Three-Dimensional MHD Flow of Casson Fluid in Porous Medium with Heat Generation

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1 Journal of Applied Fluid Mechanics, Vol. 9, No. 1, pp. 15-3, 016. Available online at ISSN , EISSN Three-Diensional MHD Flow of Casson Fluid in Porous Mediu with Heat Generation S. A. Shehzad 1, T. Hayat,3 and A. Alsaedi 3 1. Departent of Matheatics, Cosats Institute of Inforation Technology, Sahiwal 55000, Pakistan. Departent of Matheatics, Quaid-i-Aza University 4530, Islaabad 44000, Pakistan 3. Departent of Matheatics, Faculty of Science, King Abdulaziz University, P.O. Box 8057, Jeddah 1589, Saudi Arabia Corresponding Author Eail: ali_qau70@yahoo.co (Received Septeber 10, 014; accepted Noveber 17, 014) ABSTRACT The agnetohydrodynaic (MHD) three-diensional boundary layer flow of an incopressible Casson fluid in a porous ediu is investigated. Heat transfer characteristics are analyzed in the presence of heat generation/absorption. Laws of conservation of ass, oentu and energy are utilized. Results are coputed and analyzed for the velocities, teperature, skin-friction coefficients and local Nusselt nuber. Keywords: Three-diensional flow; Casson fluid; Stretching surface; Heat generation/absorption. NOMENCLATURE u, v, w velocity coponents * electrical conductivity Casson paraeter K pereability paraeter fluid density kineatic viscosity T teperature B 0 Applied agnetic field theral diffusivity Q heat generation paraeter c p specific heat T abient fluid teperature f, g diensionless velocity coponents diensionless teperature M Hartan nuber porosity paraeter ratio paraeter Pr Prandtl nuber B heat source/sink paraeter wx wall shear stress q w wall heat flux 1. INTRODUCTION The agnetohydrodynaic (MHD) flows of an electrically conducting fluid are encountered in any geophysical, astrophysical and engineering applications. Hydroagnetic flows have a key role in the fields of aeronautics, stellar and planetary agnetospheres. Magnetohydrodynaics concepts are utilized by the engineers in the design of heat exchangers, pups, theral protection, in space vehicle propulsion, control and re-entry, and in creating novel power-generating systes. The purification of olten etals fro non-etallic inclusions through the application of agnetic field is another iportant feature of MHD. All such applications of MHD give rise to investigate the probles which involves the agnetohydrodynaic effects. For exaple, Turkyilazoglu (011) studied MHD boundary layer flow of viscous fluid over a rotating sphere near an equator with heat transfer. He presented both nuerical and analytical solutions. MHD boundary layer flow of viscous fluid induced by an exponentially shrinking sheet was studied by Bhattacharyya and Pop (011). Makinde (01) exained MHD boundary layer flow of viscous fluid over flat surface with Newtonian heating and Navier slip. Hayat et al. (01a) developed the series solution for the MHD flow of an Oldroyd-B fluid passing through a porous channel. Rashidi and Mehr (014) considered proble for the series solutions of velocity and teperature. On the other hand the flows in porous edia have practical applications in heat reoval fro nuclear fuel

2 debris, underground disposal of radiative waste aterial, storage of food stuffs, paper production, oil exploration etc. Especially, the related boundary layer flows with heat transfer have received uch attention of the researchers in view of achieving industrial product of desired quality. The rates of cooling and stretching have key role in such situations. No doubt, the flow generated by the stretching of a flat surface has a great relevancy to the polyer extrusion. For exaple, the extrudite fro the die is generally drawn and siultaneously stretched into a surface in a elt spinning process, which is thereafter solidified through