Solution of the Partial Differential Equations

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Chapter 7 Solution of the Partial Differential Equations

Classes of partial differential equationsSystems described by the Poisson and Laplace equationSystems described by the diffusion equation

Greens function, convolution, and superpositionGreen's function for the diffusion equationSimilarity transformationComplex potential for irrotational flowSolution of hyperbolic systems

Classes of partial differential equationsThe partial differential equations that arise in transport phenomena are

usually the first order conservation equations or second order PDEs that areclassified as elliptic, parabolic, and hyperbolic. A system of first orderconservation equations is sometimes combined as a second order hyperbolic

PDE. The student is encouraged to read R. Courant, Methods of MathematicalPhysics, Volume II Partial Differential Equations, 1962 for a complete discussion.System of conservation laws. Denote the set of dependent variables (e.g.,

velocity, density, pressure, entropy, phase saturation, concentration) with thevariable u and the set of independent variables as t and x, where x denotes thespatial coordinates. In the absence of body forces, viscosity, thermal conduction,diffusion, and dispersion, the conservation laws (accumulation plus divergence ofthe flux and gradient of a scalar) are of the form

1

1 2 1 2

1 2 1 2

( ) ( ) ( )( )

where

( , , ) ( , , , ), Jacobianmatrix

( , , , ) ( , , , )n n

n n

g u g u f u u f uf u

t t x t g x

g g g f f f f

g u u u u u u

+ = + = +

=

i

This is a system of first order quasilinear hyperbolic PDEs. They can be solvedby the method of characteristics. These equations arise when transport ofmaterial or energy occurs as a result of convection without diffusion.

The derivation of the equations of motion and energy using convectivecoordinates (Reynolds transport theorem) resulted in equations that did not havethe accumulation and convective terms in the form of the conservation laws.

However, by derivation of the equations with fixed coordinates (as in Bird,Stewart, and Lightfoot) or by application of the continuity equation, themomentum and energy equations can be transformed so that the accumulationand convective terms are of the form of conservation laws. Viscosity and thermalconductivity introduce second derivative terms that make the system non-conservative. This transformation is illustrated by the following relations.

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( )

( )

( ) ( )

( )( )

( )( )

0, continuityequation

, for , ,or

D

Dt t

DF FF F S E

Dt t

F FFt t t

F F F F F

FDFF F F F

Dt t t

FF F

t t

DF

Dt

F

+ = + =

= + =

= +

= + = + +

= +

= + + +

=

v v

v v

v v v v v v

v v v

v v v

( )

( ), Q.E.D.

FF

t+

v

Assignment 7.1 (a) For the case of inviscid, nonconducting fluid, in the absenceof body forces, derive the steps to express the continuity equation, equations ofmotion, and energy equations as conservation law equations for mass,momentum, the sum of kinetic plus internal energy, and entropy. (b) For thecase of isentropic, compressible flow, express continuity equation and equationsof motions in terms of pressure and velocity. Transform it to a second orderhyperbolic equation in the case of small perturbations.

Second order PDE. The classification of second order PDEs as elliptic,

parabolic, and hyperbolic arise from a transformation of the independentvariables. The classification apply to quasilinear (i.e., linear in the highest orderderivatives) but we will only discuss linear equations with constant coefficientshere. Numerical solutions are needed for quasilinear systems. Again let udenote the dependent variables and t, x, y, z as the independent variables.Examples of the different classes of equations are

2 2 22

2 2 2

2 2 22

2 2 2

2 2 2 22

2 2 2 2

0 , ellipticequation

, parabolicequation

, hyperbolicequatio

u u uu

x y z

u u u uu

t x y zu u u u

u nt x y z

= + + + = +

= + + + = +

= + + + = +

.

The term represents sources. When the cgs system of units is used in

electrostatics and is the charge density, the source is expressed as 4 .

The factor 4 is absent with the mks or SI system of units. The parabolic PDEs

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are sometimes called the diffusion equation or heat equation. In the limit ofsteady-state conditions, the parabolic equations reduce to elliptic equations. Thehyperbolic PDEs are sometimes called the wave equation. A pair of first orderconservation equations can be transformed into a second order hyperbolicequation.

Convective-diffusion equation. The above equations representedconvection without diffusion or diffusion without convection. When both the firstand second spatial derivatives are present, the equation is called the convection-diffusion equation.

2

2

1u u u

t x N x

+ =

Usually a dimensionless group such as the Reynolds number, or Reynoldsnumber and Prandtl number appears as a factor to quantify the relativecontribution of convection and diffusion.

Systems described by the Poisson and Laplace equationWe saw earlier that an irrotational vector field can be expressed as the

gradient of a scalar and if in addition the vector field is solenoidal, then the scalarpotential is the solution of the Laplace equation.

2

2

, irrotational flow

0 , incompressible, irrotational flow

=

= =

= =

v

v

v

Also, if the velocity field is solenoidal then the velocity can be expressed as thecurl of the vector potential and the Laplacian of the vector potential is equal to thenegative of the vorticity. If the flow is irrotational, then the vorticity is zero and thevector potential is a solution of the Laplace equation.

