOD E , D F LTJ\ 1 !\11 C, T i 11< I t If. ' 1111' I J[) F' t I' I !3\ I, D l. ' i I' \·I' i\L pIll\)
c c *********~************************ C T II IS I IIPIJT S :~.LL Tlll: /W 1\ l L1\llLE D 1\T/\ c ********************************** c
c
llim::1s ror1 r::t. F.rtET{ IW I 5), AilODE( 1HI 1 5), on.TA( tHO, TAG c ( liE I 9) , 1\ < .1 ) , n ( 3 ) , c ( ;1 )
C I·IODAL IIIFDnlti\Tiotl IS OllTI\I:IEO c C 111E EtiTRIES OF MIOD[ i\RF. GEIIERATED. c C THE FIRST 1\I~D SECDIIIl CIITR!ES 1\RE' Fm THE C COORD I H.I\TES OF TH't II ODE i\!10 THE OTHF.Jl.S C fiRE THE OF.GI\Ef: OF Ff\UDOI1 IIUIIBEHS C AS SOC I 1\TF.D TO THE tiOOE. c
c
C'\LL GE IIODF.( 1\IIODE, II! I, tl, IIDF) \ IR I TE ( (> , ?. 0)
20 F mr tAT ( I J 1 X J I 1100 AL I !.JF OR II!\ T I OJ pH:l';-.':,·:,':l'tl'n':l': I '
11 1l·XLii.L ItJFor.:ll/l. r r or,,.,.,.,.,.'"'·:·::-.· ....... , ... , ........ ~ !OD 1\L. r ~JF or~·v\ rIot!' I 1 TJ1X, 1 1100[ 110. 1 ,11X, 1X Cllflf\0 1 ,3X, 'Y COORD' ,5X, 1 00F 1 ,
T' ilfJ. 1 ,?.X, 1 00F 110. 1 ,2X, 1 DOF 110 1 )
D 0 ?. 1 l :: 1 , I·H I
C Q.O.F. 11Uf18ERS c
c
I 1 =!\II Of) E( I, 3) I 2 =AI-IIJO r( I, If) !3=!\!100~( !,5) \F; I TF. ( 6 1 ?.2 ) I I /\, 110[) E ( I I 1 ) I /\1 JCD E ( I • 2) I I 1 ' I 2 ' I3
22 Fo:;rtAT(31 X, I5,5X,2F10.5,LfX, I5,lfX, I5,5X, !5) 21 COt!TI I!UE
C JUST LISTIIIG /\!lODE MlD DELTI\ CJt'!LY. c
GO TO 260 c C 'THE; GLOBAL COORDI!!,\TES f1IVE!I ABOVE ,'IRE FOR 1\ UtiiFDRII C RECTM!GIJLI\R IIESII. IIODIF!f:D tlEL0\-1 TD M:COUIIT FlJR TliE C TAPERIIIG OF THE DIE. X CL1Cli\OlllATES REI\,\!11 TilE S;\r~tE c
I·IR I TE ( 6 , ?. 3 ) 2 3 F nr; t \AT ( 1 11n, ' c m R E CTF n 11 oo AL I IIF mr tCI TT nr 1,·,·:.·:,·: 1 ,
T 1 ,·:·.':·.':·.'~ ,·..,·n•:;';,':;':,•:-.•..,•:;•: CDR. R F. CT F. D I I 00 AL I 11F OR i \!\ T I 011 1 I I T1 X, 1 HOOE 110, I ,lfX, 'X COORD' ,3X, 'Y CrJr1f;0 1 ,sx, 1 DOF 110, I
T ,2X, 1 00F 110. I ,?.X, 1 DOF tJO I)
T=TA I~( ALPHA) R=T/ DL DO 21+ I= 1 I lltJ 1\! I 0 0 E ( I 1 2) :: /HJ 00 F. ( I 1 2 ) ,·, ( 1 - MIOO E ( I , 1 ) •': !1) 1·1 =AI,JOOE( 1,3) I 2=AIIOD F-:( I ,Lf) l.3=AIIODE( l, 5) IN~ ITE (6 ,?.2) I I AI·IODE( I J 1), /\IIODE( I, 2)' I1, !2, !3
2Lf Cnt!TI ti!JE c C DATA ABOUT TilE ELEIIEIITS IS READ II! c C THE FlflST THREE POSIT!nt1S FOR THE 1!09AI. 11Ut1f3F.RS, TilE C FOURTH FDF\ THE VISCOSITY AIID THE FIFTH FrlR THE i\J~Et\ c
\/P ITE ( 6,?. 5) 25 FClRr\/\T( 1110, 1 ELEI\EIIT IIIFORI1ATI0!1•';,·,,·,,·..,•,,•n':''"'r-.':·.':·.';,': 1
1 T' ,., . ..,·,-:n'n':·.'; r: UJIE liT !!IF Of\:11\T I o:r• I I T1X, 1 ELE11l-:IIT 1 ,3X, 1 tlf'lfl[ ii0, 1
1 2X, 1 1100F. 110. 1 ,?.X, 1 1100[ 1l0. 1 ,
T 1 X J I v 1 s Cll s IT y I '/f X I I fir\ EA I )
190
rJO 26 r= 1,t1r: IU:AD(S,/) !1,!2,13
c C ELHIOIT VISCOSITY c
ELEI·IET( I,lt)::J.O c C tiOOAl. tHJi1BERS c
c
ELE11ET( I ,1 )::: I1 +0, 1 F. L E11 E T ( i I ~):::: I /. + 0. 1 EI.EIIET( !,3)=l_H0,1
C CEIITI~OIO Or E/\Cl-1 ELE1\EIIT IS tJ0\1 OllTAitJEO c
xr-1:::: ( MJon r: < r 1 , 1) + Ai mo H I 2, 1 ) + Armo E < r 3 I i) > /3. o YII=(;\IIODE( 1.1 ,2)+1\llf)DE( 12,2)+AIIOOE( 13_,2))/.3,0
c C LOCAL COORD Ji·IATES 1/ITH RESfJECT TO THE l.EIJTROID c
c
