low reynolds hyrdoyMICAAAAAAA.docx

8/9/2019 low reynolds hyrdoyMICAAAAAAA.docx

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System

Definition

System:A quantity of matter in space which is analyzed during a problem.

Surroundings:Everything external to the system.

System Boundary:A separation present between system and surrounding.

Classification of the system boundary:-

eal solid boundary

!maginary boundary

"he system boundary may be further classified as:-

#ixed boundary or Control $ass %ystem

$oving boundary or Control &olume %ystem

"he choice of boundary depends on the problem being analyzed.

Fig 9.1 System and Surroundings

Classification of Systems

Types of System

Control Mass System (Closed System)

1. !ts a system of fied masswith fied identity.

2. "his type of system is usually referred to as !closed system!.

3. "here is no mass transfer across the system boundary.


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4. Energy transfer may ta'e place into or out of the system.

Clic" to play t#e Demonstration

Fig 9.$ % Control Mass System or Closed System

Control &olume System ('pen System)

(. !ts a system of fied olume.

). "his type of system is usually referred to as !open systemor a !control olume!

*. $ass transfer can ta'e place across a control volume.

+. Energy transfer may also occur into or out of the system.

,. A control volume can be seen as a fixed region across which mass and energy transfers are

studied.

. Control %urface- !ts the boundary of a control volume across which the transfer of both mass andenergy ta'es place.

. "he mass of a control volume /open system0 may or may not be 1xed.

2. 3hen the net influx of mass across the control surface equals zero then the mass of the systemis fixed and vice-versa.

4. "he identity of mass in a control volume always changes unli'e the case for a control masssystem /closed system0.

(5. $ost of the engineering devices6 in general6 represent an open system or control volume.

*ample:+

7eat exchanger - #luid enters and leaves the system continuously with the transfer of heat across

the system boundary.
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8ump - A continuous flow of fluid ta'es place through the system with a transfer of mechanical

energy from the surroundings to the system.

Clic" to play t#e Demonstration

Fig 9., % Control &olume System or 'pen System

-solated System

(. !ts a system of fied masswith same identity and fied energy.

). 9o interaction of mass or energy ta'es place between the system and the surroundings.

*. !n more informal words an isolated system is li'e a closed shop amidst a busy mar'et.

Fig 9. %n -solated System
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Conseration of Mass + T#e Continuity */uation

0a of conseration of mass

"he law states that mass can neither be created nor be destroyed. Conservation of mass is inherent to acontrol mass system /closed system0.

"he mathematical expression for the above law is stated as:

m/t = 06 where m mass of the system

#or a control volume /#ig.4.,06 the principle of conservation of mass is stated as

ate at which mass enters ate at which mass leaves the region ; ate of accumulation of mass in theregion


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Clic' to play the >emonstration

Fig 9.2 % Control &olume in a Flo

Continuity */uation + Differential Form

Deriation

(. "he point at which the continuity equation has to be derived6 is enclosed by an

elementary control volume.

). "he influx6 efflux and the rate of accumulation of mass is calculated across each

surface within the control volume.
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Fig 9.3 % Control &olume %ppropriate to a 4ectangular Cartesian Coordinate System

Consider a rectangular parallelopiped in the above figure as the control volume in a rectangular

cartesian frame of coordinate axes.

9et efflux of mass along x -axis must be the excess outflow over inflow across faces

normal to x -axis.

?et the fluid enter across one of such faces A@C> with a velocity u and a density ."he

velocity and density with which the fluid will leave the face E#B7 will be

and respectively /neglecting the higher order terms in x0.

"herefore6 the rate of mass entering the control volume through face A@C> u dy

dz.

"he rate of mass leaving the control volume through face E#B7will be

/neglecting the higher order terms in dx0

%imilarly influx and efflux ta'e place in all y and z directions also.

ate of accumulation for a point in a flow field

sing6 ate of influx ate of Accumulation ; ate of Efflux


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"ransferring everything to right side

/4.)0

"his is the */uation of Continuityfor a compressible fluid in a rectangular cartesiancoordinate system.

Field

Continuity */uation + &ector Form

"he continuity equation can be written in a vector form as

or6/4.*0

where is the velocity of the point

!n case of a steady flo6

7ence Eq. /4.*0 becomes

/4.+0


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!n a rectangular cartesian coordinate system

/4.,0

Equation /4.+0 or /4.,0 represents the continuity e/uation for a steady flo.

