EFFECTS OF MASS TRANSFER ON MHD FLOW OF CASSON FLUID WITH CHEMICAL REACTION AND SUCTION

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1 Brazilian Journal of Cheical Engineering ISSN Printed in Brazil Vol. 3, No. 1, pp , January - March, 13 EFFECTS OF MASS TRANSFER ON MHD FLOW OF CASSON FLUID WITH CHEMICAL REACTION AND SUCTION S. A. Shehzad 1*, T. Hayat 1, M. Qasi and S. Asghar 1 Departent of Maeatics, Quaid-i-Aza University 453, Islaabad 44, Pakistan. E-ail: ali_qau7@yahoo.co Departent of Maeatics, COMSATS Institute of Inforation Technology, Park Road, Chak Shahzad, Islaabad 44, Pakistan. (Subitted: January 5, 1 ; Revised: May 9, 1 ; Accepted: May 14, 1) Abstract - Effect of ass transfer in e agnetohydrodynaic flow of a Casson fluid over a porous stretching sheet is addressed in e presence of a cheical reaction. A series solution for e resulting nonlinear flow is coputed. The skin friction coefficient and local Sherwood nuber are analyzed rough nuerical values for various paraeters of interest. The velocity and concentration fields are illustrated for several pertinent flow paraeters. We observed at e Casson paraeter and Hartan nuber have siilar effects on e velocity in a qualitative sense. We furer analyzed at e concentration profile decreases rapidly in coparison to e fluid velocity when we increased e values of e suction paraeter. Keywords: Casson fluid; Mass transfer; Cheical reaction. INTRODUCTION The analysis of boundary layer flow of viscous and non-newtonian fluids has been e focus of etensive research by various scientists due to its iportance in continuous casting, glass blowing, paper production, polyer etrusion, aerodynaic etrusion of plastic sheet and several oers. Nuerous studies have been presented on various aspects of stretching flows since e seinal work by Crane (197). One ay refer to recent investigations by Hayat and Qasi (1), Fang et al. (1), Khan and Pop (1), Ahad and Asghar (11), Kandasay et al. (11), Rashidi et al. (11), Hayat et al. (11), Yao et al. (11) and Makinde and Aziz (11) in is direction. On e oer hand, ass transfer is iportant due to its appearance in any scientific disciplines at involve convective transfer of atos and olecules. Eaples of is phenoenon are evaporation of water, separation of cheicals in distillation processes, natural or artificial sources etc. In addition, ass transfer wi cheical reaction has special significance in cheical and hydroetallurgical industries. The foration of sog represents a first order hoogeneous cheical reaction. For instance, one can take into account e eission of NO fro autoobiles and oer soke-stacks. Thus, NO reacts cheically in e atosphere wi unburned hydrocarbons (aided by sunlight) and produces peroyacetylnitrate, which fors a layer of photocheical sog. Cheical reactions can be treated as eier hoogeneous or heterogeneous processes. It depends on wheer ey occur at an interface or as a single-phase volue reaction (Kandasay et al., 8). A few representative studies dealing wi ass transfer in e presence of cheical reaction ay be entioned (Kandasay et al., 5; Hayat et al., 1; Ziabaksh et al., 1; Makinde, 1; Ibrahi and Makinde, 1; Bhattacharyya and Layek, 11; Hayat et al., 11; Makinde, 11). Previous studies on e topic show at little work *To who correspondence should be addressed