XI =t-t!ODF.( 11,1) ·XI·\ X2=At!ODF.( P, 1 )-XI·\ Y3=At·l11DE( 13 1 1) -X~i Yl=1\l!ODE( !1 ,2)-Ytl Y2=MIODF.( 12 ,2)-Yt-1 Y3=MIOOE( !3,2)-Ytl
C AREA OF Ai·l ELE11EIH c
c
A.REA::J,5·::(X2·::Y3-X3'''Y2) E L E!· IE T ( I I 5 ) :: J4R E A
C GEOtiETRTC CDIISTAIITS OF EACIJ ELE!1EIIT IS C OBTAI I·IEO , E/\Cfl QUMITITY IS D IV IDEO l3Y T\/TCF. THE C AREA FOf~ C OliVO! IE I ICE c
c c c
27
28 26
260 32
A(1)=(X2*Y3-X3*Y2)I(?.O*AREA) A(2)=(X3*Y1-Xl*Y3)1(2,0*AREA) A( 3)::: (X J>':Y2 -X2'''Y 1) I ( 2 .o·:: AREA) 8( 1 )::(Y2-Y3 )I(2.0>':AREA) B ( 2 ) :: ( Y 3 - Y 1 ) I ( 2 , 0>': ARE A) B(3)=(Y1-Y2)1(2,0*AREA) C ( 1 ) ::: ( X3- X2) I ( 2, 0>': ARE A) C ( 2 ) ::: ( X 1 •• X3 ) I ( 2 , 0 >'q\R E A) C (3 ) ::: ( X 2 •• >: 1 ) I ( 2 , 0 ,., ARE A) DO 27 J::1 ,3 Acc( r, J) =A( ,J) ABC( I, JH)=8(J) ABC( I, J+6)::C(J) COIITI I'IUE \.JR r T E < 6 , :~ n) I , I 1 , I 2 , I J , E L H t E r ( I 1 tf > FOR I"IA T ( 1 X 1 I 5 1lf X • I 5 ' s X I I 5 I 5 X , l5 ' 5 X J 2 F 1 0 • 5 ) emiT!. I'IUE
OOUIIDARY COIIDITIOIIS STORED Ill DELT/'.().
1·/R IT F. ( G , 3?.) FORt1AT(ltl0 1 32:<, 1 f30UIIDNW COIIDITIOiiS 1 /
T3iX, '*******************'II T31X,' DrJF 110 ',' Pr<ESSURE ')
DO 33 I=llflF+ 1, iiDF+!l DELTI\(I):-:J.O
3 ~ \ IP, I T E ( () 1 3 If ) I 1 D F. LT 1\ ( I) 3 -f F rn r lA T ( :ll :<I I~' I 5 ~<, F 1 o, s J
IU~ TURII E rID
1 91
SUBROUTINE GENODE(ANODE,NN,N,NDF) c c ************************************ C THIS GENERATES THE NODAL COORDINATES C AND THE DEGREE OF FREEDOM NUMBERS c
DIMENSION ANODE(NN,S) c C IN THIS PROCEDURE, THE UNKNOWN NODAL VALUES C ARE NUMBERED FIRST, FOLLOWED BY THE KNmt\TN C NON-ZERO VALUES AND THEN THE ZERO VALUES. c
c
NDOF=1 DO 1 I=1,9 DO 1 JJ=1,9 J=10-JJ I1=0 I2=0 I3=0
C OBTAIN NODAL NUHBER c
NNODE=1+9*(J-I) c C IF NODE LIES ALONG THE TOP OR BOTTOM C OF THE CHANNEL WHERE THE X-COMPONENT OF C VELOCITY IS ZERO, THE X-VELOCITY C D.O.F. NUMBER IS LEFT BLANK IN THE MEANTIME c
IF((J.EQ.1) .OR. (J.EQ.9)) GO TO 2 c C NODE IS NOT ALONG THE TOP OR BOTTOM OF C THE CHANNEL. ASSIGN A DEGREE OF FREEDOM NUMBER C TO THE X-VELOCITY. c
c
I1=NDOF NDOF=NDOF+1
C IF NODE LIES ALONG THE ENTRANCE OR EXIT C OF THE CHANNEL, WHERE THE Y-COMPONENT OF C VELOCITY IS ZERO, THE Y-VELOCITY D.O.F. C NUMBER IS LEFT BLANK IN THE HEANTIME. c
IF ( (I . EQ. 1 ) . OR. (I . EQ. 9) ) GO TO 2 c C NODE IS NOT ALONG THE BOUNDARIES OF THE C CHANNEL. ASSIGN A DEGREE OF FREEDOM NUMBER TO C THEY-VELOCITY. c
c
I2=NDOF NDOF=NDOF+1
C IF NODE LIES ALONG THE ENTRANCE OR EXIT C OF THE CHANNEL, WHERE THE PRESSURE IS KNmt\TN,
1 92
C THE PRESSURE D.O.F. NUMBER IS LEFT BLANK IN THE MEANTIME. c
2 IF ( (I . EQ. 1 ) . OR. (I . EQ. 9) ) GO TO 3 c C NODE IS NOT ALONG THE ENTRANCE OR EXIT C OF THE CHANNEL. ASSIGN A DEGREE OF FREEDOM C NUMBER TO THE PRESSURE. c
I3=NDOF NDOF=NDOF+1
C OATI\lll TIIF. CDOI~Ollli\TES OF{\ liODE. c
c
3 X:: ( I ~ 1 ) ,·,o. 1 :> S Y= ( ,J ~ 1 ) ·::o. 1 2 5
C STORE TilE i\BOVE VALUES Ill 1\tiODE. c
c
/\!lODE( IIIIODF., 1 )::X MIOnE( :liiDDE, /.) ::Y MIO\l E( ilil()f)l~ I 3) =I 1 +0. 1 MIODE(II!IODE,If):.:I2+0.1 AiiOfJE( I•I'IODt,:;):: !3+0, I COIIT!IllJF.
