Study of MHD boundary layer flow over a heated stretching sheet with variable viscosity

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Abstract

This paper concerns with a steady two-dimensional flow of an electrically conducting incompressible fluid over a heated stretching sheet. The flow is permeated by a uniform transverse magnetic field. The fluid viscosity is assumed to vary as a linear function of temperature. A scaling group of transformations is applied to the governing equations. The system remains invariant due to some relations among the parameters of the transformations. After finding two absolute invariants a third-order ordinary differential equation corresponding to the momentum equation and a second-order ordinary differential equation corresponding to energy equation are derived. The equations along with the boundary conditions are solved numerically. It is found that the decrease in the fluid viscosity makes the velocity to decrease with the increasing distance of the stretching sheet. At a particular point of the sheet the fluid velocity decreases with the decreasing viscosity but the temperature increases in this case. It is found that with the increase of magnetic field intensity the fluid velocity decreases but the temperature increases at a particular point of the heated stretching surface. The results thus obtained are presented graphically and discussed.

Introduction

Lie-group analysis, also called symmetry analysis was developed by Sophius Lie to find point transformations which map a given differential equation to itself. This method unifies almost all known exact integration techniques for both ordinary and partial differential equations [7]. Group analysis is the only rigorous mathematical method to find all symmetries of a given differential equation and no ad hoc assumptions or a prior knowledge of the equation under investigation is needed. The boundary layer equations are especially interesting from a physical point of view because they have the capacity to admit a large number of invariant solutions i.e. basically analytic solutions. In the present context, invariant solutions are meant to be a reduction to a simpler equation such as an ordinary differential equation (ODE). Prandtl’s boundary layer equations admit more and different symmetry groups. Symmetry groups or simply symmetries are invariant transformations which do not alter the structural form of the equation under investigation [4].

The non-linear character of the partial differential equations governing the motion of a fluid produces difficulties in solving the equations. In the field of fluid mechanics, most of the researchers try to obtain the similarity solutions in such cases. In case of scaling group of transformations, the group-invariant solutions are nothing but the well known similarity solutions [6]. A special form of Lie-group of transformations, known as scaling group, is used in this paper to find out the full set of symmetries of the problem and then to study which of them are appropriate to provide group-invariant or more specifically similarity solutions.

The study of magnetohydrodynamic (MHD) flow of an electrically conducting fluid is of considerable interest in modern metallurgical and metal-working processes. There has been a great interest in the study of magnetohydrodynamic flow and heat transfer in any medium due to the effect of magnetic field on the boundary layer flow control and on the performance of many systems using electrically conducting fluids. This type of flow has attracted the interest of many researchers due to its applications in many engineering problems such as MHD generators, plasma studies, nuclear reactors, geothermal energy extractions. By the application of magnetic field, hydromagnetic techniques are used for the purification of molten metals from non-metallic inclusions. So such type of problem, that we are dealing with, is very much useful to polymer technology and metallurgy. Crane [5] extended the work of Sakiadis [1], [2] who was the first person to study the laminar boundary layer flow caused by a rigid surface moving in its own plane. Gupta and Gupta [9] studied the problem in the light of suction or blowing. In all the above mentioned studies, fluid viscosity was assumed uniform in the flow region. But it is known from physics that with the rise of temperature, the coefficient of viscosity decreases in case of liquids whereas it increases in case of gases. Abel et al. [8] studied the visco-elastic fluid flow and heat transfer over a stretching sheet with variable viscosity.

In this paper, application of scaling group of transformation for a hydromagnetic flow over a heated stretching sheet with variable viscosity has been employed. This reduces the system of non-linear coupled partial differential equations governing the motion of fluid into a system of coupled ordinary differential equations by reducing the number of independent variables. The system remains invariant due to some relations among the parameters of the transformations. Two absolute invariants are obtained and used to derive a third-order ordinary differential equation corresponding to momentum equation and a second-order ordinary differential equation corresponding to energy equation. Using shooting method the equations are solved. Finally, analysis have been made to investigate the effect of fluid viscosity parameter, Prandtl number and magnetic parameter in the motion of an electrically conducting liquid.

Section snippets

Equations of motion

We consider the steady two-dimensional flow of a viscous incompressible electrically conducting fluid over a heated stretching sheet in the region y > 0. Keeping the origin fixed, two equal and opposite forces are applied along the x-axis which results in stretching of the sheet and a uniform magnetic field of strength B0 is imposed along the y-axis.

The continuity, momentum and energy equations governing such type of flow are written asux+vy=0,uux+vuy=1ρμTTyuy+μρ2uy2-σB02ρu,uTx+v

Numerical method for solution

The above Eqs. (35), (36) along with boundary conditions are solved by converting it to an initial value problem. We setf=z,z=p,p=z2-fp+Aqp+M2z(a+A-Aθ),θ=q,q=-Prfqand the boundary conditions aref(0)=0,f(0)=1,θ(0)=1.Since these equations are non-linear we can not superpose solutions on this problem. Furthermore, in order to integrate (38), (39) as an initial value problem we require a value for p(0) i.e. f″(0) and q(0) i.e. θ′(0) but no such values are given. The suitable guess values for f

Results and discussions

In order to analyse the results, the numerical computation has been carried out using the method described in the previous section for various values of the parameter such as fluid viscosity variation parameter A, Hartman number M and Prandtl number Pr. For illustration of the results numerical values are plotted in the figures one to six. The physical explanation of the appropriate change of parameters are given below.

Fluid viscosity and thermal conductivity (hence thermal diffusivity) play an

Conclusion

Under the assumption of temperature dependent viscosity, the present method gives solutions, for steady incompressible boundary layer flow over a heated stretching surface in the presence of uniform transverse magnetic field. The results pertaining to the present study indicate that the temperature dependent fluid viscosity plays a significant role in shifting the fluid away from the wall. The effect of transverse magnetic field on a viscous incompressible conducting fluid is to suppress the

Acknowledgments

One of the authors (S.M.) gratefully acknowledges the financial support of Council of Scientific and Industrial Research (CSIR), Delhi, India for pursuing this work.

The authors are thankful to the honourable reviewers for constructive suggestions.

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