rapid quenching or gradual cooling by direct contact water or chilled etal rolls. Closed for solution for the boundary layer flow of viscous fluid over a stretching surface was firstly constructed by Crane (1970). He assued that the stretching surface possess a linear velocity with fixed distance fro the origin. Afterwards the Crane's proble has been studied extensively through different aspects (see few recent studies for two-diensional flows (Rashidi et al. (011), Javed et al. (011), Mukhopadhyay (01) and Hayat et al. (01b)). It is well known fact that the fluids appear in industrial and engineering processes are ostly non-newtonian fluids. There are aterials like drilling uds, sugar solution, certain oils, clay coating, colloidal and suspension solution, certain oils, lubricants etc. which fall into the category of non-newtonian fluids. The properties of such aterials cannot be explored by siple Navier- Stokes equations. According to the diverse characteristics of such aterials, different fluid odels are developed in the past like second grade fluid (Jail et al. (011)), third grade fluid (Abelan et al. (009)), fourth grade fluid (Hayat et al. (010)), Maxwell and Oldroyd-B fluids (Wang and Tan (011) and Jail et al. (014)), Burgers' fluid (Jail and Fetecau (010)), Jeffrey fluid (Hayat et al. (01c)), Eyring-Powell fluid (Hayat et al. (014a)), icropolar fluid (Rashidi et al. (011)), Walters' B fluid (Hayat et al. (014b)), Casson fluid (Shahohaadi (01)) etc. The fluid odel under consideration is Casson. This odel is plastic fluid odel that exhibits the characteristics of shear thinning that quantifies the yield stress and high shear viscosity. This fluid odel is a good candidate to explore the properties of biological aterials, foas, olten chocolate, cosetics, nail polish etc. Previous literature on the topic witnesses that little has been said yet about the three-diensional flows. There are only few attepts in this direction. For exaple, Wang (1984) investigated the three-diensional boundary layer flow of viscous fluid generated by linearly stretched surface. Hydroagnetic three-diensional free convection flow over a stretching surface was studied by Chakha (1999). Ariel (003) provided the hootopy perturbation solution of the proble of Wang (1984). Shehzad et al. (01) discussed the three-diensional boundary layer flow of Jeffrey fluid with convective conditions. The ai here is to develop a atheatical odel for threediensional flow of Casson fluid over a linearly stretching surface. The agnetohydrodynaic fluid fills the half space. Further, heat transfer effects are taken into account when heat generation/absorption effects are present. The governing nonlinear partial differential equations are converted into the ordinary differential equations by eploying suitable transforations. The resulting nonlinear is coputed by a newly developed odern technique naely the hootopy analysis ethod (Liao (003), Rashidi and Erfani (01), Hayat et al. (01b), Turkyilazoglu (01), Shehzad et al. (013), Shehzad et al. (014), Hayat et al. (014c,d) and Malvandi et al. (014a,b) (HAM). Results are plotted and displayed. The iportant observations of this study are listed in the conclusions.. MATHEMATICAL FORMULATION We consider three-diensional boundary layer flow of an incopressible Casson fluid in a porous ediu. The fluid is electrically conducting under the influence of a constant applied agnetic field B 0. In addition, the induced agnetic field is not considered because of sall agnetic Reynolds nuber. Physical properties of fluid are assued constants. Effects of viscous dissipation are neglected. Matheatical forulation is given in the presence of heat generation/absorption. We denote u, v and w the x, y and z coponents of velocity, T the teperature and T the abient teperature, respectively. The resulting boundary layer equations are u v w 0, x y z u u u 1 u u v w 1 x y z z B0 u u, K v v v 1 v u v w 1 x y z z B0 v v, K T T T T u v w x y z z Q ( T T ), cp (1) () (3) (4) where B c / py is the Casson fluid paraeter, the electrical conductivity, K the pereability paraeter, c p the specific heat, the theral diffusivity of the fluid, ( B / ) the kineatic viscosity, B the plastic dynaic viscosity of Casson fluid, the density of fluid 16

