Proc. Indian Acad. Sci., Vol. 85 A, No. 5, 1977, pp Elastico-viscous flow engendered due to an impulsively started porous plane wall

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1 Proc. Indian Acad. Sci., Vol. 85 A, No. 5, 1977, pp Elastico-viscous flow engendered due to an impulsively started porous plane wall PRITI MOHAPATRA AND PRAMODINI DEVI Post-Graduate Department of Mathematics, Utkal University,. Vani Vihar, Bhubaneswar , Orissa MS received 30 July 1976; after revision 29 October 1976 ABSTRACT Approximate solution for the flow past an impulsively started infinite porous plane wall in an elastico-viscous fluid is obtained for the velocity and shearing stress. The roles of elasticity of the liquid and the suction on the velocity and the shearing stress have been studied. 1. INTRODUCTION HASIMOTOI has studied the unsteady motion of an infinite flat plate, started at time t = 0 with uniform suction or injection. Nanda 2 has considered flow formation in Couette motion in viscous fluids. In the present-day technology the non-newtonian fluids play an important role. A non-newtonian fluid with slight elastic effects in shear flow has been proposed by Walters. 3 Soundalgekar 4 has studied the impulsive motion of an infinite flat plate in Walters' B' elastico-viscous model, the detailed description of which has been given by Beard and Walters 5 In this paper our aim is to study the impulsive motion of a porous infinite flat plate in Walters' B' liquid. 2. MATHEMATICAL ANALYSIS In this problem, the x' axis is taken along the flat plate and y' axis is perpendicular to it. Uniform suction v' o has been applied at the plate in the negative direction of y' axis. As the plate is infinite in length, all variables in this problem are functions of y' and t' only. Hence from equation of continuity, we have = 0 = v = constant = v' o. (1) From the constitutive equations for Walters' liquid B' (ref. 4) we can show that the problem is governed by the following equation: 272 ^Ul ^u' ^2u' ^3u' cb3u' ' oko ly'3. (2) t^/ ( fit v'0 k0 c^y'2 c^t ' + v

2 UNSTEADY ELASTICO-VISCOUS FLOW 273 At time t' c 0, the fluid and the plate are assumed to be everywhere stationary. For t' > 0, an impulsive motion is given to the plate so that u' = U, a constant. Hence eq. (2) is to be solved under the boundary conditions u' = 0 everywhere for t' < 0 u' = U at y' = 0 1t, >0. (3) = 0 or finite as y -* 00 Introducing the following non-dimensional quantities 77 = Y'/ wz T = t';?, u = u'/ U, k = k0-7)0t v o = v' o/"3 VT, (4) where T is a reference time, eqs (2) and (3) are transformed to the following: bu `v ^u_ 2u 33U {-kv b3u (5) c^t 0 X71 2 c^y)2 t^t 0 ^ 3 T< 0 u = 0 everywhere T>0u=1 at7=0 (6) =0as7) ^o0 3. SOLUTION OF EQUATION Equation (5) is a third order differential equation when k 0 and for k = 0, it reduces to an equation governing the Newtonian fluid in the presence of suction, where k is the elastic parameter. Mathematically we need three boundary conditions to solve the third order differential equation for a unique solution. But there are two boundary conditions on u (71, T). So we adopt small perturbation technique to solve the equation. Here k, the elastic parameter, is very small and can be taken as the perturbation parameter. We can therefore take u in the form u=uo +kul. (7) We now solve eq. (5) by the method of Laplace transforms. If the Laplace transform of u (9, T) is defined as u (71, p), then u (, P) = f e-pt u (^, T) dt (p > 0). (8) 0 Taking Laplace-transforms of eqs (5) and (7), we get Z 2 3 Pu v o d ui = d^ ^ 2 Pk d2 + v ok d 3 (9) ^; = uo + ku, (10)

3 274 PRITI MOHAPATRA AND PRAMODINI DEVI The transformed boundary conditions are from (6) u= lath=0 (11) P u = 0 or finite as q -^ oo. Substituting (10) in (9) and equating the coefficients of like powers of k' we get the set of equations duo d2uo Puo v o = d7j d,12 12 ( ) dul d2u1 _ d2u0 d3uo ( ) Pui vodn = dv2 p -+vo s. (13) boundary conditions on uo and ul are from (11), uo =pu1 =0atq=0 uo =0u1 =0 asq-goo. (14) Solving (12) with the boundary conditions (14) we have ( ^/v o2 _ 4p vo uo = p exp ( { 2-1- J ^). (15) Substituting eq. (15) in (13) and solving with the boundary conditions (14), we get ul 2 3 ^vo^t4p + ^vo + 4P -r71vo + 2P 2p { iiv^.,/yo + x 4p exp ( ^vo 2 + 4p vo 2 } ^. (16) From eqs (10), (15) and (16), we have l k r 2y02 u = 1 ex ( 3voL f 4P -i- vo p p { 2 2 S l L /v o2 + 4p lip v 7vo3 -F 'qy04 l /yo2 4p F^7 0+ 2P +2PVV02+4PJ x exp ( tj^a, 02 -i- + 4p vo} 2 2 q) (17) Inverting (17), and substituting f- p = z 2, 4 = m.