2

2

2

2

, incompressibleflow

, for 0

, intwodimensions

0 , irrotational flowand 0

0, for twodimensional, irrotational, incompressible flow

w

=

= = =

=

= = =

=

v A

v w A A

v A A

Other systems, which are solution of the Laplace equation, are steady

state heat conduction in a homogenous medium without sources and inelectrostatics and static magnetic fields. Also, the flow of a single-phase,incompressible fluid in a homogenous porous media has a pressure field that is asolution of the Laplace equation.

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Systems described by the diffusion equationDiffusion phenomena occur with viscous flow, thermal conduction, and

molecular diffusion. Heat conduction and diffusion without convection aredescribed by the diffusion equation. Convection is always present in fluid flowbut its contribution to the momentum balance is neglected for creeping (low

Reynolds number) flow or cases where the velocity is perpendicular to thevelocity gradient. In this case

2 , velocityperpendicular tovelocitygradientp

t

= +

v

f v .

Greens function, convolution, and superpositionA property of linear PDEs is that if two functions are each a solution to a

PDE, then the sum of the two functions is also a solution of the PDE. Thisproperty of superposition can be used to derive solutions for general boundary,initial conditions, or distribution of sources by the process of convolution with a

Greens function. The student is encouraged to read P. M. Morse and H.Feshbach, Methods of Theoretical Physics, 1953 for a discussion of Greensfunctions.

The Greens function is used to find the solution of an inhomogeneousdifferential equation and/or boundary conditions from the solution of thedifferential equation that is homogeneous everywhere except at one point in thespace of the independent variables. (The initial condition is considered as asubset of boundary conditions here.) When the point is on the boundary, theGreens function may be used to satisfy inhomogeneous boundary conditions;when it is out in space, it may be used to satisfy the inhomogeneous PDE.

The concept of Greens solution is most easily illustrated for the solution to

the Poisson equation for a distributed source (x,y,z) throughout the volume.The Greens function is a solution to the homogeneous equation or the Laplaceequation except at (xo, yo, zo) where it is equal to the Dirac delta function. TheDirac delta function is zero everywhere except in the neighborhood of zero. Ithas the following property.

( ) ( ) ( )f x d f x

=

The Green's function for the Poisson equation in three dimensions is thesolution of the following differential equation

2 ( ) ( ) ( ) (

1( )

4

o o o

o

o

G x x y y

G

= =

=

x x

x xx x

)oz z

.

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It is a solution of the Laplace equation except at x=xo where it has a singularity,i.e., it has a point source. The solution of the Poisson equation is determined byconvolution.

( ) ( ) ( )o o o ou G dx dy dz= x x x x o

o

Suppose now that one has an elliptic problem in only two dimensions.One can either solve for the Green's function in two dimensions or just recognizethat the Dirac delta function in two dimensions is just the convolution of the three-dimensional Dirac delta function with unity.

( ) ( ) ( ) ( ) ( )o o o o ox x y y x x y y z z d

= z

Thus the two-dimensional Green's function can be found by convolution of thethree dimensional Green's function with unity.

( ) ( )2 2

( , , ) ( )

1ln

4

o o o o

o o

G x y x y G dz

x x y y

=

= +

x x.

This is a solution of the Laplace equation everywhere except at (xo, yo) wherethere is a line source of unit strength. The solution of the Poisson equation intwo dimensions can be determined by convolution.

( , ) ( , , ) ( , )o o o o o o

u x y G x y x y x y dx dy=

.

Assignment Derivation of the Greens functionDerive the Greens function for the Poisson equation in 1-D, 2-D, and 3-D

by transforming the coordinate system to cylindrical polar or spherical polarcoordinate system for the 2-D and 3-D cases, respectively. Compare the resultsderived by convolution.

Green's functions can also be determined for inhomogeneous boundaryconditions (the boundary element method) but will not be discussed here. TheGreen's functions discussed above have an infinite domain. Homogeneous

boundary conditions of the Dirichlet type (u = 0) or Neumann type (u/n = 0)along a plane(s) can be determined by the method of images.Suppose one wished to find the solution to the Poisson equation in the

semi-infinite domain, y > 0 with the specification of either u = 0 or u/n = 0 onthe boundary, y = 0. Denote as u0(x,y,z) the solution to the Poisson equation fora distribution of sources in the semi-infinite domain y > 0. The solutions for theDirichlet or Neumann boundary conditions at y = 0 are as follows.

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.0 0

0 0

( , , ) ( , , ) ( , , ), 0 0

( , , ) ( , , ) ( , , ), / 0 0

u x y z u x y z u x y z for u at y

u x y z u x y z u x y z for du dy at y

= = =

= + = =

=

S

The first function is an odd function ofy and it vanishes at y = 0. The second isan even function ofy and its normal derivative vanishes at y = 0.

Now suppose there is a second boundary that is parallel to the first, i.e. y=a that also has a Dirichlet or Neumann boundary condition. The domain of thePoisson equation is now 0


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Green's function for the diffusion equationWe showed above how the solution to the Poisson equation with

homogeneous boundary conditions could be obtained from the Green's functionby convolution and method of images. Here we will obtain the Green's functionfor the diffusion equation for an infinite domain in one, two, or three dimensions.