2 188 S. A. Shehzad, T. Hayat, M. Qasi and S. Asghar is presented regarding e effect of ass transfer on e MHD flows of non-newtonian fluids in e presence of cheical reaction. Constitutive equations of e Casson fluid odel (Nakaura and Sawada, 1988; Eldabe and Silwa, 1995; Dash et al., 1996; Boyd et al., 7) are eployed in e aeatical odeling. The rest of e paper is organized as follows. The net section copletes e proble forulation. Then, e net section develops e hootopic solutions. Convergence of e derived series solutions and e discussion of velocity and concentration fields are presented in e sequence. The last section suarizes ain points. GOVERNING PROBLEMS Consider a agnetohydrodynaic (MHD) and incopressible flow of a Casson fluid over a porous stretching surface at y=, as shown in Figure 1. We select e Cartesian coordinate syste such at e ais be taken parallel to e surface and y is perpendicular to e surface. The fluid occupies a half space y>. The ass transfer phenoenon wi cheical reaction is also retained. The flow is subjected to a constant applied agnetic field B in e y direction. The flow is taken to be steady and e agnetic Reynolds nuber is considered to be very sall so at e induced agnetic field is negligible in coparison to e applied agnetic field. The fluid properties are constant. C y-ai In e above equation π = ee ij ij and e ij denotes e (i, j) coponent of e deforation rate, π e product of e coponent of deforation rate wi itself, π c a critical value of is product based on e non-newtonian odel, μ B e plastic dynaic viscosity of e non-newtonian fluid and p y e yield stress of e fluid. The equations governing e steady boundary layer flow of e Casson fluid are (Mustafa et al., 1) u v + =, y u u u σb u + v = ν (1+ 1/ β) u, y y ρ C C C u + v = D k 1C y y along wi e following boundary conditions: () (3) (4) u = u w() = c, v = v, C = Cw at y =, (5) u, C C as y (6) in which u and v represent e velocity coponents in e and y directions, β=μb π c /py e non-newtonian Casson paraeter, ν= ( μb / ρ ) e kineatic viscosity, D e ass diffusion, C e concentration field and k 1 e reaction rate. Equations ()-(6) can be ade diensionless by introducing e following change of variables u w () = c, C w B u w () = c, -ai u = cf ( η ), v = cνf( η), c C C η= y, φ=. ν C C w (7) Figure 1: Physical sketch of e proble The rheological equation of state for an isotropic flow of a Casson fluid can be epressed as (Eldabe and Silwa, 1995): μ + π>π py ( B )e ij, c, π τ ij = p y ( μ B + )e ij, π< π π c (1) The diensionless proble satisfies: (1+ 1 / β )f + ff f Mf =, (8) φ + Scfφ Scγφ =, (9) f = S, f = 1, φ = 1 at η =, (1) f =, φ= as η, (11) Brazilian Journal of Cheical Engineering

3 Effects of Mass Transfer on MHD Flow of Casson Fluid wi Cheical Reaction and Suction 189 where Eq. () is satisfied identically, M = σb / ρ c e Hartan nuber, Sc =ν / D e Schidt nuber, γ= k 1 /c e cheical reaction paraeter and S= v / ν c e suction paraeter. The skin friction coefficient and e local Sherwood nuber can be written as: τ jw C =, Sh =, u () D(C C ) w f ρ w w (1) in which τ w is e skin friction (or shear stress along e stretching surface) and j w e ass flu fro e surface, defined by e following relations: py u C τ w = μ B+, jw D. c y = y (13) π y= y= Now Eqs. (1) and (13) give: Re C = (1 + 1/ β)f (), f Sh / Re = φ (). (14) ( ) L ˆ φ 1 p ( ;p) ( ) φ η θ η = p f( ˆ ;p), ˆ φn φ η φ( η,p), ˆ ˆ f (;p) S, f (;p) 1, f ˆ = = ( ;p) =, φ ˆ(,p) = 1, φˆ(,p) =, (19) () where p is an ebedding paraeter; N f and N φ are nonlinear operators which can be defined as: N 3 ˆ η ˆ η f [f ( η,p)] = (1 + 1/ β ) + f ( η,p) 3 N [ ˆ(,p),f( ˆ φ φη η,p)] = f( ˆ,p) f( ˆ,p) ˆf( η,p) f( η,p) M, φη ˆ(,p) ˆ ˆ φη (,p) + Scf( η,p) Sc γφˆ( η,p). By setting p = and p = 1 we have: (1) () HOMOTOPY ANALYSIS SOLUTIONS The initial guesses and auiliary linear operators for is proble are selected as follows: f( ˆ η ;) = f ( η), φˆ( η,) =φ ( η) and f( ˆ η ;1) = f( η), φˆ( η,1) =φ( η). (3) f ( η ) = S + 1 ep( η), φ( η ) = ep( η ), (15) Lf = f f, L φ = f f, (16) such at: η η f 1 3 L (C + C e + C e ) =, φ η η 4 5 L (C e + C e ) =, (17) where C i (i= 1 5) represent e arbitrary constants. Denoting e nonzero auiliary paraeters f and φ, e resulting zero order probles are developed as follows: ( ) L ˆ f 1 p f( ;p) f ( ) η η = p ˆ fn f f( η;p), (18) We observed at, when p changes fro to 1, en f ( η,p) and φη (,p) vary fro f(), η φ () η to f( η ) and φη ( ). In view of e Taylor series we can write: (4) = 1 f( η,p) = f ( η ) + f ( η)p, (5) = 1 φη (,p) =φ( η ) + φ ( η)p, 1 f( η;p) f ( η ) =,! p= 1 φ( η;p) φ( η ) =.! p= (6a) (6b) Brazilian Journal of Cheical Engineering Vol. 3, No. 1, pp , January - March, 13

