MHD Flow and Heat Transfer over an. Exponentially Stretching Sheet with Viscous. Dissipation and Radiation Effects

Transcription

1 Applied Mathematical Sciences, Vol. 7, 3, no. 4, 67-8 MHD Flow and Heat Transfer over an Exponentially Stretching Sheet with Viscous Dissipation and Radiation Effects R. N. Jat and Gopi Chand Department of Mathematics, University of Rajasthan, Jaipur 34, India. Abstract The steady two-dimensional laminar flow of a viscous incompressible electrically conducting fluid over an exponentially stretching sheet in the presence of a uniform transverse magnetic field with viscous dissipation and radiative heat flux is studied. By suitable similarity transformations, the governing boundary layer equations are transformed to ordinary differential equations and solved numerically by standard techniques. The effects of various parameters like, Magnetic and Radiation parameters, Prandtl number and Eckert number for velocity and temperature distributions have been discussed in detail with graphical representation. Keywords: Viscous dissipation; Exponentially stretching sheet; MHD; Boundary layer flow; Radiation Introduction The flow of a viscous incompressible fluid over a stretching sheet has many applications in manufacturing industries and technological process, such as, glass-fiber production, wire drawing, paper production, plastic sheets, metal and polymer processing industries and many others. The boundary layer flow over a stretching sheet was first studied by Sakiadis [4]. Later, Crane [8] extended this idea for the two dimensional flow over a stretching sheet problem. Gupta and Gupta [],Carragher and Crane [], Dutta et al. [5] studied the heat transfer in the flow over a stretching surface taking into account different aspects of the problem. Magyari and Keller [7] investigated the study of boundary layers on an

2 68 R. N. Jat and Gopi Chand exponentially stretching continuous surface with an exponential temperature distribution. Aboeldahab and Gendy [6] studied the radiation effects on MHD free convective flow of a gas past a semi-infinite vertical plate with variable thermophysical properties for higher-temperature difference. Many other problems on exponentially stretching surface were discussed by Raptis et al. [], Partha et al. [9] and Sajid and Hayat []. Jat and Chaudhary [3-6] studied the MHD boundary layer flow over a stretching sheet for stagnation point, heat transfer with and without viscous dissipation and Joule heating. Bidin and Nazar [3] investigated the Numerical solution of the boundary layer flow over an exponentially stretching sheet with thermal radiation. Recently Ishak [] investigated the thermal radiation effects on hydro-magnetic flow due to an exponentially stretching sheet. Realizing the increasing technical applications of MHD effects, the present paper studies the problem of MHD boundary layer flow over an exponentially stretching sheet with viscous dissipation and radiation effects. Formulation of the Problem Consider a steady two dimensional laminar flow of a viscous incompressible electrically conducting fluid over a continuous exponentially stretching surface. The x-axis is taken along the stretching surface in the direction of motion and y- axis is perpendicular to it. A uniform magnetic field of strength B is assumed to be applied normal to the stretching surface (fig. ). The magnetic Reynolds number is taken to be small and therefore the induced magnetic field is neglected. The surface is assumed to be highly elastic and is stretched in the x-direction with x a velocity U = U e L. All the fluid properties are assumed to be constant throughout the motion. Under the usual boundary layer approximations, the governing boundary layer equations by considering the viscous dissipation and radiation effects are: u v + = x y () u u u σ B u + v = υ u x y y ρ () T T T qr u ρcp u + v = κ + µ σ B + u x y y y y (3)

3 MHD flow and heat transfer 69 Fig. : Physical modal and coordinate system. Where u and v are the velocities in the x- and y- directions respectively, ρ is the µ density of the fluid, µ is the dynamic viscosity, ν = is the kinematic viscosity, ρ C is the specific heat at constant pressure, κ is thermal conductivity of the fluid p under consideration, q r is the radiative heat flux, T is the temperature. The boundary conditions are: x L L y = : u = U w = Ue, v =, T = T + Te y : u, T (4) Where U, T and L are the reference velocity, temperature and length respectively. The radiative heat flux qr is simplified by using Rosseland approximation (Rosseland [7]) as: 4 4α T qr = (5) 3β y Where α is the Stefan- Boltzmann constant and β is the mean absorption coefficient. This approximation is valid at points optically far from the boundary surface and it is good for intensive absorption, which is for an optically thick boundary layer. It is assumed that the temperature difference with in the flow 4 such that the term T may be expressed as a linear function of temperature. 4 Hence, expanding T by Taylor series about T and neglecting higher-order terms gives: T 4T T 3T (6) Using equation (5) and (6), equation (3) reduces to: x

