Unsteady Hydromagnetic Flow past a Stretching Surface with Dissipation and Radiation Effects

D. Vasantha Kumari¹ and S.P. Anjali Devi²

Department of Applied Mathematics, Bharathiar University, Coimbatore – 641046, Tamil Nadu

*Corresponding Author: vasanthi.sheena@gmail.com

ABSTRACT:

This paper deals with an analysis which is performed to investigate the effects of dissipation and radiation over an unsteady MHD flow past a stretching surface. The fluid is assumed to be viscous, incompressible electrically conducting and radiating. The unsteadiness in the flow is caused by the time-dependence of the velocity of the stretching surface. The governing partial differential equations of the flow are transformed into nonlinear ordinary differential equations using Similarity Transformations. Numerical solution of the non-linear differential equations are obtained using efficient shooting technique such as Runge-Kutta Fourth order based shooting method along with Nachtsheim-Swigert Iteration technique for the satisfaction of asymptotic boundary conditions. Numerical computations for velocity and temperature are carried out for different values of the non-dimensional parameters involved in the study such as Unsteadiness parameter (A), Magnetic parameter (M), Prandtl number (Pr), Eckert number (Ec), Radiation parameter (R). The Skin Friction Co-efficient and the Rate of heat transfer at the plate are also calculated numerically.

KEY WORDS: Unsteady, Stretching Surface, MHD, Viscous dissipation, Joule’s dissipation, Radiation.

INTRODUCTION:

A study of boundary layer behavior over stretching surface is an important type of flow occurring in a number of technical processes and hence it has attracted the attention of several researchers in view of its application in different areas such as continuous aerodynamic extrusion of plastic, polymer and metallic sheets or filaments from a die etc. The study of a viscous fluid flow due to a stretching surface is an important type of flow occurring in several engineering processes. Such processes are wire drawing, heat-treated materials travelling between a feed roll and a wind-up roll or materials manufactured by extrusion, glass-fiber and paper production, cooling of metallic sheets or electronic chips, crystal growing, drawing of plastic sheets and many others. In these cases, the final product of desired characteristics depends on the rate of cooling that involves in the process and the process of stretching. In special, the effect of the magnetic field on the boundary layer flow over a continuous moving surface is useful during the cooling process in the presence of an electrolyte bath.

Sakiadis (1961) studied theoretically the boundary layer on a continuous semi-infinite sheet moving steadily through an otherwise quiescent fluid environment. Erickson et al. (1966) considered the study of heat and mass transfer in the laminar boundary layer flow of moving flat surface with constant surface velocity and temperature considering the effect of suction/injection.

Tsou et al. (1967) investigated the heat transfer effects of moving solid surface having constant velocity and temperature. Crane (1970) analysed the flow over a linearly stretching sheet for the steady two dimensional problem. These types of flows usually occur in the drawing of plastic films and artificial fibers. Chakrabarti and Gupta (1979) obtained the analytical solution for linear stretching problem with hydromagnetic effect.

Soundalgekar and Ramanamurthy (1980) investigated the constant surface velocity case with power law temperature variation. Dutta et al. (1985) considered the temperature field in flow over a stretching sheet with uniform heat flux. Later, the effects of variable surface temperature and variable surface heat flux over the heat transfer characteristics of a linearly stretching sheet was analyzed by Chen and Char (1988). Vajravelu and Rollins (1992) investigated the effect of Heat transfer in an electrically conducting fluid over a stretching surface. Kumar et al. (2002) studied MHD flow and heat transfer on a continuously moving vertical plate.

Gebhart (1962) was the first who studied the problem taking into account the viscous dissipation. Vajravelu and Hadjinicalaou (1993) studied the heat transfer characteristics over a stretching surface with viscous dissipation in the presence of internal heat generation or absorption. Amin (2003) studied the combined effect of viscous dissipation and Joule heating on MHD forced convection flow over a non-isothermal horizontal cylinder embedded in a fluid saturated porous medium. Pantokratoras (2005) studied the effect of viscous dissipation in natural convection in a new way. Alam et al. (2007) considered the effect of viscous dissipation in natural convection over a sphere. Mahmoud (2007) investigated variable viscosity effects on MHD flow in presence of radiation. Copiello and Fabbri (2008) studied the effect of viscous dissipation on the heat transfer in sinusoidal profile finned dissipaters. Cortell (2008) studied the effects of viscous dissipation and radiation on the thermal boundary layer over a nonlinearly stretching sheet. Samad and Mohebujjaman (2009) investigated the case along a vertical stretching sheet in the presence of magnetic field and heat generation.

With regard to unsteady flows, Elbashbeshy and Bazid (2004) have presented similarity solutions of the boundary layer equations, which describe the unsteady flow and heat transfer over an unsteady stretching sheet. Recently Pal and Talukdar (2010) presented perturbation analysis of unsteady MHD mixed convective heat and mass transfer in a boundary layer flow with thermal radiation and chemical reaction effects. However, there are no attempts in literature so far to consider the effects of thermal radiation on unsteady hydromagnetic flows with dissipation effects and heat transfer past a continuous moving surface and hence the present work is considered.

2. FORMULATION OF THE PROBLEM:

Consider the two dimensional, unsteady, hydromagnetic flow of a viscous, incompressible, electrically conducting and radiating fluid with dissipation and radiation effects over a stretching surface which issues from a slot at the origin. The fluid is considered to be gray, absorbing, emitting but non-scattering medium and the Rosseland approximation is used to describe the radiative heat flux in the energy equations. The flow is assumed to be in the x-direction, which is taken along the moving surface and the y-axis is taken to be normal to the surface. The non-uniform transverse magnetic field is imposed along the y-axis. Since the magnetic Reynolds number is very small for most fluids used in industrial applications, the induced magnetic field is assumed to be negligible. A schematic representation of the physical model and coordinates system is depicted in Fig.(i). The stretching surface aligned with the temperature distribution T_w(x, t) varies both along the surface and with time.

