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1 Available online at Advances in Applied Science Research,, (5:85-96 ISSN: CODEN (USA: AASRFC Hdroagnetic natural convection flow of an incopressible viscoelastic fluid between two infinite vertical oving and oscillating parallel plates S. Sreekanth *, S. Venkataraana **, G. Sreedhar Rao*** and R. Saravana** * Departent of Matheatics, Sreenivasa Institute of Technolog and Manageent Studies, Chittoor, A.P. India ** Departent of Matheatics, Sri Venkateswara Universit, Tirupati, A.P. India *** Departent of Matheatics and Coputer Sciences, The Universit of The West Indies, St.Augustine, Trinidad and Tobago _ ASTRACT In this paper, we stud the theral and ass diffusion on incopressible viscoelastic fluid flow (Rivlin-Ericksen between two infinite vertical parallel plates oving in opposite direction, while one plate is oscillating in tie and agnitude about a constant non-zero ean under the influence of a unifor transverse agnetic field. The suction velocit at the plate fluctuates with the tie haronicall fro a constant ean velocit. We have evaluated velocit distribution, teperature distribution, concentration distribution, phase and aplitude of the skin friction. The effects of agnetic paraeter M, visco-elastic paraeter S, Schidt nuber S c, Grashof nuber G r, odified Grashof nuber G c, and Prandtl nuber P r on the above said phsical quantities are discussed. _ INTRODUCTION The echanical behaviour of a large nuber of aterials of industrial iportance such as snthetic fibers, high poler solutions and an other highl viscous fluids cannot be explained full b the classical linear stress-strain relation. Attepts in the past especiall in the last two decades, have been ade to forulate ore general non-linear theories which could take full into account the observed behaviour of the non-newtonian and elastic viscous fluids. Consequentl large nuber of flow probles has been solved b using such general theories either for soe theoretical interest or fro the point of view of coparing the experiental results with the prediction of various theories. As the general stress-strain relations are expressed b highl coplicated non-linear differential equations, to work out solutions for such a class of fluids even for sall flows is not an earl task. Generalizing the stress-strain velocit relations of classical hdrodnaics the rheological behaviour of the non-newtonian liquids has been studied b Rivlin [] and Reiner [].On account of the non-linear nature of the equation of state of even the siplest elastic viscous fluid, it is alost ipossible to obtain the exact solution of the equation of otion and one has to resort to the approxiate ethods. In ost of the investigations of elastic viscous fluids, the flow has 85

2 S. Sreekanth et al Adv. Appl. Sci. Res.,, (5:85-96 been considered slow and paraeters characterizing the elastic properties of the fluid have been assued sall. In fact the increase eergence of non-newtonian fluids such as olten plastics, pulps, eulsions, aqueous solutions of polacrlaid and polisobutlene etc., as iportant raw aterials and cheical products is a large variet of industrial processes has stiulated a considerable attention in recent ears to the stud of non- Newtonian fluids and their related transport process. The two diensional unstead flow of an incopressible viscous fluid, when the free strea is oscillating about a non-zero constant ean, has been studied b Karan-pohlausen ethod b Lighthill [8].Taking the free strea oscillations into account, Stuart [6] has analzed the forced flow and heat transfer fro an infinite porous plate. Soundalgekar [] has studied the free convection effects on the oscillator flow of an incopressible viscous fluid past an infinite vertical plate with variable section. Krishnanlal [7] has investigated the application of fluctuating section to free strea lainar flow past a porous vertical wall. Sastri and hadra [3] have studied hdroagnetic convective heat transfer in vertical pipes. Georgantopoules et al. [5] have discussed the effects of free convection and ass transfer in a conducting liquid when the fluid is subjected to a transverse agnetic field. Raptis [] has studied free convection and ass transfer effects on the flow past an infinite oving vertical porous plate with constant suction and heat sources when free strea velocit is an oscillator function of tie. Agarwal et al. [3] have discussed the cobined buoanc effects of theral and ass diffusion on MHD convection flows. Vajravelu [7] has studied the proble of free convection heat transfer between two long vertical plates oving in opposite direction. Agarwal and kishore [] studied theral and ass diffusion on MHD natural convection flow between two infinite vertical oving and oscillating porous parallel plates. Johri [6] has considered the flow of visco-elastic fluid induced b elliptic haronic oscillations of a disc. He also studied the unstead channel flow of an elastic viscous liquid. Recentl, Asghar et al. [] discussed the flow of a non-newtonian fluid induced due to the oscillations of a porous plate. Ogulu et al. [9] studied the Heat transfer to unstead agneto-hdrodnaic flow past an infinite oving vertical plate with variable suction and Abbas et al. [] studied Hdroagnetic flow in a viscoelastic fluid due to the oscillator stretching surface In this paper, we stud the theral and ass diffusion on incopressible visco-elastic fluid flow (Rivlin-Ericksen between two infinite vertical parallel plates oving in opposite direction, while one plate is oscillating in tie and agnitude about a constant non-zero ean under the influence of a unifor transverse agnetic field. The suction velocit at the plate fluctuates with the tie haronicall fro a constant ean velocit. We have evaluated velocit distribution, teperature distribution, concentration distribution, phase and aplitude of the skin friction. The effects of agnetic paraeter M, visco-elastic paraeter S, Schidt nuber S c, Grashof nuber G r, odified Grashof nuber G c, and Prandtl nuber P r on the above said phsical quantities are discussed in section 3.. Rheological equations of the state: The constitute equation for a Rivlin-Ericksen fluid is T PI + ø d + ø b + ø 3 d (. where T T ij, T ij is stress tensor I δ ij, δ ij is Kroneckar delta 86

