Vidyasagar et al., International Journal of Advanced Engineering Technology E-ISSN A.P., India.

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Research Paper MHD CONVECTIVE HEAT AND MASS TRANSFER FLOW OVER A PERMEABLE STRETCHING SURFACE WITH SUCTION AND INTERNAL HEAT GENERATION/ABSORPTION G.Vidyasagar 1 B.Ramana P.

1 Research Paper MHD CONVECTIVE HEAT AND MASS TRANSFER FLOW OVER A PERMEABLE STRETCHING SURFACE WITH SUCTION AND INTERNAL HEAT GENERATION/ABSORPTION G.Vidyasagar 1 B.Ramana P. Bala Anki Raddy 3 Address for Correspondence 1 Assistant Professor, Department of BS&H, Siddharth College of Engineering and Technology, Puttur, Chittoor, A.P., India. Department of Applied Mathematics, Sri Padmavati Mahila University, Tirupati, Chittoor (Dt), A.P., India 3 Fluid Dynamics division, Dept of SAS, VIT University, Vellore, Tamilnadu ABSTRACT In the present paper e consider a convective heat and mass transfer in a porous medium of an incompressible viscous conducting fluid over a permeable stretching surface ith suction and internal heat generation/absorption. Using a similarity transformation the governing equations of the problem are converted into simultaneous linear differential equations of first order. The governing boundary layer equations are solved numerically by using shooting technique. In order to further study the behavior of the non linear differential equations for various values of the physical parameters. The numerical results to bring out the effects of the Grashof number, modified Grashof number, suction parameter, porosity parameter, heat generation/absorption, stretching parameter, Prandtl number and Schmidt number. The effectiveness of porosity on stagnation point flo toards a stretching surface ith heat generation/absorption. KEYWORDS: Magnetic field; Porous medium; Stagnation point flo; Permeable stretching surface; Heat generation/ absorption and Heat and Mass transfer. INTRODUCTION Flo of an incompressible viscous fluid over a stretching surface is a classical problem in fluid dynamics and important in various process. It is used to create polymers of fixed cross-sectional profiles, cooling of metallic and glass plates. Aerodynamics shaping of plastic sheet by forcing through die and boundary layer along a liquid film in condensation processes are among the other areas of application. The production of sheeting material, hich includes both metal and polymer sheets arises in a number of industrial manufacturing processes. The pioneering ork of Sakiadis [8] extensive literature is available on this topic for a linearly stretching sheet. A broad effort has been made to gain information regarding the stretching flo problems in various situations. Such situations include consideration of non- Netonian fluids, MHD fluid, heat transfer; mass transfer, porous medium, slip effects, etc. A vast body of literature is no available on the topic. Some very recent attempts in this direction have been made in the investigations. Crane [15] had studied flo past a stretching plate. Chakrabarti and Gupta [6] have discussed hydromagnetic flo and heat transfer over a stretching sheet. Carragher and Crane [5] have discussed heat transfer on a continuous stretching sheet. Rajagopal et al. [3] studied flo of a viscoelastic fluid over a stretching sheet. Dutta et al. [16] have discussed temperature field in flo over a stretching sheet ith uniform heat flux. In this study, the viscous dissipation as considered in the energy equation. Andersson [3] is studied MHD flo of a viscoelastic fluid past a stretching surface. Liao [19] investigation on the proposed homotopy analysis techniques for nonlinear problems and its application. Chiam [8] is studied stagnation-point flo toards a stretching plate. Chiam [9] is studied hydromagnetic flo over a surface stretching ith a poer-la velocity. Vajravelu [9] studied viscous flo over a nonlinearly stretching sheet. Ray Mahapatra and Gupta [4] investigation heat transfer in stagnationpoint flo toards a stretching sheet. Khaled and Vafai [18] is studied the role of porous media in modeling flo and heat transfer in biological tissues. Nazar et al. [1] had studied stagnation point flo of IJAET/Vol. IV/ Issue I/Jan.