Computational treatment of free convection effects on perfectly conducting viscoelastic fluid

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Abstract

We introduced a magnetohydrodynamic model of boundary-layer equations for a perfectly conducting viscoelastic fluid. This model is applied to study the effects of free convection currents with one relaxation time on the flow of a perfectly conducting viscoelastic fluid through a porous medium, which is bounded by a vertical plane surface. The state space approach is adopted for the solution of one-dimensional problems. The resulting formulation together with the Laplace transform technique is applied to a thermal shock problem and a problem for the flow between two parallel fixed plates, both without heat sources. Also a problem for the semi-infinite space in the presence of heat sources is considered. A discussion of the effects of cooling and heating on a perfectly conducting viscoelastic fluid is given. Numerical results are illustrated graphically for each problem considered.

Introduction

The phenomenal growth of energy requirements in recent years been attracting considerable attention all over the world. This has resulted in a continuous exploration of new ideas and avenues in harnessing various conventional energy sources. Such as tidal waves, wind power, geo-thermal energy, etc. It is obvious that in order to utilize geo-thermal energy to a maximum, one should have a complete and precise knowledge of the amount of perturbations needed to generate convection currents in geo-thermal fluid. Also, knowledge of the quantity of perturbations that are essential to initiate convection currents in mineral fluids found in the earth’s crust helps one to utilize the minimal energy to extract the minerals. For example, in the recovery of hydro-carbons from underground petroleum deposits. The use of thermal processes is increasingly gaining importance as it enhances recovery. Heat is being injected into the reservoir in the form of hot water or steam or burning part of the crude in the reservoir can generate heat. In all such thermal recovery processes, fluid flow takes place through a conducting medium and convection currents are detrimental.

There is extensive literature on the through a porous media which is governed by the generalized Darcy’s law. Yamamoto and Iwamura [2] expressed the equations of flow through a highly porous medium. Raptis et al. [3], [4], using these equations, studied the influences of free convective flow and mass transfer on flow through a porous medium. Raptis and Predikis [5], also studied influences on the oscillatory flow through a porous medium. Newtonian fluids were discussed in the above references. In technological fields another important class of fluids, called non-Newtonian fluids, are also being studied extensively because of their practical applications, such as fluid film lubrication, analysis of polymers in chemical engineering, etc. One such fluid is called viscoelastic fluid and Walters [6] and Beard and Walters [7] deduced the governing equations for the boundary layer flow for a prototype viscoelastic fluid, which they have designated as liquid B, when this liquid had a very short memory. The flow of viscoelastic incompressible and electrically conducting fluid past an infinite plate in the presence of a transverse magnetic field, when the plate executes simple harmonic motion parallel to itself, has been discussed by Sherief and Ezzat [8]. Sen [9] studied the behavior of unsteady free convection flow of a viscoelastic fluid past an infinite porous plate with constant suction. The effects of suction, free oscillations and free convection currents on flow have been studied by Soundalgekar and Patil [10]. Singh [11] have studied the magnetohydrodynamic flow of viscoelastic fluid past an accelerated plate. In most of the above applications, the method of solution due to Lighthill [12] and Stuart [13] is utilized. This method has a severe drawback in that it is applicable only to problems of simple harmonic vibrations. This prompted many authors to use other methods of solution when dealing with the problems of a non-vibrating fluid. Gupta [14] and Riely [15] used an approximate pholhausen method, Wilks and Hunt [16] used the method of similarity solution, Saponkoff [17] and Vajravelu and Sastri [18] used a perturbation method to solve problems of hydromagnetic flows.

In this work, we use a more general model of magnetohydrodynamic free convection flow, which also includes the relaxation time of heat conduction and the electric permeability of the electromagnetic field. The inclusion of the relaxation time and electric permeability modify the governing thermal and electromagnetic equations, changing them from parabolic to hyperbolic type, and thereby eliminating the unrealistic result that thermal disturbance is realized instantaneously everywhere within a fluid [19], [20].

The solution is obtained using a state space approach [21], [22]. The importance of state space analysis is recognized in field where the time behavior of physical process is of interest.

