R. K. BHATNAGAR: Flow of a Visco-Elastic Fluid between two Parallel Walls. . . 377

Flow of a Visco-Elastic Fluid between two Parallel Walls in Relative Motion with Uniform Suction at the Stationary Wall

By R. K. BHATNAGAR*)

Es werden Ldsungen fur die stationure Stromung einer R i v l i n - Ericksen-Pliissigkeit zwischen zwei ebsnen par- allelen Wanden angegeben, wobei die obere Wand entweder in Ruhe ist oder mit konstanter Ueachwindigkeii bewegt wird und an der unteren Wand eine gleichmiiJige Abaaugung erfolgt. Man findet, daJ die Druckverteilung fCr eine iiivkoelavtisehe Flussigkeit - iin Gegensutz zu derjenigen einer N e w t onschen Fliissigkeit - vom Wandabstmil ab- kiingt. Dus Geuchu.indigkeitsfel(1, die Wundreibung und der Durchsatz werden iin einzelnen diskutiert.

The solutions are obtainedfor the steady flow of a R i v l i n - Ericksenj luid between two plane parallel m l l s when the upper wall is held at rest or moves with a constant velocity and an uniform suction i s applied at the lower wall. The pressure distribution for a viseoelastic fluid is fourui? to depend on the transverse distance as well i n contrast to that for a Newtonian fluid. A detailed discuseion ofthe velocity field, skin friction and the flow coefficient has been given.

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1. lntroducliori

The problem of COUETTE flow of a viscous incompressible fluid between two plane parallel walls with unil'orm suction a t the lower wall has been discussed by SINHA and CHOUUIIARY [l] by taking the pressure- betwern the walls as uniform. VEEMA and BANSAL [2J later pointed out this to be a w r y restrict,cd assumption in view of the fact, that the pressure gradicnt normal to the wall can not bc neglected even if the uniform suction applied on it may be sufficiently small and reconsidered the problem by taking the flow to be due to shear and the suction a t the stationary wall to be uniform and small. They have shown that the application of suction at the stationary wall develops an adverse pressure gradient which is responsible for causing back-flow at large distances from the mouth of the channel. The skin friction and the flow coefficient decrease with suction para- meter. In the absence of suction, as expected, the results reduce to those for plane COUETTE flow discussed by SCIILICHTING [3].

The present investigation extends the above mentioned problem to the case of a visco-elastic fluid and discusses the effects of small suction and visco-elasticity on the velocity and pressure distributions as well as on the skin friction and the coefficient of discharge. Attention has been focused to two distinct cases namely (i) both the walls are a t rest but there is constant suction a t the lower wall and (ii) the upper wall moves with a constant velocity U in the direction of flow and the lower wall has constant suction applied on it. Although all the conclusions have been drawn on the basis of numerical results which take into account effects up to second order only, the solutions are obtained by taking complete third-order effects to the order of approxima- tion considered.

The cross-viscosity of the fluid does not affect the velocity field but only modifies the yressurc rlistribu- tion, a result in agreement with a previously proved general theorem [4]. In the presence of suction, visco- elasticity of the fluid decreases the longitudinal velocity as well as the transverse velocity in comparison to that for a NEWToNiali fluid. At the stationary wall, for a fixed value of suction parameter, the pressure in- crease in the main flow direction is less for a visco-elastic fluid than that for a NEwTONian fluid. It is interesting to note that due to the presence of uniform suction, the pressure distribution for a visco-elastic fluid depends on transverse distance as well in contrast to that for a NEwToNian fluid. Further the coefficient of skin friction is found to decrease up to a certain distance from the mouth of the channel beyond which it increases as the fluid becomes more and more visco-elastic. The flow coefficient, however, decreases with respect to visco- elasticity of the fluid and the distance from the mouth of the channel.

We take the constitutive equations as the ones proposed by RIVLIN and ERICKSEN [5] up to terms of third order in the form (in a Cartesian frame of reference):

*) This investigation was completed when the author worked as a post-doctoral fellow at the Depnrtnient, of Xpplied Physics, Farbenfabriken Beyer AG, Leverkusen, West Germany.