3 S. Sreekanth et al Adv. Appl. Sci. Res.,, (5:85-96 d d ij, d ij (/ (W i, j + W j, i deforation rate tensor b b ij, b ij a i, j + a j, i + W, i W, j viscoelastic paraeter with Wi ai + W i, j W j acceleration vector t The ø, ø, ø 3 are coefficients of viscosit, visco-elasticit and cross-viscosit respectivel and these are in general functions of teperature, aterial properties and invariants of d, b, d. For an liquids aqueous solutions of polacralaid and polbutlene, the coefficients ø, ø, ø 3 a be taken as constant. 3. Forulation and solution of the proble: We consider the flow of theral and ass diffusion on viscoelastic (Rivlin-Ericksen fluid between two infinite parallel plates oving in opposite direction, while one place is oscillating in tie and agnitude about a constant non-zero ean, under the influence of a unifor transverse agnetic field. The x-axis is taken along an infinite flat plate oving verticall upwards and a straight line perpendicular to that as -axis. The agnetic field of sall intensit H o is introduced in the -direction. Since the fluid is slightl conducting, the agnetic Renolds nuber is uch less than unit and hence the induced agnetic field is neglected in coparison with the applied agnetic field (Sparrow and Cess [5]. In the absence of an input electric field, the equations governing the otion of the conducting Rivlin-Ericksen fluid are given b u u u u u σµ e H + v gβ *( T Ts + gβ **( C Cs + υ + β + v u t t ρ v p + ( β + γ u u t ρ v + v t ρc T T K T P C C d C + v D t (3. (3. (3.3 (3. (3.5 where u, v are the velocit coponents along the axes of coponents along the axes of coordinates respectivel, g the acceleration due to gravit, β* the coefficient of theral expansion, β** the voluetric coefficient of expansion with concentration, T the teperature of the plate, T s the teperature at second plate, υ the kineatic viscosit, β the visco- elasticit, t the tie, σ the electrical conductivit of the fluid, µ e the agnetic pereabilit,ρ the densit of the fluid, P the fluid pressure, γ the kineatic cross viscosit, C P the specific heat at constant pressure and K the theral conductivit of the fluid. The continuit equation (3.3 shows that v is a function of tie onl. In order to obtain a stead state solution of the boundar laer tpe it is known that v ust be a negative non-zero constant v.in the unstead case also we shall ake this restriction (Stuart [6]. The boundar conditions are u U, T T w, C C w at (3.6a 87

4 S. Sreekanth et al Adv. Appl. Sci. Res.,, (5:85-96 ( i ω t u U t U + e, T T C C at d (3.6b S, Fro the continuit equation, we get (3.3 v + v (3.7 We now introduce the following non-diensional quantities * v dv * v t * υ w,, t, w υ υ υ v (3.8 * u * v * T Ts * C CS u, v, θ, C U v T T C C w s W S S In view of the equation (3.8, the equation (3. to (3.5 reduce to (dropping the super scripts * u u u u G u + rθ + GcC + S Mu + t t (3.9 P P + θ t (3. r r S C + S C C c t c where P r µc p / K S c υ /D G r [vgβ* (T w -T s ] / (U v Prandtl nuber Schidt nuber Grashof nuber βv S Visco-elastic paraeter υ σµ H v e M Magnetic paraeter ρ v ( ω S ( Gc υgβ ** C C / U v Modified Grashof nuber (3. S - S is the visco-elastic paraeters S for Rivlin- Ericksen second order fluid the visco-elasticit paraeter S is necessaril negative. The non diensional boundar conditions are u, θ, C at (3.a i t + e ω, θ, C at (3.b u - Substituting equations (3.3 to (3.5 in to (3.9 to (3. and coparing the haronic and nonharonic ter we obtain 88

