Mixed Convection Flow of Casson Nanofluid over a Stretching Sheet with Convectively Heated Chemical Reaction and Heat Source/Sink

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1 Journal of Applied Fluid Mechanics Vol. 8 No. 4 pp Available online at ISSN EISSN DOI: /acadpub.jaf Mixed Convection Flow of Casson Nanofluid over a Stretching Sheet with Convectively Heated Cheical Reaction and Heat Source/Sink T. Hayat 1 M. Bilal Ashraf 3 S. A. Shehzad 4 and A. Alsaedi 1 Departent of Matheatics Quaid-i-Aza University 4530 Islaabad Pakistan Nonlinear Analysis and Applied Matheatics (NAAM) Research Group Faculty of Science King Abdulaziz University P. O. Box 8003 Jeddah 1589 Saudi Arabia 3 Departent of Matheatics Cosats Institute of Inforation Technology Wah Cantt Pakistan 4 Departent of Matheatics Cosats Institute of Inforation Technology Sahiwal Pakistan Corresponding Author Eail: bilalashraf_qau@yahoo.co (Received May 8 014; accepted July ) ABSTRACT The present study addresses the ixed convection flow of non-newtonian nanofluid over a stretching surface in presence of theral radiation heat source/sink and first order cheical reaction. Casson fluid odel is adopted in the present study. Magnetic field contribution is incorporated in the oentu equation whereas the aspects of nanoparticles are considered in the energy and concentration equations. Convective boundary conditions for both heat and ass transfer are utilized. Siilarity transforations are eployed to reduce the partial differential equations into ordinary differential equations. Series solutions of the resulting proble are obtained. Ipacts of all the physical paraeters on the velocity teperature and concentration fields are analyzed graphically. Nuerical values of different involved paraeters for local skin friction coefficient local Nusselt and Sherwood nubers are obtained and discussed. Keywords: Casson nanofluid; Mixed convection flow; Theral radiation; Magnetic field; Cheical reaction; Heat source/sink. NOMENCLATURE u v and w velocity coponents M Hartan nuber py yield stress of fluid R radiation paraeter k e ean absorption coefficient Nb Brownian otion paraeter c specific heat of nanoparticles DB brownian diffusion coefficient T theral expansion coefficient T f and C f hot fluid teperature and concentration electrical conductivity Gr x The local Grashof nuber density of fluid Pr Prandtl nuber kineatic viscosity Nt therophoretic paraeter density of nanoparticles B0 applied agnetic field ixed convection paraeter g gravitational acceleration β casson fluid paraeter Q heat generation/absorption C concentration expansion coefficient h h and heat and ass transfer theral diffusivity coefficients N concentration buoyancy kineatic viscosity

2 T. Hayat et al. / JAFM Vol. 8 No. 4 pp s Stefan-Boltzann constant 1 heat source/sink paraeter 1 Heat transfer Biot nuber Mass transfer Biot nuber 1. INTRODUCTION Now a days the energy efficiency is an extreely iportant topic in view of theral conductivity enhanceent aongst the researchers. For this purpose the researchers considered the involveent of nanoparticles in the base fluid. Originally Masuda et al. (1996) reported the liquid dispersions of subicron particles or nanoparticles. After that first tie nanofluid ter is used by Choi (1995). In coparison to the base fluids theral conductivity of nanofluid is too high that's why these have been used in any energetic systes such as cooling of nuclear systes radiators natural convection in enclosures etc. The odel proposed by Buongiorno (006) studies the Brownian otion and the therophoresis on the heat transfer characteristics. Recently the analytical solutions for the lainar axisyetric ixed convection boundary layer nanofluid flow past a vertical cylinder is obtained by Rashidi et al. (01a). Stagnation point flow of nanofluid near a pereable stretched surface with theral convective condition is provided by Alseadi et al. (01). Mustafa et al. (013) discussed the boundary layer flow of nanofluid over an exponentially stretching sheet with convective boundary conditions. Rashidi et al. (014b) presented the analytical solutions of transport phenoena in nanofluid adjacent to a nonlinearly porous stretching sheet. Sheikholeslai