378 It. K. BHATNAGAR: Flow of a Visco-Elastic Fluid between two Parallel Walls . . .

where Ti, denotes the st,ress tensor, ui the velocity vector, b,, the KRONECKER delta, p the isotropic pressure; @, represents the viscosity, @, the visco-elasticity, @, the cross-viscosity and Q4, UD,, UD6 the third-order con- stants for the fluid under Consideration').

The equations (1) and (2) are to be solved along with the familiar equations of continuity and motion for steady, incompressible flow of a fluid of density g :

2. Forinrilntion of the prohlem

Consider tlie two-dimensional flow of a ~VLIN-ERICICSEN fluid between a pair of plane parallel walls defined by y = 0 and y = yo in a rectangular Cartesian frame of reference. Let x-axis be taken along the wall y = 0 which is held a t rest and y being measured perpendicular to it and to tlie wall y = yo which is taken to move with a constant velocity U . We assume that a uniform constant suction is applied a t the stationary wall.

If u and v represent the components of velocity in the increasing directions of x and y respectively and V , the constant suction velocity, the boundary conditions for the problem under consideration become

U = O , '0 = - 1' a t y = o , u = U o r O , ' v - o a t y = y o . (5)

In general, we introduce the following nondiinensionaI quantities

U = u / U , u = v / V o , P = ule u2 E = "/Yo 9

11 = Y/Yo > 1 = e V , yo/@, I? = B u YO/@I , K = @,/e YE , (6) 8 = y i , I<, = 2 @, QD,/p2 2/: ) K, = 2 @, @,/p2 ~t , K3 = @, CD5/p2 Y: ,

in view of' which equations (3) itnd (4) take the form

wherc all the coefficients of f i , IF, K,, K,, K3 can be casily calculated with the help of the equations ( 1 ) and (2). Thc equations (7)-(9) Ii;ivu to bc solved under the boundary conditions

u = o , v = - 1 a t 11 = 0 , ,U = 1 or 0 , V = 0 a t q = l .

l) These equations are, as is well known, identical with third-order-approximation of the constitutive equations for a simple jluul with jading memory according to a so-called approximation for slow flows.

R. K. BEATNAQAR: Flow of a Visco-Elastic Fluid between two Parallel Walls. . . 379

The above equations reduce to those for a NEwToNian fluid as obtained by VERMA and BANSAL [2] if K = S = Kl = K, = K3 = 0. Following them, we write

where u*, p * are known quantities for the flow in the absence of suction:

q , p * = constant , (12) u* =

and ul, vl, p1 take into account the inertial and the non-NEwToNian effects in the presence of suction. Since it is clear that, to the order of approximation under consideration, these effects will involve only zero and first order solution for which the transverse velocity V will be purely function of q, equations (8) and (9) can be sim- plified considerably. Further, equation (7) allows to introduce a stream function !P(t, q ) in the form

In view of above considerations and eliminating the pressure p l ( [ , q) from the equations (8) and (9) after sub- stituting from equations (ll), (12) and (13) in them, it can be easily seen that Y([ , q) is determined by the fol- lowing partial differential equation:

X, , X,, X3, X , are given by

+3--

The boundary conditions satisfied by Y(8,q) are

- = 1 a t q = O , ay a t - = o , aul

a17

( 1 ti)

The cross-Viscosity of the fluid does not affect the flow field since the equation (14) is free from parameter S. For two-dimensional flows a general result of this nature has been proved earlier [4].

380 R. K. BEATNAGAX: Flow of a Visco-Elastic Fluid between two Parallel Walls. . .

3. Perturbation equations and thoir solutions

We seek a solution of the equation (14) satisfying the boundary conditions (17) in the form

where the boundary conditions satisfied by yo, yl, y2, y3, y4, y5 can bc easily obtained with the help of the equation (17).

Y'(5, r ~ ) = yo + (1 ~i + R yz) + (A2 ~3 + R 1 ~4 + R2 ~ 5 ) + * - * ) . (18)

Zero-order so lu t ion Substituting from equation (18) into (14), it, is seen that yo is given by

V4y0 = 0 . Solution of this equation satisfying the conditions

is found to be

so tha.t yo(5,11) = 5 (1 - 3 + 2 ?/3) ,

This result is in agreement with that of VERMA and BANSAL [2] , gained by a different method for a NEWTONian f h id .