4 19 S. A. Shehzad, T. Hayat, M. Qasi and S. Asghar The convergence of e series is strongly dependent upon f and φ.we select f and φ in such a way at e series converge at p = 1 and hence: f( η ) = f ( η ) + f ( η ), (7) = 1 φη ( ) =φ( η ) + φ ( η ). (8) = 1 The -order deforation equations are obtained by differentiating e Equations (18)-() ties wi respect to p and en putting p= to obtain: approiation in order to find e range of adissible values of f and φ. Fig. shows at e range of adissible values of f and φ are.7 f.1 and.8 φ.3. The series solutions converge in e whole region of η when f = φ =.5. Table 1 shows e convergence of our series solutions for different orders of approiation. It is very clear fro is table at 1 order deforations are enough for e velocity whereas15 order deforations are required for e concentration. f 1 f f L [f ( η) χ f ( η )] = R ( η), (9) φ 1 φ φ L [ φ ( η) χ φ ( η )] = R ( η), (3) f () = f () = f ( ) =, φ () = φ ( ) =, (31) R f 1 ( η ) = (1+ 1/ β)f ( η) 1 + f 1 kfk f 1 kf k Mf 1( η), k= 1 φ η =φ 1+ 1 kφk γφ 1 k= (3) R ( ) Sc f Sc, (33) χ, 1, = 1, > 1. Our general solutions can be epressed in e for: η η 1 3 (34) f ( η ) = f ( η ) + C + C e + C e, (35) η η 4 5 φ ( η ) =φ ( η ) + C e + C e, (36) in which f and φ represent e special solutions. CONVERGENCE ANALYSIS The developed series solutions Eqs. (4) and (5) contain f and φ.the convergence and rate of approiation for e constructed series solutions depend upon ese auiliary paraeters. Therefore e curves have been plotted for e -order of Figure : curves for e functions β and θ Table 1: Convergence of e hootopy solution for different orders of approiation when β =.6,M =.5, S =.5, Sc =.6, γ =.3, and f = =-.5. θ Order of approiation -f () φ () RESULTS AND DISCUSSION The velocity ( f ) and concentration ( φ ) fields are shown graphically in Figs Figs. 3-5 show e effects of e Casson paraeter β, Hartan nuber M and e suction paraeter S, respectively, on e velocity profile f ( η ). Fro Fig. 3, we observed at e velocity field decreases when β increases. An increase in β leads to an increase in plastic dynaic viscosity at creates resistance in e flow of fluid and a decrease in fluid velocity is observed. The Brazilian Journal of Cheical Engineering

5 Effects of Mass Transfer on MHD Flow of Casson Fluid wi Cheical Reaction and Suction 191 effects of Hartan nuber M and e suction paraeter S on f( η ) are seen in Figs. 4 and 5. These figs. show at bo M and S decrease e velocity f( η ). This is due to e fact at e applied agnetic field noral to e flow direction induces e drag in ters of a Lorentz force which provides resistance to flow; suction is an agent which causes resistance to e fluid flow and e fluid velocity also decreased. Figs. 6-1 show e plots of e effects of e Casson paraeter β, Hartan nuber M, suction paraeter S, Schidt nuber Sc and cheical reaction paraeter γ on e concentration field φ( η ). The concentration field and associated boundary layer ickness increase when β increases (Fig. 6). It is also noticed fro Figs. 3 and 6 at e Casson paraeter β has quite opposite effects on e velocity and concentration profiles. Fig. 7 depicts at, by increasing e Hartan nuber, bo e concentration profiles and boundary layer ickness increase. Thus, Hartan nuber here decreases e resistive force when M increases. The influence of e suction paraeter on e concentration profile is seen in Fig. 8. The concentration profile is a decreasing function of S.This is in accordance wi e fact at e fluid eperiences a resistance upon increasing e friction between its layers. As a consequence, ere is a decrease in concentration. Effects of e Schidt nuber on φη ( ) are displayed in Fig. 9. Here bo e concentration profile and e boundary layer ickness decrease when e Schidt nuber Sc increases. Fro a physical point of view, e Schidt nuber is dependent on ass diffusion D and an increase in Schidt nuber corresponds to a decrease in ass diffusion and e concentration profile reduced. When γ=, ere is no cheical reaction. An increase in e cheical reaction paraeter corresponds to an increase in e reaction rate paraeter and an increase in e reaction rate paraeter caused a reduction in concentrarion. Fro Fig. 1, one can see at an increase in e value of e cheical reaction paraeter γ decreased e concentration field φ( η ). Figs. 11 and 1 are sketched to visualise e influence of key paraeters at are used in e present probles for Re C f. The influence of M against β is described in Fig. 11. It is obvious at Re C f is an increasing function of M. Siilar effects can be seen in Fig. 1, which shows e influence of β against M. Figs. 13 and 14 are shown to present e influence of sundry paraeters on Sh / Re. Fig. 13 describes e influence of β vs γ on Sh / Re. This figure confirs at e Sherwood nuber is a decreasing function of β and e effects on e Sherwood nuber of γ are e opposite (see Fig. 14). Table shows e skin friction coefficient for e different values of β, M, and S. By increasing e values of β, e value of e skin friction coefficient decreases, but it increases upon increasing M and S Table 3 shows e nuerical values of e local Sherwood nubers for e paraeters β, M, S, Sc, and γ. This table concludes at e values of e local Nusselt nuber decrease upon increasing β and M, but increase upon increasing S, Sc and γ. Table 4 shows e coparison wi e previous liited studies in e literature. Fro is table one can see at our series solutions are in ecellent agreeent wi e previous studies, validating e present series solutions. Figure 3: Influence of β on f ( η ) Figure 4: Influence of M on f ( η ) Brazilian Journal of Cheical Engineering Vol. 3, No. 1, pp , January - March, 13