and Ganji (013a) studied the heat transfer of Cu-water nanofluid flow between the parallel plates. Turkyilazoglu (013) studied the unsteady ixed convection flow of nanofluids over a oving vertical flat plate with heat transfer. Sheikholeslai et al. (013b) deterined free convection flow of nanofluid. Hayat et al. (014) presented the ixed convection peristaltic flow of agnetohydrodynaic (MHD) nanofluid in presence of Brownian otion and therophoresis. Casson fluid odel is one of the base fluids which exhibits yield stress. However such fluid behaves like a solid when shear stress less than the yield stress is applied and it oves if applied shear stress is greater than yield stress. Exaples of Casson fluid include jelly soup honey toato sauce concentrated fruit juices blood and any others. In fact several substances like protein fibrinogen and globin in an aqueous base plasa huan red cells for a chain like structure known as aggregates or rouleaux. If the rouleaux behaves like a plastic solid then there exists a field stress that can be identified with the constant yield stress in Casson fluid by Dash et al. (1996). Recently Mukhopadhyay (013a) provided the boundary layer flow of Casson fluid over a non-linearly stretching sheet. Soe of the recent studies about flow of Casson fluid are [Shehzad et al. (013) Mukhopadhyay and Vajravelu (013b) Hayat et al. (01a)]. Mixed convective flows along with theral 804 radiations are coonly encountered in any environental and scientific developents for instance in aeronautics fire research heating and cooling of channels etc. Thus it is of great worth to study the radiative convective flow. Makinde (005) discussed the transient free convection interaction with theral radiation of an absorbing eitting fluid along oving vertical pereable plate. Hayat et al. (010) explored the MHD radiative ixed convection boundary layer stagnation point flow through a porous ediu. No doubt the MHD flow has gained considerable interest due to its fundaental iportance in the industrial and technological applications such as in coating of etals crystal growth electroagnetic pups MHD generators and reactor cooling. The Lorentz force interacts with the buoyancy force in governing the flow and teperature fields. The effect of Lorentz force is known to reduce the velocities. Heat and ass transfer characteristics in the agnetohydrodynaic (MHD) viscous flow over a pereable stretching surface is studied by Turkyilazoglu (011). Also Motsa et al. (01) obtained the solutions for flow of upper- convected Maxwell fluid over porous stretching sheet in presence of agnetic field by using successive Taylor series linearization ethod. In several natural processes the fluids experiencing exotheric or endotheric cheical reactions. Hence it is iportant to discuss the effects of heat source or sink. Occurrence of heat source or sink ay change the teperature distribution in the fluid which disturbs the particle deposition rate in systes such as nuclear reactors electronic chips and seiconductor wafers. Hayat et al. (01b) presented the radiative flow of Jeffery fluid in a porous ediu with power law heat flux and heat source. Recently Kandasay et al. (011) discussed the cobined effect of theral diffusion and diffusion thero in free convective heat and ass transfer flow over a porous stretching surface in the presence of therophoresis particle deposition and heat source/sink. The researchers at present are also very keen to discuss the significance of convective boundary conditions by Aziz (009) in the flows under various aspects. The present study deals with the convective boundary conditions in the ixed convection flow of nanofluid over a stretching sheet. Proble forulation is ade in presence of theral radiation heat source/sink and first order cheical reaction. Casson fluid is taken as a base fluid. Boundary layer partial differential equations are reduced into set of ordinary differential equations by using appropriate transforations. Convergent solutions of the resulting probles are obtained by using hootopy analysis ethod [Liu et al. (013) Hayat et al. (013) Abbasbandy et al. (013) Zheng et al. (01) Rashidi et al. (01b) Turkyilazoglu (01) Rashidi et al. (014)].