F i r s t - o r d e r so lu t ions Equating coefficient8s o f 1 on both sides of equation (14) and using equation (20) in it, we see that

V 4 y 1 = 125(q - 1) (4v2 - 2 7 + 1 ) , (22) which gives y , ( t , q ) satisfying the conditions

This results in t.he velocity components

1 70

z - - 14 ?/7 - 14 76 + 21 ?/5 - 36 ?/4 + 43 73 - 19 72

y z ( t , 7) satisfies the equation

Since right hand side in the above equation is function of 7 only, y2(5 Thus

once integration of which leads to

7) must also be so.

aY2 at

It may be noted that the conditions - = 0 a t

only two boundary conditions - = 0 a t 7 = 0,l.

termined:

= 0, l are identically satisfied and we are thus left with aY2 a7

These allow to integrate equat.ion (26) without involving any additional constants but c1 remains uiider-

R. K. BHATNA~AR: Flow of a Visco-Elastic Fluid between two Parallel Walls. . . 38 1

It shall be seen in 9 3.1 that constant ti, may be determined by imposing some condition on pressure distribu- tion.

Second-o rde r so lu t ions

Equating coefficients of A2 on both sides of equation (14) leads to the following differential equation determining Y3(5*,):

where Xz0 and X, , correspond to X , and X , in equation (16) with Y replaced by yo. Substituting in the above equation for yo and y1 from equations (20) and (23) respectively, it reduces to

Q4 y3 = 5 + t3 Y2(q) 9 (29) where

6 70

k ; ( r / ) = - -(64 ti7 - 224 rls + 168 q5 + 385 q* - 764 773 + 544 q2 - 216 q + 43) -

- 36 K (4q3 - 97% + 611 - 1) - 2160(K, + K,) (24q3 - 36q2 + 147 - 1) + 72 K3 (64 T / ~ - 96 q2 + 48 T/ - 7) ,

sl,(7l) = - 5184 (A', + K,) (2 7 - 1) . In order to separate 5 and q dependence in equation (29) we let

yA5, 7) = 5 Gl(,) + t3 G,(q) *

Solution of the resulting equations for G, and G2 satisfying the boundary conditions that

gives 1

- g4m0(448q11 - 2464'1" + 3080 qs + 1 2 7 0 5 ~ ~ - 60424,' + 83776q6 -

K 70

- 99792 q5 + 99330 q4 - 64378 7j3 + 17719 q 2 ) + - (- 12 7j7 + 63 q6 - 126 q6 + 18 175

+ 106 114 - 24 q3 - 6 T / ' ) - ~ ( 9 1 + K 2 ) (360 8' - 1260 q6 + 1442 q5 - 455 q4 -

(33)

I 3 35

- 176 q3 + 89 172) + - K3 (64 q7 - 224 q 6 + 336 35 - 245 q4 + 58 q3 + 11 q2) + -1-

216 t3 (K, -t K,) (- 2 715 + 5 ria - 4 q3 + ?la) . 3

The equation determining y4([, q ) is

where LJq) = 72 (12q3 - 18q2 + 5 q ) , L,(T/) = 432 (2 7 - 1) , La(,) = -12 (32 q3 - 33 q2 + 9 7 + 1) .

Writing Y4(t> 9 ) = Ml(,) + t2 MA,) 9

(35)

(36) 20

382 R. K. BHATRAOAR: Flow of a Visco-Elastic Fluid between two Parallel Walls. . .

gives the differential equations determining the functions Ml and M , in the form

to be solved under the boundary conditions

Substituting for L2(q) in equation (38) from equation (35) and integrating using the boundary conditions (39) it can be easily seen that

M,(q) = - 36 (K, + K,) (ria - 2 rl3 + q2) . (40)

Having known M,(q), the equation (37) for M,(q) can be immediately integrated but it may be noted that, similar to that for p2, we have only two boundary conditions to be satisfied.

