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Mathematical Problems in Engineering
Volume 2011, Article ID 731876, 23 pages
http://dx.doi.org/10.1155/2011/731876
Research Article

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

Department of Mathematics, Faculty of Girls, Ain Shams University, Cairo 11757, Egypt

Received 2 August 2010; Revised 28 December 2010; Accepted 23 February 2011

Academic Editor: Sergio Preidikman

Copyright © 2011 Angail A. Samaan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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.

1. 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].

2. Formulation of the Problem

Let a constant magnetic field of strength 𝐻0 act in the direction of the 𝑦-axis. Due to the effect of this magnetic field, there arises in the medium an induced magnetic field and an induced electric field 𝐸. All the considered functions will depend on 𝑦 and the time 𝑡 only.

The electromagnetic quantities satisfy Maxwell’s equations [26] curl=𝐽+𝜀𝑜̇𝐸,(2.1)curl𝐸=𝜇𝑜̇,(2.2)div=0,div𝐸𝐵=0,(2.3)=𝜇𝑜𝐻0+,𝐷=𝜀𝑜𝐸,(2.4) where 𝐽 is the electric current density, 𝜇𝑜 and 𝜀𝑜 are the magnetic and electric permeabilities, respectively, and 𝐵, 𝐷 are the magnetic and electric induction vectors, respectively, and dotte is the time variable.

These equations are supplemented by Ohm’s law𝐽=𝜎𝑜𝐸+𝜇𝑜𝑣×𝐻0,(2.5) where 𝑣=(𝑢,0,0) is velocity vector of the fluid and 𝜎𝑜 is the electric conductivity.

As mentioned above, the applied field 𝐻𝑜 and the induced magnetic field has the components 𝐻0=0,𝐻𝑜,0,=(,0,0).(2.6) The vector 𝐸 and 𝐽 will have nonvanishing components only in the 𝑧-direction. That is,𝐸=(0,0,𝐸),𝐽=(0,0,𝐽),(2.7) where from (2.5) 𝐽=𝜎𝑜𝐸+𝜇𝑜𝐻𝑜𝑢.(2.8) The vector (2.1) and (2.2) reduce to the following scalar equations ,𝑦=𝐽+𝜀𝑜̇𝐸,𝐸(2.9),𝑦=𝜇𝑜̇.(2.10) Eliminating 𝐽 between (2.8) and (2.9), we obtain,𝑦𝜎=𝑜𝐸+𝜀𝑜̇𝐸𝜎𝑜𝜇𝑜𝐻𝑜𝑢.(2.11) The pondermotive force 𝐹=𝐽×𝐵 has one nonvanishing component in the 𝑥-direction𝐹𝑥=𝜇𝑜𝐻𝑜,𝑦+𝜀𝑜̇𝐸,(2.12) where 𝜎𝑜 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 [2830]𝜌̇𝑢=𝜌𝜈𝑢,𝑦𝑦𝜈𝜌𝐾𝛼𝑢+𝜌2𝐻𝑜,𝑦+𝜀𝑜̇𝐸+𝜌𝑔𝛽𝑇𝑇,,𝑦𝜎=𝑜𝐸+𝜀𝑜̇𝐸𝜎𝑜𝜇𝑜𝐻𝑜𝐸𝑢,,𝑦=𝜇𝑜̇,𝜌𝑐𝑝̇𝑇=𝜆𝑇,𝑦𝑦𝜌𝑐𝑝𝜐𝑜̈𝑇+𝒬𝜐𝑜̇𝒬.(2.13) And the constitutive equation 𝜌𝜌=𝜌𝛽𝑇𝑇.(2.14) 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, 𝑐𝑝 is specific heat at constant pressure, 𝜆 is the thermal diffusivity, 𝜐𝑜 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 variables𝑦=𝑦𝛼𝜈,𝑡=𝑡𝛼2𝜈,𝑢=𝑢𝛼,𝐸=𝐸𝜇𝑜𝐻𝑜𝛼,=𝐻𝑜,𝜐𝑜=𝛼2𝜐𝑜𝜈,𝐾=𝐾𝛼2𝜈2,𝜃=𝑇𝑇𝑇𝑜𝑇,𝐺𝑟=𝑇𝜈𝛽𝑔𝑜𝑇𝛼3,𝒬=𝑣2𝒬𝜆𝛼2𝑇𝑜𝑇.(2.15) In view of these transformations, (2.13)–(2.14) becomė𝑢=𝑢,𝑦𝑦1𝐾𝑢+,𝑦̇+𝑎𝐸+𝐺𝑟𝜃,,𝑦̇𝐸=𝜂(𝐸+𝑢)𝑎𝐸,𝑦̇𝜃=,,𝑦𝑦𝑝𝑟̇𝜃+𝜐𝑜̈𝜃=𝒬𝜐𝑜̇𝒬,(2.16) where 𝐺𝑟 is the Grashof number, 𝑐 is the speed of light given by 𝑐2=1/𝜀𝑜𝜇𝑜, 𝑎=𝛼2/𝑐2, 𝜂=𝑣𝜇𝑜𝜎𝑜 is a measure of the magnetic viscosity, and 𝑝𝑟=𝜌𝑐𝑝𝜐/𝜆 is the Prandtl number. From now on, we will consider a heat source of the form𝒬=𝒬𝑜𝛿(𝑦)𝐻(𝑡),(2.17) 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 𝒬𝑜 is a constant.