3 T. Hayat et al. / JAFM Vol. 8 No. 4 pp Ipacts of all ebedding paraeters are analyzed graphically for the teperature concentration and flow fields. Nuerical values of skin-friction coefficient local Nusselt and Sherwood nubers for different paraeters are calculated and analyzed.. MATHEMATICAL MODELING Consider the ixed convection boundary layer flow of Casson nanofluid over a stretching surface with heat source/ sink. Flow is considered in presence of an applied agnetic field theral radiation and first order cheical reaction. Convective heat and ass conditions are taken at surface of the sheet. The rheological equation of state for an isotropic and incopressible flow of a Casson fluid is [ ]: In above expression (1) ee ij ij and e ij is the ( i j ) th coponent of the deforation rate the product of the coponent of deforation rate with itself c a critical value of this product based on the non-newtonian odel B the plastic dynaic viscosity of non- Newtonian fluid and p y the yield stress of fluid. The velocity field is taken as V [ uxy ( ) vxy ( )0] () where u and v denote the velocity coponents in the x - and y -directions. The governing equations for flow can be put into the fors: u v 0 (3) x y u u 1 u u v 1 g ( ) T T T x y y B0 gc ( CC ) u (4) 3 T T T 16 st T u v x y 3 y kecp y c p C T D T T D B (5) cp y y T y Q T T c p C C C u v DB x y y DT T k 1( C C ) T y (6) where is the Casson fluid paraeter T the theral expansion coefficient C the concentration expansion coefficient the electrical conductivity B 0 the agnitude of applied agnetic field the density of fluid g the gravitational acceleration ( B / ) the kineatic viscosity the dynaic viscosity B the theral diffusivity T the fluid teperature s the Stefan-Boltzann constant k e the ean absorption coefficient c p the specific heat of fluid the density of nanoparticles c the specific heat of nanoparticles Q the unifor voluetric heat generation/absorption C the concentration field and D B the Brownian diffusion coefficient. The boundary conditions can be expressed as follow: u Uw( x) cx v 0 T k h( Tf T) (7) z C D h ( Cf C) at y 0 z u0 v0 (8) T T C C as y where subscript w corresponds to the wall condition h the heat transfer coefficient h the concentration transfer coefficient T f the hot fluid teperature and C f the hot fluid concentration. By using siilarity transforations c y u cxf ( ) v c f ( ) (9) T T ( ) CC ( ) Tf T Cf C equation (1) is identically satisfied and Eqs. ()-(8) give: 1 (1 ) f ff ( f ) (10) ( N) Mf 0 4 (1 R) Pr f Pr Nb Nt 3 (11) Pr 1 0 N t Scf Sc 0 (1) Nb f 0 f 1 1(1 (0)) (13) (1 (0)) at 0 f as (14) where is the ixed convection paraeter Gr x the local Grashof nuber N the concentration buoyancy paraeter M the Hartan nuber R the radiation paraeter Pr the Prandtl nuber N b the Brownian otion paraeter which is the irregular jiggling sort of oveent exhibited by a sall particle suspended in a fluid N t the therophoretic paraeter which is observed in ixtures of obile particles where the different particle types exhibit different responses to the force of a teperature gradient 1 the heat 805

4 T. Hayat et al. / JAFM Vol. 8 No. 4 pp source/sink paraeter Sc the Schidt nuber the cheical reaction paraeter 1 the heat transfer Biot nuber and the ass transfer Biot nuber. These can be defined in the fors 3 Gr gt( Tf T ) x x Gr x Rex C( Cw C ) N ( T T ) T f 3 4 T 0 ( B R ) M kk e cdb( Cw C ) Pr Nb c cdt( Tw T ) Nt ct Q k 1 1 Sc cp D c h h 1. k a D a (15) Non-diensional local skin friction coefficient local Nusselt and Sherwood nubers are 1 1/ Cf Rex 11/ f(0) (16) 1/ /Re x (0) Nu (17) 1/ /Re x (0) Sh (18) cx in which Rex is the local Reynold nuber. 