It may be easily checked that

Thus, integrating equation (37) once with respect to 11, we get

Simplifying the right hand side of the above equation using eyuitbions (35), (40), (G help of boundary conditions (39), we have (after absorbing one constant of integration in M I )

) and integrating H

3 - s (K , + K,) (4 710 - 12 '15 + 6 714 + 8 ? ] Z ) 4-

The remarks macle about trhe constant c, hold for c2 as well. Finally the equations (36), (40) and (43) determine y4(&, q) completely. Equating coefficients of R2 on both sides of equation (14), the following equation determining ye(E, y)

is obtained:

(44)

The right hand side of the above equation simplifies to zero and thus its solution under the prescribed boundary conditions leads to

(45)

a a a4yl

a t a t aq4 Q4y5 = T ~ - - Q ~ Y , - K ~ 1 - Q 4 w Z - 3 ( K , + K,) -2.

y&E, ?/I = 0 *

Finally equations ( l l ) , (13) and (18) give

which are identical with the equations (3.15), (3.16) derived by VERMA and BANSAL [2] for a NEwI'ONian fluid.

R. K. BHATNAQAR: Flow of a Visco-Elastic Fluid between two Parallel Walls. . . 383

3.1 Pressu re d i s t r i b u t i o n For the sake of demonstration of the nature of pressure field and coefficient of skin friction we shall neglect the effects of K,, K, and K3 in equation (46). Then U and V can be written as

(47) A6 -

u = ?I + zf(?I) + P(V) 9 21 = -f(V) 2

where expressions for f(7) and F(7) can be easily found from equation (46). It may be noted that in the absence of suction we have taken the flow to be due to shear only given by

equation (12) in which the pressure gradient is zero. In the presence of suction, this will not be a real restric- tion for it would only determine the origin of the coordinate system with respect to t : 5' = 0 determines that cross-section in which the zero-order flow is due to shear whereas in every other cross-section it is superposi- tion of flows due to shear as well as pressure gradient. Correspondingly if c1 = c, = 0, then there is no pressure gradient in this cross-section for a NEWToNian flow. In the paper cited earlier [2], the calculations have been carried out for the above mentioned special case and thus in order to visualize the differences when the fluid is viscoelastic, we also restrict ourselves only to it.

Substituting from equations (47) in equations (8) and (9) the pressure distribution is found to be (after some laborious calculations)

A t - p' --P - --$' (1 + F') - ~ la t 2 f"] , (48) 1 2 R 2 Ra

where tlie dashes denote differentiation with respect to 7, and C is given by 2929 24

C = l 2 - A (ii - + 3 6 K ) $1 z ( 5 3 9 0 0 + ~ K ) ' ~

In particular, it is interesting to calculate the increase of pressure in &direction determined by A C @(E,q) - p ( 0 , q ) = 2 R i 6 2 + 2 ( 2 K + s )

(49)

Froni above it is verified that for a NEwToNian fluid, the increase of pressure in the main flow direction is pro- portional to 6 2 but does not depend on q. However, this is no longer true when we take into account the visco- elastic properties of the fluid. Then, the two terms giving additional contributions to the pressure-increase both depend on q and there is not only a term proportional to but also to E [cf. ref. [6] and [7] also]. I n equation (50) the coefficient of 6 in the lowest order in I is given by 6 (2 7 - 1) so that its average over the cross-section vanishes. As will be seen from this, setting c1 = c, = 0 means that for 6 = 0 a t least the mean value of the pressure gradient is zero.

At the wall on which suction is applied,

P( t ) = " 9 q ) - P(0, 7)lt ,=o

(51) A C 2 R2 - - - - 5 2 + 2 (2 K + S) f'(0) (P'(0) + 1) +

with

f'(0) = - 6 + + A 2 ( - -___ 17719 +".)I, [ 35 323400 17

Sbilar ly the coefficient of skin friction at the wall 7 = 0 is given by

c -

q - 0

= , [ 1 + , , A + m i 2 - - 2 11 97 6 + r L + - 17719 "( R :: 323400 257 11 47 9 + Kil. 3 + -A + - Aa + 15 K 1 + 70 KA2 + { 70 20

27 32189 "( R 7 53900 35 + 18K'A'- - 12 + -A f- (53)

26 *

384 R. K. BHATNAQAB: Flow of a Visco-Elastic Fluid between two Parallel Walls . . .

If Q determines the discharge per unit breadth of the wall, t,hen 1

Q=Uyo]i idq= U y o (54) 0

Denoting the discharge in the absence of suction by Qo = U yo/2, equation (54) gives the coefficient of discharge as v

2 1 5 11 13 C ~ = Q / Q o = 1 - ~ + - 1 + - 1 ' + K GO 400 (55)

It may be noted that Cf and C, do not depend on cross-viscosity of the fluid which only modifies the pressure distribution as is seen by equation (48).