Fig.1 Dimensionless velocity profiles for various A

Fig.2 Dimensionless velocity profiles for various M

4. NUMERICAL SOLUTION:

Equation (15) and (16) constitute a boundary value problem of third and second order differential equations. In order to solve this, we develop a most efficient numerical shooting technique called Nachtsheim-Swigert shooting technique with fourth order Runge- Kutta integration algorithm so as to satisfy the asymptotic boundary conditions.

In order to solve these equations we need a set of five initial conditions. The guess values of the two non- prescribed initial conditions f ´´ (0) and θ´ (0) are obtained by employing the shooting technique.

Fig.3 Dimensionless temperature profiles for various A

It must be noted that convergence of the scheme largely depends on good guesses of the initial conditions in the above mentioned shooting technique.

5. RESULTS AND DISCUSSION:

Computations through employed numerical scheme has been carried out for values of the parameters involved such as Unsteadiness parameter (A), Eckert number (Ec), Magnetic interaction parameter (M), Prandtl number (Pr), Radiation parameter (R). For illustrations of the results, numerical values are plotted by means of figures. In the absence of the magnetic field and the dissipation effects, the results of the problem are compared with that of Mohamed Abd El-Aziz (2009). From Fig. (c) it is noted that ther results are in excellent agreement with his results.

In Fig. 1, velocity profiles are shown for different values of the unsteadiness parameter A. It is seen that the dimensionless velocity decreases with the increase of unsteadiness parameter A. It is evident from this figure that the thickness of the boundary layer decreases with the increasing values of A. Fig. 2 displays the nature of velocity for various values of M. It shows that the velocity is getting decelerated for the increasing values of M. This is because of the introduction of transverse magnetic field normal to the flow direction which has the tendency to create a drag due to the Lorentz force which tends to resist the flow and, hence the velocity boundary layer decreases.

Fig.5 Effect of M over the dimensionless temperature profiles

Fig.6 Dimensionless temperature profiles for various Pr

Fig. 7 Dimensionless temperature profiles for different values of R

Fig.8 Variation of M over f ″(0)

Fig.9 M variation over – θ ′(0)

Fig.10 Variation of Ec over – θ ′(0)

Fig.11 Pr variation over – θ ′(0)

The effect of the unsteadiness parameter A over the dimensionless temperature is displayed through Fig. 3. It is clear that for all values of A considered, θ is found to decrease with the increase of η. Effect of the Eckert Number over the dimensionless temperature is shown through Fig. 4. Even though the effect of the Ec is less prominent over the temperature field it has a decreasing effect for its higher values. Fig. 5 shows the temperature distribution for different values of M. It is observed that the effect of magnetic parameter is to increase the temperature profile in the boundary layer. This is due to the Lorentz force which has the tendency to slow down the motion of the fluid in the boundary layer and to increase its temperature. And also the effect on the flow and thermal fields become more when the strength of the magnetic field increases.

Fig. 12 Effect of R over – θ ′(0)

The effect of the Prandtl number Pr over the dimensionless temperature is demonstrated through Fig.6. It is found that the effect of Prandtl number is to decrease the temperature in the boundary layer. This is because there would be a decrease of the boundary layer thickness with the increase in the values of Prandtl number. Fig.7 displays the dimensionless temperature distribution for the different values of the radiation parameter R. The temperature gets decreases for the increasing values of R and it is clearly shown through the figure. Influence of the magnetic parameter M over the local skin-friction coefficient is shown in Fig. 8. In the presence of magnetic field, the fluid velocity is found to be decreased, associated with a reduction in the velocity gradient at the wall, and thus the local skin-friction coefficient decreases.

The effect of the magnetic parameter M over the local Nusselt number is portrayed through Fig. 9. It is clear from the figure that the magnetic parameter decreases the rate of heat transfer. Fig.10 portrays the effect of Eckert number over the Nusselt number. The rate of heat transfer decreases for the increasing values of Ec. Fig.11 displays the effects of Prandtl number Pr over the local Nusselt number and Fig.12 explains the effects of radiation parameter R over the rate of heat transfer. The local Nusselt number increases for the higher values of both Pr and R.

6. CONCLUSION

In the present work the governing equations of the unsteady hydromagnetic flow of an viscous, incompressible, electrically conducting and radiating fluid over a stretching surface with dissipation and radiation effects are numerically solved. Numerical evaluations have been performed and graphical results are obtained to illustrate the details of flow and heat transfer characteristics and their dependence on some of the physical parameters. In the absence of the magnetic field and the dissipation effects the results are identical to that of Mohamed Abd El-Aziz (2009). Results and discussions presented above lead to the following conclusions.

Ø Both the dimensionless velocity as well as the temperature get decreased for the increasing values of the unsteadiness parameter A

Ø Eckert Number Ec has less significant influence over the temperature distribution.

Ø In the presence of the magnetic field, the fluid velocity is found to be decreased. In contrary to this, the magnetic parameter M increases the dimensionless temperature for its increasing values. Just like its effect on velocity, the magnetic parameter decreases the skin-friction coefficient and the non-dimensional rate of heat transfer.

Ø Prandtl number decreases the dimensionless temperature but increases the non-dimensional rate of heat transfer

Ø Radiation parameter has a decreasing effect over the dimensionless temperature distribution, but enhances the non-dimensional rate of heat transfer.

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Received on 13.01.2013 Accepted on 11.02.2013

Research J. Science and Tech 5(1): Jan.-Mar.2013 page 177-183