One-Dimensional Problem of a Conducting Viscous Fluid with One Relaxation Time

We introduce a magnetohydrodynamic model of boundary-layer equations for conducting viscous fluids. This model is applied to study the effects of free convection currents with thermal relaxation time on the flow of a viscous conducting fluid. The method of the matrix exponential formulation for these equations is introduced. The resulting formulation together with the Laplace transform technique is applied to a variety problems. The effects of a plane distribution of heat sources on the whole and semispace are studied. Numerical results are given and illustrated graphically for the problem.


Introduction
The modification of the heat-conduction equation from diffusive to a wave type may be affected either by a microscopic consideration of the phenomenon of heat transport or in a phenomenological way by modifying the classical Fourier law of heat conduction.
Many authors have considered various aspects of this problem and obtained similarity solutions. Samaan 1 investigated steady oscillating magnetohydrodynamic flow in a circular pipe. Analytical and numerical methods for the momentum and energy equations of a viscous incompressible fluid along a vertical plate have been considered by Samaan 2 . Chamkha 3 studied the magnetohydrodynamic flow of a uniformly stretched vertical permeable surface in the presence of heat generation/absorption and chemical reaction. Ishak et al. 4 investigated theoretically the unsteady mixed convection boundary layer flow and heat transfer due to a stretching vertical surface in a quiescent viscous and incompressible fluid.
Many authors presented some mathematical results, and a good amount of references can be found in the papers by Liao and Pop 5 and Nazar et al. 6 . Further, the stagnation region encounters the highest pressure, the highest heat transfer, and the highest rate of mass deposition studied by Wang 7 . Singh et al. 8 investigated the problem of heat transfer in the flow of an incompressible fluid. Samaan 9 investigated the heat and mass transfer over an accelerating surface with heat source in presence of suction and magnetic field. The flow of an unsteady, incompressible magnetohydrodynamics MHDs viscous fluid with suction is investigated by Muhammad et al. 10 . Heat and mass transfer over an accelerating surface with heat source in presence of magnetic field is derived by Samaan 11 . Recently, Samaan 12 studied the effects of variable viscosity and thermal diffusivity on the steady flow in the presence of the magnetic field. variable viscosity effects on hydrodynamic boundary layer flow along a continuously moving vertical plate were done by Mostafa 13 . Concerning the studies of state space formulation for MHDs and free convection flow with two relaxation times and the free convection effective perfectly conducting couple stress fluid, we may refer to Samaan 14 ,Ezzat et al. 15 ,and Hayat et al. 16 . Using differential transform method and Pade approximate for solving MHDs flow in a laminar liquid film from a horizontal stretching surfaces investigated by Rashidi et al. 17 . In the present work, we use a more general model of magnetohydrodynamic free convection flow, which also includes the relaxation time of heat conduction and the electric displacement current 18,19 . An attempt to account for the time dependence of heat transfer, Cattaneo 20 and Vernotte 21 independently modified Fourier's law to include the relaxation time of the system. Generalized thermoelasticity stands for a hyperbolic thermoelasticity in which a thermomechanical load applied to a body is transmitted in a wave-like manner throughout the body, only transient thermoelastic waves are included in the survey by Hertnarski and Ignaczak 22 . The inclusion of the relaxation time and the electric displacement current modifies the governing thermal and electromagnetic field equations, changing them from the parabolic to a hyperbolic type, and thereby eliminating the unrealistic result that thermal and electromagnetic disturbances are realized instantaneously within a fluid.
The solution is obtained using a state-space approach. The importance of state-space analysis is recognized in fields where the time behavior of physical process is of interest. The first writer to introduce the state space approach in magnetohydrodynamic free convection flow was Ezzat 23,24 . His works dealt with free convection flow in the absence of the applied magnetic field or when there are no heat sources. The present work is an attempt to generalize these results to include the effects of heat sources. The results obtained are used to solve a problem for the whole space with a plane distribution of heat sources. The solutions obtained are utilized in combination with the method of images to obtain the solution for a problem with heat sources distributed situated inside a semispace whose surface bounded by an infinite vertical plate.
The Laplace transform techniques is applied to one-dimensional problem. The inversion of the Laplace transforms is carried out using a numerical technique 25 . where σ o is the electric conductivity. We investigate the free convective heat transfer in an incompressible hydromagnetic flow past an infinite vertical plate. The x-axis is taken along the plate in the direction of the flow and the y-axis normal to it. Let u be the component of the velocity in the x direction. All the fluid properties are assumed constant, except that the influence of the density variation with temperature is considered only in the body force term. In the energy equations, terms representing viscous and Joule's dissipation are neglected, as they are assumed to be very small in free convection flow 27 . Also, in the energy equation, the term representing the volumetric heat source is taken as a function of the space variables. In view of the assumptions, the equations that govern unsteady one-dimensional free convection flow in an incompressible conducting fluid through a porous medium bounded by an infinite nonmagnetic vertical plate in the presence of a constant magnetic field are 2.10 -2.12 , and the equations describing the flow in the boundary layer reduce to 28-30