3.2 Case R = 0 i.e. U = 0 The solutions for the case U = 0 can be immediately written down from equations (46) by taking V , as the characterist,ic velocity in the forin

If we represent the characteristic pressure by Vo/yo then equation (50) gives C F ( E , 17) - P(0,rl) = 3 5, + 1 ( 2 K + J') t 2 f J 2 ,

from which the increase of pressure a t the wall on which suction is applied can be easily calculated. To the order A2, Cf at q = 0 is given by

648

19 Tz Y - - 5 6 + - - 1 + - c,=+ @ I ~ O / Y O '1=0 - [ 35 323400 3204 G234

35 Similarly

1

VOYO 0

which is independent of the non-NEwToNian parameters of the fluid.

(57)

(58 )

(59)

4. Discussion It may be remarked that for sake of convenience we shall derive the conclusions on the basis of the results which would neglect the third order parameters of the fluid i.e. Kl = K, = K3 = 0. This will essentially give t,he broad features of the nature of the flow field. Since in equation (18), R occurs only in its first power to the order of approximation under consideration, we shall treat only 1 to be small. To be able to coinpare the results with those for a NEwToNian fluid (2), we further assume that the ratio of the suction velocity V , to the velocity of upper wall is small and accordingly choose AIR = Vo]U to vary from 0.0005 to 0.05.

Pig. 1. 1.ungitudiiial velocity profiles for the case when the upper wall ie held at rest slid fur suction parameter A = 0.1

R. K. BHATNAQAR: Flow of a Visco-E'astic Fluid between two Parallel Walls. . . 385

Fig. 2. Longitudhal velocity profiles for the case when the upper wall is held at rest and for suction parameter a = 1.0

-30 -25 -20 -15 -I0 -05 0 u*: _u

v,

Figs. 1 and 2 depict the profiles of longitudinal velocity u**( t , 7) for L = 0.1, 1.0 and various values of the parameter K. When L = 0.1, these profiles for a NEwToNian fluid are parabolic a t all distances from the mouth of the channel E = 0 with maximum occuring on the mid-plane. For a visco-elastic fluid they no more remain absolutely parabolic and maximum on them shifts towards the upper wall. As suction increases e.g. when A = 1.0, the NEWTONian profiles show a slight shift of the maximum towards the lower wall (Fig. 2) while, the profiles for a visco-elastic fluid show a further shift towards the upper wall in comparison to those for 3, = 0.1. The effect of visco-elasticity is to increase the longitudinal velocity up to a certain distance from the lower wall beyond which it decreases in comparison to the NEWTONian Velocity.

The above character of longitudinal velocity shows some changes when upper wall starts moving i.e. R + 0. From Fig. 3, which depicts the profiles for R = 2,L = 0.01 and K = 0, - 10, we note that for a NEW- ToNian as well as for a visco-elastic fluid the velocity is completely positive up to a certain distance Eo from the mout,h of the channel (20 < 5, < 40 for L = 0.01) and that this distance decreases with increase in L for a fixed

Fig. 3. Longitudinal velocity profiles for the case when the upper wall moves with a constant velocity ( R = 2,1 = 0.01)

K= 0 K = -04 K = -0.8

--

I I

-075 -052 -025 0 025 050

Fig. 4. Longitudinal velocity profiles for the cBse when the upper wall move8 with a con- stant velocity ( R = 60,1 = 0.1)

386 R. K. BEATNAQAR: Flow of a Visco-Elastic Fluid between two Parallel Walls. . .

R. When ,$ > tC, the velocity is both positive and negative and the point where it vanishes shifts towards the upper wall as increases. Further, a t the mouth of the channel, as il increases, the velocity for both NEW- TONirtn and visco-elastic fluid increases while away from the mouth it decreases. The effect of visco-elasticity is found to decrease the longitudinal velocity.

Figs. 4 and 5 show the effect of variation of R whereas Figs. 5 and 6 that of variation o f l . For a fixed &, as (Kl increases, the point where the longitudinal velocity becomes positive shifts more and more towards the upper wall. From Fig. 4 for 1 = 0.1 and R = 50, 5, < 100 while in Fig. 5 for R = 100, 100 < 5, < 200. Thus tC increases as R increases. On the other hand, 6, decreases considerably if 3, is increased (compare Figs. 5 and 6 for1 = 0.1 and 1 = 0.5 respectively).