The Green's function is for the parabolic PDE

2 2 uu at

=

where the parameter a2 represents the ratio of the storage capacity and theconductivity of the system and is a known distribution of sources in space and

time. The infinite domain Green's function gn(x,t xo,to) is a solution of thefollowing equation

2 2( , , )

( , , ) ( ) ( )n o o

n o o o

g t t

g t t a t tt

=

x x

x x x x o

The source term is an impulse in the spatial and time variables. The form of theGreen's function for the infinite domain, for n dimensions, is (Morse andFeshbach, 1953)

2 2( /4 )

2

1, 0

( , , ) ( , ) 2

0 , 0

where

n

a R

n o o n

o

o

ae

g t t g R a

t t

R

> = =


8/32

2 2[ / 2(2 / )]2

2

1, 0

( , ) 2 (2 / )

0 , 0

n

R a

n

ea g R a

>

= 0 is the step response function. It hasclassical solutions in one and two dimensions. The unit step function orHeaviside function is the integral of the Dirac delta function.

1, 0( ' ) ' ( )0, 0

t o

o o

o

t tt t dt S t tt t

> = =


9/32

( )

( )

2 2

2 2

2 2

' ( ) ( ' ) '

'

' (

( )

where

'

t t

nn o

t

nt

n o

nn o o

t

n n

gg a dt t t dt

t

g dt

g dt a S t tt

UU a S t t

t

U g dt

=

=

=

=

x x

x x

x x

)

o

o

In one dimension, the step response function that has a unit flux at x=0 is(R. V. Churchill, Operational Mathematics, 1958) (note: source is

)( ) (2 0x S t )0

2

2flux 1 2 24 /1 2 2

2 erfc4 /

x

t axt

U a e a x ta t a

= =

, 0>

For comparison, the function that has a value of unity at x = 0 (Dirichlet boundarycondition) is

11

2erfc , 0

2 /

xU t

t a

= = >

.

In two dimensions, the unit step response function for a continuous linesource is (H. S. Carslaw and J . C. J aeger, Conduction of Heat in Solids, 1959)

2 2

2 2

2 2 2

Ei , 04 4 /

( ) ( )

Ei( )

expint( ), MATLAB function

o o

u

x

a RU t

t a

R x x y y

ex du

ux

= >

= +

=

=

.

For large times this function can be expressed as

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2 2 2approx.2 2 2

4 / 4ln , for 100

4 4

0.5772

a t a a tU

R a R

=

=

2>

.

In three dimensions, the unit step response function for a continuous pointsource is (H. S. Carslaw and J . C. J aeger, Conduction of Heat in Solids, 1959)

2

32

1/22 2

erfc , 04 4 /

( ) ( ) ( )o o o

a RU t

R t a

R x x y y z z

= >

= + + 2

2

.

NOTE! The a2 .factor has the units of time/L2. If time is made

dimensionless with respect to and R with respect to2 / oa R oR , then the factor will

disappear from the argument of the erfc.

Assignment 7.3 Plot the profiles of the response to a continuous source in 1, 2,and 3 dimensions using the MATLAB code contins.m and continf.m in the diffusesubdirectory. From the integral of the profiles as a function of time, determinethe magnitude, spatial and time dependence of the source. Note: Theexponential integral function, expint will give error messages for extreme valuesof the argument. It still computes the correct values of the function.

Convective-Diffusion EquationThe convective-diffusion equation in one dimension will be expressed in

terms of velocity and dispersion,2

2

( ,) 0, 0

(0, ) 1, 0

u uv K

t x x

u x x

u t t

+ =

= >

= >

u

The independent variables can be transformed from (x,t) to a spatial coordinatethat translates with the velocity of the wave in the absence of dispersion, (y,t).

y x vt= This transforms the equation to the diffusion equation in the transformedcoordinates.

2

2

u uK

t y

=

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To see this, we will transform the differentials fromx to y.

1

yv

t

y

x

=

=

The total differentials expressed as a function of (x,t) or (y,t) are equal to eachother.

x t

y t

u udu dt dx

t x

u udu dt dy

t y

= +

= +

The total differentials expressed either way are equal. The partial derivatives in tand x can be expressed in terms of partial derivatives in t and y by equating thetotal differentials with either dt or dx equal to zero and dividing by the non-zerodifferential.

2 2

2 2

and

x y t

y t

t tt

t

t t

u u u y

t t y

u uv

t y

u u y

x y x

u

y

u u

x y

= +

=

=

=

=

xt

Substitution into the original equation results in the transformed equation. Thisresult could have been derived in fewer steps by using the chain rule but wouldnot have been as enlightening.

The boundary condition at x = 0 is now at changing values ofy. We will

seek an approximate solution that has the boundary condition u(y-) = 1. Asimple solution can be found for the following initial and boundary conditions.

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1, 0

( ,0) 1/2, 0

0, 0

( , ) 1

( , ) 0

y

u y y

y

u y t

u y t

=

=This system is a step with no dispersion at t = 0. Dispersion occurs for t > 0 asthe wave propagates through the system. The solution can be found with asimilarity transform, which we will discuss later. For now, the approximatesolution is given as

1 12 2erfc erfc

4 4

y xu

K t K t

= =

vt.