!n case of an incompressible flow6

constant

7ence6

$oreover

"herefore6 the continuity e/uation for an incompressi5le flobecomes

/4.0

/4.0

!ncylindrical polar coordinates eq.4. reduces to

Eq. /4.0 can be written in terms of the strain rate componentsas

/4.20
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Continuity */uation + % Closed System %pproac#6e "no t#at t#e conseration of mass is in#erent to t#e definition of aclosed system as Dm7Dt 8 (#ere m is t#e mass of t#e closed system).oeer; t#e general form of continuity can 5e deried from t#e 5asic

e/uation of mass conseration of a system.Deriation :+

0et us consider an elemental closed system of olume & and density


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Continuity */uation + % Closed System %pproac#6e "no t#at t#e conseration of mass is in#erent to t#e definition of aclosed system as Dm7Dt 8 (#ere m is t#e mass of t#e closed system).oeer; t#e general form of continuity can 5e deried from t#e 5asic

e/uation of mass conseration of a system.Deriation :+

0et us consider an elemental closed system of olume & and density


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3e 'now that the conservation of mass is inherent to the definition of a closed system as >mD>t 5/where m is the mass of the closed system0.

7owever6 the general form of continuity can be derived from the basic equation of mass conservation of asystem.

Deriation :+

?et us consider an elemental closed system of volume & and density .

9ow /dilation per unit volume0

!n vector notation we can write this as

"he above equations are same as that formulated from Control &olume approach


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Stream Function

?et us consider a two-dimensional incompressible =ow parallel to the x - y plane in a rectangular cartesian coordinatesystem. "he =ow 1eld in this case is de1ned by

u = u(x, y, t)v = v(x, y, t)

w = 0

"he equation of continuity is

/(5.(0

!f a function (x, y, t)is defined in the manner

/(5.)a0

/(5.)b0

so that it automatically satisfies the equation of continuity /Eq. /(5.(006 then the function is 'nown asstream function.9ote that for asteady flo; = is a function of to aria5les and y only.

Constancy of = on a Streamline

%ince is a point function6 it has a value at every point in the =ow 1eld. "hus a change in the streamfunction can be written as

"he equation of a streamline is given by

!t follows that d 5 on a streamline."his implies the value of is constant along a streamline. "herefore6 theequation of a streamline can be expressed in terms of stream function as

(x, y) =constant /(5.*0

'nce t#e function = is "non; streamline can 5e dran 5y >oining t#e same alues of = in t#e flofield.


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Stream function for an irrotational flo

!n case of a two-dimensional irrotational flow

Conclusion dran:For an irrotational flo; stream function satis?es t#e 0aplace@s e/uation

A#ysical Significance of Stream Funtion =

#igure (5.( illustrates a two dimensional flow.


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Fig 1.1 A#ysical -nterpretation of Stream Function

?et A be a fixed point6 whereas 8 be any point in the plane of the flow. "he points A and 8 areFoined by the arbitrary lines A@8 and AC8. #or an incompressible steady flow6 the volume flowrate across A@8 into the space A@8CA /considering a unit width in a direction perpendicular tothe plane of the flow0 must be equal to that across AC8. A number of different paths connecting

A and 8 /A>86 AE86...0 may be imagined but the volume flow rate across all the paths would bethe same. "his implies that therate of flo across any cure 5eteen % and A dependsonly on t#e end points % and A.

%ince A is fixed6 the rate of flow across A@86 AC86 A>86 AE8 /any path connecting A and 80 is afunction only of the position 8. "his function is 'nown as the stream function =.

"he value of at 8 represents the volume flow rate across any line Foining 8 to A."he value of at A is made arbitrarily zero. !f a point 8G is considered /#ig. (5.(b0688G beingalong a streamline6 then the rate of flow across the curve Foining A to 8G must be the same asacross A86 since6 by the definition of a streamline6 there is no flow across 88H

"he value of thus remains same at 8G and 8. %ince 8G was ta'en as any point on the


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streamline through 86 it follows that is constant along a streamline. "hus the flow may berepresented by a series of streamlines at equal increments of .

!n fig /(5.(c0 moving from A to @ net flow going past the curve A@ is

"he stream function6 in a polar coordinate system is defined as

"he expressions for &rand &Iin terms of the stream function automatically satisfy the equationof continuity given by

Stream Function in T#ree Dimensional and Compressi5le Flo

Stream Function in T#ree Dimensional Flo

-n case of a t#ree dimensional flo; it is not possi5le to dra a streamlineit# a single stream function.

%n aially symmetric t#ree dimensional flo is similar to t#e to+dimensional case in a sense t#at t#e flo field is t#e same in eery planecontaining t#e ais of symmetry.