C TilE I<.IJO\t:l 11011-ZERO lln~)I\L VAI.lJES 1\RE ti0\-1 C 1\SS!GiiF.D DEGf1EI:-:E OF F1~F.EOOI\ IILHICERS, c
c
!1 0 L, J :: 1 1 9 1\!IC8E( 1 +'),'<( J-1) 1 5) ::'IOOF+O, 1 IIDiJI~:: i IOOF + 1
L; COIITI ~.JUE 1\:.:llOF+Il+ 1
C THE ZERO I·IOOAL VALUES ARE 1101·/ ASS IGtiED C DEC1f1EE OF FREEDOII I!LH1BERS, c
005!=1,9 DO S J::l ,9 !JilODE:: !+C);'<( J-1) !1 =AIIfJOE( IIIIODE,3) !2=1\IIODE( I~IIODE:,I+) I 3:: A liDO E (I if I onE, 5) IF(I1,GE.1) GO TO 6 AI' I il D E ( i I i I 0 D E , 3 ) :::I i~ 0 • 1 il=l\1· 1
6 IF(I2,GE.1) GO TO 7 AliOfJE( IF lODE, 1;)::!·1+0, 1 1·\=11+1
7 IF(I3.GF..1) GO TO 5 AIIODE( I·I!IODE,5)=!1+0. I 1·1=11+ 1
~; COIITI I·IUE RE TlJR!\1 END
193
194
sun R n lJT r 1 IE /\S :; E llll ( 1-:. 1 I( E , r:: LEt 1 E I' , 1\ll onE, n E u /\ , c c , AD c I 1~ 1 , K 2 I IlK, t 1 E , lH 1, T I I D F , I ·I 1 1 !31 I , fJ L, II , I I , A l. PI It\, I T ER)
c c ~************************************~****************** C TillS CI\LClJL/\TES TilE ST!FFIIESS 111\TRI:< or EI\CII ELEt\EIIT AIIO C IW,ERTS IT lllTrl THE SYST!.-:t\ STIFF!IESS i\1\HUX. TilE C 1\1\Tf~I:< iS n/\tiOI:O 1\ilD THE L\ltiDI<It; \iF.CTr1r. DflTAI!IED Ill C T II [ r fW C E S S c ***********************~********************~*********** c
c
D I! \ E I IS I 0 II E I. [11 E T ( ! l E 1 ) ) , /\ ll 11 0 [( II ! I , S ) , D C L T 1\ ( ! I 1\ ) , C [ ( II D F , 1 ) 1 ~l\l3C(!lE 1 CJ) 1 !111ELT(C))
f, ':: 1\ L V, ( l I[) F 1 I ?,I) , K!: ( C) 1 ~)) 1 I( 1 ( 6 1 r; ) 1 \(2 ( 6 1 3 ) fJ 'l I.L f) I:: 1 ' : ) D F J .1 I~O J = 1 , I ~)\I l<(l 1 J)=O.O Ct.( l, l )=0.0
L:-0 CDtiTI iJUE
C Ei\CH ELE!\EIH IS BROUGHT HI FOR PROCESSIIJG c
c
D () If 1 I:: 1 I I·IE D 0 Lf 2 I c :: 1 I 9 DO LL2 J::1 1 9 !( E ( I c I j) ::o. 0
1+2 CCJ!ITI IHJE
C OllTAIII THE I\1~TfHCES K1 f\'10 K?. FD>\ ELEtlF.iiT (I) c
Ci~LL 111\TRIX (1:1,K2,i\flCd,!·IE) IF ( ITER. GT , 1 ) CALl. UP D 1\ TE ( E L Ell E T 1 AWl:l E 1 11 E L T A 1 K 1 1 1(7. , I 1 ~IE , I II I, !·IK)
c C EUJ101T/\l. ~.lTIFFIIF.SS 1\f\TRI:< ·Is C/\LCUL1~TED
c DO !13 IC=1 1 6 DO 43 J::l ,() 1\ E ( I c , J ) = 1-: 1 ( I c I J ) ,., E L E r 1 E T < 1 1 1; ) ,., 2 • u ,., E LEt 1 ET ( T , s )
4 3 Cot I Tl t IU l:: DO 4L; IC::1,6 on L!L; J=l 3 K E ( I C 1 J + 6 ) = K 2 ( I C , ,J) ,., 2 , 0 ,., EL El \ E T ( I , 5 )
/+4 COIITI i·IUE Wl L; 5 I C::: 1 ,3 on 1f5 J::l 16 I< E ( I C+6 , J FI(E ( .J, I C+ 6)
L;5 COIITI IIUE
C ELD\EilTAL ST! FFI!F.SS ~ll\TrUX IS /\Vi\. !L/\!.lLE; !10\/ c C 0 8 T 1\1 1-1 Tfl E ll. 0, F, ~lOS, c
c
I I ::[LEIIO( I I 1) I?.::: r:: L Ell ET ( ! , 2 ) I3=ELE11CT( !,.3)
1,1 0 F. LT ( 1 ) ::: f\! HJ 0 F ( I l , 3 ) . llOELr(2)=1\llflnt:( !?. 1 3)
II!' E LT ( 3 ) :: i\ W!D F. ( I3 , 3 ) i I D E LT ( 4 ) ::: J\tllJIJ F. ( II , If ) I I[) F. L T ( 5 ) = t\ II 11 D E ( I ?. , If ) IPlF.LT(f)):::,\1/nOE( I3,lf) lllFLT(7 )=t\IHlOF.( II ,5) I JOEL T( rl) ::J\tiODr::( I2, 5) :!DEL T( 9) =1\lliJDF:( !3 1 5)