The boundary condition at x = 0 will be approximately satisfied after a small timeunless the Peclet number is very small.

Similarity transformationIn some cases a partial differential equation and its boundary conditions

(and initial condition) can be transformed to an ordinary differential equation withboundary conditions by combining two independent variables into a singleindependent variable. We will illustrate the approach here with the diffusionequation. It will be used later for hyperbolic PDEs and for the boundary layerproblems.

The method will be illustrated for the solution of the one-dimensional

diffusion equation with the following initial and boundary conditions. Theapproach will follow that of the Hellums-Churchill method.

2

2, 0,

( ,0)

(0, )

IC

BC

u uK t x

t x

u x u

u t u

= >

=

=

0>

.

The PDE, IC and BC are made dimensionless with respect to referencequantities.

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2

2 2

*

*

*

* *

* *

*( *,0) 0

*(0, *) 1,

IC

o

o

o

o

o

BC ICo BC I

o

u uu

u

tt

t

xxx

K tu u

t x x

u x

u uu t u u u

u

=

=

=

= =

= = =

C

There are three unspecified reference quantities and two dimensionless groups.

The BC can be specified to equal unity. However, the system does not have acharacteristic time or length scales to specify the dimensionless group in thePDE. This suggests that the system is over specified and the independentvariables can be combined to specify the dimensionless group in the PDE toequal 1/4.

2

1isdimensionless

4 4

( , ) ( )

o

o

K t x

x K t

u x t u

= =

=

The partial derivatives will now be expressed as a function of the derivatives ofthe transformed similarity variable.

2 2

2 2

1

4

2

2

1

4

1

4

x K t

t t

u du du

t d t t d

u du

x dK t

u d u

x K t d

=

=

= =

=

=

The PDE is now transformed into an ODE with two boundary conditions.

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2

2

* *2 0

*( 0) 1

*( ) 0

d u du

d d

u

u

+ =

= =

=

( )

2

2

2

2

2

2

1

1 20

2

10

1

0

0

0

*Let

2 0

ln 2

*

*( 0) 1 1

*( ) 0 1

1

*( ) 1 erfc( )

* , erfc4

duv

d

dvv

d

dvd v d

v

v C e

u C e d C

u C

u C e

Ce d

e du

e d

xu x tK t

d

=

+ =

= =

=

= +

= = =

= = +

=

= + =

=

Therefore, we have a solution in terms of the combined similarity variable that isa solution of the PDE, BC, and IC.

Complex potential for irrotational flowIncompressible, irrotational flows in two dimensions can be easily solved

in two dimensions by the process of conformal mapping in the complex plane.First we will review the kinematic conditions that lead to the PDE and boundaryconditions. Because the flow is irrotational, the velocity is the gradient of avelocity potential. Because the flow is solenoidal, the velocity is also the curl of avector potential. Because the flow is two dimensional, the vector potential hasonly one non-zero component that is identified as the stream function. Thekinematic condition at solid boundaries is that the normal component of velocityis zero. No condition is placed on the tangential component of velocity at solidsurfaces because the fluid must be inviscid in order to be irrotational.

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( )

( )

2

3 3

2

, ,0 , ,

0

( , ,0)

, ,0 , ,

0

, ,0 , ,0

or

and

x y z

x y z

v v vx y

A A

y x

v v vy x

w

x y y x

x y

y x

= = =

= =

= =

= =

= = =

= =

=

=

=

v

v

v A

v

v A

Both functions are a solution of the Laplace equation, i.e., they areharmonic and the last pair of equations corresponds to the Cauchy-Riemann

conditions if and are the real and imaginary conjugate parts of a complexfunction, w(z).

( ) ( ) ( )

[ ]

[ ]

1/22 2

( ) ( ) ( )

cos sin , , arctan /

or

( ) real ( )

( ) imaginary ( )

i

w z z i z

z x i y

r e r i r z x y y x

z w z

z w z

= +

= += = + = = + =

=

=

The Cauchy-Riemann conditions are the necessary and sufficient condition forthe derivative of a complex function to exist at a point zo , i.e., for it to beanalytical. The necessary condition can be illustrated by equating the derivative

ofw(z) taken along the real and imaginary axis.

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( ) ( )( )

Re '( ) Im '( )

lim0

lim 0

and , Q.E.D.

dw zw z i w z

dz

ii

x i x x

i iii y i y y

x y

x y

= +

+ = = +

+ +

= = +

=

=

i

Also, if the functions have second derivatives, the Cauchy-Riemannconditions imply that each function satisfies the Laplace equation.

2 2

2 2

2 2

2 2

0

0

x y

x y

+ =

+ =

The Cauchy-Riemann conditions also imply that the gradient of thevelocity potential and the stream function are orthogonal.

( ) ( )0

x x y y

= + =

If the gradients are orthogonal then the equipotential lines and the streamlinesare also orthogonal, with the exception of stagnation points where the velocity iszero.