-March., 13/41-45 a micropolar fluid toards a stretching sheet. Ray Mahapatra and Gupta [5] had studied stagnationpoint flo of a viscoelastic fluid toards a stretching surface. Cortell [1] reported as flo and heat transfer of a fluid through a porous medium over a stretching surface ith internal heat generation/absorption and suction/bloing. Sadeghy et al. [7] studied realistic viscoelastic fluid models such as a Maxell model should be invoked in the analysis. Indeed, this fluid model has recently been used to study the flo of viscoelastic fluids. Xu [3] had studied an explicit analytic solution for convective heat transfer in an electrically conducting fluid at a stretching surface ith uniform free stream. Cortell [11] had studied the effects of viscous dissipation and ork done by deformation on the MHD flo and heat transfer of a viscoelastic fluid over a stretching sheet, and considered ith internal heat generation or absorption. Liao [] had studied unsteady boundary-layer flos caused by an impulsively stretching plate. Chen [7] is studied marangoni effects on forced convection of poer-la liquids in a thin film over a stretching surface. Cortell [13] studied viscous flo and heat transfer over a nonlinearly stretching sheet. Abbas and Hayata [1] studied radiation effects on MHD flo in a porous space. Cortell [14] studied the effects of viscous dissipation and radiation on the thermal boundary layer over a nonlinearly stretching sheet. El-Aziz [17] is studied thermal-diffusion and diffusion-thermo effects on combined heat and mass transfer by hydromagnetic three-dimensional free convection over a permeable stretching surface ith radiation. On one side, the effects of thermal radiation are included in the energy equation; on the other hand, the prescribed all heat flux. Afify [] have investigated similarity solution in MHD effects of thermal diffusion and diffusion thermo on free convective heat and mass transfer over a stretching surface considering suction or injection. Puvi Arasu et al. [] have discussed lie group analysis for thermal diffusion and diffusion-thermo effects on free convective flo over a porous stretching surface ith variable stream conditions in the presence of thermophoresis particle deposition. Robert et al. [6]

2 has discussed convective heat transfer in a conducting fluid over a permeable stretching surface ith suction and internal heat generation/ absorption. From above, there are still not considering the free convection couple ith radiation effect over a nonlinearly stretching sheet. In the present investigation a study of convective heat and mass transfer in a conducting fluid over a permeable stretching surface ith suction and internal heat generation/absorption. The basic equations governing the flo are in the form of partial differential equations and have been reduced to a set of non-linear ordinary differential equations by applying suitable similarity transformations. The equations governing the flo are solved numerically by shooting technique. The expressions for velocity, temperature and concentration are obtained. The effects of Grashof number (Gr), modified Grashof number (Gc), Suction parameter (S), Porosity parameter (K), heat generation\ absorption parameter (B), Prandtl number (Pr), Stretching parameter (C) and Schmidt number (Sc) are studied. Formulation of the problem We consider the steady to dimensional stagnation point flo of a viscous incompressible electrically conducting fluid near a stagnation point at a surface coinciding ith the plane y =, ith the flo being restricted to y >. To equal and opposing forces are applied along the x-axis so that the surface is stretched (hile keeping the origin fixed). The potential flo that arrives from the y-axis (impinges on the flat all at y = ), divides into to streams on the all and leaves in both directions. The flo is through a porous medium here the Darcy model is assumed. The viscous flo must adhere to the all, hereas the potential flo slides along it. We denote the components of the fluid velocity by (u, v) at any point (x, y) for the viscous flo, hile (U, V) denote the velocity components for the potential flo. We consider the case in hich there may be a suction velocity (-W) on the stretching surface. Also, e denote the fluid temperature by T. The velocity distribution of the frictionless flo in the neighborhood of the stagnation point is becomes = = (1) U( x) ax, V ( x) ay here the parameter a > is proportional to the free stream velocity. The continuity and momentum equations for the to dimensional steady flo, using the usual boundary layer approximations reduces to u u + = () x y u u du u µ * σb ρ u U ( U u) g ( T T ) g ( C C ) o + ν = + µ + + β + β u x y xx y K ρ (3) T T T (4) ρ c p u + v = k + Q ( T T) x y y C C C u + v = D (5) x y y here u and v are the velocity components along the x and y axes respectively, µ is the coefficient of viscosity of the fluid, K is the Darcy permeability, ν is the kinematics viscosity, * β, β are the thermal and concentration expansion coefficient respectively, σ electric conductivity, B is the uniform magnetic IJAET/Vol. IV/ Issue I/Jan.-March., 13/41-45 field, ρ is the density, T is the temperature inside the boundary layer,, T is the temperature for aay from the plate, C is the species concentration in the boundary layer, C Species concentration of the ambient fluid, c p is the specific heat at constant pressure, k is the thermal conductivity, and Q is the volumetric rate of heat generation or absorption, D is the is the molecular diffusivity of the species concentration. We also have the boundary conditions u= cx, v=, T = T, C= C at y= u= ax, T T, C C as y here c >. We introduce the folloing non-dimensional variables: c 1 ν η=, u ( x, y ) = cxf ( η), v ( x, y ) = cv f ( η), M =, Gr = g β( T T ), ν y * T T C C W ρνc p ν Gc= gβ ( C C ),,, s,pr, Sc, B Q θ= φ= = = = = (7) T T C C cν k D cρcp In vie of (7), the Equations (3) - (5) take the form f ( η) ( f ( η) ) f η f ( η) M( C f ( η) ) C Grθ η Gcφ η (8) 11 1 θ ( η) + Pr f ( η) θ ( η) + Pr Bθ ( η) = (9) φ + Sc fφ = (1) + ( ) + + ( ) ( ) = here the primes denote the differentiation ith respect to η, M is the magnetic parameter, C = a/c > is the stretching parameter B is the dimensionless heat generation or absorption parameter, Pr is the Prandtl number and Sc is the Schmidt number. The corresponding boundary conditions are f = 1, f = s, θ = 1, φ= 1 at η= (11) f = C, θ =, φ= as η SOLUTION OF THE PROBLEM The governing boundary layer equations (8) - (1) subject to boundary conditions (11) are solved numerically by using shooting method. First of all higher order non-linear differential equations (8) - (1) are converted into simultaneous linear differential equations of first order and they are further transformed into initial value problem by applying the shooting technique. RESULTS AND DISCUSSION In order to get a physical insight into the problem, a representative set of numerical results is shon graphically in Figs.1-, to illustrate the influence of physical parameters viz., the grashof number(free convection parameter) Gr, modified grashof number Gm, suction parameter S, porosity parameter M, heat generation/absorption parameter B, Prandtl number Pr, stretching parameter C and Sc is the Schmidt number on the velocity f ( η), temperature θ ( η) and concentration φ ( η). The effects of the grashof number (free convection parameter) Gr on the velocity, temperature and concentration fields are shon in Figs The velocity in the y direction decreases in magnitude ith an increase in the free convection parameter Gr for air and an increase in Gr yields a uniform increase in temperature and concentration profiles. Figs. 4-6 sho the dimensionless velocity, temperature and concentration profiles for different ck