We will assume that the initial state of the medium is quiescent. Taking the Laplace transform, defined by the relation𝑓(𝑠)=0𝑒𝑠𝑡𝑓(𝑡)𝑑𝑡,(2.18) of both sides of (2.16), we obtain𝜕2𝑢𝜕𝑦21𝐾+𝑠𝑢=𝐺𝑟𝜕𝜃𝜕𝑦𝑎𝑠𝜕𝐸,(2.19)𝜕𝑦=(𝜂+𝑎𝑠)𝐸𝜂𝜕𝑢,(2.20)𝐸𝜕𝑦=𝑠𝜕,(2.21)2𝜕𝑦2𝑝𝑟𝑠1+𝜏𝑜𝑠𝜃=𝒬𝑜1+𝜏𝑜𝑠𝑠𝛿(𝑦).(2.22) Eliminating 𝐸 between (2.19)-(2.20), we get 𝜕2𝜕𝑦2𝐹𝑢=𝐺𝑟𝜕𝜃𝑛,𝜕𝜕𝑦2𝜕𝑦2𝜉=𝜂𝜕𝑢,𝜕𝑦(2.23) where 1𝐹=𝑠+𝐾+𝑎𝑠𝜂𝜂𝜂+𝑎𝑠,𝑛=𝜂+𝑎𝑠,𝜉=𝑠(𝜂+𝑎𝑠).(2.24)

We will choose as state variables the temperature 𝜃, the velocity component 𝑢, and induced magnetic field and their gradients. Equations (2.22)–(2.23) can be written as 𝜕𝜃=𝜕𝑦𝜃,𝜕𝑢=𝜕𝑦𝑢,𝜕=𝜕𝑦,𝜕𝜃𝜕𝑦=𝑝𝑠𝜃𝒬𝑜𝛿(𝑦)1+𝜐𝑜𝑠𝑠,𝜕𝑢𝜕𝑦=𝐺𝑟𝜃+𝐹𝑢𝑛,𝜕𝜕𝑦=𝜂𝑢+𝜉,(2.25) where 𝑝=𝑝𝑟(1+𝜐𝑜𝑠). The above equations can be written in matrix form as𝑑𝑣(𝑦,𝑠)𝑑𝑦=𝐴(𝑠)𝑣(𝑦,𝑠)+𝐵(𝑦,𝑠),(2.26) where𝐴(𝑠)=000100000010000001𝑝𝑠00000𝐺𝑟,𝐹000𝑛00𝜉0𝜂0𝑣(𝑦,𝑠)=𝜃(𝑦,𝑠)𝑢(𝑦,𝑠)(𝑦,𝑠)𝜃(𝑦,𝑠)𝑢(𝑦,𝑠),(𝑦,𝑠)𝐵(𝑦,𝑠)=𝒬𝑜𝛿(𝑦)1+𝜐𝑜𝑠𝑠000100.(2.27) In order to solve the system (2.26), we need first to find the form of the matrix exp(𝐴(𝑠)𝑦).

