Lie group analysis for the effects of temperature-dependent fluid viscosity and chemical reaction on MHD free convective heat and mass transfer with variable stream conditions

Abstract

This paper concerns with a steady two-dimensional flow of an electrically conducting incompressible fluid over a vertical stretching surface. 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 three absolute invariants a third-order ordinary differential equation corresponding to the momentum equation and two second-order ordinary differential equation corresponding to energy and diffusion equations are derived. The equations along with the boundary conditions are solved numerically. It is found that the decrease in the temperature-dependent 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. Impact of chemical reaction in the presence of thermal radiation plays an important role on the concentration boundary layer. 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, Oberlack (1999). 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. 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, Bluman and Kumei (1989). 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, Pakdemirli and Yurusoy (1998). 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.

Newton's law of viscosity states that shear stress is proportional to velocity gradient. Fluids that obey this law are known as Newtonian fluids. Amongst Newtonian fluids we can cite water, benzene, ethyl alcohol, hexane and most solutions of simple molecules. There are numerous fluids that violate Newton's law of viscosity. Combined heat and mass transfer problems with chemical reaction are of importance in many processes and have, therefore, received a considerable amount of attention in recent years. In processes such as drying, evaporation at the surface of a water body, energy transfer in a wet cooling tower and the flow in a desert cooler, heat and mass transfer occur simultaneously. Possible applications of this type of flow can be found in many industries. For example, in the power industry, among the methods of generating electric power is one in which electrical energy is extracted directly from a moving conducting fluid.

We are particularly interested in cases in which diffusion and chemical reaction occur at roughly the same speed. When diffusion is much faster than chemical reaction, then only chemical factors influence the chemical reaction rate; when diffusion is not much faster than reaction, the diffusion and kinetics interact to produce very different effects. The study of heat generation or absorption effects in moving fluids is important in view of several physical problems, such as fluids undergoing exothermic or endothermic chemical reaction. Due to the fast growth of electronic technology, effective cooling of electronic equipment has become warranted and cooling of electronic equipment ranges from individual transistors to main frame computers and from energy suppliers to telephone switch boards and thermal diffusion effect has been utilized for isotopes separation in the mixture between gases with very light molecular weight (hydrogen and helium) and medium molecular weight.

There are two types of reactions such as (i) homogeneous reaction and (ii) heterogeneous reaction. A homogeneous reaction occurs uniformly throughout the given phase, whereas heterogeneous reaction takes place in a restricted region or within the boundary of a phase. The effect of a chemical reaction depends on whether the reaction is heterogeneous or homogeneous. A chemical reaction is said to be first order, if the rate of reaction is directly proportional to concentration itself. In many industrial process involving flow and mass transfer over a moving surface, the diffusing species can be generated or absorbed due to some kind of chemical reaction with the ambient fluid which can greatly affect the flow and hence the properties and quality of the final product. These processes take place in numerous industrial applications, such as the polymer production and the manufacturing of ceramics or glassware. Thus we are particularly interested in cases in which diffusion of the species and chemical reaction occurs at roughly the same speed in analyzing the mass transfer phenomenon. The flow of a fluid past a vertical stretching sheet is of fundamental importance since this type of flow constitutes a general and wide class of flows in which the free stream velocity is proportional to the power of the length coordinate measured from the stagnation point. Kandasamy et al. (2005) studied the effects of chemical reaction, heat and mass transfer along a vertical stretching sheet with heat source and concentration in the presence of suction or injection. Kandasamy et al. (2008) analyzed the thermophoresis and chemical reaction effects on non-Darcy mixed convective heat and mass transfer past a porous wedge with variable viscosity in the presence of suction or injection. Mingchun et al. (2007) analyzed the effect of non-thermal equilibrium model of the coupled heat and mass transfer in strong endothermic chemical reaction system of porous media. The recent advances in understanding physics of flows and computational flow modeling (CFM) can make tremendous contributions in chemical engineering. Elperin and Fominykh (1994) discussed the effect of exact analytical solution of a convective diffusion from a wedge to a flow with a first order chemical reaction at the surface. Chamkha and Ali (2003) investigated the effects of MHD flow of a uniformly stretched vertical permeable surface in the presence of heat generation/absorption and a chemical reaction. Marle (1982) analyzed the impact of macroscopic equations governing multiphase flow with diffusion and chemical reactions in porous media.

The effect of radiation on MHD flow and heat and mass transfer problem has become more important industrially. At high operating temperature, radiation effect can be quite significant. Many processes in engineering areas occur at high temperature and knowledge of radiation heat transfer becomes very important for the design of the pertinent equipment. Nuclear power plants, gas turbines and the various propulsion devices for aircraft, missiles, satellites and space vehicles are examples of such engineering areas. 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, which we are dealing with, is very much useful to polymer technology and metallurgy. Crane (1970) extended the work of Sakiadis, 1961a, Sakiadis, 1961b 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 (1977) 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. (2002) studied the visco-elastic fluid flow and heat transfer over a stretching sheet with variable viscosity.

The effect of cooling on the viscosity is represented by the temperature-dependent fluid viscosity. A variety of spreading experiments have been performed within many international programs. The cooling process will lead to variable flow properties, e.g. variable fluid viscosity and, consequently, to a coupled system of temperature, concentration and flow equations. Spent fuel nuclear storage systems must guarantee sufficient cooling to prevent excessive air and solid temperatures in the system and therefore avoid the possibility of catastrophic accidents. Because of the complex physical processes dictating flow conditions and heat and mass transfer characteristics inside the unit, analysis requirements must include the ability to accurately predict free and forced convection, conductive and radiative heat transfer, and turbulence. In all of the above mentioned studies, fluid viscosity was assumed to be constant. However, it is known that the physical properties of fluid may change significantly with temperature. For lubricating fluids, heat generated by the internal friction and the corresponding rise in temperature affects the viscosity of the fluid and so the fluid viscosity can no longer be assumed constant. The increase of temperature leads to a local increase in the transport phenomena by reducing the viscosity across the momentum boundary layer and so the heat transfer rate at the wall is also affected. Therefore, to predict the flow behavior accurately it is necessary to take into account the viscosity variation for incompressible fluids. Gary et al. (1982) and Mehta and Sood (1992) showed that, when this effect is included the flow characteristics may changed substantially compared to the constant viscosity assumption. Mukhopadhyay et al. (2005) investigated the MHD boundary layer flow with variable fluid viscosity over a heated stretching sheet. Mukhopadhyay and Layek (2008) studied the effects of thermal radiation and variable fluid viscosity on free convective flow and heat transfer past a porous stretching surface.

In this paper, application of scaling group of transformation for a hydromagnetic flow over a vertical stretching sheet with chemical reaction in the presence of variable stream condition 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. Three absolute invariants are obtained and used to derive a third-order ordinary differential equation corresponding to momentum equation and two second-order ordinary differential equation corresponding to energy and diffusion equations. Using Runge Kutta Gill scheme with shooting method, the equations are solved. Finally, analysis has been made to investigate the effect of thermal radiation parameter, temperature-dependent fluid viscosity parameter and magnetic parameter in the motion of an electrically conducting liquid.