C CIIECK ALOIIG THE Rnll OF t\ATRIX, C IF IT IS f1F.Yr1~1D THE f~i\1 !GE OF ( K) I\Er1LECT IT C IF IT IS IIITflll·l THE r{1\IIGE, GO i\LOIIG TilE UJLltrHJ c
00 2 1<!=1,9 c C IF BEYOIIO 1< 1 S RAIIGE, TRY !IEXT Rm/ c
c c
c
IF(IlDELT(I\I) ,GT .tiOF)GO TO 2 I i K I :: I l 0 E L T ( 1\I )
DO 3 KJ=l ,9 I·IKJ=I,IOELT(K,J)
C IF BEYGriD f: 1 s RMIGE, OBT/1!11 THE LOMIIIG VECTOR c
IF (IIIU.GT,I!OF)GO TO 4 c C \.f!THHJ K'S RIHIGE. 110\/ !~I OAIJDEO FORI1 c
IJKJ 1 =NI\J- HK I+ 1 IF(Ili~J1.LT.1)GO TO 3
K (I If\ I 1 I·JI(J 1 ) =I< (Ill< I, I 11\J 1 ) +KE ( K I, KJ) G'l TCl 3
I~ C E ( II K I , 1 ) =C E ( I II\ I , 1 ) - K E ( 1\ I , 1\J )":DE LT 1\ ( I II< .I) 3 CDIHII,IlJE 2 CotiTI HUE
Lf 1 C 0 l'lT I IIU E RETURIJ E r JD
195
SUf1ROUTJ fiE f IMR I X(~~ 1 I I<<' i\[JC, l I liE) c c *********~*********~************************ C THIS 08TATfiS THE IV\TRJCES 1\1 A:IO !(2 FOR EACH C E L. E: 1 E. JT ( I ) c ******************************************** c
c
o I r 1 o1s r n 11 A r1 c <t 'E I 9 > , .l'l. C3 > , G < 3 ) I c < 3 ) REAL K1(6,6),1<2(6,3)
C GE011ETR!C Cm!STAIITS OF EACH El.E!1EilT c
on. L~9 J=l ,3 fl( J) =M3C( I, J) G ( J) =A f3 C ( I , J+ 3 ) C ( J) =t\BC( T, J+6)
1~-9 COIITI IJUE c C 0 f3 T A T f I 111\T H I X K 1 c
DO 50 11=1 ,3 DO 50 J::J,3 K 1 ( I I , J ) ::: 2 • 0 ~·, 8 ( I ·1) ~·: f3 ( J ) + C ( f 1) ~·, C ( J ) I( 1 ( H+ 3 , J + 3 ) :: 2 ~ 0 ,., C ( I 1) ,., C ( J ) + f3 ( 11) ,., f3 ( J ) I< 1 ( f 1, J+ 3 ) :: t ( II; ~·: fl ( J ) I< 1 ( Ill-] I J ) =B ( 11) ..... c ( j)
50 C Dl 1 TI ! JU F. c C OBTAHI f·iATRTX K2. c
00 51 1·1=1 ,3 DO 51 J=l,3 l( 2 ( I 1, J ) :: !\( II) ,., :1 ( J ) 1~ 2 ( II~ 3 1 J ) =A ( 11) ~·, C ( J)
51 fJl' IT I 1\UE f;[TUfUI El\0
196
197
SUBROUTINE UPDATE(ELEMET,ANODE,DELTA,K1,K2,I,NE,NN,NK) c c ********************************************* C THIS SUBROUTINE UPDATES THE VISCOSITY OF EACH C ELEMENT c ********************************************* c
c
DIMENSION ELEMET (NE, 5) , ANODE (NN, 5) , DELTA (NK) , NDELTA ( 6) REAL K 1 ( 6 , 6 ) , K 2 , ( 6 , 3 ) REAL MU
C OBTAIN THE NODAL NOS. OF EACH ELEMENT c
c
I1=ELEMET (I,1) I2=ELEMET(I,2) I3=ELEMET(I,3)
C OBTAIN THE REQUIRED D.O.F. NOS. OF EACH NODE c
c
NDELTA( 1) =ANODE (I1, 3) NDELTA(2)=ANODE(I2,3) NDELTA(3)=ANODE(I3,3) NDELTA(4)=ANODE(I1,4) NDELTA(5)=ANODE(I2,4) NDELTA(6)=ANODE(I3,4)
C CALCULATE 4I2(FOURI2) NOW. c
c
FOURI2=0.0 DO 60 IC=1,6 DO 60 J=1,6 FOURI2=FOURI2+K1 (IC,J)*DELTA(NDELTA(IC))*DELTA(NDELTA(J))
60 CONTINUE R=SQRT(FOURI2)
C CALCULATE THE VISCOSITY (MU) IF THE FLmiJ BEHAVIOUR C INDEX (FBI) OF THE POLYMER IS KNOV'JN c
FBI=0.5 MU=(R**(FBI-1.0)) ELEMET (I, 4) =~1U RETURN END
c c c c c c
c c ,.. ....