3. SERIES SOLUTIONS Initial guesses and auxiliary linear operators for series solutions are chosen in the fors 1 exp( ) f0( ) 1 e 0( ) 1 1 (19) exp( ) 0( ) 1 Lf f f L L (0) with the following properties Lf ( C1Ce C3e ) 0 L ( C4e C5e ) 0 (1) L ( C6e C7e ) 0 where C i ( i 17) are the arbitrary constants. Zeroth order deforation probles are 1 pl ˆ( ; ) 0( ) f f p f () p ˆ( ; ) ˆ( ; ) ˆ( ; ) f N f f p p p 1 pl ˆ( ; ) 0( ) p (3) p fˆ ( ; p) ˆ( ; p) ˆ( ; p) N 1 p L ˆ ( ; p) ( ) 0 p ˆ( ; ) ˆ( ; ) ˆ( ; ) N f p p p ˆ(0; ) 0 ˆ (0; ) 1 ˆ f p f p f ( ; p ) 0 ˆ ˆ ( p) 0 (0 p) 1[1 (0 p)] ˆ (0 p) [1 ˆ(0 p)] ˆ( p) 0 N [ ˆ( ; ) ˆ f f p ( ; p) ˆ ( ; p)] 3 1 fˆ( p) ˆ ˆ f( p) 1 f( p) 3 fˆ ( p ) ˆ( p ) ˆ( ; ) N p fˆ( p ) M N [ ˆ( ; ) ˆ( ; ) ˆ f p g p ( ; p) ˆ ( ; p)] 4 ˆ ( p) ˆ ˆ ( p) 1 R Pr( f( p) 3 Pr N b ˆ ( p) ˆ ( p) ˆ( p ) Pr N Pr ˆ t 1 ( p) N [ ˆ( ; ) ˆ( ; ) ˆ f p g p ( ; p) ˆ ( ; p)] (4) (7) (5) (6) ˆ ( ; p) ˆ ( ; p) Scfˆ( p ) N ˆ( ) t p Scˆ( ; p) N b (8) where p is an ebedding paraeter f and are the non-zero auxiliary paraeters and N f N and N are the nonlinear operators. Taking p 0 and p 1 we have f ˆ ( ;0) f ( ) ˆ ( 0) ( ) ˆ ( ;0) ( ) fˆ (;1) f() ˆ (1) () ˆ (;1) (). (9) As p varies fro 0 to 1 then f ( p) ( p) and ( p) differ fro f 0 ( ) 0 ( ) and 0 ( ) to f ( ) ( ) and ( ). Using Taylor's expansion we obtain f ( p) f0( ) f( ) p 1 (30) 1 f( ; p) f( )! p p0 ( p) 0 ( ) ( ) p 1 1 ( ; p) ( )! p p0 (31) 806

5 T. Hayat et al. / JAFM Vol. 8 No. 4 pp ( p) 0( ) ( ) p 1 1 ( ; p) ( ). (3)! p p0 The convergence of above series strongly depends upon f g and. Considering that f and are selected properly so that Eqs. (30) (3) converge at p 1 therefore f( ) f0 ( ) f ( ) 1 ( ) 0 ( ) ( ) 1 ( ) 0 ( ) ( ). 1 (33) (34) (35) The final solution expressions can be expressed as follows: ( ) f f ( ) C C e C e (36) 1 3 ( ) ( ) Ce 7 Ce 8 ( ) ( ) Ce 9 C10e f (37) (38) where the special solutions are. g and Fig. 1. H-curves for the function f() () and (). Fig. 1a. Residual error for f(). 4. CONVERGENCE ANALYSIS AND DISCUSSION It is obvious that the series solutions (33) (35) consist of the auxiliary paraeters f and. These auxiliary paraeters are very iportant in order to adjust and control the convergence of the series solutions. We have sketched the curves at 18 th order of approxiations to deterine the convergence region where the curves becoe parallel to horizontal axis. Appropriate range of auxiliary paraeters f and are 0.9 f and (see Fig.1). Figs. 1a 1c are drawn to predict the best values for auxiliary paraeters. Fro these Figs. we noticed that the optial values are f 0.7. Table 1 concludes that the series solutions converge in the whole region of when f 0.7. Effects of Casson fluid paraeter Hartan nuber M ixed convection paraeter and concentration buoyancy paraeter N on the velocity profile f ( ) are analyzed in the Figs. -5. Fig. reveals that the velocity profile f ( ) and oentu boundary layer thickness decrease with an increase in Casson paraeter. Ipact of Hartan nuber M on the velocity profile f ( ) is displayed in Fig. 3. With an increase in M the velocity profile f ( ) decreases. Fig. 1b. Residual error for (). Fig. 1c. Residual error for (). Also oentu boundary layer thicknesses is a decreasing functions of M. This is due to the reason that with an increase in M the Lorentz force increases which resist the flow. Fig. 4 is plotted to analyze the influence of ixed convection paraeter on the velocity profile f ( ) in both assisting and opposing flows. It is observed that the velocity profile f ( ) and oentu boundary layer thickness increase when 0 (assisting flow) while opposite behavior is noted for 0 (opposing flow). 807