Since the derivative

0

lim

z

dw w

dz z

=

is independent of the direction of the differential z in the (x, y) plane, we mayimagine the limit to be taken with z remaining parallel to the x-axis (z = x)giving

x y

dwi v i

dz x xv

= + =

.

Now choosing z to be parallel to the y-axis (z = i y),

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1

y x x

dwi v v v i v

dz i y yy

= + = + =

These equations are a restatement that an analytical function has a derivative

defined in the complex plane. Moreover, we see that the real part ofw'(z) isequal to vxand the imaginary part ofw'(z) is equal to vy. Ifv is written for the

magnitude ofv and for the angle between the direction ofv and the x-axis, theexpression for dw/dz becomes

or

real

imaginary

ix y

x

y

dwv i v ve

dz

dwv

dz

dwv

dz

= =

=

=

.

Flow Fields. The simplest flow field that we can imagine is just a constanttranslation, w =(U-iV) z where U and V are real constants. The components ofthe velocity vector can be determined from the differential.

( ) ( ) ( )( ) ( )

( )

,

,

x y

x y

w z U iV z U iV x i y Ux Vy i Vx Uy i

dwU iV v i v

dz

v U v V

Ux Vy Vx Uy

= = + = + + + = +

= =

= == + = +

Another simple function that is analytical with the exception at the origin is

1

( )

cos sin

thus

cos

sin

n n i n

n n

n

n

nx y

w z Az Ar e

Ar n i Ar n

i

Ar n

Ar n

dwnAz v i v

dz

= =

= +

= +

=

=

= =

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where A and n are real constants. The boundary condition at stationary solidsurfaces for irrotational flow is that the normal component of velocity is zero orthe surface coincides with a streamline. The expression above for the stream

function is zero for all rwhen = 0 and when = /n. Thus these equationsdescribe the flow between these boundaries are illustrated below.

Earlier we discussed the Green's function solution of a line source in twodimensions. The same solution can be found in the complex domain. A functionthat is analytical everywhere except the singularity at zo is the function for a linesource of strength m.

Fig. 6.5.1 (Batchelor, 1967) Irrotational flow in theregion between two straight zero-flux boundaries

intersecting at an angle /n.

( ) ln( ), linesource2

dw 1

dz 2 ( )

o

x y

o

mw z z z

mv i v

z z

=

= =

This results can be generalized to multiple line sources or sinks by superpositionof solutions. A special case is that of a source and sink of the same magnitude.

( ) ln( ), multiplelinesources

2

( ) ln , source-sink pair2

ii

i

o

o

mw z z z

z zmw z

z z

=

= +

The above flow fields can be viewed with the MATLAB code corner.m,linesource.m, and multiple.m in the complex subdirectory.

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Assignment 7.4 Line Source Solution For zo at the origin, derive expressions forthe flow potential, stream function, components of velocity, and magnitude of

velocity for the solution to the line source in terms of r and . Plot the flowpotentials and stream functions. Compute and plot the flow potentials andstream function for the superposition of multiple line sources corresponding to

the zero flux boundary conditions at y=+1 and 1 of the earlier assignment.

The circle theorem. (Batchelor, 1967) The following result, known as thecircle theorem (Milne-Thompson, 1940) concerns the complex potentialrepresenting the motion of an inviscid fluid of infinite extent in the presence of asingle internal boundary of circular form. Suppose first that in the absence of thecircular cylinder the complex potential is

( )ow f z=

and that f(z) is free from singularities in the region |z|a, where a is a real length.

If now a stationary circular cylinder of radius a and center at the origin boundsthe fluid internally, the flow is modified to the following complex potential:

( ) ( )1 2 /w f z f a z= +

We show that the surface of the cylinder, z=a, is a streamline.

2a z= z ,

( )1 2( ) ( / )

( ) ( / )

( ) ( )

2Real ( ) 0

z az a z a

z a z a

z a z a

z a

z a z a

w z f z f a z

f z f zz z

f z f z

f z i

i

== =

= =

= =

=

= =

= +

= +

= +

= +

= +

A complex potential of this form thus has |z| =a as a streamline, = 0; and it hasthe same singularities outside |z| =a as f(z), since ifz lies outside |z| =a,a2/z liesin the region inside this circle where f(z) is known to be free from singularities.

Consequently the additional term 2( / )f a z in the equation represents fully the

modification to the complex potential due to the presence of the circular cylinder.It should be noted that the complex potentials considered, both in the absence

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and in the presence of the circular cylinder, refer to the flow relative to axis suchthat the cylinder is stationary.

The simplest possible application of the circle theorem is to the case of acircular cylinder held fixed in a stream whose velocity at infinity is uniform withcomponents (-U, -V). In the absence of the cylinder the complex potential is

, it is singular at infinity and the circle theorem shows that, with thecylinder present,

(U iV z )

.2( ) ( ) ( ) /w z U i V z U iV a z= +

The potentials and streamlines for the steady translation of an inviscidfluid past a circular cylinder can be viewed with the MATLAB code circle.m.