c c c
c c c
c
1000
250 260 200
2000
28S 290
300
400
198
SLJBRDUT r ltE s y'IISOL < 11111 ;v1, 110 !1-1, 1·:t<K I A, o)
8ArJOED S'OFIUI\ff. 1\1\TI(fX EQlJATintl SOLVCF: CROUT llETIIOD ******~~**********************************************************
D Ir·IEI~S lUll li(:) I fl( 1)
L oc < r I J) = r + ( J -I ) qJo HI GO TO (1000 1 2000);1\Kf<.
REDUCE 11/\TR I X
DO 280 tJ::J 1 1·lt~ 1·11 =LOC ( 1! 1 l) DO ?.60 L=2tlll-1 ~IL:: LOC (If, LJ C=A( IlL) /A( Ill) I=!I+L-1 lF(NN.LT.I) GO TO 260 J::O 0 0 2 50 K =L I ~tl J=J+l I J::: l.OC ( I, J) IIK=I.OC ( f·l 1 K) A( I j) =A (I J) -c~·: A( IlK) ,'\(IlL) =C COIITI I·HJE GO TO 500
REDUCE VECTOR
DO 290 t-1=1 INN III=LOC(Il,l) DO 285 L=2 1 1·i'1 tiL=LOC( 1~, L) I==I·I+L-1 IF(tiii.LT. f) GO TO 290 B (I ) ::: R ( I ) - A ( !·I L ) ~·• B ( I· I) B( 1·1) =B( II) /A( Nl)
BACK SUBSTITUTION
r·r=~rN H=t·l-1 IF(N.EQ.O) GO TO 500 Oil h,OO K=2,1·1t1 111\=LOC (H. K) L:::fi+K-1 IF(Ilt~.LJ,L) GO TO lfOO B ( t I) ::: f3 ( ! I) - 1\ ( NK) ~·• 8 ( L) CmiTI I·IUE GO TO 300
500 RETURN EtiO
c c c c
c
s !JEll\ ourr 1 IE our PUT c r TE F 1 o E LT A, MiuD E 1 r: L Er 1 u I o, u: I r ~~~I r 11: I~~~ i, 110 F 1 'I)
TillS OUTPlJT TilE SC1U!T!,1!1 Fm IIEI!T:l'III\11 FUJII.
11 Ir IE liS I ()II [) E LT 1\ ( II f() I /\' :oo E ( ~ 1: I, 5 ) I E UTE T ( ':E , 5 ) j Q ( S1) J c E (t: K I I ) , T 0 r: L T ( I I+ 3 )
1 QC) \ /'( I Tf~ ( f) I ~1 )
199
9 r:rr:'li\T(11!Cr, 'THIS IS Tllr:: UiJSC/\LED-SDI.UT!Dil Ff!i,: :!E\JTO:IrAt! Fl.n\/'''''"'n':,·: 1 ITEr •. uiml 11n.=1 • 11 tx,•:::n~:·:.-·:,·::rw. • ,sx, 'Y.~VEUJC ITY' ,;i~<. 'Y-VE!.OC ITY' , 25'(, 1 PRES~~IJI\E 1 )
C I iiSER.T V.\LlJES OF UIIKIIOIJ!!S l llTO DELTA c
c
rn 1 o r -= 1 , ' : ~J F D [ u 'I ( l) =C E ( I I I )
1 0 Ul' I TI l·!lJF: 00 11 1=1 1 1·1N
C 0, 0. F o ! llJI!G ER S OF A ! I DOE. c
c
J 1 ::f\!<!OD E ( I I 3 ) J2=/1!10fll:( I,lf) J3=1\:noE( I, 5)
11 1 '~ ITE c r, I 1 ~) 1 I o E L r A ( .J 1 ) I n E L r /\ c J?.) , o E L r 1\ c .J3) 1 2 F m 1 lA r c 1 x, 15 I 1 ox I F 1 o. :; Is xI F 1 o. 5 , 5 x 1 F 1 o • s )
C FlJJ'.IRATE RF.r!UIRED. c
CALL FIJJRET(QtOELTi\,1\:!0fJE 1 IIK 1 iHJ) 1./R I H ( (-, , 1 3 )
13 Ff1\!1AT( iH0 1 'FLOI/RATE AT Vi\fUOUS SECT!nt-!S OF (r IF.' II6X 1 'SECTTOtl' ,8X, 1 I FLO\IRfl TE I) IIRITE(6,JI+)(I ,Q( I), I:::J 1 J)
14 FORi1i\T(8:<, I2 1 BX 1 F10.5) \/R I n: ( () , C) 0 )