4 7 R. N. Jat and Gopi Chand 3 T T 6αT T u ρcp u + v = κ + + µ σ B u + x y 3β y y (7) Analysis The equation of continuity () is identically satisfied if we choose the stream function ψ such that ψ ψ u =, v = (8) y x The momentum and energy equations can be transformed into the corresponding ordinary differential equations by introducing the following similarity transformations: where x L ( x, y) = U Le f ( ) ψ ν T T T = e x L θ ( ) x U e L y = (9) ν L Then, the momentum and energy equations () and (7) are transformed to: ''' ' '' ' f ( f ) + ff Mf = () 4 '' ' ' '' ' + R Pr ( f f Ec( f ) MEc( f ) ) 3 θ + θ θ + + = () The corresponding boundary conditions are: = : f = ' f =, θ = ' : f θ () Where prime ( ) denote the differentiation with respect to and dimensionless parameters are: σ B L M = x (Magnetic parameter) L ρu e U Ec = (Eckert number) T c p µc Pr = p κ (Prandtl number) 3 4αT R = βκ (Radiation parameter) (3)

5 MHD flow and heat transfer 7 The physical quantities of interest are the skin-friction coefficient c f and heat transfer rates i.e. the Nusselt number Nu are: u µ y τ w y= c f = = ρu w ρu w '' c f = f () (4) Re and T x y y= o Nu = T T where w ' Re () Nu = θ (5) Re U L ν = (Reynold number) (6) Result and Discussion The set of nonlinear ordinary differential equations () and () with boundary conditions () were solved numerically using Runge - Kutta forth order algorithm with a systematic guessing of f and θ by the shooting technique until the boundary conditions at infinity are satisfied. The step size. is used while obtaining the numerical solution and accuracy up to the seventh decimal place i.e., which is very sufficient for convergence. In this method, we choose suitable finite values of, say, which depend on the values of the parameter used. The computations were done by a program which uses a symbolic and computational computer language Matlab. The shear stress which is proportional to f and the rate of heat transfer which is proportional to θ are tabulated in Table. for different values of Parameters. It is observed from the table that the shear stress decreases and heat transfer rate increases as Magnetic Parameter increases. Also the Nusselt number decreases for increasing value of Ec for a given Pr, whereas it is increases for increasing value of Pr for a given value of Ec. The velocity profile f ' ( ) for different values of the magnetic parameter M is shown in fig.. It is observed that velocity boundary layer thickness decrease with the increasing values of M. It is obvious, because the increasing value of M tends to the

6 7 R. N. Jat and Gopi Chand increasing of Lorentz force, which produces more resistance to the transport phenomena. The temperature profiles for different values of M, Pr, Ec and R are presented in figure 3 to figure 3. It is observed from the figures that the boundary conditions are satisfied asymptotically in all the cases, which supporting the accuracy of the numerical results obtained. All the figures shows that increasing value of any parameter except the Prandtl number, result is increase the thermal boundary layer, whereas increase in Prandtl number is to decrease the thermal boundary layer. Acknowledgements This work has been carried out with the financial support of C.S.R.I in the form of J.R.F awarded to one of the author (Gopi Chand). References. A.Ishak, MHD boundary layer flow due to an exponentially stretching sheet with radiation effect, Sains Malaysiana, 4(), A.Raptis, C. Perdikkis and H.S. Takhar, Effect of Thermal Radiation on MHD Flow, Int. J. Heat Mass Transfer, 53(4), B.Bidin and R. Nazar, Numerical solution of the boundary layer flow over an exponentially stretching sheet with thermal radiation, European journal of scientific research, 33(9), B.C. Sakiadis, Boundary-layer behaviour on continuous solid surfaces: I. Boundary-layer equations for two-dimensional and axi-symmetric flow, Am Inst Chem Eng Journal, 7(96), B.K. Dutta, P. Roy and A.S. Gupta, Temperature field in flow over a stretching surface with uniform heat flux, Int. Comm. Heat Mass Transfer. (985), E.M. Aboeldahab and M.S. EI Gendy, Radiation effect on MHD free convective flow of a gas past a semi-infinite vertical plate with variable thermo physical properties for high-temperature difference, Can. J. Phys., 8(), E. Magyari and B. Keller, Heat and transfer in the boundary layers on an exponentially stretching continuous surface, J. Phys. D: Appl.Phys., 3(999), L.J. Crane, Flow past a stretching sheet, Z. Angew. Math. Phys., (97), M.K. Partha, P.V.S.N. Murthy and G.P. Rajasekhar, Effect of viscous dissipation on the mixed convection heat transfer from an exponentially stretching surface, Heat Mass Transfer, 4(5),