Conformal Transformation (Batchelor, 1967). We now have the complexpotential flow solutions of several problems with fairly simple boundary

conditions. These solutions are analytical functions whose real and imaginaryparts satisfy the Laplace equation. They also have a streamline that coincideswith the boundary to satisfy the condition of zero flux across the boundary.Conformal transformations can be used to obtain solutions for boundaries thatare transformed to different shapes. Suppose we have an analytical function

w(z) in the z = x + iy plane. This solution can be transformed to the = + iplane as another analytical function provided that the relation between these two

planes, = F(z) is an analytical function. This mapping is a connection between

-3 -2 -1 0 1 2 3-3

-2

-1

0

1

2

3

x

y

Potentials and Streamlines for Flow Past Cylinder

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the shape of a curve in the z - plane and the shape of the curve traced out by the

corresponding set of points in the - plane. The solution in the - plane is

analytical, i.e., its derivative defined, because the mapping, = F (z) is ananalytical function. The inverse transformation is also analytical.

( )( ) ( )

( )

( ) ( ) ( )

d w zdw d w z dz dz

dF zd dz d

dz

d w z dw dF z

dz d dz

= =

=

w() is thus the complex potential of an irrotational flow in a certain region of the

- plane, and the flow in the z plane is said to been 'transformed' into flow in

the - plane. The family of equipotential lines and streamlines in the z plane

given by (x,y) = const. and (x,y) = const. transform into families of curves inthe - plane on which and are constant and which are equipotential linesand streamlines in the - plane. The two families are orthogonal in theirrespective plane, except at singular points of the transformation. The velocity

components at a point of the flow in the - plane are given by

( )x ydw dw dz dz

v iv v ivd dz d d

= = = .

This shows that the magnitude of the velocity is changed, in the transformation

from the z plane to the - plane, by the reciprocal of the factor by which lineardimensions of small figures are changed. Thus the kinetic energy of the fluidcontained within a closed curve in the z plane is equal to the kinetic energy of

the corresponding flow in the region enclosed by the corresponding in the -plane.

Flow around elliptic cylinder (Batchelor, 1967). The transformation of theregion outside of an ellipse in the z plane into the region outside a circle in the

- plane is given by

( )

2

1/22 21 1

2 24

z

z z

= +

= +

where is a real constant so that

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2 2

2 21 , 1x y

= + =

.

This converts a circle of radius c with center at the origin in the - plane into the

ellipse

2 2

2 21

x y

a b+ =

in the z plane, where

( )1/2

2 21

2a b =

If the ellipse is mapped into a circle in the - plane, it is convenient to use polarcoordinates (r,), especially since the boundary corresponds to a constant radius.The radius that maps to the elliptical boundary is (ellipse.m in the complexdirectory)

1

2logo

a br

a b

+ =

The transformation fromthe polar coordinates to thez plane is defined by

2 cosh

where

z

r i

=

= +

The polar coordinates (r,),transform to an orthogonalset of curves which areconfocal ellipses andconjugate hyperbolae.

This transformationcan be substituted into the complex potential expression for the flow of a fluidpast a circular cylinder.

-2 -1 0 1 2

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

x

y

Confocal Ellipses and Conjugate Hyperbolae, a =1 b =0.2

Transformation of cylindrical polar coordinates intoorthogonal, elliptical coordinates

( ) ( ) ( )1

2

o or rw a b U iV e U iV e = + + +

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It is convenient to write - for the angle which the direction of motion of theflow at infinity makes with the x axis so that

( )1/2

2 2 iU iV U V e + = +

The complex potential now becomes

( ) ( ) (1/2

2 2 cosh ow U V a b r i ) = + + +

The corresponding velocity potential and stream function are

( ) ( ) ( ) (

( ) ( ) ( ) (

1/22 2

1/22 2

cosh cos

sinh sin

o

o

U V a b r r

U V a b r r

)

)

= + + +

= + + +

The velocity potentials and streamlines are illustrated below for flow pastan elliptical cylinder (fellipse.m in the complex directory). Note the stagnationstreamlines on either side of the body. These two stagnation points are regionsof maximum pressure and result in a torque on the body. Which way will itrotate?

-1.5 -1 -0.5 0 0.5 1 1.5

-1.5

-1

-0.5

0

0.5

1

1.5

x

y

Potentials and Streamlines for Flow Past Ellipse

Flow past an ellipse of an inviscid fluid that is in steadytranslation at infinity.

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Pressure distribution. When an object is in a flow field, one may wish todetermine the force exerted by the fluid on the object, or the 'drag' on the object.Since the flow field discussed here has assumed an inviscid fluid, it is notpossible to determine the viscous drag or skin friction directly from the flow field.It is possible to determine the 'form drag' from the normal stress or pressure

distribution around the object. However, one must be critical to determine if thecalculated flow field is physically realistic or if some important phenomena suchas boundary layer separation may occur but is not allowed in the complexpotential solution.

The Bernoulli theorems give the relation between the magnitude ofvelocity and pressure. We have assumed irrotational, incompressible flow. If inaddition we assume the body force can be neglected, then the quantity, H, mustbe constant along a streamline.

2

2

1

2

1

2

constant

=- constant, since is also constant

pH v

p v

= + =

+

The pressure relative to some datum can be determined by the square of themagnitude of velocity. This is easily calculated from the complex potential.