6 T. Hayat et al. / JAFM Vol. 8 No. 4 pp Table 1 Convergence of series solutions for different order of approxiations when 0.5 Pr 1.0 Sc 0.7 Nt Nb 1 0. N M R1 0.3 and f 0.7. order of f (0) (0) (0) approxiations It is exained that the oentu boundary layer thickness and velocity profile f ( ) increase with an increase in N (see Fig. 5). Fig. 5. Variation of N on f(). Figs are displayed to see the ipacts of different paraeters on the teperature ( ) and concentration ( ). Figs. 6 and 7 show the variation of therophoretic paraeter N t on the teperature ( ) and concentration ( ). As therophoresis causes the sall particles to be driven away fro a hot surface towards a cold one thats why teperature ( ) and concentration profiles ( ) increase with an increase in therophoretic paraeter N t. Fig.. Variation of on f(). Fig. 6. Variation of N t on (). Fig. 3. Variation of M on f(). Fig. 7. Variation of N t on (). Fig. 4. Variation of on f(). Also the associated boundary layer thicknesses are increasing functions of N t. Figs. 8 and 9 are plotted to see the variation of Brownian otion paraeter N b on the teperature ( ) and concentration ( ). Theral boundary layer 808

7 T. Hayat et al. / JAFM Vol. 8 No. 4 pp thickness and teperature ( ) enhance when N b increases. This is due to the reason that Brownian otion is a zig zag otion in which the kinetic energy of the particles increases which in results shows an increase in particle collision. It is also noticed that the concentration ( ) and associated boundary layer thickness reduces with an enhanceent in N b. Fig. 10 exhibits the effect of Prandtl nuber Pr on the teperature ( ). Both conduction and convection happen in fluids. As heat transfer through both processes reduces the teperature difference they can be considered as copeting against each other in transferring heat. There are any different types of fluids such as air water oil or ercury. The rates of conduction and convection vary in different fluids. Soeties conduction doinates. Other ties convection doinates. The Prandtl nuber is a paraeter that can be used to roughly deterine which process will win. As Pr is inversely proportional to theral diffusivity so due to an increase in Pr the theral diffusivity decreases which reduces both the theral boundary layer thickness and teperature ( ). Variation of Schidt nuber Sc on the concentration profile ( ) is seen in Fig. 11. With an increase in Sc the ass diffusivity reduces which in turn decreases the concentration boundary layer thickness. Influence of heat source/sink paraeter 1 on the teperature ( ) is analyzed in Fig. 1. Theral boundary layer thickness and teperature ( ) are increasing functions of 1 0 (internal heat source) while reverse is observed in case of 1 0 (internal heat sink). Fig. 13 is designed to see the influence of theral radiation R on the teperature ( ). It is noticed that teperature ( ) enhances due to an increase in radiation paraeter R. Fig. 8. Variation of N b on (). Fig. 11. Variation of Sc on (). Fig. 9. Variation of N b on (). Fig. 1. Variation of 1 on (). Fig. 10. Variation of Pr on (). Fig. 13. Variation of R on (). 809

8 T. Hayat et al. / JAFM Vol. 8 No. 4 pp It is because of the fact that for larger R the ean absorption coefficient k e decreases which enhances the divergence of the radiative heat flux. Hence the rate of radiative heat transfer to the fluid will rise and consequently the fluid teperature increases. Figs. 14 and 15 are drawn to see the effects of heat transfer Biot nuber 1 and ass transfer Biot nuber on the teperature ( ) and concentration ( ) respectively. With an enhanceent in heat transfer Biot nuber 1 the theral boundary layer thickness and teperature ( ) increase. Also the associated boundary layer and concentration ( ) are increasing functions of ass transfer Biot nuber. Sherwood nubers decrease. Also with an increase in 1 and N b the skin-friction coefficient and Sherwood nuber increases while local Nusselt nuber reduces. In case of assisting flow ( 0) and with an enhanceent in N the local Nusselt and Sherwood nubers enhance. The skin-friction coefficient reduces while reverse is the ipact in case of opposing flow ( 0). Sherwood nuber rises with an increase in ass transfer Biot