2.13
And the constitutive equation In these equations, K is the permeability of the porous medium, α is the Alfven velocity, g is the acceleration due to gravity, β is the coefficient of volume expansion, T is the temperature distribution, T ∞ is the temperature of the fluid away from the plate, c p is specific heat at constant pressure, λ is the thermal diffusivity, υ o is the relaxation time, ρ ∞ is the density of the fluid far from the surface, and ρ is the density of the fluid. Let us introduce the following nondimensional variables

2.15
Mathematical Problems in Engineering 5 In view of these transformations, 2.13 -2.14 becomė where G r is the Grashof number, c is the speed of light given by vμ o σ o is a measure of the magnetic viscosity, and p r ρc p υ/λ is the Prandtl number. From now on, we will consider a heat source of the form where δ y and H t are the Dirac delta function and the Heaviside unit step function, respectively, Q is the strength of the applied heat source, and Q o is a constant. We will assume that the initial state of the medium is quiescent. Taking the Laplace transform, defined by the relation of both sides of 2.16 , we obtain Eliminating E between 2.19 -2.20 , we get

2.27
In order to solve the system 2.26 , we need first to find the form of the matrix exp A s y .
Mathematical Problems in Engineering 7 The characteristic equation of the matrix A has the form

2.29
The roots ±k 1 , ±k 2 , and ±k 3 of 2.28 satisfy the relations

2.30
One of the roots, say k 2 1 , has a simple expression given by ps.

2.31
The other two roots k 2 2 and k 2 3 satisfy the relation Fξ.

2.32
The Taylor series expansion of the matrix exponential has the form exp A s y ∞ n 0 1 n! A s y n .

2.33
Using the well-known Cayley-Hamilton theorem, the infinite series can be truncated to the following form:

Mathematical Problems in Engineering
where I is the unit matrix of order 6 and b o -b 5 are some parameters depending on s and y.
The characteristic roots ±k i , i 1, 2, 3 of the matrix A must satisfy the equations

2.35
The solution of this system of linear equations is given by

2.37
Substituting for the parameters b o -b 5 from 2.36 into 2.34 and computing A 2 , A 3 , A 4 , and A 5 , we get the elements L ij 1, 2, 3, 4, 5, 6 of the matrix L s, y to be

2.38
It is worth mentioning here that 2.32 have been used repeatedly in order to write the above entries in the simplest possible form. We will stress here that the above expression for the matrix exponential is a formal one. In the actual physical problem, the space is divided into two regions accordingly as y ≥ 0 or y ≺ 0. Inside the region 0 ≤ y ≤ ∞, the positive exponential terms, not bounded at infinity, must be suppressed. Thus, for y ≥ 0, we should replace each sinh ky by − 1/2 exp −ky and each cosh ky by 1/2 exp −ky . In the region y ≤ 0, the negative exponentials are suppressed instead.
We will now proceed to obtain the solution of the problem for the region y ≥ 0. The solution for the other region is obtained by replacing each y by −y.
The formal

2.42
Gauss's divergence theorem will now be used to obtain the thermal condition at the plane source. We consider a short cylinder of unit base, whose axis is perpendicular to the plane source of heat and whose bases lie on opposite sides of it. Taking the limit as the height of the cylinder tends to zero and noting that there is no heat flux through the lateral surface, we get We will use the generalized Fourier's law of heat conduction in the nondimensional form 31 , namely,

2.50
Also, substituting from 2.49 into 2.21 the induced electric field is given by

2.51
Equations 2.47 -2.51 determine completely the state of the fluid for y ≥ 0. We mention in passing that these equations give also the solution to a semispace problem with a plane source of heat on its boundary that constitutes a rigid base. As mentioned before, the solution for the whole space when y ≺ 0 is obtained from 2.47 -2.51 , by taking the symmetries under considerations. We will show that the solution obtained above can be used as a set of building blocks from which the solutions to many interesting problems can be constructed. For future reference, we will write down the solution to the problem in the case when the source of heat is located in the plane y c, instead of the plane y 0. In this case, we have

2.55
where the upper plus sign denotes the solution in the region y ≤ c, while the lower minus sign denotes the solution in the region y c.