The characteristic equation of the matrix 𝐴 has the form𝑘6𝑍1𝑘4+𝑍2𝑘2𝑍3=0,(2.28) where𝑍1𝑍=𝐹+𝜉+𝑝𝑠+𝜂𝑛,2𝑍=𝐹𝜉+𝑝𝑠(𝐹+𝜉)+𝜂𝑛𝑝𝑠,3=𝑝𝑠𝐹𝜉.(2.29) The roots ±𝑘1, ±𝑘2, and ±𝑘3 of (2.28) satisfy the relations𝑍1=𝑘21+𝑘22+𝑘23,𝑍2=𝑘21𝑘22+𝑘22𝑘23+𝑘21𝑘23,𝑍3=𝑘21𝑘22𝑘23.(2.30) One of the roots, say 𝑘21, has a simple expression given by𝑘21=𝑝𝑠.(2.31) The other two roots 𝑘22and𝑘23 satisfy the relation𝑘22+𝑘23𝑘=𝐹+𝜉+𝜂𝑛,22𝑘23=𝐹𝜉.(2.32) The Taylor series expansion of the matrix exponential has the form[]=exp𝐴(𝑠)𝑦𝑛=01[]𝑛!𝐴(𝑠)𝑦𝑛.(2.33)

Using the well-known Cayley-Hamilton theorem, the infinite series can be truncated to the following form:[]exp𝐴(𝑠)𝑦=𝐿(𝑠,𝑦)=𝑏𝑜𝐼+𝑏1𝐴+𝑏2𝐴2+𝑏3𝐴3+𝑏4𝐴4+𝑏5𝐴5,(2.34) where 𝐼 is the unit matrix of order 6 and 𝑏𝑜𝑏5 are some parameters depending on 𝑠 and 𝑦. The characteristic roots ±𝑘𝑖, 𝑖=1,2,3 of the matrix 𝐴 must satisfy the equationsexp±𝑘1𝑦=𝑏𝑜±𝑏1𝑘1+𝑏2𝑘21±𝑏3𝑘31+𝑏4𝑘41±𝑏5𝑘51,exp±𝑘2𝑦=𝑏𝑜±𝑏1𝑘2+𝑏2𝑘22±𝑏3𝑘32+𝑏4𝑘42±𝑏5𝑘52,exp±𝑘3𝑦=𝑏𝑜±𝑏1𝑘3+𝑏𝑎2𝑘23±𝑏3𝑘33+𝑏4𝑘43±𝑏5𝑘53.(2.35) The solution of this system of linear equations is given by 𝑏𝑜𝑘=𝑅22𝑘23𝐶1+𝑘21𝑘23𝐶2+𝑘22𝑘21𝐶3,𝑏1𝑘=𝑅22𝑘23𝑆1+𝑘23𝑘21𝑆2+𝑘21𝑘22,𝑏2𝑘=𝑅22+𝑘23𝐶1+𝑘23+𝑘21𝐶2+𝑘21+𝑘22𝐶3,𝑏3𝑘=𝑅22+𝑘23𝑆1+𝑘23+𝑘21𝑆2+𝑘21+𝑘22𝑆3,𝑏4𝐶=𝑅1+𝐶2+𝐶3,𝑏5𝑆=𝑅1+𝑆2+𝑆3,(2.36) where 1𝑅=𝑘21𝑘22𝑘22𝑘23𝑘23𝑘21,𝐶1=𝑘22𝑘23𝑘cosh1𝑦,𝑆1=𝑘22𝑘23𝑘1𝑘sinh1𝑦,𝐶2=𝑘23𝑘21𝑘cosh2𝑦,𝑆2=𝑘23𝑘21𝑘2𝑘sinh2𝑦,𝐶3=𝑘21𝑘22𝑘cosh3𝑦,𝑆3=𝑘21𝑘22𝑘3𝑘sinh3𝑦.