2 2

thus

x y

x y

x y

dwv i v

dz

dwv i v

dz

dw dwv v v

dz dz

=

= +

= + = 2

There are some theorems that facilitate the integration of pressure aroundbodies in the complex plane, but they will not be discussed here. The pressureand tangential velocity profiles for the inviscid flow around an object are neededfor calculation of the viscous flow in the boundary layer between the solidboundary and the inviscid outer flow.

Assignment 7.5 Pressure profiles Calculate the pressure field or the square ofthe velocity field for the flow in or around a corner and the flow past a circularcylinder. Look at the expression for the corner flow. Under what conditions isthere a flow singularity? Show the pressure or velocity squared pseudo-color for

wall angles of/2, , 3/2, and 2. Which cases are physically realistic and whatdo you think happens in the unrealistic cases? What is special about thepressure profile around the circular cylinder? What value of form drag will it

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predict? Is it realistic and if not, why not? Add the following code to the code forcorner flow and flow around a circular cylinder.pause% Cal cul at e pr essur e di st r i but i on f r om squar e of vel oci t y( your code her e t o cal cul at e pr essur e f i el d)

pcol or ( x, y, p)col ormap( hot )shadi ng f l ataxi s i mage

Solution of hyperbolic systemsThe conservation equations for material, momentum, and energy reduce

to first order PDE in the absence of diffusivity, dispersion, viscosity, and heatconduction. In thin films, viscosity may be a dominant effect in the velocity profilenormal to the surface but the continuity equation integrated over the filmthickness will have only first order spatial derivatives unless the effects ofinterfacial curvature become important. In one dimension, the system of firstorder partial differential equations can be calculated by the method ofcharacteristics [A. J effrey (1976), H.-K. Rhee, R. Aris, and N. R. Amundson(1986, 1987)]. Here we will only consider the case of a single dependentvariable with constant initial and boundary conditions. Denote the dependentvariable as S and the independent variables as x and t. The differential equationwith its initial and boundary conditions are as follows.

( )0, 0, 0

( ,0)

(0, )

IC

BC

S f St x

t x

S x S

f t f

+ = >

=

=

>

The dependent variable can be normalized such that the initial condition is equalto zero and the boundary condition is equal to unity. Thus, henceforth it isassumed such a transformation has been made. The dependent variable will becalled 'saturation' and the flux called 'fractional flow' to use the nomenclature formultiphase flow in porous media. However, the dependent variable could be filmthickness in film drainage or height of a free surface as in water waves. ThePDE, IC, and BC are rewritten as follows.

( )

0, 0, 0

( ,0) 0

(0, ) 1

S df S S

t xt dS x

S x

f t

+ = >

=

=

>

The differential, df/dS is easily calculated since there is only oneindependent saturation. If there were three or more phases this differential would

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be a J acobian matrix. The locus of constant saturation will be sought by takingthe total differential ofS(x,t).

0S S

dS dt dxt x

= + =

( )

0dS

S

dx S t

dt S x

dfS

dS

v

=

=

=

=

This equation expresses the velocity that aparticular value of saturation propagatesthrough the system, i.e., the saturation

velocity, vS is equal to the slope of thefractional flow curve. It is also the slope ofa trajectory of constant saturation (i.e.,dS=0) in the (x,t) space. Since we areassuming constant initial and boundaryconditions, changes in saturation originateat (x,t)=(0,0). From there the changes insaturation, called waves, propagate intrajectories of constant saturation. Weassume that df/dS is a function ofsaturation and independent of time ordistance. This assumption will result in the trajectories from the origin beingstraight lines if the initial and boundary conditions are constant. The trajectoriescan easily be calculated from the equation of a straight line.

x

t

( )( )df S

x S tdS

=

Wave: A composition (or saturation) change that propagates through thesystem.

Spreading wave: Awave in which neighboringcomposition (or saturation)values become more distantupon propagation.

a a

b b

S S

x

t1

t2>t1

a bS S

dx dx

dt dt


27/32

Indifferent waves: A wavein which neighboringcomposition (or saturation)

values maintain the samerelative position uponpropagation.

a

b

S S

x x

t1 t2>t1a

b

a bS S

dx dx

dt dt

=

Step Wave: An indifferent wavein which the compositions changediscontinuously.

Self Sharpening Waves: A wavein which neighboringcompositions (saturations)become closer together uponpropagation.

a

b

S S

x

t1 t2>t1 a

b

a bS S

dx dx

dt dt >

Shock Wave: A wave ofcomposition (saturation) discontinuity that results from a self sharpening wave.

aa

bb

SS

xx

t1 t2>t1

aa

bb

S

t1 t2a

b

t3t3t3

Rule: Waves originating from the same point (e.g., constant initial and boundaryconditions) must have nondecreasing velocities in the direction of flow. This isanother way of saying that when several waves originate at the same time, theslower waves can not be ahead of the faster waves. If slower waves from

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compositions close to the initial conditions originate ahead of faster waves, ashock will form as the faster waves overtake the slower waves. This isequivalent to the statement that a sharpening wave can not originate from apoint; it will immediately form a shock.