3 and Q the voluetric heat generation/absorption coefficient. The corresponding boundary conditions are: u ax, vby, w0, T Tw at z 0, (5) u 0, v0, T T as z, (6) where the fluid teperature of the wall is T w. Eploying the transforations (Wang (1984) and Shehzad et al. (01)): u axf ( ), v ayg( ), T T a w a ( f( ) g( )) ( ), z Tw T (7) one obtains 1 1 f ( f g) f 1 f M f 0, 1 1 g ( f g) g 1 g M g 0, (8) (9) Pr( f g) Pr B 0, (10) f 0, g 0, f 1, g, 0 at 0, (11) f0, g0, 0 as, (1) where Eq. (1) is satisfied autoatically, M B 0 / a the Hartan nuber, Ka the porosity paraeter, b is a ratio paraeter, a Pr the Prandtl nuber, B Q/ ac p the heat generation/absorption coefficient and prie depicts differentiation with respect to. If C fx and C fy are the skin-friction coefficients and Nu x is the local Nusselt nuber, then we have wy C wx fx, C, fy uw uw xqw Nux, kt ( w T ) (13) where w and q w are the wall shear stress and the wall heat flux, respectively. The above equation in diensionless for can be written as 1/ 1 Rex Cfx 1 f(0), 1/ 1 Rex Cfy 1 g (0), 1/ Re x Nux (0), (14) where the definition of Reynolds nuber is Re x uw( x) x/. 3. HOMOTOPY SOLUTIONS According to HAM, the functions f, g and by a set of base functions k { exp( n), k 0, n 0} can be expressed as k k f ( ) a, n exp( n), (15) n0k 0 k k g ( ) b, n exp( n), (16) n0k 0 k ( ) c, n k exp( n ), (17) n0k 0 k k k in where a n,, b n, and c n, are the coefficients. Initial approxiations and auxiliary linear operators are given by f0( ) 1 e, g0( ) 1 e, 0( ) e, Lf f f, Lg g g, L. (18) The operators have the following properties Lf ( C1Ce C3e ) 0, Lf ( C4 C5e C6e ) 0, L ( C7e C8e ) 0, (19) in which constants. C i ( i 1 8) show the arbitrary The zeroth order deforation probles can be obtained as: pl ˆ f f p f0 1 ( ; ) ( ) p ˆ f N f f( ; p), gˆ ( ; p), (0) p Lg gˆ p g0 1 ( ; ) ( ) p ˆ g N g f( ; p), gˆ ( ; p), (1) pl ˆ p 0 1 ( ; ) ( ) p ˆ( ; ), ˆ( ; ), ˆ N f p g p (, p), () 17

4 ˆ ˆ (0; ) 0, (0; ) 1, ˆ f p f p f ( ; p) 0, ˆ(0; ) 0, ˆ (0; ), ˆ g p g p g( ; p) 0, ˆ (0, p) 1, ˆ (, p) 0, 3 ˆ ˆ 1 f (, p) N f [ f(, p), gˆ (, p)] 1 3 ˆ f (, p ) ˆ ˆ (, ) ( f(, p) gˆ (, p)) f p fˆ(, p ), M 3 ˆ 1 gˆ (, p) N g [ gˆ (, p), f(, p)] 1 3 (3) (4) g ˆ(, p ) ˆ ˆ(, ) ( f(, p) gˆ (, p)) g p gˆ(, p) M, (5) ˆ ˆ ˆ(, p ) N [ (, p), f(, p), gˆ (, p)] ˆ ˆ(, p ) Pr( f (, p) gˆ (, p)) Pr Bˆ (, p). (6) In above equations, p is an ebedding paraeter, f, g and the non-zero auxiliary paraeters and N f, N g and Setting p 0 and p 1 one has fˆ ( ;0) f ˆ 0( ), (,0) 0( ) N the nonlinear operators. and fˆ ( ;1) f( ), ˆ (,1) ( ), (7) and when variation of p is taken fro 0 to 1 then f (, p), g(, p) and (, p) vary fro f 0 ( ), g0( ), 0( ) to f ( ), g( ) and ( ). Taylor's series expansion yields f (, p) f0( ) f( ) p, (8) 1 g(, p) g0( ) g( ) p, (9) 1 (, p) 0( ) ( ) p, (30) 1 1 f ( ; p) f ( ),! p 0 1 g( ; p) g ( ),! p 0 1 ( ; p) ( ).! p 0 Considering that f, g (31) Note that the convergence in the above series strongly depends upon f, g and. and are selected properly so that (8)-(30) converge at p 1. Therefore we have f( ) f0( ) f( ), (3) 1 g( ) g0( ) g( ), (33) 1 ( ) 0( ) ( ). (34) 1 We write the general solutions as follows f() f() C1Ce Ce 3, (35) g( ) g( ) C4 C5e C6e, (36) ( ) ( ) Ce 7 Ce 8, (37) where f, g and are the special solutions. 