(2.37) Substituting for the parameters 𝑏𝑜𝑏5 from (2.36) into (2.34) and computing 𝐴2, 𝐴3, 𝐴4, and 𝐴5, we get the elements (𝐿𝑖𝑗=1,2,3,4,5,6) of the matrix 𝐿(𝑠,𝑦) to be𝐿11𝑘=𝑅21𝑘22𝑘23𝑘21𝐶1,𝐿12=𝐿13𝐿=0,14𝑘=𝑅21𝑘22𝑘23𝑘21𝑆1,𝐿15=𝐿16𝐿=0,21=𝐺𝑟𝑅𝑘21𝐶𝜉1+𝑘22𝐶𝜉2+𝑘23𝐶𝜉3,𝐿22𝑘=𝑅21𝑘22𝐹𝑘23𝐶2+𝑘21𝑘23𝐹𝑘22𝐶3,𝐿23𝑘=𝑛𝜉𝑅22𝑘21𝑆2+𝑘23𝑘21𝑆3,𝐿24=𝐺𝑟𝑅𝑘21𝑆𝜉1+𝑘22𝑆𝜉2+𝑘23𝑆𝜉3,𝐿25𝑘=𝑅22𝑘𝜉21𝑘22𝑆2+𝑘23𝑘𝜉21𝑘23𝑆3,𝐿26𝑘=𝑛𝑅22𝑘21𝐶2+𝑘23𝑘21𝐶3,𝐿31=𝐺𝑟𝑘𝜂𝑅21𝑆1+𝑘22𝑆2+𝑘23𝑆3,𝐿32𝑘=𝜂𝐹𝑅22𝑘21𝑆2+𝑘23𝑘21𝑆3,𝐿33=𝑅𝑛𝑘23𝑘21𝑘22𝐶2+𝑛𝑘22𝑘21𝑘23𝐶3,𝐿34=𝐺𝑟𝐶𝜂𝑅1+𝐶2+𝐶3,𝐿35𝑘=𝜂𝑅22𝑘21𝐶2+𝑘23𝑘21𝐶3,𝐿36𝑘=𝑅22𝑘𝐹21𝑘22𝑆2+𝑘23𝑘𝐹21𝑘23𝑆3,𝐿41=𝑅𝑘21𝑘21𝑘22𝑘21𝑘23𝑆1,𝐿42=𝐿43𝐿=0,44𝑘=𝑅21𝑘22𝑘21𝑘23𝐶1,𝐿45=𝐿46𝐿=0,51𝑘=𝐺𝑅21𝑘21𝑆𝜉1+𝑘22𝑘22𝑆𝜉2+𝑘23𝑘23𝑆𝜉3,𝐿52𝑘=𝐹𝑅21𝑘22𝑘21𝑆𝜉1+𝑘21𝑘23𝑘23𝑆𝜉3,𝐿53=𝑛𝜉𝑅𝜂𝐿35,𝐿54=𝐿21,𝐿55𝑘=𝑅21𝑘22𝑘22𝐶𝑛2+𝑘21𝑘23𝑘23𝐶𝑛3,𝐿56𝑘=𝑛𝑅22𝑘21𝑘22𝑆2+𝑘23𝑘21𝑘23𝑆3,𝐿61=𝜂𝐺𝑟𝑅𝑘21𝐶1+𝑘22𝐶2+𝑘23𝐶3,𝐿62=𝜂𝐹𝑛𝐿26,𝐿63=𝜉𝐿36,𝐿64=𝐿31,𝐿65=𝜂𝑛𝐿56,𝐿66𝑘=𝑅22𝑘𝐹21𝑘22𝐶2+𝑘23𝑘𝐹21𝑘23𝐶3.(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 𝑦0 or 𝑦0. Inside the region 0𝑦, the positive exponential terms, not bounded at infinity, must be suppressed. Thus, for 𝑦0, we should replace each sinh(𝑘𝑦) by (1/2)exp(𝑘𝑦) and each cosh(𝑘𝑦) by (1/2)exp(𝑘𝑦). In the region 𝑦0, the negative exponentials are suppressed instead.

We will now proceed to obtain the solution of the problem for the region 𝑦0. The solution for the other region is obtained by replacing each 𝑦 by 𝑦.