I -04

Fig. 5. Longitudinal velocity profiles for the case when theupperwallmoves witha constant velocity (R = 100,A = 0.1)

Fig. 6. Longitudinal velocity profiles for the case when the upper wall moves with a constant velo- city ( R = 100,A = 0.5)

-25 -20 -15 -I0 -05 0 0 5 10

Li =t

Fig. 7. Transverse velocity profiles plotted as a function of q for various values of snction and viseoelastic P R ~ meters

R. K. BHATNAQAR: Flow of a Visco-Elastic Fluid between two Parallel Walls. . . 387

Equation (47) shows that the transverse velocity V does not depend on R and 6, the distance along the wall. From the profiles of - i in Fig. 7 for various values of 1 and K we note that for a NEwToNian fluid the transverse velocity decreases with increase in 1. The contribution of visco-elasticity to - i is always positive for a fixed A. In other words, the transverse velocity for a visco-elastic fluid is less than the corresponding for a NEwToNian fluid.

The coefficient of skin friction C, evaluated using equation (53) is represented in Fig. 8 for R = 100, A = 0.1, 0.3, 0.5 and various values of K . Both for NEwToNian and visco-elastic fluid, Cf decreases steadily with 5. On comparing the curves of K = -0.8 for A = 0.1 and 0.3 it is seen that 0, decreases everywhere else except in a certain neighbourhood of the mouth of the channel where it is seen to increase. The same is true for a NEwToNian fluid as seen by the plots for1 = 0.1,0.3 and 0.5. If we fix1 and increase lKl then CJ decreases up to a certain beyond which it increases.

0 OL

0 02

Cf

0

- 001

-004

-OM

K- 0 K=-O.4 K= - 0 8

_- _ _

Fig. 8. Variation of eoeffiriciit of friction versus longitudinal distance for H = 100,1 = 0.1, 0.3, 0.5 and R = 0;-0.4, -0.8

1.0

0 5

c,

0

- 0 5

-10

-15

K - - 0 4 K = O

K - - 08 _ _

1

400 600 c 200

Fig. 9. Variatlon of flow coefficient versus longitudinal distance for 1 = 0.1, 0.2; R = 100, 200 and K = 0, -0.4, -0.8

The flow coefficient C, is represented in Fig. 9 for R = 100, 200; 1 = 0.1, 0.2 and K = 0, -0.4 and -0.8. It is seen to decrease steadily with 5 for NEwToNian as well as a visco-elastic fluid. For a fixed 1 and R, the effect of visco-elast,icity is also to decrease C,. Further, for a given fluid, it decreases with increase in A everywhere else except in a certain neighbourhood of 5 = 0. With increase in R, the behaviour is exactly opposite to this.

5. Ackriowlndgen~ent The author expresses his thanks to ALEXANDER VON HUMBOLDT-Stiftung for the grant of a research fellowship which allowed this work to be done. He is also grateful to Prof. Dr. H. GIESERUS for giving valuable help and suggestions during the prepa- ration of this paper.

Iieferoneos 1 K. D. SINHA and R. C. CBOUDHARY, Proc. Ind. Acad. Sci. 61, 308 (1965). 2 P. D. VERMA and J. L. BANSAL, Proc. Ind. Acad. Sci. 64, 385 (1966). 3 H. SCHLICHTINQ, Boundary-Layer Theory, McGraw-Hill Book Co. Inc., 1960. 4 P. L. BHATNAQAR and R. K. BHATNAGAR, C. R. Acad. Sc., Paris 261, 3041 (1965). 5 R. S. RIVLIN and J. L. ERICKSEN, J. Rat. Mech. Anal. 4, 323 (1955). 6 R. K. BHATNAQAR, J. Ind. Inst. Sci. 46, 126 (1963). 7 S. C. RAJVANSHI, Ind. Jour. Pure and Appl. Phys. 6, 512 (1968-69).

Manuskripteingang: 3.7. 1970

Anschrijt: R. K. BHATNAQAR, Department of Mathematics, Indian Institute of Technology, Powai, Bombay-70 (India)