Viscous Dissipation Effect on Steady free Convection and Mass Transfer Flow past a Semi-Infinite Flat Plate

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1 Journal of Computer Science 7 (7): 3-8, 0 ISSN Science Publications Viscous Dissipation Effect on Steady free Convection and Mass Transfer Flo past a Semi-Infinite Flat Plate Palanisamy Geetha and Muthamperumal Bhavathikannu Krishna Moorthy Department of Mathematics, Bannari Amman Institute of Technology, Sathyamangalam, , India Department of Mathematics, Institute of Road and Transport Erode, , India Abstract: Problem statement: A steady to-dimensional free convection and mass transfer flo past a continuously moving semi-infinite flat plate is studied theoretically, by taking into account the viscous dissipation effect. Approach: The governing equations are transformed into a set of simultaneous ordinary differential equation by using suitable similarity transformations. The coupled differential equations are integrated using the Runge-Kutta Gill method together ith the shooting technique. Results: Numerical results ere presented for the distribution of velocity, temperature and concentration profiles ithin the boundary layer. Conclusion: The effects of varying parameter Gb, the Gebhart number, Sc, Schmidt number and Pr, Prandtl number on the velocity, temperature and concentration profiles ere displayed graphically for different values of parameters entering into the problem. Significant changes ere observed in heat and mass transfer coefficient, due to viscous dissipation in the medium. In addition, the skin-friction coefficient, Nusselt number and Sherood number ere shon in a tabular form. Key ords: Heat transfer, mass transfer, moving surface, prandtl number, gebhart number, schmidt number, viscous dissipation, skin-friction coefficient, sherood number INTRODUCTION Oing to their numerous applications in industrial manufacturing process, the problem of heat and mass transfer in the boundary layers of a continuously moving semi-infinite flat plate has attracted the attention of researchers for the past 3 decades. Some of the application areas are hot rolling, study production, metal spinning, draing plastic films, glass bloing, continuous casting of metals and spinning of fibers. Annealing and thinning of copper ire is another example. In all these cases, the quality of the final product depends on the rate of heat and mass transfer at the moving surface. By draing the strips in an electrically conducting fluid subjected to a magnetic field the rate of cooling can be controlled and the final products of desired characteristic might be achieved. Flo in the boundary layer of a viscous fluid on a moving continuous solid surfaces as investigated by Sakiadis (96a). It as observed that the boundary layer groth is in the direction of motion of the continuous solid surface and is different from that of the Blasius flo past a flat plate. Still the boundary layer theory and the basic differential equations are applicable. Many authors have attacked this problem, hoever, this attack has been limited to some constrains on the surface velocity and temperature distribution. In addition to these investigations, experimental and theoretical studies of the flo and temperature fields in the boundary layer on a continuous moving surface have been made by Tsou et al. (967) for different values of the Prandtl number. Measure of the laminar velocity field ere in excellent agreement ith the analytical predictions. The heat transfer of above problem of different physical situations have been studied. Viscous dissipation hich, appears as a source term in the fluid flo generates appreciable temperature, gives the rate at hich mechanical energy is converted into heat in a viscous fluid per unit volume. This effect is of particular significant in natural convection in various devices that are subjected to large variation of gravitational force or that operate at high rotational speeds, pointed by Gebhart (96) in his study of viscous dissipation on natural convection in fluids. Similarity solutions for the same problem ith exponential variation of all Corresponding Author: P. Geetha, Department of Mathematics Bannari Amman Institute of Technology, Sathyamangalam, , India 3