Applications
We will now consider the problems of a semispace with a plane source of heat located inside the medium at the position y c and subject to the following boundary conditions.
i The shearing stress and the induced magnetic field are vanishing at the wall y 0 , ∂u 0, t ∂y 0, or ∂u 0, s ∂y 0, h 0, t 0, or h 0, s 0.

3.1
ii The temperature is kept at a constant value T ∞ , which means that the temperature increment θ satisfies θ 0, t 0 or θ 0, s 0.

3.2
This problem can be solved in a manner analogous to the outlined above though the calculations become quite messy. We will instead use the reflection method together with the solution obtained above for the whole space. This approach was proposed by Nowacki in the context of coupled thermoelasticity 31 . The boundary conditions of the problem can be satisfied by locating two heat sources in an infinite space, one positive at y c and the other negative of the same intensity at y −c. The temperature increment θ is obtained as a superposition of the temperature for both plane distribution. Thus, θ θ 1 θ 2 , where θ 1 is the temperature due to the positive heat source, given by 2.52 and θ 2 is the temperature due to the negative heat source and is obtained from 2.52 by replacing c with −c, and noting that for all points of the semispace, we have y c 0. Thus, θ 2 is given by Combining 2.52 and 3.2 , we obtain

3.4
Clearly, this distribution satisfies the boundary condition 3.2 . We turn now to the problem of finding the distributions velocity, the induced magnetic field, and the induced electric field. Unfortunately, the above procedure of superposition cannot be applied to these fields as in the temperature fields. We define the scalar stream function ψ by the relation u ∂ψ ∂y .

3.5
By integration 2.53 and using 3.5 , we obtain the stream function due to the positive heat source at the position y c as where the upper sign is valid for the region 0 ≤ y ≺ c and the lower sign is valid for the region y ≥ 0. Similarly, the stream function for the negative heat source at y −c is given by

3.7
Since ψ is a scalar field, we can use superposition to obtain the stream function for the semispace problem as 3.8 Using 3.8 and 3.5 , we obtain the velocity distribution

3.9
Differentiating 3.9 and using the resulting expressions together with 2.28 , we obtain

Inversion of the Laplace Transform
In order to invert the Laplace transforms in the above equations, we will use a numerical technique based on Fourier expansions of functions.
Let g s be the Laplace transform of a given function g t .  Taking Δy π/t 1 , we obtain g t e dt t 1 For numerical purposes, this is approximated by the function where N is a sufficiently large integer chosen such that where η is a preselected small positive number that corresponds to the degree of accuracy to be achieved, Formula 3.11 is the numerical inversion formula valid for 0 ≤ t ≤ 2t 1 22 . In particular, we choose t t 1 , getting g N t e dt t 1 2 g d Re

Numerical Results
The constants of the problem were taken as ε o 0.003, α 0.3, G r 4, K 1.2, and c 2. All constants are given in SI units. The computations were carried out for the two different values of time, namely, t 0.7 and 1. The functions θ, u, h, and E are evaluated. The results are shown in Figures 1, 2 The important phenomenon observed in all computations is that the solution of any of the considered functions vanishes identically outside a bounded region of space surrounding the heat source at a distance from it equal to x * t , and say that x * t is a particular value of y depending only on the choice of t and is the location of the wave front. This demonstrates clearly the difference between the solution corresponding to using classical Fourier heat equation υ o 0.03 and to using the non-Fourier case υ o 0.4 . In the first and older theory, the waves propagate with infinite speeds, so the value of any of the functions is not identically zero though it may be very small for any large value of y. In the non-Fourier theory, the response to the thermal and mechanical effects does not reach infinity instantaneously but remains in a bounded region of space given by 0 ≺ y ≺ y * t for the semispace problem and by Min 0, y * t − c ≺ y ≺ y y * t for the whole space problem.
We notice that results for all functions considered in the semispace problem when the relaxation time is appeared in heat equation are distinctly different from those when the relaxation time disappeared.
We also notice that for small values of time, the solution is localized in a finite region near the plane of heat sources. This region grows with increasing time until it fills the whole boundary-layer region.