7 MHD flow and heat transfer 73. M. Sajid and T. Hayat, Influence of thermal radiation on the boundary layer flow due to an exponentially stretching sheet, Int. Comm. Heat Mass Transfer, 35(8), P. Carragher and L.J. Crane, Heat transfer on a continuous stretching sheet, Z. Angew. Math. Mech., 6(98), P.S. Gupta and A.S. Gupta, Heat and mass transfer on stretching sheet with suction or blowing, Can. J. Chem. Eng., 55(977), R.N. Jat and S. Chaudhary, Magnetohydrodynamic boundary layer flow near the stagnation point of a stretching sheet, IL NUOVO CIMENTO, 3(8), R.N. Jat and S. Chaudhary, MHD flow and heat transfer over a stretching sheet, Applied Mathematical Science, 3(9), R.N. Jat, S. Chaudhary, Unsteady magnetohydrodynamic boundary layer flow over a stretching surface with viscous dissipation and joule heating, IL NUOVO CIMENTO, 4(9), R.N. Jat and S. Chaudhary, Radiation effects on the MHD flow near the stagnation point of a stretching sheet. Z. Angew. Math. Phys., 6(), S. Rosseland, Theoretical Astrophysics. New York: Oxford University, (936).

8 74 R. N. Jat and Gopi Chand Ec Pr R M.5.. f '' () θ ' ()

9 MHD flow and heat transfer M=,.5,.,. f ' ( ) Fig. : Velocity profile against for various values of Magnetic parameter M M=,.5,.,. θ ( ) Fig. 3: Temperature distribution against for various values of Magnetic parameter M for Pr =., Ec=,R=.

10 76 R. N. Jat and Gopi Chand M =,.5,.,..6 θ ( ) Fig. 4: Temperature distribution against for various values of Magnetic parameter M for Pr =., Ec =, R = M =,.5,.,..6 θ ( ) Fig. 5: Temperature distribution against for various values of Magnetic parameter M for Pr =., Ec =, R =..

11 MHD flow and heat transfer M =,.5,.,. θ ( ) Fig. 6: Temperature distribution against for various values of Magnetic parameter M for Pr =., Ec=.,R= M =,.5,.,. θ ( ) Fig. 7: Temperature distribution against for various values of Magnetic parameter M for Pr =., Ec=.,R=.5.

12 78 R. N. Jat and Gopi Chand M=,.5,.,. θ ( ) Fig. 8: Temperature distribution against for various values of Magnetic parameter M for Pr =., Ec =., R = θ ( ).5 M =,.5,., Fig. 9: Temperature distribution against for various values of Magnetic parameter M for Pr = 7., Ec =., R =.

13 MHD flow and heat transfer M =,.5,.,. θ ( ) Fig. : Temperature distribution against for various values of Magnetic parameter M for Pr = 7., Ec =., R = M =,.5,.,. θ ( ) Fig. : Temperature distribution against for various values of Magnetic parameter M for Pr = 7., Ec =., R =..

14 8 R. N. Jat and Gopi Chand R =,.5,..6 θ ( ) Fig. : Temperature distribution against for various values of Radiation parameter R for Pr =., Ec =., M = Ec =,.,.5.6 θ ( ) Fig. 3: Temperature distribution against for various values of Eckert number Ec for Pr =., M =., R =.. Received: September,