2 J. Computer Sci., 7 (7): 3-8, 0 temperature as obtained by Gebhart and Mollendorf (969). It is observed that the effect of viscous dissipation is more predominant in vigorous natural convection and mixed convection processes. They also have shon the existence of similarity solution for the external flo over an infinite flat vertical surface having an exponential variation of surface temperature. Vajravelu and Hadjinicolaou (993) studied the heat transfer characteristic in the laminar boundary layer of a viscous fluid over a linearly stretching surface ith variable all temperature subject to suction or bloing. They considered the effects of viscous dissipation and internal heat generation. The effect of chemical reaction, heat and mass transfer on a laminar flo along a semi infinite horizontal plate have been studied by Anjali and Kandasamy (999). Studied the effect of viscous dissipation on heat transfer in flo past a continuously moving semi-infinite flat plate. Natural convection boundary layer flo over a sphere of a viscous incompressible electrically conducting fluid in the presence of magnetic field and heat generation ith the effects of viscous dissipation has been investigated by Alam et al. (007) and Subhas Abel et al. (0) studied the effect of viscous and Joules dissipation on MHD flo over a porous nonlinear vertical stretching sheet ith partial slip. Since the pioneering ork of Sakiadis (96b) hich studied the moving plate flo problem, herein various aspects of the problem have been investigated by Cortell (006). Cortell (007a) has orked on viscous flo and heat transfer over a nonlinearly stretching sheet. Cortell (008) further x y investigated on the effects of viscous dissipation and u v + = 0 radiation on the thermal boundary layer, over a nonlinear stretching sheet. Moreover, Cortell (007b) u + v = ν u u u x y y studied the viscous flo and heat transfer over a nonlinearly stretching surface. Raptis and Perdikis (006) studied viscous flo near a non-linear stretching T T k T ν u sheet in the presence of a chemical reaction and u + v = + x y ρcp y cp y magnetic field. Kumar (009) investigated the study of radiation and viscous dissipation effects over a stretching surface subjected to variable heat flux in C C C u + v = D presence of transverse magnetic field. Kishan and x y y Amrutha (00) studied the to-dimensional steady nonlinear MHD boundary layer flo of an incompressible, viscous, electrically conductive and Along ith the boundary conditions: Boussinesq fluid floing over a vertical stretching surface in the presence of uniform magnetic field by u = U 0; v = 0; T = T ; C = C at y = 0 taking into account the viscous dissipation ith heat, u = 0; T T ; C C at y mass transfer chemical reaction and thermal stratification effects. A study on MHD free convective Here: flo of an incompressible viscous dissipative fluid in a = Thermal diffusivity constant 4 an infinite vertical oscillating plate ith constant heat flux has been carried by Kishan et al. (006). Kairi et al. (0) investigated the combined effect of viscous dissipation and radiation on natural convection in non- Darcy porous medium saturated ith non-netonian fluid of variable viscosity. El-Arabay (009) studied the effects of suction/injection and chemical reaction on mass transfer over a stretching surface. Adegbie and Alao (007). investigated the steady-state flo of Netonian liquid ith exponential temperaturedependent viscosity and substantial viscous heat generation beteen symmetrically parallel heated alls ith alls at different temperatures. The aim of the present study is to study the effect of viscous dissipation on heat transfer and mass transfer in flo past a continuously moving semi-infinite flat plate. The analysis shoed that the viscous dissipation have significant influence on the non-dimensional heat and mass transfer coefficients. Mathematical formulation: Consider the todimensional, flo past a continuously moving semiinfinite flat plate. The x-axis is assumed to be taken along the plate and the y-axis normal to the plate. u,v are the velocity components in the x and y directions. No, the governing equations under the boundary layer and Boussinesq approximations may be ritten as: () () (3) (4) (5)