4. CONVERGENCE ANALYSIS AND DISCUSSION Eqs. (8)-(1) are solved by hootopy analysis ethod (HAM). HAM solution contains the auxiliary paraeters f, g and. These auxiliary paraeters f, g and can adjust and control the convergence of the constructed series solutions. In order to find the range of adissible values of f, g and, the curves are potrayed for 19th -order of approxiations. Fig. 1 clearly shows that the range of adissible values of f, g and are 1.15 f 0.05, 1. g 0.05 and The series converges in the whole region of for f g 0.7 and 0.9. Figs. and 3 are presented to analyze the influences of Hartan nuber M and Casson fluid paraeter on the velocities f ( ) and g ( ). Figs. a and b describe the effects of Hartan nuber on the fluid velocities f ( ) and g ( ). The fluid velocities and their associated oentu boundary layer thicknesses are reduced with an increase in Hartan nuber. Haran nuber has siilar effects for the velocities f ( ) and g ( ). An increase in the Hartan nuber leads to a stronger Lorentz force. 18

5 Stronger Lorentz force creates a resistance in the fluid flow that appears in the reduction of velocities (see Figs. a and b). Fig. 1. curves for the functions f, g and when M 0.6, 1.5, 0.5,.0, Pr 0.9 and B 0.3. Figs. 3a and 3b show that the larger values of Casson paraeter caused a decrease in the velocities f ( ) and g ( ). Fro these Figs. we analyzed that the fluid velocities and their associated oentu boundary layer thicknesses are decreasing functions of Casson paraeter. Fig. 3. Influence of on f ( ) and for g ( ) when M 0.6,.0 and 0.5. Fig.. Influence of M on f ( ) and for g ( ) when 1.5,.0 and 0.5. To see the influences of ratio paraeter, Prandtl nuber Pr, heat generation/absorption paraeter B and porosity paraeter on the teperature ( ), Figs. 4 and 5 are sketched. Fig. 4a shows that the teperature ( ) and theral boundary layer thickness are increasing functions of ratio paraeter. Fig. 4b presents the variations of Prandtl nuber on the teperature. In this Fig. we have seen that the teperature and theral boundary layer thickness are reduced for the larger Prandtl nuber. In fact the fluids with lower Prandtl nuber have higher theral diffusivity. Higher theral diffusivity gives rise to a decrease in teperature and lowers the theral boundary layer thickness. The role of Prandtl nuber is to control the rate of cooling in conducting fluid. Fig. 5a is sketched to analyze the variations of heat generation/absorption paraeter on the teperature. We have seen that an increase in B enhances the teperature and theral boundary layer thickness. When heat generation paraeter is increased, ore heat is produced in the fluid that causes to a higher teperature and stronger theral boundary layer thickness. Fig. 5b depicts the variation of porosity paraeter on the teperature. An increase in porosity paraeter shows a decrease in the teperature. 19

6 and lead a reduction in the skin-friction coefficients. Fro Figs. 9a and 9b, we observed that the local Nusselt nuber is an increasing function of, and Pr. Fig. 4. Influence of on ( ) when M 0.6, 1.5,.0, Pr 0.9 and B 0.4. Influence of Pr on ( ) when M 0.6, 0.5, 1.5,.0 and B 0.4. Figs. 6 and 7 are plotted to see the effects of different physical paraeters on the skin-friction coefficients 1 1/ f (0), 1 1/ g(0) and local Nusselt nuber (0). Fig. 6a shows the effects of Hartan nuber vs Casson paraeter on 1 1/ f (0). By increasing M and, the skin-friction coefficients increases. It can be seen for Fig. 6b that Hartan nuber and Casson paraeter have siilar effects on 1 1/ g (0). A coparison of Figs. 6a and 6b shows that the skin-friction coefficient 1 1/ f (0) at the wall are greater than the skin-friction coefficient 1 1/ g(0). A decrease in the local Nusselt nuber is noticed corresponds to the larger values of Hartan nuber vs Casson paraeter (see Fig. 7a). Fro Fig. 7b we see that an increase in heat generation/absorption paraeter leads to an increase in the local Nusselt