T#e e/uation of continuity in t#e cylindrical polar coordinate system foran incompressi5le flo is gien 5y t#e folloing e/uation


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For an aially symmetric flo (t#e ais r 8 5eing t#e ais of symmetry);

t#e term 8 ;and simplified e/uation is satisfied 5y functionsdefined as

/(5.+0

T#e function = ; defined 5y t#e */.(1.) in case of a t#ree dimensionalflo it# an aial symmetry; is called t#e sto"es stream function.

Stream Function in Compressi5le Flo

For compressi5le flo; stream function is related to mass flo rateinstead of olume flo rate 5ecause of t#e etra density term in t#e

continuity e/uation (unli"e incompressi5le flo)

T#e continuity e/uation for a steady to+dimensional compressi5le flois gien 5y

ence a stream function = is de?ned #ic# ill satisfy t#e a5oee/uation of continuity as

Jwhere0is a reference density]/(5.,0


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t#e slo motion of a iscous fluid past a sp#ere. -n #is analysis;Sto"es neglected t#e inertia terms of aier+Sto"es e/uations.

%oiding details; integrating t#e pressure distri5ution and t#e

s#earing stress oer t#e surface of a sp#ere of radius 4 ; Sto"esfound t#at t#e drag D of t#e sp#ere; #ic# is placed in a parallel

stream of uniform elocity ; is gien 5y

/).(0

T#is is t#e ell+"non Sto"es e/uation for t#e drag of a sp#ere.

-t can 5e s#on t#at one t#ird of t#e total drag is due to pressure

distri5ution and t#e remaining to t#ird arises from frictionalforces. -f t#e drag coefficient is defined according to t#e relation

/).)0

#ere is t#e frontal area of t#e sp#ere; t#en

or/).*0

% comparison 5eteen Sto"es drag coefficient in */. ($.,) and

eperiments is s#on in Fig. $.1. T#e approimatesolution due to Sto"es is alid for 4e E 1.


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F- $.1 + Comparison 5eteen Sto"es drag coefficient and eperimentaldrag coefficient

%n important application of Sto"es la is t#e determination of

iscosity of a iscous fluid 5y measuring t#e terminal elocity of afalling sp#ere. -n t#is deice; a sp#ere is dropped in a transparentcylinder containing t#e fluid under test. -f t#e specific eig#t of t#esp#ere is close to t#at of t#e li/uid; t#e sp#ere ill approac# asmall constant speed after 5eing released in t#e fluid. o e canapply Sto"es la for steady creeping flo around a sp#ere #eret#e drag force on t#e sp#ere is gien 5y */. ($.1).

6it# t#e sp#ere; falling at a constant speed; t#e acceleration is

Gero. T#is signifies t#at t#e falling 5ody #as attained terminalelocity and e can say t#at t#e sum of t#e 5uoyant force and dragforce is e/ual to eig#t of t#e 5ody.

/).+0

#ere


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acceleration #as ceased.

T#eory of ydrodynamic 0u5rication

T#in film of oil; confined 5eteen t#e interspace of moing parts;

may ac/uire #ig# pressures up to 1 MAa #ic# is capa5le ofsupporting load and reducing friction. T#e salient features of t#istype of motion can 5e understood from a study of slipper 5earing(Fig. $.$). T#e slipper moes it# a constant elocity I past t#e5earing plate. T#is slipper face and t#e 5earing plate are not

parallel 5ut slig#tly inclined at an angle of . % typical 5earing #asa gap idt# of .$2 mm or less; and t#e conergence 5eteen t#ealls may 5e of t#e order of 172. -t is assumed t#at t#e sliding

surfaces are ery large in transerse direction so t#at t#e pro5lemcan 5e considered to+dimensional.

Fig $.$ + Flo in a slipper 5earing

For t#e analysis; e may assume t#at t#e slipper is at rest and t#e

plate is forced to moe it# a constant elocity I .

T#e #eig#t #() of t#e edge 5eteen t#e 5loc" and t#e guide isassumed to 5e ery small as compared it# t#e lengt# l of t#e5loc".

T#e essential difference 5eteen t#is motion and t#at discussed in

0ecture $3 (Couette flo) is t#at #ere t#e to alls are inclined at
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an angle to eac# ot#er.

Due to t#e gradual reduction of narroing passage; t#e conectie

acceleration is distinctly not Gero.

For all practical purposes; inertia terms can 5e neglected as

compared to iscous term. T#is can 5e >ustified in folloing ay

T#e inertia force can 5e neglected it# respect to iscous force if t#emodified 4eynolds num5er;

T#e e/uation for motion in y direction can 5e omitted since

t#e component of elocity is ery small it# respect to u .

