Heat and Mass Transfer Flow of MHD Viscous Dissipative Fluid in a Channel with a Stretching and Porous Plate

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1 Appl. Math. Inf. Sci. Lett. 5, No. 3, 8-87 (27) 8 Applied Mathematics & Information Sciences Letters An International Journal Heat and Mass Transfer Flow of MHD Viscous Dissipative Fluid in a Channel with a Stretching and Porous Plate Kiran Kumari and Mamta Goal Department of Mathematics, Universit of Rajasthan, Jaipur, India Received: 7 Nov. 26, Revised: 9 Apr. 27, Accepted: 4 Ma 27 Published online: Sep. 27 Abstract: Present analsis has been carried out to stud the two dimensional stead hdromagnetic convective fluid flow between two parallel plates with heat and mass transfer in the presence of thermal radiation and viscous dissipation. Here the lower plate is stretching and porous. Governing equations are solved b forth order Runge - Kutta Method along with shooting technique. The effects of the flow parameters such as magnetic parameter, radiation parameter, suction parameter, Renolds number, Eckert number, Prandtl number and Schmidt number on the velocit, temperature and concentration profile are examined and the effects are explained graphicall. Kewords: Stretching sheet, suction, viscous dissipation, MHD. Introduction The heat and mass transfer due to a stretching surface in a channel, used in a present work, has important application in polmer technolog and metallurg. For example, a number of technical processes concerning polmers involve the cooling of continuous strips extruded from a die b drawing them through a quiescent stretched. Other examples are continuous casting of metals, manufacture of plastic and rubber sheets, glass blowing, the extruded material issues through die and spinning of fibers. In view of these applications, Afif [] studied Similarit 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. Ali [2] presented the thermal boundar laer on a power law stretched surface with suction or injection. Borkakoti and Bharali [4] have studied hdromagnetic flow and heat transfer between two horizontal plates. Chakrabarti and Gupta [5] reported on the hdromagnetic flow and heat transfer over a stretching sheet. The effect of heat transfer of a continuous, stretching surface with suction or blowing has been investigated b Chen and Char [6]. Crane [7] discussed the flow past a stretching plate. Cortell [8] trailed the investigation of viscous dissipation and radiation on the thermal boundar laer over a nonlinearl stretching sheet. Heat and mass transfer on a moving continuous flat plate with suction or injection has been studied b Erickson et al. [9]. Giterere et al. [] analzed MHD flow in porous media over stretching surface in a rotating sstem with heat and mass transfer. Guptaand Gupta [] inspected the effect of heat and mass transfer on a stretching sheet with suction or blowing. Hazem [2] discussed the effect of suction and injection on unstead Couette flow with variable properties.boundar laer flow and heat transfer over an unstead stretching vertical surface is analzed b Ishak et al. [4]. The effects of MHD flow past a stretching permeable sheet are examined b Kumaran [5]. Kung and Srivastava [6] analzed analtic transient solutions of a clindrical heat equation with oscillating heat flux. Several researches used different approaches and find some applications [3, 3, 7, 8, 9, 2, 2, 22] Mukhopadha [23] considered the unstead boundar laer flow and heat transfer past a porous stretching sheet in presence of variable viscosit and thermal diffusivit. Subhas and Mahesha [24] estimated the heat transfer in MHD viscoelastic fluid flow over a stretching sheet with variable thermal conductivit, non-uniform heat source and radiation. Sheikholeslami [25] inspected the rotating MHD viscous flow and heat transfer between stretching and porous surfaces using analtical method. The purpose of present stud is to examine the effects of mass transfer,viscous dissipation and suction parameter on two dimensional stead hdromagnetic viscous fluid flow between two parallel plates in the presence of Corresponding author kiran.kumariprajapati@gmail.com c 27 NSP