3 J. Computer Sci., 7 (7): 3-8, 0 c p = Specific heat v = Kinematic viscosity of the ambient fluid The above set of partial differential equations are converted into ordinary differential equations using the folloing similarity transformation: U 0 η = y (6) vx ψ (x, y) = ν xu0 f (7) T T θ= T T C C ϕ= C C Where: η = The similarity variable (8) (9) θ = The dimensionless stream function depends on η only Let ψ be the stream function defined such that ψ u = v y = ψ x and so that it automatically The non-dimensional numbers are defined as the viscous dissipation parameter knon as the Gebhart U number, given by Gb =, Schmidt number c (T T ) p given by Sc= v/d and the Prandtl number given by Pr = v/a. MATERIALS AND METHODS The set of Eq. 0- together ith the boundary conditions () have been solved numerically by applying shooting technique along ith Runge-Kutta Gill method.from the process of numerical computation, the skin-friction coefficient, the local Nusselt number and the local Sherood number, hich are respectively proportional to f (0), - θ (0) and - φ (0) are also sorted out and their numerical values are presented in a tabular form. The parameters for the present problem are the local Nusselt number and the local Sherood number, hich indicate physically the rate of heat transfer and the rate of mass transfer respectively. No the heat flux (q ) and the mass flux (M ) at the all are given by: T U = = θ y νx 0 ' q k( ) y= 0 k (T T ) (0) And: C y 0 ' M = D = D (C C ) ϕ (0) y= 0 U νx satisfies the continuity Eq.. The final transformed equation and boundary conditions are: + = (0) ''' '' f f f 0 Pr f GbPr (f ) 0 '' ' '' θ + θ + = () " Sc φ + f φ ' = 0 () With the boundary and initial conditions as: ( ) ( ) ( ) ( ) f 0 = 0;f' 0 = ; θ 0 = ; φ 0 = at η = 0 f'( ) = 0; θ( ) = 0; φ( ) = 0 at η Hence the Nusselt number (Nu) and Sherood number (Sh) are obtained as: x q k T Nu = = Re θ'(0) ( T ) ( ) i.e. Nu Re ( ) ( ) ( ) = θ'(0) and xm Sh = = ( Re ) ϕ'(0) D C C i.e.sh Re = ϕ'(0) here, Re=U 0 x/ν is the Reynold s number. The ordinary differential Eq. 0- along the boundary condition Eq. 3 are solved by giving approximate initial guess values for the missing initial conditions of f '' (0), θ (0), φ (0) and these values are (3) matched ith the corresponding boundary conditions at f () θ () and φ (). Extensive calculations have been 5

4 performed to obtain the flo and temperature fields for a ide range of parameters 0 < Pr 0, 0 < Sc and Gb 0. RESULTS Table shos numerical values of viscous dissipation effects on f (0), θ (0) and φ (0). The profiles for velocity, temperature and concentration are shon in Fig. -7 respectively ith various values of the parameters. J. Computer Sci., 7 (7): 3-8, 0 Table : Viscous dissipation effects on f (0), θ (0) and φ (0) Sc Pr Gb f "(0) θ (0) φ (0) Fig. 3: Effect of viscous dissipation par θ Fig. : Effect of viscous dissipation parameter Gb on non-dimensional velocity f ' Fig. 4: Effect of viscous dissipation parameter Gb on non-dimensional temperature θ Fig. : Effect of viscous dissipation parameter Gb on Fig. 5: Effect of viscous dissipation parameter Gb on non-dimensional temperature θ non-dimensional concentration φ 6

5 J. Computer Sci., 7 (7): 3-8, 0 The non-dimensional heat transfer coefficient and mass transfer coefficient are plotted against the parameter Gb. The heat transfer coefficient and mass transfer coefficient are increased ith the Prandtl number and also as the value of the flo governing parameter is increased. Table presents the effect of Sc, Pr and Gb on the skin friction coefficient f (0), the Nusselt number -θ (0) and the Sherood number - φ (0). The results reveal that the Nusselt number and the Sherood number increases as Sc, Pr and Gb increases. Fig. 6: Effect of viscous dissipation parameter Gb on non-dimensional concentration φ CONCLUSION An increase in the viscous dissipation parameter Gb resulted: Fig. 7: Effect of viscous dissipation parameter Gb on non-dimensional concentration φ An increase in temperature A fall in concentration An increase in heat transfer coefficient and mass transfer coefficient ACKNOWLEDGEMENT The research render their heartfelt thanks to Dr. T. Govindarajulu former Professor and Head of Department of Mathematics, Anna University, Chennai for his generous help. The authors ish to thank the Director IRT Chennai and the Principal IRTT, Erode for their generous help. The authors ish to thank the Chairman, Director, Chief Executive, Principal, Bannari Amman Institute of Technology for their support to do this study. DISCUSSION REFERENCES No e discuss the results. The velocity, temperature and concentration