5 S. Sreekanth et al Adv. Appl. Sci. Res.,, (5:85-96 C C e e Sc e S u u + u + Mu G + G C sc sc ( 3.6 rθ c ( 3.7 ( ω / ( ω / rθ c ( 3.8 ( i '" " ' S u + S i u + u + M + i u G + G C "' " ' θ P θ 3.9 " ' r θ P θ ω / P θ 3. " ' r r In view of (3.3 to (3.5, the boundar conditions (3. reduces to u, u, θ, θ, C, C at (3.a u -, u -, θ, θ, C, C at (3.b Following Lighthill [8] we assue the solution in the for u u + S u + S ( ( 3. ( S ( u u + S u + S a θ θ + S θ + θ θ + S θ + S 3.b Substituting (3. in equations (3.7 to (3. and equating the coefficients of S, we (neglecting the ters of ( S and higher ter. " ' Gc sc sc u + u + Mu Grθ + ( e e ( 3.3 s c e "' '' ' u u + u + Mu G θ 3. r u u T u G θ " '' r ( 3.5 " iw " '' ' u + u u + u + Tu Grθ " ' θ Pθ 3.7 r " ' r ( 3.6 θ Pθ 3.8 " ' iw θ Pr θ Pr θ 3.9 " ' iw θ Pr θ Pr θ 3.3 obtain where T M + ( iω / In view of (3. the boundar conditions (3. reduce to u, u, θ, θ, u, u, θ, θ at 3.3a u, u, θ, θ, u, u, θ, θ at 3.3b 89

6 S. Sreekanth et al Adv. Appl. Sci. Res.,, (5:85-96 Solving equations (3.3 to (3.3 b using the boundar condition (3.3, we obtain the velocit and the teperature distributions as (, + ( ω ω ( 3.3 u t u M Cos t M Sin t where M + im u ( r θ θ Pr e i pr pr ( ( e e The equations for u and u are TY 5 r r C 5 + T Y 6 + P Y S Y i u T e T e T e T e T A Y AY u ( e ( K3 Cos ( b / K Sin ( b / e ( K5 Cos ( b / + K6 Sin ( b / A Y AY + i [ e ( K Cos ( b / + K Sin ( b / + e ( K Cos ( b / + K Sin ( b / ] ( The various constants are given in Appendix before the reference section of the text. Transient velocit can be deduced fro equation (3.3, when ωt π /, u u M (3.3 i Skin Friction The non-diensional for of the shear stress at the plates is τ u u iωt + e iωt A ( τ e e ( K7Sin( b / K8Cos ( b / τ + + A ( 9 ( / 3 ( / { ( 8 ( / A e K Sin b K Cos b i e K Sin b + + A + K7Cos b / + e ( K3Sin b / K9Cos b / } where ean shear stress is T T5 Pr τ TT5 e + T5T 6e + T7 Pr e + T8 Sce Sc (3.35 (3.36 (3.37 The aplitude of the skin-friction at the plates is given b + i r i A ( 7 ( / 8 ( / ( 7 ( / 8 ( / + + A e K Sin b K Cos b e K Sin b K Cos b + i { e ( K Sin b / + K Cos b / + e ( K Sin b / K Cos b / } ] A A where K A K + K b / K A K K b / ( / ( / K K b A K K K b + A K (3.38 9

7 S. Sreekanth et al Adv. Appl. Sci. Res.,, (5:85-96 The phase of the skin friction is Tan α i / r CONCLUSION In fig., the fluctuating part of the velocit profile M i is drawn against for different values of agnetic paraeter M. We have observed that M i increases with the increase in M. Further we have also seen that M i increases with the increase in. Fro table, we conclude that the fluctuating part of the velocit profile M r decreases with the increase in M. In fig., we have drawn M i against for different values of visco-elastic paraeter S.We have noticed that M i increases with the increase in S. Fro table, we conclude that M r increases with the increase in S. Further we have also seen that M r first increases as increases up to and then the trend gets reversed. In figures (3, (, (5, (6, (7 and table (3 velocit distribution u is drawn against for different values of agnetic paraeter M, Grashof nuber G r, odified Grashof nuber G c, Schidt nuber S c, Prandtl nuber P r and visco-elastic paraeter S respectivel. We have noticed that u increases first and then decreases with the increases in. We have also seen that the velocit increases with the increase in G r or G c or S c or P r or S, whereas the velocit decreases with the increase in M. Fro fig.8, we conclude that the teperature distribution profile T increases with the increase in P r. Further we noticed that T decreases with the increase in. In fig.9, the concentration distribution profile C is drawn against for different values of S c.we have observed that C increases with the increase in S c, where as it decreases with increase in. In fig., the aplitude of the skin friction is drawn against S for different values of M. We have seen that increases with the increase in M, whereas it decreases with the increase in S.in fig., the phase lead of the skin friction Tan α at the plate is drawn against S for different values of M. We have noticed that Tan α decreases with the increase in M, whereas it increases with the increase in S 5. GRAPHS 9