90 FIJf~!1i\T(lll0,27X, 1 Til!S IS TilE SCALED-SfJLUTIOii.FOR IIEI.JTOII!Atl FL0\1 1 SC 1/ll[-'[) SOI.lJTintl FOR t!E'.JTn:l!.~.!l FLmi. 1 I/1!SX, 1 SOI.llrir1!1 !II F10,5 Fmrti\T 1
2,50X 11 SOI.UTI011 Til E12.1f FOI\11111 II1X, 1 !10DE~':h':IID. 1 ,5X, 1 X··VE'l.OCITY 1 ,5
3X, 'Y-VEI.OCITY' lsx, 'PRESSURE' 120X, 'X-VEI.OC ITY' ,~)X, 'Y-VELOC IT¥ ,sx, I !f P f\ E S S lJ R E ' )
~I::! lfJ F +I I 0 0 1 5 I ::: 1 , Ht I
C D.O.F. tllHlllERS. c
c
J1=/\IIODF.( I ~.3) J 2 =1\JI()\1 [( I I lf) J3=A.IIODE( 1,5)
C TilE 1100/l.l. VAR!Al3LF.S ARF. 1101/ SCt\LED fW THE FI.O\IR/\TE. c c C lluLTlPL!CiiTirJil r1Y ZERO IS AVOIDED flY TilE IF :.>T/\TUIEIITS c
!F(J1.GTJ1) GO TO 20 DE: L T ( J 1 ) =DEL i'.l\ ( J 1 ) /Q ( 1 )
20 IF(J2.GT,II) c,r) TO 30 OF.LT( J2)=DELTt\(J2)/Q(1)
30 fF( J3 .GT .11) r11l TO 15 DELT(J3)=0F.LTA(J3)/Q(I)
1 5 I, If\ I TE ( G , ~15 ) I I f) E LT ( J I ) I f) F. LT ( J2) ' 0 E LT ( J 3 ) , 1 f) E LT ( j I ) , [) E L T ( .J 2 ) I f) E !. T ( j 3 )
200
9 5 F oRr II\ T C 1 >: , !5 , 1 o ~< I F 1 o. 5, 5 x, F 1 o • 5 , 6 x 1 F 1 o. 5 , 1 8 xI E 1 2 • ,,, L1 x, F. 1 2 • L1 I 1, x 1 E 1 2 1 • L1)
999 RETURII EIID
c c c c c c c c c
c
SU[JHOUT !I W FLOH F.T ( <l, [) El. T J\, AI lfJO E, IlK 1 Htl)
Tl!IS CJ\l.CUL!\TES Til[ Fl.O\!Ri\TE AT 0 lFFUtEI!T SCCTI OtJ:; OF TliE fJ IE
D l11EI!S !Oil Q(S') ,DELTA( II!(), A!!DDE( 1111 1 5)
SET 110. OF SE::TIOIIS(I1) OVER TilE DIE.
11::::9 DO 2 I:: 1 , 1·1
2 O( l) =o.o OJ 3 l:: 1 , I l DO 3 J::J ,.1·1-1
C IIOOF. 11UIIf3ER. c c C D,O,F. liD. COf1RESPOIIO!IIG TO TilE X-VELOCITY. c
c
J I =!HIOO 1:( IHJOOF., 3) J2::AIIOD[~( IHJODF.+II,3) D= :\IIOD E (I ll-lr1D F.+ 1'1, 2) -1\ 1 HJO E ( IHiOD E, 2)
C IIITERGRATE (Lb':OY) ACROSS THE CliAI:iiEL TO OBTJ\ fil THE C F LOI·/R/\ TE • c
3 Q( I) =Q( !)+(DELTA( Jl )+DELTA( ~1 2) )'':D D 0 Lf I:: 1 11
Lf Q ( I) ::Q ( r) n. 0 R ETURIJ Ei·IO
201
SUBROUTINE CONVEG(DELTA,CE,NDF,KODE,NK) c c ******************** C TEST FOR CONVERGENCY c ******************** c
c
DIMENSION DELTA(NK) ,CE(NDF) KODE=1 DO 34 I=1, NDF Y=DELTA(I)
C VERY SMALL NODAL VALUES ARE NOT C TESTED FOR CONVERGENCY. c
IF(Y.LT.0.00001) GO TO 34 E=ABS(CE(I)-DELTA(I))-0.001*ABS(CE(I)) IF(E.GT.O.O)KODE=KODE+1
34 CONTINUE RETURN END
202
203
';·,,· .. -,':!JTI '.p·. '"I '''-l i'( .'-l'f\ Ll.-1 T .,, )"r· "' r·•·-1· ·., .. "" ·•r· "'"' "\.1 ITl- 1') ' ... . I I " \ .. ... v.. ~ l I '·. ~ ' \ • I \ I • } .:. , ...... t.. . • • I l I\ ' ' i J I 1 ... , ' . ' I , ' • ' • I ....