2 82 K. Kumari, M. Goal: Heat and mass transfer flow of MHD... thermal radiation, where the lower plate is stretching and porous.in this stud Runge-Kutta forth order method successfull applied to solve the governing equations. 2 Mathematical Formulation Consider stead two dimensional flow of an electricall conducting incompressible dissipative fluid between two vertical parallel plates when the lower plate is a permeable and stretching plate and two equal opposite forces are applied along the x - axis, so that the wall is stretched (a >) keeping the origin fixed. The flow is assumed to be in the x - direction which is taken along the plates and - axis is normal to it. A uniform magnetic field B is acting along - axis. The lower plate at = and upper plate at = h of the channel are kept at temperature T w and T h. The lower plate is subjected to a constant flow suction with a velocit v. Under these assumptions, the Geometr and governing equations of the problem are: ρ is the base fluid densit, υ is the dnamic viscosit, k is the thermal conductivit, c p is the specific heat and D is the chemical molecular diffusivit. The boundar conditions applicable to the present channel flow are { u=ax,v= v,t = T w,c= C w at = u=, v=,t = T h,c= C h at =h B using the Rosseland approximation consider the radiative heat flux for opticall thick fluid is given b q r = 4σ T 4 3k where is the Stefan-Boltzmann constant and k is the mean absorption coefficient. Assuming that the difference in temperature within the flow are sufficientl small such that T 4 can be expressed as a linear function of the temperature, we expand T 4 in a Talor s series about T h and neglecting higher order terms, thus (5) (6) T 4 = 4Th 3 T 3T h 4 Let us introduce the following similarit transformations η = h, u=ax f (η), v= ah f(η), (η)= T T h T w T h, (η)= C C h C w C h (7) and non-dimensional parameters M = σb2 h 2 ρυ, Re= ah2 Ec= (ax)2 C p (T w T h ), S= v ah, Sc= υ D υ kk, R=, Pr= µc p 4σTh 3 k, (8) Fig. : Geometr of the Problem u x + v = () where M is the magnetic parameter, Re is the Renolds number, R is the radiation parameter, Pr is the Prandtl number, Ec is the Eckert number, S suction parameter (S > ), Sc Schmidt number and a prime denotes differentiation with respect to η. The above equations (6)-(8) reduces the equations ()-(4) into the following sstem of non-dimensional equations u u x + v u = υ 2 u 2 σb2 u ρ (2) f Re f 2 + Re f f M f = (9) u T x + v T = k ρc p 2 T 2 ρc p q r + υ ρc p ( u )2 (3) u C x + v C = D 2 C 2 (4) Here u and v are the velocities in the x, directions respectivel, T is the temperature, C is the concentration, (3R+4) + 3RePr ( R f + Ec f 2) = () + ReSc f = () Non- dimensional boundar conditions are { f =, f = S, =, = at η = f =, f =, =, = at η = (2) c 27 NSP

3 Appl. Math. Inf. Sci. Lett. 5, No. 3, 8-87 (27) / 83 The local skin-friction coefficient (τ) on the lower stretching wall is given b τ = τ w ρ(ax) 2 Rex h τ = f () where the wall shear stress ma be written as ( ) u τ w = µ = The rate of heat transfer (Nusselt number) and rate of mass transfer (Sherwood number) on the lower stretching wall are defined below ( T ) Nu=h Nu= () T h T w Sh=h ( C ) C h C w = = Sh= () 3 Numerical results and discussions Using a similarit transformation, the governing equations ()-(4) are transformed into a sstem of non-dimensional ordinar differential equations (9)-(), which are solved numericall using forth order Runge - Kutta method along with shooting technique with the help of MATLAB (R2a) software and throughout the computations we emplo M =.5,Re = 2,R =.8,Pr = 3,Ec=.8,Sc=.3,S= M =.5, 3., 7.,.6.8 Fig. 3: Temperature profiles for various values of M with Re = 2, R=.8 Pr=3,Ec=.8, S= M =.5, 3., 7., Fig. 4: Concentration profiles for various values of M with Re= 2, Sc=.3, S= f M =.5,., 2., Fig. 2: Velocit profiles for various values of M with Re = 2, S=.5. The results are illustrated graphicall in figures (2)-(4) for the non-dimensional parameters, namel the magnetic parameter M, suction parameter S, Renolds number Re, radiation parameter R, Prandtl number Pr, Eckert number Ec,Schmidt number Sc on velocit, temperature and concentration profiles. Also, the skin friction coefficient, Nusselt and Sherwood numbers are discussed and given in tabular form. For the case of different values of magnetic field parameter M, the velocit, temperature and concentration profiles are shown in Figures (2)-(4). It is noticed from figure (2) that the velocit decreases with increase the magnetic parameter M. This is due to the fact that an increase in transverse magnetic field develops the opposite force to the flow direction (Lorentz force), which results in retarding force on the velocit field. This force has the tendenc to reduce the velocit boundar laer and increase the thermal boundar laer thickness. It is clear from the Figures (3) and (4), that an increase in the magnetic field parameter enhances the temperature and concentration profile. Influence of suction parameter S over velocit, temperature and concentration profiles are displaed in Figures (5)-(7). We know that the effect of suction is to bring the fluid closer to the surface and, therefore, to c 27 NSP