3 values of modified grashof number Gm. It can be seen that the velocity profiles decrease ith the increase of modified grashof number Gm. It is noticed that the temperature and concentration profiles increase ith the increase of modified grashof number Gm. We plot the various values of the suction/injection parameter s profiles for velocity, temperature and concentration in Figs We find that such an increase in s results in a uniform decrease in the profiles for velocity, temperature and concentration. Fig.5. Temperature profile for different values of Gc hen K = 1., S =, B =.1, C =.5, Gr =.1, Pr = Fig.1. Velocity profile for different values of Gr hen K = 1., S =, B =.1, C =.5, Gc =.1, Pr =.71, Sc = Fig.6. Concentration profile for different values of Gc hen K = 1., S =, B =.1, C =.5, Gr =.1, Pr = Fig.: Temperature profile for different values of Gr hen K = 1., S =, B =.1, C =.5, Gc =.1, Pr = Fig.3. Concentration profile for different values of Gr hen K = 1., S =, B =.1, C =.5, Gc =.1, Pr = Fig.4. Velocity profile for different values of Gc hen K = 1., S =, B.1, C =.5, Gr =.1, Pr =.71, Sc = Fig.7. Velocity profile for different values of S hen K = 1., B =.1, C =.5, Gr =.1 = Gc, Pr =.71, Sc = The effects of the porosity parameter M on velocity, temperature and concentration is shon in Figs It is obvious that an increase in the permeability parameter M results in a decrease in the velocity, hile the temperature and concentration profiles increase. Figs sho the dimensionless velocity, temperature and concentration profiles for different values on B. It can be seen that the velocity profiles increase ith the increase of B. It is noticed that the temperature and concentration profiles increase ith the increase of B. The influence of the Prandtl number Pr on velocity, temperature and concentration fields is shon in Figs It is noticed that the velocity or concentration increases, hile the temperature profiles decrease ith the increase of Prandtl number Pr. e plot the change in the profiles for velocity, temperature and concentration on the stretching parameter C is shon in Figs An increase in the stretching parameter results in an increase in the velocity. Meanhile, an increase in the stretching parameter results in a decrease in the temperature or concentration profiles. The effects of the Schmidt number Sc on concentration is shon in Fig.. It is noticed that the concentration decreases ith the increase of Schmidt number Sc. IJAET/Vol. IV/ Issue I/Jan.-March., 13/41-45

4 Fig.8. Temperature profile for different values of S hen K = 1., B =.1, C =.5, Gr =.1 = Gc, Pr = Fig.13. Velocity profile for different values of C hen K = 1., S =, B =.1, Pr =.71, Gr =.1 = Gc, Sc =.6 and M= Fig.9. Concentration profile for different values of S hen K = 1., B =.1, C =.5, Gr =.1 = Gc, Pr = Fig.14. Temperature profile for different values of C hen K = 1., S =, B =.1, Pr =.71, Gr =.1 = Gc, Sc =.6 and M= Fig.1. Velocity profile for different values of M hen S =, B =.1, C =.5, Gr =.1 = Gc, Pr =.71, Sc = Fig. 15. Concentration profile for different values of C hen K = 1., S =, B =.1, Pr =.71, Gr =.1 = Gc, Sc =.6 and M= Fig.11. Temperature profile for different values of B hen K = 1., S =, C =.5, Gr =.1 = Gc, Pr = Fig. 16. Concentration profile for different values of Sc hen K = 1., S =, B =.1, C =.5, Pr =.71, Gr =.1 = Gc and M= Fig.1. Temperature profile for different values of Pr hen K = 1., S =, B =.1, C =.5, Gr =.1 = Gc, Sc =.6 and M= IJAET/Vol. IV/ Issue I/Jan.-March., 13/41-45 Fig.17. Velocity profile for different values of M hen K = 1., S =, B =.1, Pr =.71, Gr =.1 = Gc, Sc =.6 and C =.5

5 Fig.18. Temperature profile for different values of M hen K = 1., S =, B =.1, Pr =.71, Gr =.1 = Gc, Sc =.6 and C =.5 Fig. 19. Concentration profile for different values of M hen K = 1., S =, B =.1, C =.5, Pr =.71, Gr =.1 = Gc, Sc =.6 CONCLUSION In this chapter e study MHD Convective heat and mass transfer flo over a permeable stretching surface ith suction and internal heat generation/absorption. The expressions for the velocity, temperature and concentration distributions are the equations governing the flo are numerically solved by shooting technique. It can be seen that the velocity decreases ith the increase of magnetic parameter M. It is noticed that the temperature and concentration increases ith the increase of magnetic parameter M. An increases in the stretching parameter results in an increase in the velocity. Meanhile, an increase in the stretching parameter results in a decrease