Besides; in t#e +momentum e/uation; can 5e neglected

as compared it# 5ecause t#e former is smaller t#an t#e

latter 5y a factor of t#e order of . 6it# t#ese simplificationst#e e/uations of motion reduce to

/).0

T#e e/uation of continuity can 5e ritten as :

/).0

T#e 5oundary conditions are:at y 8 ; u 8 I at 8 ; p 8 pat y 8 #; u 8 and at 8 l; p 8 p($.J)


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-ntegrating */. ($.3) it# respect to y ; e o5tain

%pplication of t#e "inematic 5oundary conditions (at y8; u 8I and

y 8 #; I8); yields

/).40

ote t#at is constant as far as integration along y is concerned;

5ut p and ary along +ais.

%t t#e point of maimum pressure; 8 #ence

/).(50

*/uation ($.1) depicts t#at t#e elocity profile along y is linear at

t#e location of maimum pressure. T#e gap at t#is location may 5edenoted as #K.

Su5stituting */. ($.9) into */. ($.J) and integrating; e get

or

/).((0

#ere

-ntegrating */. ($.11) it# respect to ; e o5tain


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/).()a0

or

/).()b0

#ere is a constant

Since t#e pressure must 5e t#e same (p 8 p); at t#e ends of t#e

5earing; namely; p 8 p at 8 and p 8 p at 8l; t#e un"nons int#e a5oe e/uations can 5e determined 5y applying t#e pressure5oundary conditions. 6e o5tain

6it# t#ese alues inserted; t#e e/uation for pressure distri5ution

($.1$) 5ecomes

or

/).(*0

-t may 5e seen from */. ($.1,) t#at; if t#e gap is uniform; i.e. # 8

#18#$; t#e gauge pressure ill 5e Gero. Furt#ermore; it can 5e saidt#at ery #ig# pressure can 5e deeloped 5y "eeping t#e filmt#ic"ness ery small.

Figure $.$ s#os t#e distri5ution of pressure t#roug#out t#e

5earing.

...T#eory of ydrodynamic 0u5rication... cont from preious slide


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T#e total load 5earing capacity per unit idt# is

%fter su5stituting #8#1 + L it# L8(#1+#$)7 l in t#e a5oe e/uation andperforming t#e integration;

/).(+0

T#e s#ear stress at t#e 5earing plate is

/).(,0

Su5stituting t#e alue of from */. ($.1) and t#en ino"ing t#e alueof in */. ($.12); t#e final epression for s#ear stress 5ecomes

T#e drag force re/uired to moe t#e loer surface at speed I isepressed 5y

/).(0

Mic#ell t#rust 5earing; named after %..M. Mic#ell; or"s on t#e

principles 5ased on t#e t#eory of #ydrodynamic lu5rication . T#e>ournal 5earing (Fig. $.,) deelops its force 5y t#e same action;ecept t#at t#e surfaces are cured.


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F- $., ydrodynamic action of a >ournal 5earing

Blasius Flo 'er % Flat Alate

T#e classical pro5lem considered 5y . Blasius as

1. To+dimensional; steady; incompressi5le flo oer a flatplate at Gero angle of incidence it# respect to t#e uniform

stream of elocity .

$. T#e fluid etends to infinity in all directions from t#e plate.

T#e p#ysical pro5lem is already illustrated in Fig. $J.1

Blasius anted to determine

(a) t#e elocity field solely it#in t#e 5oundary layer;

(5) t#e 5oundary layer t#ic"ness ;(c) t#e s#ear stress distri5ution on t#e plate; and(d) t#e drag force on t#e plate.

T#e Arandtl 5oundary layer e/uations in t#e case under

consideration are

/)2.(,0


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T#e 5oundary conditions are

/)2.(0

ote t#at t#e su5stitution of t#e term in t#e original

5oundary layer momentum e/uation in terms of t#e free stream

elocity produces #ic# is e/ual to Gero.

ence t#e goerning */. ($J.12) does not contain any pressure+

gradient term.

oeer; t#e c#aracteristic parameters of t#is pro5lem are

t#at is;

T#is relation #as fie aria5les .

-t inoles to dimensions; lengt# and time.

T#us it can 5e reduced to a dimensionless relation in terms of (2+$)

8, /uantities ( Buc"ing#am Ai T#eorem)

T#us a similarity aria5les can 5e used to find t#e solution

Suc# flo fields are called self+similar flo field .


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0a of Similarity for Boundary 0ayer Flos

!t states that the u component of velocity with two velocity profiles of u(x,y) at

differentx locations differ only by scale factors in u and y .

"herefore6 the velocity profiles u(x,y) at all values ofx can be made

congruent if they are plotted in coordinates which have been madedimensionless with reference to the scale factors.