c c . ~************~***************************** c T'!I:-; :~-r:·'.TIT!;:r: :J:n:,r··.; T'l[ rr··:.t. '~:·:s~r:.r C ::.\C'I t·q:;:)t..\CE·.;:·:T 1 IS :ii'!~:-:.\T?.fl \!PT! !lY T!IF. C /\PPf\'JDrd.\TE VJ\IJJF. (1F TW FI.Cli/R;\1'E c ~****************************************** c
c
o r 1 E' 1s 1 'l: 1 :J E 1. T,\ < rn~) , .\ • 1o:J E ( u: 1, :>) I E t. Et lET< 11c Is:. , Q < 9) , r.: EL T ,, (1 1 K) R E,\1. lllfl0flf:: ( 81 ) 01 500 f::!IOF+l ,IIOF+!I
C K!Ja·/11 VI\LlJES OF TilE SnLUTr Otl. c·
c
CELT!\( 1)::1,0 500 DEI.TI\( I)::j .0
ll=iiOF+t-1 In. I E ( (j , 1 ) I TE R FflRII/\ T ( 1 I iO, 1 I TEP-1H I OWt·.'t~·.·~·t :lU! W ER ~·:·:"·:·:t ITER A TI n:J·::-::·::·:: l lUll l3ER'''''d:·::y TERA T I 0
T """·:,·,,·: r '' J: 1 n ER ,., ... ".,,., I r E·r:un r 'l' ,,."·"·,,·:r 1u • Fl ER =' , !51 1 T1X, ·~Jnf)E·:,,·,·!fl. ',:;x, 'X-Vt:L'!:: tTY'"'"''':-'.'!::LrJC ITY' 1 !;x, T' Y- VE Loc r yy,·:,·:~·,y- VE 1. nc I TY' , s x, 'PREs suc:;r·::·::~:~:·::·:"'' PRES suRE' 1 T18X: 1 PRF.SEilT1 ,SX, 1 PRt-:VTOI_JS 1 ,flX, 1 PRC:SE:Il"f ,SX, T'PREVIOU~' ,7:<, 1 PRESEtiT1 ,2X, 1 PREV IOUS 1 ).
DO 2 I=1 ,uti
C D.O.F. llliiiElERSS. OF 1\ tJODE c
c
J1=Ail0DE( !,3) J2=AIIODE (I 1 4) J 3 =A II ODE ( I , 5)
C OUTPUT TilE VAH TABLES OF TH/\T t!ODE. c
c
2 \-/R IT E ( 6 , 3 ) I , D E L T A ( J I ) , C E L TA ( J 1 ) , 0 E LT A ( .12 ) , C F. L T A ( J 2 ) , TDELTA(J3),CELTA(J3)
3 F m t 1AT ( L<, I 5 ~ I zx·, F 1 0. 5 , 2 X, F 1 0. 5 , 6 X, F I 0. 5 , I X, F 1 0. 5 ,3X , F 1 0. 5, F 1 0. 5)
C CALCU.LATE AilD PRI tiT THE Fl.OI-/RATE THROUGH THF. C DIE. c
c
99 CALL FLORET(Q 1 DELTA, AIIOOE, tiK,NN) \·IR I TE ( 6 , 1 3) ·
13 FClf.Uii\T(1110, 1 FLOI-/RATE AT VARIOUS SECTIQ:Js OF OIE 1 // T6X, 'SECTIOII' ,8X, 'FLOI-/RI\TE')
HR I TE ( 6 I JLf) ( I, Q ( I} I l= 1 '9) 14 FORt lA T ( 8 X, I 2 , 1 2X, F 1 0. 5)
204
C FiliAL SOLUTIOtl IS Wl\1 OflT/\TflEO. c c
c
I IR 1 T F. ( 6 I '1 ()()) 7CXJ FIYU11\T(JII0,27X, 1 Till~; r;, TltF: SCALED<;()!JITJOI) Fm~ i·lCHI-IIEI/TotiJi\tl FLO\-/
1, ~;CJ\1 ED SOLlJTtDll FDf.: IIO!In!I[I/TOilli\!1 FUl\1 1 //1i)){ 11 SOLUT!Oil Ill F10,5
2Fnf:l\t\T 1 1SOX 1 1 SOLUTIITI Ill El?.,l1 t='flH1/\TI//1X 1 1 110DE·.':·:.,·:tl(), 1 1 ~iX 1 1 X··VEI.O :; c 1 TY I ' s X ' I y - v F UJC IT y I I 5 X ' I p 1\ E s s l )[{ E f I :< I) :< I f :< - \1 r: L 0 c 1 TY I I 5 X ' I y - 1/ E L 0 c I T 11 Y' I s :< , ' r n E s s u ~~ ~~ ' )
DO (,Q I::J,tlll J 1 ::J\IIDDE:( I, 3) J2::1'dl0 11[( I 1 l1) JJ::J\,;Jf)DE( I ,5) F8! ::o. 5
C THE 11001\L '/AI~!Al3LES 1\RE l!G\/ SCALED flY TilE C F LOI /k A TF.. c c c ~IULTIPL!CATim' BY zr-.:Hn IS tWOIDEO nY THE 11 IF STATEitF.IITS 11
c IF(Jl,GT .11) GO TO 20 0 E :_ T 1\ ( .J 1 ) ::DE I. T 1\ ( .Jl ) IQ ( 1 )
20 IF(J2,13T,11) Gfl TO 30 0 F.LTt1 ( J~·) ::DELTA ( J2) IQ ( 1)
30 IF(J3.GT.t1) GO TO ()0 · D E U A ( J 3 ) :: 1J f:: LT/\ ( .J 3 ) I ( Q ( l ) .,..,., F 8 I)
6 0 \ c I T E ( () ' 7 2 ) I I [) E LT A ( J 1 ) ' [) E LT 1\ ( J 2 ) I [) E LT /\ ( .I J ) ' [) F.l.T J\ ( .J 1 ) I [) E L T /\ ( .) 2 ) ' !DELTA( J3)
72 F m 11 exT ( 1 :<, Is , s :<, F 1 o. 5 , s x IF 1 o • 5 1 6 y, F 1 o .. L~ , 1 8 x, E 1 2 ,If , 1f x. E 1 7. ,11 , '1 x, E 1 2. Jll)
600 CCli ITI flUE o o n r: r ;: 1 tlE
7 2 5 EL UlE'r ( I , If~ ::: E L E I 1£ T ( I , If) I ( Q ( 1 ) ,·,·: ( F ll i - 1 • 0) ) c C VISCOSITY OF EACH IIODE CALCUU\TED !\tiD PRIIHED. c
C 1\ L L V I S r 0 ( l ·IU II 0 D E , E L Ell E T , I ill , t ·IE ) \IHITE(6,/50)
?SO FO:-::['\T(lli0,30X, 1 IIODE VISCOSITY') \ lit I H ( G , R 0 G ) ( ( I 1 IIU t I 0 D E ( r) ) , I:: 1 , I ,p 1)
110 0 FOR' 1/\ T ( 1 :<, I 6, 3 X , F 1 0 , 5 ) Rf.TIJRII EIID
S IJ!1 r; OUT I I IE V l SCi) ( t IU! lf)fl E, E LEt 1C T , tIt I 1 I IE ) c c ***************~********~****~**************************** C T'llS U\LCULIHES THE VISCOSITY llF Ei\C!l :HJDE IW /\!! t\VFJ~i\GI:H3
C Pf\OC E ;JI HU: c ***************************~******~*********************** c
c
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THIS READS Ill ALL THF. I:!FDR~VITTO!l REI!tl!RED,OfJT/\ll!S THE SYSTEil IIATRIX EQUATIOII,TIIE LOI\OPIG VECTOR ArlO SOLVES TilE EQlJt\.IIOIIS.