nuber while skin-friction coefficient and local Nusselt nuber decrease with an increase in. Fig. 16 exhibits the effect of generative/destructive cheical reaction on the concentration ( ). It is found that the associated boundary layer thickness and concentration profile ( ) enhance with generative cheical reaction ( 0) while opposite behavior is noted for destructive cheical reaction ( 0). Fig. 16. Variation of on (). Fig. 14. Variation of 1 on (). Table Nuerical values of local Nusselt nuber (0) and Sherwood nuber (0) for different values of paraeters N N when RM 0.0 N Pr 1.0 Sc 0.7 and f 0.7. N N (0) (0) t b t b CONCLUSIONS Fig. 15. Variation of on (). Tables -4 are prepared to explore the ipacts of Casson fluid paraeter therophoretic paraeter N t Brownian otion paraeter N b ixed convection paraeter concentration buoyancy paraeter N heat transfer Biot nuber 1 and ass transfer Biot nuber on skinfriction coefficient and local Nusselt and Sherwood nubers. With an enhanceent in N t and the Skin-friction coefficient and local Nusselt and The present investigation discusses the ixed convection flow of nanofluid over a stretching surface in presence of heat source/ sink and first order cheical reaction. Main findings of present study are entioned below. a) Moentu boundary layer thickness and velocity profile f ( ) enhance with the increase in and M while these are reduced via N. b) In case of assisting flow 0 both the velocity profile f ( ) and oentu boundary layer thickness shoot up while reverse behavior is observed in case of 810

9 T. Hayat et al. / JAFM Vol. 8 No. 4 pp opposing flow 0. c) With an enhanceent in therophoretic paraeter N t the teperature ( ) and concentration profiles ( ) increase. d) Influence of Brownian otion paraeter N b on the theral boundary layer thickness and teperature ( ) is qualitatively opposite to that of concentration boundary layer thickness and concentration profile ( ). e) Theral boundary layer thickness reduces with an increase in Prandtl nuber Pr and internal absorption paraeter 1 0 while it increases with the increase in theral radiation R internal heat generation 1 0 and heat transfer Biot nuber 1. f) Concentration profile ( ) and associated boundary layer thickness are increasing functions of ass transfer Biot nuber and generative cheical reaction 0 while reverse behavior is noted for Schidt nuber Sc and destructive cheical reaction 0. g) Skin-friction coefficient and local Nusselt and Sherwood nubers reduce through enhanceent in Casson fluid paraeter and therophoretic paraeter N t. h) Rise in Brownian otion paraeter N b increases the skin-friction coefficient and Sherwood nuber while it decreases the local Nusselt nuber. i) In assisting flow 0 the skin-friction coefficient decreases while opposite behaviors found for opposing flow 0. Table 3 Nuerical values of local Nusselt nuber (0) and Sherwood nuber (0) for different values of paraeters 1 N when 1 RM 0.0 Nt Nb Pr 1.0 Sc 0.7 and f 0.7. N 1 (0) (0) j) Enhanceent in concentration buoyancy paraeter N heat transfer Biot nuber 1 and ass transfer Biot nuber reduces the skin-friction coefficient. k) Local Nusselt and Sherwood nubers are increasing functions in case of assisting flow 0 and decreasing functions for opposing flow 0. Table 4 Nuerical values of skin-friction 1 1 f (0) for different values of paraeters Nt Nb 1 N when 1 RM 0.0 Pr 1.0 Sc 0.7 and f 0.7. coefficient 1 Nt Nb N 1 1 f (0) Also the associated boundary layer thicknesses are increasing functions of N t. ACKNOWLEDGEMENT The suggestions of reviewer regarding an earlier version are appreciated. REFERENCES Abbasbandy S. M. S. Hashei and I. Hashi (013). On convergence of hootopy analysis ethod and its application to fractional integro-differential equations. Quaestiones Matheaticae 36(1)

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