profiles in the boundary Adegbie, K.S. and F.I. Alao, 007. Flo of layer are plotted for selected values of the governing temperature-dependent viscous fluid beteen parameters. The non-dimensional velocity across the parallel heated alls: Exact analytical solutions in boundary layer is plotted for varying values of Gb the presence of viscous dissipation. J. Math. Stat., (,5,0) ith Prandtl number Pr = 0.7, 7, 0 and 3: -4. DOI: /JMSSP Schmidt number Sc = 0.3, 0.6, 0.78 to depict the flo Alam, M.M., M.A. Alim and M.M.K. Chodhury, field. The magnitude of the velocity remains same as the 007. Viscous dissipation effects on MHD natural value of the flo governing parameter increases hich is convection flo over a sphere in the presence of shon in Fig.. heat generation. Nonlinear Analysis: Model. The non-dimensional temperature distribution Control, : inside the boundary layer are presented in Fig. -4. The thermal boundary layer thickness is increased ith the Anjali, S.P. and R. Kandasamy, 999. Effects of increasing of Gb. chemical reaction, heat and mass transfer on The non-dimensional concentration distribution laminar flo along a semi infinite horizontal plate. inside the boundary layer are presented in Fig The Heat. Mass Trans. 35: DOI: concentration decreases ith the increase of Gb /S

6 J. Computer Sci., 7 (7): 3-8, 0 Cortell, R., 006. 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: DOI: 0.06/J.PHYSLETA Cortell, R., 007a. MHD flo and heat transfer of an electrically conducting fluid of second grade in a porous medium over a stretching sheet subject ith chemically reactive species. Chemical Eng. Process., 46: DOI: 0.06/J.CEP Cortell, R., 007b. Viscous flo and heat transfer over a non-linearly stretching sheet. Applied Math. Comput., 84: DOI: 0.06/J.AMC Cortell, R., 008. Effects of viscous dissipation and radiation on the thermal boundary layer over a nonlinearly stretching sheet. Phys. Lett. A, 37: DOI: 0.06/J.PHYSLETA El-Arabay, H.A.M., 009, Exact solution of mass transfer over a stretching surface ith chemical reaction and suction/injection. J. Math. Stat., 5: DOI: /JMSSP Gebhart B., 96. Effect of viscous dissipation in natural convection. J. Fluid. Mech., 4: 5-3. DOI: 0.07/S Gebhart, B. and Mollendorf, 969. Viscous dissipation in external natural convection flos. J. Fluid. Mech., 38: DOI: 0.07/S Kairi, R.R., P.V.S.N. Murthy and C.O. Ng, 0. Effect of viscous dissipation on natural convection in a non-darcy porous medium saturated ith nonnetonian fluid of variable viscosity. Open Transport Phenomena J., DOI: 0.74/ Kishan, N. and P. Amrutha, 00. Effects of viscous dissipation on MHD flo ith heat and mass transfer over a stretching surface ith heat source, thermal stratification and chemical reaction. J. Naval Archit. Marine Eng., 7-8. DOI: 0.339/jname.v7il e/vie/354 Kishan. N., Srihari and Rao, J.A., 006. MHD free convective flo of an incompressible viscous dissipative fluid in an infinite vertical oscillating plate ith constant heat flux. J. Energy Heat Mass Trans., 8: &ETOC=RN&from=searchengine Kumar, H., 009. Radiative heat transfer ith hydromagnetic flo and viscous dissipation over a stretching surface in the presence of variable heat flux. Thermal Sci., 3: DOI: 0.98/TSCI09063K Raptis, A. and C. Perdikis, 006. Viscous flo over a non-linearly stretching sheet in the presence of a chemical reaction and magnetic field. Int. J. Non- Linear Mech., 4: DOI: DOI: 0.06/J.IJNONLINMEC Sakiadis, B.C., 96a. Boundary layer behavior on continuous moving solid surfaces i. boundary layer equations for to-dimensional and axi-symmetric flo. AIChE J., 7: 6-8. DOI: 0.00/aic Sakiadis, B.C., 96b. Boundary layer behavior on continuous solid surfaces: II the boundary layer on a continuous flat surfaces. AIChE J., 7: -5. DOI: 0.00/aic Subhas Abel. M., K.A. Kumar and R. Ravikumara, 0. MHD flo and heat transfer ith effects of buoyancy, viscous and joules dissipation over a nonlinear vertical stretching porous sheet ith partial slip. Sci. Res., 3: DOI: 0.436/eng Tsou, F.K., E.M. sparro and R.J. Goldstein, 967. Flo and heat transfer in the boundary layers on a continuous moving surface. Int. J. Heat Mass Trans., 0: DOI: 0.06/ (67) Vajravelu, K. and A. Hadjinicalaou, 993. Heat transfer in a viscous fluid over a stretching sheet ith viscous dissipation and internal heat generation. Int. Comm. Heat. Mass Trans., 0: DOI: 0.06/ (93)9006-R 8