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CONTROL INFORMATION*******CONTROL INFORMATION
INITIAL NO. OF D.O.F. NK= 243 NO. OF ELEMENTS NE= 128 NO OF NODES NN= 81 UNKNOWN NO. OF D.O.F. NDF= 175 NON ZERO NO. OF D.O.F. N= 9 BANDWIDTH IBW= 29 DIE LENGTH DL= 1. 00000 HEIGHT OF DIE AT INLET H= 1. ooooo WIDTH OF DIE W= 1. 00000 ANGLE OF TAPER OF DIE ALPHA 0.00000 RADIAN
NODAL INFORMATION**********NODAL INFORMATION**********NODAL INFORMATION
NODE NO. X COORD Y COORD DOF NO. DOF NO. DOF NO, 1 0.00000 0.00000 185 186 176 2 0.12500 0.00000 196 197 30 3 0.25000 0.00000 200 201 53 4 0.37500 0.00000 204 205 76 5 0.50000 0.00000 208 209 99 6 0.62500 0.00000 212 213 122 7 0.75000 0.00000 216 217 145 8 0.87500 0.00000 220 221 168 9 1.00000 0.00000 224 225 226
10 0.00000 0.12500 7 187 177 11 0.12500 0.12500 27 28 29 12 0.25000 0.12500 50 51 52 13 0.37500 0.12500 73 74 75 14 0.50000 0.12500 96 97 98 15 0.62500 0.12500 119 120 121 16 0. 75000 0.12500 142 143 144 17 0.87500 0.12500 165 166 167 18 1. 00000 0.12500 175 227 228 19 0.00000 0.25000 6 188 178 20 0.12500 0.25000 24 25 26 21 0.25000 0.25000 47 48 49 22 0.37500 0.25000 70 71 72 23 0.50000 0.25000 93 94 95 24 0.62500 0.25000 116 117 118 25 0. 75000 0.25000 139 140 141
208
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50 o.~;oooo O,IS2500 ?1 L1 85 86 5'1 0.62500 o. 62 r;oo 107 lOB 109 52 0.75000 0,62500 130 1 31 132 53 0,8750~) 0.62500 15 3 151f 15 5 51~ 1,00000. O,fi2500 i 71 235 ?.36 55 o. ooo on 0,75000 ...,
I. 19?. 182 5f} 0.12500 0.75000 12 13 ,_, 57 0.25000 0.75000 35 J(; 3'1 58 0,37500 0. 7 5000 58 59 60 59 0,50000 0.75000 81 82 83 60 O,li2500 0. 7 5000 1 OLf 105 106 61 0.7SOOO 0. 7 5000 127 128 129 62 O,fl7500 0,75000 150 15 1 13 2 63 1. 00000 0. 7 5000 170 237 238 61} 0 .ooooo 0,87500 1 193 183 65 0.12500 0.8 7 500 9 10 11 66 0.25000 0,87500 32 33 3 L; 67 0,37500 0.87500 55 % 57 68 0.50000 0.87500 78 79 80 69 0.62500 0,87500 101 102 103 70 0.75000 0.87500 1 ?.1;. 1 25 126 71 0,87500 0,87500 147 11f8 Jlf9 72 1 .ooooo 0,87500 169 239 2110 73 0,00000 1.00000 194 195 184 ?If o. 1 2500 1. 00000 198 199 8 75 (). 2 5 00 () 1. 00000 202 203 31 76 0.37500 1. 00000 ?.06 207 51; 77 o.soooo 1.00000 210 211 77 78 0.62SOO 1,00000 214 215 iOO 79 0.75000 1. 00000 218 219 1 23 80 0.81500 1 ,OGOOO 222 223 11~6 81 1. 00000 1 ,00000 2lf 1 ?.If 2 2lf3
n flU liD 1\r{Y COil[) 1 T I OilS
OOF NO 176 177 178 179 lflO llll 182 183 li1Lf
PI\ES SUHE 1 • 00000 1 ,00000 ~ • 00000 1. 00000 1. 00000 1 .ooooo 1.00000 i. 00000 I .00000
2.09