nuber. Figs. 8a and 8b show the variations in skin-friction 1 1/ f (0) and coefficients 1 1/ g(0) for different values of and. Here we exained that the increasing values of Fig. 5. Influence of B on ( ) when M 0.6, 1.5,.0, 0.5 and Pr 0.9. Influence of on ( ) when M 0.6, 1.5, B 0.3, 0.5 and Pr 0.9. Table 1 is coputed to analyze the nuerical values of f (0), g (0) and (0) for the fixed values of involved paraeters. Here we have seen that the solutions for velocities converge fro 10th order of approxiation whereas the solution for teperature converges fro 40th order of deforations. Table is coputed for the coparison values of f (0), g (0), f ( ) and g( ) for different values of when and M 1/ 0. This Table shows that our solutions in liiting situations have an excellent agreeent with the previous study (Wang (1984). 5. CONCLUSIONS The ain results of present study can be suarized as follows: Hartan nuber M has quite opposite effects on the velocities and teperature. Casson fluid paraeter has siilar behavior for the velocities f ( ) and g ( ) in a qualitative sense. Teperature increases by increasing in heat generation paraeter B. 0

7 Increase in porosity paraeter decreases the fluid teperature. Skin-friction coefficient is increased for larger Casson fluid paraeter. Local Nusselt nuber is an increasing function of B vs. Fig. 7. Effects of M vs on (0) when 0.5,.0, B 0.3 and Pr 0.9. Effects B vs on (0) when M 0.6,.0, 1.5 and Pr 0.9. Fig. 6. Effects of M vs on 1 and for 1 1 g (0) when f (0) and Fig. 8. Effects of vs on 1 and for 1 1 g(0) when f (0) M and 1

8 Table 1 Convergence values of hootopic solutions for different order of deforations when 1.5,.0, M 0.6, 0.5, Pr 0.9, B 0.3, f g 0.7 and 0.9. Order of deforations f(0) g (0) (0) Fig. 9. Effects of vs on (0) when M 0.5, 1.5, B 0.3 and Pr 0.9. Effects vs Pr on (0) when M 0.5, 1.5 and B 0.3. REFERENCES Abelan, S., E. Mooniat and T. Hayat (009). Couette flow of a third grade fluid with rotating frae and slip condition. Nonlinear Analysis: Real World Appl. 10, Ariel, P. D (003). Generalized three diensional flow due to a stretching surface. Z. Angew. Math. Mech. 83, Bhattacharyya, K. and I. Pop (011). MHD boundary layer flow due to an exponentially shrinking sheet. Magnetohydrodynaics 47, Chakha, A. J. (1999). Hydroagnetic threediensional free convection on a vertical stretching surface with heat generation or absorption. Int. J. Heat Fluid Flow 0, Crane, L. J. (1970). Flow past a stretching plate. Z. Angew. Math. Physk. 1, Hayat, T., S. Asad and A. Alsaedi (014d). Flow of variable theral conductivity fluid due to inclined stretching cylinder with viscous dissipation and theral radiation. Appl. Math. Mech. 35, Table Nuerical values of f (0), g (0), f ( ) and g( ) for different values of when, and M f (0) f (0) Wang (1984) Present results g (0) g (0) f ( ) 1 f ( ) g( ) g( ) Hayat, T., S. Asad M. Mustafa and A. Alsaedi (014a). Radiation effects on the flow of Powell-Eyring fluid past an unsteady inclined stretching sheet with non-unifor heat source/sink. Plos One 9, e Hayat, T., S. Asad, M. Mustafa and H. H. Alsulai (014b). Heat transfer analysis in the flow of Walters'B fluid with a convective boundary condition. Chin. Phys. B 3, Hayat, T., R. Naz and M. Sajid (010). On the hootopy solution for Poiseuille flow of a fourth grade fluid. Coun. Nonlinear Sci. Nuer. Siulat. 15, Hayat, T., S. A. Shehzad and A. Alsaedi (01c). Soret and Dufour effects in agnetohydrodynaic (MHD) flow of Casson fluid Appl. Math. Mech. 33, Hayat, T., S. A., Shehzad, S. Al-Mezel and A. Alsaedi (014c). Three-diensional flow of an Oldroyd-B fluid over a bidirectional stretching surface with prescribed surface teperature and prescribed surface heat flux. J. Hydrol. Hydroech. 6,