"he local free stream velocity U(x) at sectionx is an obvious scale factor

for u6 because the dimensionless u(x) varies between zero and unity with yatall sections.

"he scale factor for y 6 denoted by g(x) 6 is proportional to the local boundary

layer thic'ness so that y itself varies between zero and unity.

&elocity at two arbitraryx locations6 namely x(and x)should satisfy the

equation

/)2.(0

9ow6 for @lasius flow6 it is possible to identify g/x0 with the boundary layers

thic'ness K we 'now

"hus in terms of x we get

i.e.6


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/)2.(20

whereor more precisely6

/)2.(40

"he stream function can now be obtained in terms of the velocity components as

or

/)2.)50

where > is a constant. Also and the constant of integration is zeroif the stream function at the solid surface is set equal to zero.

9ow6 the velocity components and their derivatives are:

/)2.)(a0

or


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/)2.)(b0

/)2.)(c0

/)2.)(d0

/)2.)(e0

%ubstituting /)2.)0 into /)2.(,06 we have

or6

where

/)2.))0


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and

Contd. from Areious Slide

"he boundary conditions as in Eg. /)2.(06 in combination with Eg. /)2.)(a0 and

/)2.)(b0 become

at 6 therefore

at therefore /)2.)*0

Equation /)2.))0 is a t#ird order nonlinear differential e/uation .

@lasius obtained the solution of this equation in the form of series expansion through

analytical techniques

3e shall not discuss this technique. 7owever6 we shall discuss a numerical technique

to solve the aforesaid equation which can be understood rather easily.

9ote that the e/uation for does not contain .

Boundary conditions at and merge into t#e

condition . T#is is t#e "ey feature of similarity solution.

3e can rewrite Eq. /)2.))0 as three first order differential equations in the following


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way

/)2.)+a0

/)2.)+b0

/)2.)+c0

?et us next consider the boundary conditions.

(. "he condition remains valid.

). "he condition means that .

*. "he condition gives us .

ote that the equations for f and G have initial values. 7owever6 the value for 7/50 is not'nown. 7ence6 we do not have a usual initial-value problem.

S#ooting Tec#ni/ue

3e handle this problem as an initial-value problem by choosing values of and solving

by numerical methods 6 and .

!n general6 the condition will not be satisfied for the function arising from thenumerical solution.

3e then choose other initial values of so that eventually we find an which results

in ."his method is called the shooting technique.

!n Eq. /)2.)+06 the primes refer to differentiation wrt. the similarity variable . "he

integration steps following unge-Lutta method are given below.

/)2.),a0

/)2.),b0

/)2.),c0


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values of k, l and m are as follows.

#or generality let the system of governing equations be

!n a similar way K*6 l*6 m*and k+6 l+6 m+mare calculated following standard formulae for theunge-Lutta integration. #or example6 K*is given by

"he functions #(6 #)and #*are

B6 7 6 -f H / 2respectively. "hen at a distance from the wall6 we have

/)2.)a0

/)2.)b0

/)2.)c0

/)2.)d0

As it has been mentioned earlier is un'nown. !t must be

determined such that the condition is satisfied.

"he condition at infinity is usually approximated at a finite value of /around 0. "he

process of obtaining accurately involves iteration and may be calculated using theprocedure described below.

#or this purpose6 consider #ig. )2.)/a0 where the solutions of versus for two

different values of are plotted.


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"he values of are estimated from the curves and are plotted in #ig. )2.)/b0.

"he value of now can be calculated by finding the value at which the line

(-) crosses the line @y using similar triangles6 it can be said

that . @y solving this6 we get .

9ext we repeat the same calculation as above by using and the better of the

two initial values of . "hus we get another improved value . "his process

may continue6 that is6 we use and as a pair of values to find more

improved values for 6 and so forth. "he better guess for 7 /50 can also be

obtained by using the 9ewton aphson $ethod. !t should be always 'ept in mind that

for each value of 6 the curve versus is to be examined to get the

proper value of .

"he functions and are plotted in #ig. )2.*."he velocity

components6 u and v inside the boundary layer can be computed from Eqs /)2.)(a0and /)2.)(b0 respectively.

A sample computer program in #ownload the program
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Fig $J.$ Correcting t#e initial guess for (')

Fig $J., f; and distri5ution in t#e 5oundary layer


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$easurements to test the accuracy of theoretical results were carried out by many

scientists. !n his experiments6 M. 9i'uradse6 found excellent agreement with the

theoretical results with respect to velocity distribution within the boundarylayer of a stream of air on a flat plate.

!n the next slide weHll see some values of the velocity profile

shape and in tabular format.

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