The state space approach is more general than the classical Laplace and Fourier transform techniques. Consequently, state space theory is applicable to all systems that can be analyzed by integral transforms in time, and is applicable to many systems for which transform theory breaks down.

In this approach, the governing equations are written in matrix form using a state vector that consists of the Laplace transforms in time of the temperature, the velocity and the induced magnetic field and their gradients. Their integration, subjected to zero initial conditions, is carried out means of matrix exponential method. Influence functions in the Laplace transform domain are explicitly developed.

The inversion of the Laplace transform is carried out using a numerical technique see [1], [23].

Section snippets

Formulation of the problem

In our consideration of one-dimensional problems of hydromagnetic free convection flow, we shall make two important restrictions. First, we assume that the medium under consideration is perfectly conducting fluid and, secondly, that the initial magnetic field of uniform strength is applied transverse to the infinite vertical surface plane.

We investigate the free convective heat transfer in a perfectly conducting viscoelastic hydromagnetic flow past an infinite vertical surface plane through a

State space formulation

We shall choose as state variables the temperature increment θ, the velocity component w and their gradients. Eqs. (15), (18) can be written as follows∂θ¯∂y=θ¯′,∂w¯∂y=w¯′,∂θ¯′∂y=nθ¯-(1+τ0s)Q¯,∂w¯′∂y=mw¯-Graθ¯,where n = prs(1 + τ0s) and m=ba.

The above equations can be written in matrix form asdf¯(y,s)dy=A(s)f¯(y,s)+B(y,s),wheref¯(y,s)=θ¯(y,s)w¯(y,s)θ¯′(y,s)w¯′(y,s),A(s)=00100001n000-Gram00,andB(y,s)=-(1+τ0s)Q¯(y,s)0010.

The formal solution of Eq. (23) can be expressed asf¯(y,s)=exp[A(y,s)y]f¯(0,s)+∫0y

Applications

Problem 1 Thermal shock problem

We shall consider the magnetohydrodynamic free convection flow of a perfectly conducting viscoelastic fluid occupying a semi-infinite region y ⩾ 0 of the space bounded by an infinite vertical plane surface (y = 0) through porous medium, with the conditionw(0,t)=-U0H(t),where U0 is a constant.

We assume that a thermal shock of the formθ(0,t)=θ0H(t),is applied to the plane surface at time t = 0, where H(t) is Heaviside unit step function. All initial conditions are assumed to be zero.

We now apply the

Numerical inversion of the Laplace transform

In order to invert the Laplace transform in the above equations, we adopt a numerical inversion method based on a Fourier series expansion [23]. In this method, the inverse g(t) of the Laplace transform g¯(s) is approximated by the relationg(t)=ectt112g¯(c)+Re∑k=0∞eikt/t1g¯(c+ik/t1)0⩽t⩽2t1,where N is a sufficiently large integer representing the number of terms in the truncated infinite Fourier series. N must chosen such thatectRe[eiNt/t1g¯(c+iN/t1)]⩽ε,where ε is a persecuted small positive

Results and discussion

The function θ and u are evaluated for different values of time. These results are shown in Fig. 1, Fig. 2 for Problem 1, Fig. 5, Fig. 6 for Problem 2 and Fig. 7, Fig. 8 for Problem 3. In these figures solid lines represent the solution corresponding to using the Non-Fourier equation of heat conduction, while dotted lines represent the solution corresponding to using the classical Fourier heat equation (τ0 = 0.0). In our computations the values θ0 and U0 are taken as unity.

It is clear from these

Concluding remarks

Many metallic materials are manufactured after they have been refined sufficiently in the molten state. Therefore, it is a central problem in metallurgical chemistry to study the free convection effects on liquid metal, which is a perfect electric conductor. For instance, liquid sodium Na (100 °C) and liquid potassium K (100 °C) exhibit very small electrical resistivity (ρL(exp) = 9.6 × 10−6 Ω cm.) and (ρL(exp) = 12.97 × 10−6 Ω cm.) [23].

The effects of Grashof number, Alfven velocity, elastic parameter and

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