8 S. Sreekanth et al Adv. Appl. Sci. Res.,, (5:

9 S. Sreekanth et al Adv. Appl. Sci. Res.,, (5:85-96 TALE M r against for different M M E-9.573E E E E-.356E E E E E E E E-.3889E-8.875E E E-.7978E E E- -.5 TALE M r against for different S S E-9.587E E E E-9.567E E E E-9.388E E E E-9.89E E E E-9.573E E E

10 S. Sreekanth et al Adv. Appl. Sci. Res.,, (5:85-96 TALE 3 u against for different S S REFERENCES [] Abbas, Z. Wang, Y, Haat, T. and Oberlack, M. International Journal of Non-Linear Mechanics, 3(8, 8, 783. doi:.6/j.ijnonlinec.8..9 [] Agarwal, A.K. and Kishore,. Ind.J.Technol., 6, 998, -. [3] Agarwal, H.L., Ra, P.C. and Singh, S.S. Can. J. Che.Engg.,58, 98, 3 [] Asghar, S., Mohuddin, M.R., Haat. T., Siddiqui A.M.,,, [5] Georgatopoulos, G.A., Coullies, J., Gouda, C.L. and Cougenis,C. Astrophs Space Sci. 7, 98, 357 [6] Johri, A.K. Indian J Pure and Appl. Maths, 9(5, 978, 8. [7] Krishnanlal, proceedings of the 3 th Congress on theoretical and applied echanics, Durgapur, 968. [8] Lighthill, M.J.Proc. Roal. Soc., A, 95,. [9] Ogulu, A. Prakash, J. Phs. Scr. 7, 6, [] Raptis, A.A. Astrophs Space Sci, 86, 98, 3. [] Reiner, M.Quart. J. Mech. and Appl. Maths, 6, 956, 6. [] Rivlir, R.S. Proc. of Roal Soc., 93A, 98,6. [3] Shastri and hadara. Appl.Sci. Res., 3,, 978, 7. [] Soundalgekar, V.M. Proc.Ind. Acad. Sci., 86A,, 977, 37. [5] Sparrow, E.M. and Cess, R.D. Trans. ASME, J. Appl. Mech. 9(, 96, 8. [6] Stuart. Proc. of Roal Soc. 3A, 955, 6. [7] Vajravelu, K. Ind.J.Technol. 6, 978, 397. Appendix: The list of the constants appearing in the text. Gr Gc T, T P 3 r Pr e e T T + + M + M, T T T3, T P P M S S M r r c c Pr Sc C T T ( T e T e T M T T M , 9 C T T P T T M P P M r, T 5 5 r r T 9

11 S. Sreekanth et al Adv. Appl. Sci. Res.,, (5:85-96 T S T T C T T C T 3 7 c, 3 + 9, + Sc Sc M T C + S T, T C + S T, T T + S T T8 T7 + ST Pr c C T6 e e + T T 7 e e + T8 T5 e e e + e T T5 ( e e T 5 T5 S T5 T5 P r T Sc T T T ( ( C T6 e e + T7 e e + T8 e + e T T5 ( e e T 5 T T5 P T r 5 Sc C3 T9 e e + T e e + T e e T T P T S T C T e e + T e e + T e e 5 r c T T5 ( e e A A A /, A / T e e Cos b T e e Sin b T 9 9 T9 + T T9 + T T, T L + M T a L + L + W b L + W L A + a /, A a /, A 3 b /, A 3 b /, 3 ( ω A b /8, A A A A ba /, ( ω ( ω ( ω ( ω / ( ω /6, A A A A + ba /, A A A A + A / b /6, A A A A + A + b A T A T A A T A + T A, A T A + T A, A T A T A, A A ( A b /, A5 ( A M / ( A, A6 ω / ( A ( A b ω A7 + A + M A M ( A 6 ω ( M A b A8, A9 ( A ( A7 + A8 A A b A, A, A A A 5 6 A7 + A8 A7 + A8 A A M ω, A ( 3 A A 95

12 S. Sreekanth et al Adv. Appl. Sci. Res.,, (5:85-96 ( A b ω A5 + A + M A M ( A 6 ω ( M A b A6, A7 A ( A ( A5 + A6 A, A A5 + A6 A5 + A K K T + K T 3 9 K K T K T K K 9 K T T9 + T K T K T + T K T9 + T K K K + K 7 5 K K + K K K K K K K K 7 K K K 6 K K K 5 K T + S K A 3 o K T S K o 9 K T S K 5 o K T S K 6 o K 5 K K K K 5 7 K K K + 8 K K K 9 K K + K 3 K K + K 5 8 K K K