SpreadingWave

SharpeningWave

ShockWave

Wave does

notexist

S

BC

IC

( ) ( )0dS

x dx df S v S

t dt ds=

= = =

Mass Balance Across Shock

We saw that sharpening wavemust result in a shock but that does nottell us the velocity of a shock nor thecomposition (saturation) change acrossthe shock. To determine these wemust consider a mass balance acrossa shock. This is sometimes called anintegral mass balance as opposed tothe differential mass balance derivedearlier for continuous composition(saturation) changes.

S

X

S2

S1

t t+dt

dx

7-28


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f

S

S1

S2

shock

B( )2 1Accumulation: A x S S

( )2 1input output: u A t f f

A x S u A t f =

D

D S

dx f

dt S

=

f/S is the cord slope of the fversus S curvebetween S1 and S2.

The conservation equation for the shockshows the velocity to be equal to the cord slope

between S1 and S2 but does not in itselfdetermine S1and S2. To determine S1 and S2,

we must apply the rule that the waves must havenon-decreasing velocity in the direction of flow.The following figure is a solution that is notadmissible. This solution is not admissiblebecause the velocity of the saturation values(slope) between the IC and S1 are less than thatof the shock and the velocity of the shock (cordslope) is less than that of the saturation valuesimmediately behind the shock.

f

S

S2

Not dmissible

IC

shock

S1

BC

This solution is admissible in that the velocity innondecreasing in going from the BC to the IC.However, it in not unique. Several values ofS2 will give admissible solutions. Suppose thatthe value shown here is a solution. Alsosuppose that dispersion across the shockcauses the presence of other values of Sbetween S1 and S2. There are some values ofS that will have a velocity (slope) greater thanthat of the shock shown here. These values of

S will overtake S2 and the shock will go thethese values of S. This will continue until thereis no value of S that has a velocity greater thanthat of the shock to that point. At this point thevelocity of the saturation value and that of theshock are equal. On the graphic construction ,the cord will be tangent to the curve at thispoint. This is the unique solution in the

f

S

S1

S2

shock

BC

IC

admissible butnotunique

7-29


30/32

presence of a small amount of dispersion.

f

S

S1

S2

shock

BC

IC

unique

7-30


31/32

Composition (Saturation) Profile The

composition (saturation) profile is thecomposition distribution existing in the systemat a given time.

Composition or Flux History: Thecomposition or flux appearing at a given pointin the system, e.g., x=1.

Summary of Equations

The dimensionless velocity that asaturation value propagates is given by the

following equation.

S

x

t=to

f

x=1

( )0dS

dx df S

dt dS=

=

With uniform initial and boundary conditions, the origin of all changes insaturation is at x=0 and t=0. Iff(S) depends only on S and not on x or t, then thetrajectories of constant saturation are straight line determined by integration ofthe above equation from the origin.

( )

( )

0( )

dS

S

dx df

x S tdt dS

dx f

S t

x S tdt S

=

= =

= = t

These equations give the trajectory for a given value ofS or for the shock. Byevaluating these equations for a given value of time these equations give thesaturation profile.

The saturation history can be determined by solving the equations for twith a specified value of x, e.g. x=1.

( )

( )( )

, 1

, 1

xt S x

f

S

xt S x

dfS

dS

= =

= =

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The breakthrough time, tBT, is the time at which the fastest wave reaches

x=1.0. The flux history (fractional flow history) can be determined by calculatingthe fractional flow that corresponds to the saturation history.

x

x=1.0

t=to

I.C.

B.C.

Summary of Diagrams

The relationship between thediagrams can be illustrated in adiagram for the trajectories. Theprofile is a plot of the saturation at t=to.

The history at x=1.0 is the saturationor fractional flow at x=1. In thisillustration, the shock wave is thefastest wave. Ahead of the shock is aregion of constant state that is the

same as the initial conditions.

New References

Carslaw, H. S. and J aeger, J . C., Conduction of Heat in Solids, Oxford, (1959).

Churchill, R. V., Operational Mathematics, McGraw-Hill, (1958).

Courant, R. and Hilbert, D., Methods of Mathematical Physics, Volume II PartialDifferential Equations, Interscience Publishers, (1962).

Hellums, J . D. and Churchill, S. W., "Mathematical Simplification of BoundaryValue Problems," AIChE. J. 10, (1964) 110.

J effrey, A., Quasilinear Hyperbolic Systems and Waves, Pitman, (1976)

Lax, P. D., Hyperbolic System of Conservation Laws and the MathematicalTheory of Shock Waves, SIAM, (1973).

LeVeque, R. J ., Numerical Methods for Conservation Laws, Birkhauser, (1992).

Milne-Thompson, L. M., Theoretical Hydrodynamics, 5th ed. Macmillan (1967).

Morse, P. M. and Feshbach, H., Methods of Theoretical Physics, (1953)

Rhee, H.-K., Aris, R., and Amundson, N. R., First-Order Partial DifferentialEquations: Volume I & II, Prentice-Hall (1986, 1989).

Documents

Solution of the Partial Differential Equations