4 84 K. Kumari, M. Goal: Heat and mass transfer flow of MHD f.6 S =.5,.,.5, 2..6 S =.5,.,.5, Fig. 5: Velocit profiles for various values of S with M =.5, Re= Fig. 7: Concentration profiles for various values of S with M =.5, Re=2, Sc= S =.5,.,.5, 2.6 f Re = 2, 4, 6, Fig. 6: Temperature profiles for various values of S with M=.5, Re=2, R=.8 Pr= 3,Ec= Fig. 8: Velocit profiles for various values of Re with M =.5, S=.5. reduce the thermal boundar laer thickness. Thus, these plots show that velocit, temperature and concentration profiles decreases significantl with increasing suction parameter. Figures (8) - () depict the effects of the Renolds number Re on velocit, temperature and concentration profiles, respectivel. It is observed from the figures that an increase in the Renolds number decreases the velocit, temperature and concentration flow profile. Figure () demonstrates the variations of temperature distribution for different value of Eckert number Ec. This figure reveals that an increase in the value of the Eckert number indicates that there is an increase in the viscous dissipation and thereb increases the temperature of the fluid as the dissipative force adds energ to the fluid. The temperature distribution for different values of radiation parameter R and Prandtl number Pr are shown in Figure (2)-(3). From this figures it is noticed that temperature decreases with the increase in radiation parameter and Prandtl number. From Figure (2), it is observed that increase in the radiation parameter R decreases the temperature distribution in the thermal boundar laer. This is because large values of radiation parameter correspond to an increase the conduction over radiation, thereb decreasing the thickness of the thermal boundar laer. From Figure (3), The Prandtl number defines the ratio of momentum diffusivit to thermal diffusivit. Therefore, an increase in the Prandtl number means slow rate of thermal diffusion. Figure (4) illustrates the concentration profiles for different values of the Schmidt number Sc. The Schmidt number defined as the ratio of viscosit and molecular diffusivit, and it is phsicall related to the relative thickness of the mass-transfer boundar laer. It is clear that the concentration of the fluid reduces with an increase of the Schmidt number. As can be expected, the concentration boundar laer thickness decreases as Schmidt number increases, with all other parameters fixed. All these phsical behaviors are due to the increase of Schmidt number means a decrease of molecular diffusion. c 27 NSP

5 Appl. Math. Inf. Sci. Lett. 5, No. 3, 8-87 (27) / Re = 2, 4, 6, Ec =.8,.5, 2., Fig. 9: Temperature profiles for various values of Re with M =.5, R=.8, Pr=3, Ec=.8, S=.5. Fig. : Temperature profiles for various values of Ec with M =.5, Re=2, R=.8, Pr=3,S=.5..8 Re = 2, 4, 6, R =.8,.5, 2, Fig. : Concentration profiles for various values of R with M =.5, Re=2, S=.5. Fig. 2: Temperature profiles for various values of R with M =.5, Re=2, Pr=3, Ec=.8, S=.5. Table : Effects of various phsical parameters on local skin friction coefficient f and Nusselt number at the lower plate. M Re S R Pr Ec f () () To assess the results of the analtical method, we have tabulated our local skin friction coefficient, Nusselt Table 2: Effects of various phsical parameters on Sherwood number at the lower plate. M Re S Sc () number () and Sherwood number () for different values of involving phsical parameters at the lower stretching plate in Table and Table 2. It can be seen that the skin friction coefficient decreased b increasing the magnetic parameter M, Renolds number c 27 NSP