6 19 S. A. Shehzad, T. Hayat, M. Qasi and S. Asghar Figure 5: Influence of S on f( η ) Figure 6: Influence of β on φη ( ) Figure 7: Influence of M on φη ( ) Figure 8: Influence of S on φη ( ) Figure 9: Influence of Sc on φη ( ) Figure 1: Influence of γ on φη ( ) Figure 11: Influence of M on Re C vs β Figure 1: Influence of β on Re C vs M. f f Brazilian Journal of Cheical Engineering

7 Effects of Mass Transfer on MHD Flow of Casson Fluid wi Cheical Reaction and Suction 193 Figure 13: Influence of β on Sh / Re vs γ. Figure 14: Influence of γ on Sh / Re vs β Table : Nuerical values of e skin-friction coefficient (1+1 / β)f () for different values of β, M and S. β M S -(1+1 / β )f () Table 3: Nuerical values of e local Sherwood nuber - φ () for different values of β, M, S, Sc and γ. β M S Sc γ - φ () Table 4: Coparison of values of - φ () for different values of γ when β Brazilian Journal of Cheical Engineering Vol. 3, No. 1, pp , January - March, 13 and M = S =.. γ Sc Salee ans El Aziz Andersson et al. Present results (8) (1994)

8 194 S. A. Shehzad, T. Hayat, M. Qasi and S. Asghar CONCLUSIONS Effects of ass transfer on e MHD boundary layer flow of a Casson fluid odel wi cheical reaction are addressed. The present analysis leads to e following observations. The Casson paraeter β and Hartan nuber M have siilar effects on e velocity profile f( η ). β has opposite effects on e velocity and concentration profiles. The concentration field φη ( ) as well as e boundary layer ickness increase upon increasing e Hartan nuber M. An increase in e Schidt nuber Sc causes a decrease in e concentration profile and e boundary layer ickness. When γ =, ere is no cheical reaction. An increase in γ decreases φη ( ). REFERENCES Ahad, A. and Asghar, S., Flow of a second grade fluid over a sheet stretching wi arbitrary velocities subject to a transverse agnetic field. Applied Ma. Letters 4, p. 195 (11). Andersson, H. I., Hansen, O. R. and Holedal, B., Diffusion of a cheically reactive species fro a stretching sheet. Int. J. Heat Mass Transfer, 37, p. 659 (1994). Bhattacharyya, K. and Layek, G. C., Slip effect on diffusion of cheically reactive species in boundary layer flow over a vertical stretching sheet wi suction or blowing. Cheical Eng. Coun., 198, p (11). Boyd, J., Buick, J. M. and Green, S., Analysis of e Casson and Carreau-Yasuda non-newtonian blood odels in steady and oscillatory flow using e lattice Boltzann eod. Phys. Fluids, 19, p. 93 (7). Crane, L. J., Flow past a stretching plate. Z. Angew. Ma. Mech., 1, p. 645 (197). Dash, R. K., Mehta, R. K. and Jayaraa, G., Casson fluid flow in a pipe filled wi a hoogenous porous ediu. Int. J. Eng. Sci., 34, p (1996). Eldabe, N. T. M. and Salwa, M. G. E., Heat transfer of MHD non-newtonian Casson fluid flow between two rotating cylinders. J. Phys. Soc. Japan, 64, p. 41 (1995). Fang, T., Zhang, J. and Yao, S., A new faily of unsteady boundary layers over a stretching surface. Appl. Ma. Coput., 17, p (1). Hayat, T. and Qasi, M., Influence of eral radiation and Joule heating on MHD flow of a Mawell fluid in e presence of erophoresis. Int. J. Heat Mass Transfer, 53, p. 478 (1). Hayat, T., Qasi, M., Abbas, Z. and Hendi, A. A., Magnetohydrodynaic flow and ass transfer of a Jeffery fluid