4 We have UNSTEADY ELASTICO-VISCOUS FLOW 275 u (q, T) = exp ( mt 7 /m) C(1 4kgm 312) x 2^ri J a,, exp (z 2 T zi) z2 zdz 4kgm i f exp (z 2T z'q) dz IT Br, rk f exp (z 2 T z71) (z 2 m) dz Br, 1 2ki A/m IT!f exp (z2t z'q) zdz Br, 4kijm2 ai f exp (z 2T zq) z2 dz m, (18) ar, where Bra is the Bromwich path defined in McLachlan. 6 f It is worked out in McLachlan's that Ii exp ( QT qz) Bra exp (mt "11/ m) erfc 71 ± 2 T ( z 1^m ) 1/ We observe that 1, fexp (z 2 T 'z) dz = del ^i f exp (z2t iz) dz) and Bry ery j (19) Similarly Sri J exp (z=t qz) zdz = dt {! f exp (z2t z) dz). Br, Drs i f exp ( Z 2T 71z) z$ dz = dt {-- f exp (z 2 T 71z) dz}. Hence all integrals in (18) can be evaluated by the help of (19), and we obtain

5 276 PRITI MOHAPATRA AND PRAMODINI DEVI u (I, T) = exp ( vo17) ( 1 -lvo 3k) erfc 71 2TT o qk { - erfc Ty 2 ^/T -/irt x ( 2+ " T 8T2 4T The shearing stress at the plate is--given ^u' $u' ly -o = ['Jo ^Y, ko which by the help of eq. (4) reduces to exp { Cv 4T + v2 ofl + 4T^^. (20) by + vo ke, 2u,^ Y v'=o, 1/y Ir Tay = P, x.y. 1 Y'_O r u tau +v btu ^^i QJa-v (21) From eqs (20) and (21), 7v 2 T exp (--- - ) x1 V7,T L 1 8 4T 32 8 T v Qk + k 61 v s kq + 13 yo zka g a i+ r o (4 v o4k2)11 ± erf y 2 T. (22) 4. DISCUSSION OF RESULTS The results are shown by means of graphs and tables. Bigure 1 shows the erect of the suction parameter v o razed the nondimensional time T on the velocity field. It is observed that the velocity at any point decreases as v increases, when T is constant, and it increases with an increase in T, when v is constant. Figure 2 shows the velocity fbr different values of the elastic parameter k, when v = 0.0 and v = 1.0. It is observed that for fixed value of T elasticity increases the velocity when suction is absent, whereas it decreases the velocity when suction velocity v = 1.0. Table 1 shows the velocity for different values of k, when T = 0.5 and ua-= 0-5. It is seen,that the vloci4y increases with k in the ncigl%bourhood of the plate, whereas an opposite effect is observed when i) > 0.4.

6 UNSTEADY ELASTICO-VISCOUS BLOW h , Figure 1. Velocity dtstributlon fvr different values - of v o and T, k Vo =O v0 = t7s 1.0 Sri I Figure 2. Velocity distribution for different values of k, T 0 5. k= k= k X0.1 k-0.2 X X---X-

7 278 PRITI MOHAPATRA AND PRAMODINI DEvI Table 1. Velocity distribution for different values of k and T = 0. 5, ya = 0 5,ilk Table 2. Values of the skin-friction for different values of T, k and v Tlk v, Table 2 shows the skin-friction for different values of T, k and v,. It is seen that the skin-friction decreases with an increase in T, for fixed values of k and vo. And for vo = 0.0 and ve = 0.5, the skin-friction increases as k increases, for fixed values of T, whereas an opposite effect is observed for vo = 1.0. Again the skin-friction increases as vo increases, when k and T are fixed. ACKNOWLEDGEMENT The authors are grateful to Dr. S. P. Mishra for his constant encouragement and advice,

8 UNSTEADY I;LASTICO-VISCOUS FLOW 279 REFERENCES 1. Hasimoto, H., J. Phys. Soc. Jap (1957). 2. Nanda, R. S., J. Phys. Soc. Jap (1958). 3. Walters, K., J. Mechanique 1474 (1962). 4. Soundalgekar, V. M., Rheol. Acta (1974). 5. Beard, D. W. and Walters, K., Proc. Camb. Phil. Soc (1964). 6. McLachlan, N. W., Complex Variable Theory and Transform Calculus, Cambridge University Press (1963). A6 May 77

Effect of sudden suction on the hydromagnetic natural convection from a vertical plate

Proc. Indian Acad. Sci., Vol. 85 A, No. 5, 1977, pp. 280-288. Effect of sudden suction on the hydromagnetic natural convection from a vertical plate V. V. RAMANA RAO AND V. BALA PRASAD Department of Applied