6 86 K. Kumari, M. Goal: Heat and mass transfer flow of MHD Pr =.7,.5, 2.3, Fig. 3: Temperature profiles for various values of Pr with M =.5, Re=2, R=.8, Ec=.8, S= Sc =.3,.,.5, Fig. 4: Concentration profiles for various values of Sc with M =.5, Re=2, S=.5. concentration. The findings of the numerical results can be summarized as follows: It is found that velocit, temperature and concentration decreases with the increase of variable suction parameter and Renolds number. Also, the magnetic parameter has the effect of decreasing the velocit profile. An increase in magnetic field parameter, Eckert number and Prandtl number enhances the temperature profile. However, it decreases with the increase of radiation parameter. In the case of magnetic parameter and Schmidt number increases, the concentration field increases. A rise in the values of magnetic parameter, Renolds number and suction parameter reduces the local skin friction for stretching wall and enhance the local skin friction for upper wall. Nussult number increases with decreasing values of magnetic parameter, Eckert number and increases with increasing values of Renolds number, suction parameter, radiation parameter, Prandtl number for the stretching wall. Renolds number, suction parameter, Schmidt number have the tendenc to enhance the rate of mass transfer (Sherwood number) and magnetic parameter have the tendenc to reduce the Sherwood number for the stretching wall. Acknowledgement Authors are ver much thankful to the reviewers for giving the constructive suggestion which helped us in improving the qualit of the manuscript. Re and suction parameter S. Also it is noticed that the Nusselt number increases b increasing the Renolds number Re, suction parameter S, radiation parameter R and Prandtl number Pr, whereas it decreased with increasing the magnetic parameter M and Eckert number Ec. From Table 2, it is observed that the Sherwood number increased b increasing Renolds number Re, suction parameter S and Schmidt number Sc, whereas it decreases with the increasing value of the magnetic parameter M.. 4 Conclusion A theoretical analsis is performed to stud the heat and mass transfer flow in the presence of thermal radiation and viscous dissipation between two parallel plates, where the lower plate is stretching and porous. The Runge - Kutta Method along with shooting technique is emploed to find the results of velocit, temperature and References [] A.A Afif, Communication in Nonlinear Science and Numerical Simulation 4(5), (29). [2] M.E Ali, International Journal of Heat and Fluid Flow 6, (995). [3] I. Ahmad, V.N Mishra, R. Ahmad and M. Rahaman, Springer Plus 5, -6 (26). [4] A.K Borkakoti and A. Bharali,, Quarterl of Applied Mathematics 4(4), (983). [5] A. Chakrabarti and A. S. Gupta Quarterl of Applied Mathematics 37(), (979). [6] C.K and M.I Char, Journal of Mathematical Analsis and Applications 35(2), (988). [7] L.J Crane, Zeitschrift fur angewandte Mathematik und Phsik 2(4), (97). [8] R. Cortell, Phsics Letters A 372(5), (28). [9] L.E Erickson, L.T Fan and V.G Fox, Industrial and Engineering Chemistr Fundamentals 5(), 9-25 (966). [] K. Giterere, M. Kinanjui, and S. Uppal, International Journal of Pure and Applied Mathematics 4, 9-32 (22). c 27 NSP

Appl. Math. Inf. Sci. Lett. 5, No. 3, 8-87 (27) / www.naturalspublishing.com/journals.asp 87 [] P.S Gupta and A.S Gupta, The Canadian Journal of Chemical Engineering 55, 744-746 (977). [2] A.

Pop, Meccanica 44(4), 369-375 (29). [5] V. Kumaran, A.K Banerjee, A.V Kumar and K. Vajravelu, Applied Mathematics Computation 2(), 26-32 (29). [6] K.Y Kung and H.

7 Appl. Math. Inf. Sci. Lett. 5, No. 3, 8-87 (27) / 87 [] P.S Gupta and A.S Gupta, The Canadian Journal of Chemical Engineering 55, (977). [2] A. Attia Hazem, Kragujevac Journal of Science 32, 7-24 (2). [3] S. Husain, S. Gupta and V.N Mishra, Fixed Point Theor and Applications 34, -2 (23). [4] A. Ishak, R. Nazar and I. Pop, Meccanica 44(4), (29). [5] V. Kumaran, A.K Banerjee, A.V Kumar and K. Vajravelu, Applied Mathematics Computation 2(), (29). [6] K.Y Kung and H.M Srivastava, Applied Mathematics Computation 95(2), (28). [7] L.N Mishra and R.P Agarwal, Dnamic Sstems and Applications 25, (26). [8] L.N Mishra, R.P Agarwal and M. Sen, Progress in Fractional Differentiation and Applications 2(3), (26). [9] L.N Mishra and M. Sen, Applied Mathematics and Computation 285, (26). [2] L.N Mishra, H.M Srivatava and M. Sen, International Journal of Analsis and Applications (), - (26). [2] V.N Mishra, Thesis, Indian Institute of Technolog, Roorkee, Uttarakhand, India, (27). [22] V.N Mishra and L.N Mishra, International Journal of Contemporar Mathematical Sciences 7(9), (22). [23] S. Mukhopadha, Heat Mass Transfer 52, (29). [24] A.M Subhas and N. Mahesha, Applied Mathematical Modelling 32(), (28). [25] M. Sheikholeslami, H.R Ashornejad, D.D Ganji and A. Kolahdooz, Mathematical Problems in Engineering 2, - 7 (2). Kiran Kumari working as an research scholar in Department of Mathematics, Universit of Rajasthan, Jaipur (India). She is pursuing research work under the supervision of Mamta Goal. Her area of research are Fluid Dnamics, MHD, Heat and Mass Transfer in Channels and mathematical modeling with numerical method. Mamta Goal is currentl working as Associate Professor, in Department of Mathematics, Universit of Rajasthan, Jaipur (India). Her field of research are: Fluid Dnamics, Boundar Laer theor of Newtonian and Non-Newtonian fluids, MHD, Heat and Mass Transfer in Channels. She has published man research papers in national and international journals. c 27 NSP

Magnetic Field and Chemical Reaction Effects on Convective Flow of

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