The formal solution of system (2.26) can be written in the form 𝑣(𝑦,𝑠)=exp(𝐴(𝑠)𝑦)𝑣(0,𝑠)+𝑦0exp(𝐴(𝑠)𝑧)𝐵(𝑧,𝑠)𝑑𝑧.(2.39) Evaluating the integral in (2.39) using the integral properties of the Dirac delta function, we obtain 𝑣(𝑦,𝑠)=𝐿(𝑦,𝑠)𝑣(0,𝑠)+𝐻(𝑠),(2.40) where𝐻(𝑠)=𝒬01+𝜐𝑜𝑠12𝑠2𝑘1𝐺𝑟𝑘1𝑘2𝑘3𝑘+𝜉1+𝑘2+𝑘32𝑘1𝑘2𝑘3𝑘1+𝑘2𝑘1+𝑘3𝑘2+𝑘30120𝜂𝐺𝑟2𝑘1+𝑘2𝑘1+𝑘3𝑘2+𝑘3.(2.41) Equation (2.40) expresses the solution of the problem in the Laplace transform domain for 𝑦0 in terms of the vector 𝐻(𝑠) representing the applied heat source and the vector 𝑣(0,𝑠) representing the conditions at the plate 𝑦=0. To evaluate the components of this vector, we note first, due to the symmetry of the problem, that the velocity component and induced magnetic field component vanish at the plane source of heat, thus (0,𝑡)=0,or(0,𝑠)=0,𝑢(0,𝑡)=0,or𝑢(0,𝑠)=0.(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 𝒬𝑞(0,𝑡)=𝑜2𝐻(𝑡),or𝒬𝑞(0,𝑠)=𝑜2𝑠.(2.43) We will use the generalized Fourier’s law of heat conduction in the nondimensional form [31], namely,𝑞+𝜐𝑜𝜕𝑞𝜕𝑡=𝜕𝜃𝜕𝑦.(2.44) Taking the Laplace transform of both sides of this equation and using (2.43), we obtain 𝜃(0,𝑠)=𝒬𝑜1+𝜐𝑜𝑠2𝑠.(2.45) Equation (2.42) and (2.45) give three components of the 𝑣(0,𝑠). To obtain the remaining three components, we substitute 𝑦=0 on both sides of (2.40) getting a system of linear equations whose solution gives 𝜃(0,𝑠)=𝒬𝑜1+𝜐𝑜𝑠2𝑠𝑘1,𝑢(0,𝑠)=𝑤𝐺𝑟𝒬01+𝜐𝑜𝑠2𝑠𝑘1𝑘1+𝑘2𝑘1+𝑘3𝑘2𝑘3,+𝜉(0,𝑠)=𝜂𝐺𝑟𝒬𝑜1+𝜐𝑜𝑠2𝑠𝑘1𝑘1+𝑘2𝑘1+𝑘3𝑘2𝑘3.+𝜉(2.46) Inserting the values from (2.42) and (2.46) into the right-hand side of (2.40) and performing the necessary matrix operations, we obtain𝒬𝜃(𝑦,𝑠)=𝑜1+𝜐𝑜𝑠2𝑠𝑘1𝑒𝑘1𝑦,(2.47)𝑢(𝑦,𝑠)=𝐺𝑟𝒬𝑜1+𝜐𝑜𝑠𝑘2𝑠𝛽𝜉2𝑘3𝐴1𝑒𝑘1𝑦+𝑘3𝑘1𝐴2𝑒𝑘2𝑦+𝑘1𝑘2𝐴3𝑒𝑘3𝑦,(2.48)(𝑦,𝑠)=𝐺𝑟𝜂𝒬𝑜1+𝜐𝑜𝑠𝑘2𝑠𝛽2𝑘3𝑘2+𝑘3𝑒𝑤𝑘1𝑦+𝑘3𝑘1𝑘1+𝑘3𝑒𝑤𝑘2𝑦+𝑘1𝑘2𝑘1+𝑘2𝑒𝑤𝑘3𝑦,(2.49) where𝑤=𝑘1𝑘2𝑘3𝑘+𝜉1+𝑘2+𝑘3,𝛽=𝑘1𝑘21𝑘22𝑘23𝑘21𝑘2𝑎3𝑘3𝑎2,𝐴1𝑘=𝑤𝜉1𝑘2𝑎2𝑎3𝜉𝑘3𝑘1+𝑘3,𝐴2=𝑎2𝑎3𝜉𝑘3𝑘1+𝑘3,𝐴3𝑘=𝑤𝜉2𝑘3𝑎2𝑎3𝜉𝑘3𝑘1+𝑘3,𝑎2=𝑘22𝜉,𝑎3=𝑘23𝜉.(2.50) Also, substituting from (2.49) into (2.21) the induced electric field is given by 𝐺𝐸(𝑦,𝑠)=𝑟𝜂𝒬𝑜1+𝜏𝑜𝑠𝑘2𝛽2𝑘3𝑘2+𝑘3𝑒𝑤𝑘1𝑦𝑘1+𝑘3𝑘1𝑘1+𝑘3𝑒𝑤2𝑦𝑘𝑘2+𝑘1𝑘2𝑘1+𝑘2𝑒𝑤𝑘3𝑦𝑘3.(2.51) Equations (2.47)–(2.51) determine completely the state of the fluid for 𝑦0. 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 𝑦0 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 have𝒬𝜃(𝑦,𝑠)=𝑜1+𝜐𝑜𝑠2𝑠𝑘1𝑒±𝑘1(𝑦𝑐),(2.52)𝑢(𝑦,𝑠)=𝐺𝑟𝒬𝑜1+𝜐𝑜𝑠𝑘2𝑠𝛽𝜉2𝑘3𝐴1𝑒±𝑘1(𝑦𝑐)+𝑘3𝑘1𝐴2𝑒±𝑘2(𝑦𝑐)+𝑘1𝑘2𝐴3𝑒±𝑘3(𝑦𝑐),(2.53)𝐺(𝑦,𝑠)=𝑟𝜂𝒬𝑜1+𝜐𝑜𝑠𝑘2𝑠𝛽2𝑘3𝑘2+𝑘3𝑒𝑤±𝑘1(𝑦𝑐)+𝑘3𝑘1𝑘1+𝑘3𝑒𝑤±𝑘2(𝑦𝑐)+𝑘1𝑘2𝑘1+𝑘2𝑒𝑤±𝑘3(𝑦𝑐),(2.54)𝐺𝐸(𝑦,𝑠)=𝑟𝜂𝒬𝑜1+𝜐𝑜𝑠𝑘2𝛽2𝑘3𝑘2+𝑘3𝑒𝑤±𝑘1(𝑦𝑐)𝑘1+𝑘3𝑘1𝑘1+𝑘3𝑒𝑤±𝑘2(𝑦𝑐)𝑘2+𝑘1𝑘2𝑘1+𝑘2𝑒𝑤±𝑘3(𝑦𝑐)𝑘3,(2.55) where the upper (plus) sign denotes the solution in the region 𝑦𝑐, while the lower (minus) sign denotes the solution in the region 𝑦𝑐.