in the temperature or concentration. It can be seen that the velocity decreases ith the increase of magnetic parameter M. It is noticed that the temperature and concentration increases ith the increase of magnetic parameter M. It is observed that an increases in the permeability parameter K results in a decrease in the velocity. REFERENCES 1. Abbas Z and Hayata T. Radiation effects on MHD flo in a porous space, International Journal of Heat and Mass Transfer, Volume 51, Issues 5-6, March (8), Pages Ariel P.D, Hayat T and Asghar S. The flo of an elasto-viscous fluid past a stretching sheet ith partial slip, Acta Mech. 187 (6), pp Aang Kechil S and Hashim I. Series solution of flo over nonlinearly stretching sheet ith chemical reaction and magnetic field, Physics Letters A, In Press, (7). 4. Carragher P and Crane L.J. Heat transfer on a continuous stretching sheet, ZAMM 6 (198) Chiam T.C. Stagnation-point flo toards a stretching plate, J. Phys. Soc. Jpn. 63 (1994) Cortell R. Flo and heat transfer of a fluid through a porous medium over a stretching surface ith internal heat generation/absorption and suction/bloing, Fluid Dyn. Res. 37 (5), pp Cortell R. Effects of viscous dissipation and ork done by deformation on the MHD flo and heat transfer of a viscoelastic fluid over a stretching sheet, Phys. Lett. A 357 (6), pp Cortell Bataller R. Similarity solutions for flo and heat transfer of a quiescent fluid over a nonlinearly stretching surface, Journal of Materials Processing Technology, In Press, (7). 9. Cortell R. Viscous flo and heat transfer over a nonlinearly stretching sheet, Applied Mathematics and Computation, Volume 184, Issue, 15 January (7), Pages Cortell R. Effects of viscous dissipation and radiation on the thermal boundary layer over a nonlinearly stretching sheet, Physics Letters A, Volume 37, Issue 5, 8 January (8), Pages Crane L.J. Flo past a stretching plate, ZAMP 1 (197) Dutta B.K, Roy P and Gupta A.S. Temperature field in flo over a stretching sheet ith uniform heat flux, Int. Commun. Heat and Mass Transfer. 1 (1985) Khaled A.R.A and Vafai K. The role of porous media in modeling flo and heat transfer in biological tissues, Int. J. Heat Mass Transf. 46 (3) Liao S.J. On the proposed homotopy analysis techniques for nonlinear problems and its application, Shanghai Jiao Tong University, (199). 15. Liao S.J. An analytic solution of unsteady boundarylayer flos caused by an impulsively stretching plate, Commun. Non-linear Sci. Numer. Simul. 11 (6), pp Nazar R, Amin N and Filip D. I. Pop, Stagnation point flo of a micropolar fluid toards a stretching sheet, Int. J. Nonlinear Mech. 39 (4) Rajagopal K.R, Na T.Y and A.S. Gupta. Flo of a viscoelastic fluid over a stretching sheet, Rheol. Acta. 3 (1984) Ray Mahapatra T and Gupta A.S. Heat transfer in stagnation-point flo toards a stretching sheet, Heat Mass Transf. 38 () Ray Mahapatra T and Gupta A.S. Stagnation-point flo of a viscoelastic fluid toards a stretching surface, Int. J. Nonlinear Mech. 39 (4) Robert A. Van Gorder, K. Vajravelu Convective heat transfer in a conducting fluid over a permeable stretching surface ith suction and internal heat generation/absorption, Elsevier applied mathematics and computation 17 (11), pp Sadeghy K, Najafi A.H and Saffaripour M. Sakiadis flo of an upper-convected Maxell fluid, Int. J. Non- Linear Mech. 4 (5), pp Sakiadis B.C. Boundary-layer behavior on continuous solid surfaces, AIChE J. 7 (1961), pp Vajravelu K. Viscous flo over a nonlinearly stretching sheet Applied Mathematics and Computation, Volume 14, Issue 3, (1), pp Xu H. An explicit analytic solution for convective heat transfer in an electrically conducting fluid at a stretching surface ith uniform free stream, Int. J. Eng. Sci. 43 (5), pp IJAET/Vol. IV/ Issue I/Jan.-March., 13/41-45

ON THE EFFECTIVENESS OF HEAT GENERATION/ABSORPTION ON HEAT TRANSFER IN A STAGNATION POINT FLOW OF A MICROPOLAR FLUID OVER A STRETCHING SURFACE

5 Kragujevac J. Sci. 3 (29) 5-9. UDC 532.5:536.24 ON THE EFFECTIVENESS OF HEAT GENERATION/ABSORPTION ON HEAT TRANSFER IN A STAGNATION POINT FLOW OF A MICROPOLAR FLUID OVER A STRETCHING SURFACE Hazem A.