9 Hayat, T., S. A. Shehzad, A. Alsaedi and M. S. Alhothuali (01b). Mixed convection stagnation point flow of Casson fluid with convective boundary conditions. Chin. Phys. Lett. 9, Hayat, T., Shehzad, S. A., M. Mustafa and A. A. Hendi (01a). MHD flow of an Oldroyd-B fluid through a porous channel. Int. J. Che. Reactor Eng. 10, A8. Hayat, T., S. A. Shehzad, M. Qasi and A. Alsaedi (01c). Radiative flow with variable theral conductivity in porous ediu. Z. Naturforschung A 67a, Jail, M. and C. Fetecau (010). Soe exact solutions for rotating flows of a generalized Burgers' fluid in cylindrical doains. J. Non- Newtonian Fluid Mech. 165, Jail, M., N. A. Khan and S. Nazish (013). Fractional MHD Oldroyd-B fluid over an oscillating plate. Theral Sci. 17, Jail, M., A. Rauf, C. Fetecau and N. A. Khan (011). Helical flows of second grade fluid due to constantly accelerated shear stresses. Coun. Nonlinear Sci. Nuer. Siulat. 16, Javed, T., M. Sajid, Z. Abbas and N. Ali (011). Non-siilar solution for rotating flow over an exponentially stretching surface. Int. J. Nu. Methods Heat Fluid Flow 1, Liao, S.J (003). Beyond perturbation: Introduction to hootopy analysis ethod. Chapan and Hall, CRC Press, Boca Raton. Makinde, O. D. (01). Coputational odelling of MHD unsteady flow and heat transfer toward a flat plate with Navier slip and Newtonian heating, Braz. J. Che. Eng. 9, Malvandi, A., F. Hedayati and M. R. H. Nobari (014a). An analytical study on boundary layer flow and heat transfer of nanofluid induced by a non-linearly stretching sheet. J. Appl. Fluid Mech. 7, Malvandi, A., F. Hedayati and M. R. H. Nobari (014b). An HAM analysis of stagnation-point flow of a nanofluid over a porous stretching sheet with heat generation. J. Appl. Fluid Mech. 7, Mukhophadhyay, S. (01). Heat transfer analysis of the unsteady flow of a Maxwell fluid over a stretching surface in the presence of a heat source/sink. Chin. Phys. Lett. 9, Rashidi, M. M. and E. Erfani (01). Analytical ethod for solving steady MHD convective and slip flow due to a rotating disk with viscous dissipation and Ohic heating. Engin. Coput. 9, Rashidi, M. M. and N. F. Mehr (014). Series solutions for the flow in the vicinity of the equator of an MHD boundary-layer over a porous rotating sphere with heat transfer. Theral Sci. 18, Rashidi, M. M., A. J. Chakha and M. Keianesh (011). Application of ulti-step differential transfor ethod on flow of a second-grade fluid over a stretching or shrinking sheet. Aerican J. Coput. Math. 6, Rashidi, M. M., S. A. M. Pour and S. Abbasbandy (011). Analytic approxiate solutions for heat transfer of a icropolar fluid through a porous ediu with radiation, Coun. Nonlinear Sci. Nuer. Siulat. 16, Shahohaadi, H. (01). Analytic study on non- Newtonian natural convection boundary layer flow with variable wall teperature on a horizontal plate. Meccanica 47, Shehzad, S. A., A. Alsaedi and T. Hayat (01). Three diensional flow of Jeffery fluid with convective boundary conditions. Int. J. Heat Mass Transfer 55, Shehzad, S. A., A. Alsaedi and T. Hayat (013). Hydroagnetic steady flow of Maxwell fluid over a bidirectional stretching surface with prescribed surface teperature and prescribed surface heat flux. Plos One 8, e Shehzad, S. A., T. Hayat, M. S. Alhuthali and S. Asghar (014). MHD three-diensional flow of Jeffrey fluid with Newtonian heating. J. Cent. South Uni. 1, Turkyilazoglu, M. (011). Nuerical and analytical solutions for the flow and heat transfer near the equator of an MHD boundary layer over a porous rotating sphere. Int. J. Theral Sci. 50, Turkyilazoglu, M. (01). Solution of the Thoas-Feri equation with a convergent approach, Coun. Nonlinear Sci. Nuer. Siulat. 17, Wang, C.Y (1984). The three-diensional flow due to a stretching flat surface. Phys. Fluids 7, Wang, S. and W. C. Tan (011). Stability analysis of soret-driven double-diffusive convection of Maxwell fluid in a porous ediu. Int. J. Heat Fluid Flow 3,