over a nonlinear stretching surface. Z. Naturforsch A, 64a, p (1). Hayat, T., Shehzad, S. A. and Qasi, M., Mied convection flow of a icropolar fluid wi radiation and cheical reaction. Int. J. Nu. Meods Fluids, 67, p (11). Hayat, T., Shehzad, S. A., Qasi, M. and Obaidat, S., Steady flow of Mawell fluid wi convective boundary conditions. Z. Naturforsch., 66a, p. 417 (11). Ibrahi, S. Y. and Makinde, O. D., Cheically reacting MHD boundary layer flow of heat and ass transfer past a oving vertical plate wi suction. Scientific Research Essays, 5, p. 875 (1). Kandasay, R., Loganaan, P. and Arasu, P. V., Scaling group transforation for MHD boundarylayer flow of a nanofluid past a vertical stretching surface in e presence of suction/injection. Nuclear Engineering and Design, 41, p. 53 (11). Muhaiin, I., Kandasay, R., Hashi, I., Therophoresis and cheical reaction effects on non-darcy MHD ied convective heat and ass transfer past a porous wedge in e presence of variable strea condition. Che. Eng. Research Design, 87, p. 157 (9). Kandasay, R., Periasay, K. and Prabhu, K. K. S., Cheical reaction, heat and ass transfer on MHD flow over a vertical stretching surface wi heat source and eral stratification effects. Int. J. Heat Mass Transfer, 48, p (5). Khan, W. A. and Pop, I., Boundary flow of a nanofluid past a stretching sheet. Int. J. Heat Mass Transfer, 53, p. 477 (1). Makinde, O. D. and Aziz, A., Boundary layer flow of a nano fluid past a stretching sheet wi convective boundary conditions. Int. J. Theral. Sci., 5, p. 136 (11). Makinde, O. D., MHD ied-convection interaction wi eral radiation and n order cheical reaction past a vertical porous plate ebedded in a porous ediu. Che. Eng. Coun., 198, p. 59 (11). Makinde, O. D., Siilarity solution of hydroagnetic heat and ass transfer over a vertical plate wi a convective surface boundary condition. Int. J. Physical Sci., 5, p. 7 (1). Brazilian Journal of Cheical Engineering

9 Effects of Mass Transfer on MHD Flow of Casson Fluid wi Cheical Reaction and Suction 195 Mustafa, M., Hayat, T., Pop, I. and Hendi, A. A., Stagnation-point flow and heat transfer of a Casson fluid towards a stretching sheet. Z. Naturforsch., 67a, p.7 (1). Nakaura, M. and Sawada, T., Nuerical study on e flow of a non-newtonian fluid rough an aisyetric stenosis. ASME J. Bioech. Eng., 11, p.137 (1988). Rashidi, M. M., Mohianian pour, S. A. and Abbasbandy, S., Analytic approiate solutions for heat transfer of a icropolar fluid rough a porous ediu wi radiation. Coun. Nonlinear Sci. Nuer. Siulat., 16, p (11). Sale, A. M. and El-Aziz, M. A., Effect of Hall currents and cheical reaction on hydroagnetic flow of a stretching vertical surface wi internal heat generation/absorption. Appl. Maeatical Modelling, 3, p. 136 (8). Yao, S., Fang, T. and Zhong, Y., Heat transfer of a generalized stretching/shrinking wall proble wi convective boundary conditions. Coun. Nonlinear Sci. Nuer. Siulat., 16, p. 75 (11). Ziabakhsh, Z., Doairry, G., Bararnia, H. and Babazadeh, H., Analytical solution of flow and diffusion of cheically reactive species over a nonlinearly stretching sheet iersed in a porous ediu. J. Taiwan Institute Cheical Eng., 41, p. (1). Brazilian Journal of Cheical Engineering Vol. 3, No. 1, pp , January - March, 13