3. 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,or𝑢(0,𝑠)𝜕𝑦=0,(0,𝑡)=0,or(0,𝑠)=0.(3.1)(ii)The temperature is kept at a constant value 𝑇, which means that the temperature increment 𝜃 satisfies𝜃(0,𝑡)=0or𝜃(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=𝜃1+𝜃2, where 𝜃1 is the temperature due to the positive heat source, given by (2.52) and 𝜃2 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+𝑐0. Thus, 𝜃2 is given by 𝜃2𝒬(𝑦,𝑠)=𝑜1+𝜐𝑜𝑠2𝑠𝑘1𝑒𝑘1(𝑦+𝑐).(3.3) Combining (2.52) and (3.2), we obtain 𝒬𝜃(𝑦,𝑠)=𝑜1+𝜐𝑜𝑠2𝑠𝑘1𝑒𝑘1𝑦sinh𝑘1𝑐for𝑦𝑐,𝒬𝜃(𝑦,𝑠)=𝑜1+𝜐𝑜𝑠2𝑠𝑘1𝑒𝑘1𝑐sinh𝑘1𝑦for𝑦𝑐.(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 relation𝑢=𝜕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 as𝐺𝜓=𝑟𝒬𝑜1+𝜐𝑜𝑠𝑘2𝑠𝛽𝜉2𝑘3𝐴1𝑒±𝑘1(𝑦𝑐)𝑘1+𝑘3𝑘1𝐴2𝑒±𝑘2(𝑦𝑐)𝑘2+𝑘1𝑘2𝐴3𝑒±𝑘3(𝑦𝑐)𝑘3,(3.6) where the upper sign is valid for the region 0𝑦𝑐 and the lower sign is valid for the region 𝑦0. Similarly, the stream function for the negative heat source at 𝑦=𝑐 is given by 𝐺𝜓=𝑟𝒬𝑜1+𝜐𝑜𝑠𝑘2𝑠𝛽𝜉2𝑘3𝐴1𝑒𝑘1(𝑦+𝑐)𝑘1+𝑘3𝑘1𝐴2𝑒𝑘2(𝑦+𝑐)𝑘2+𝑘1𝑘2𝐴3𝑒𝑘3(𝑦+𝑐)𝑘3.(3.7) Since 𝜓 is a scalar field, we can use superposition to obtain the stream function for the semispace problem as𝐺𝜓=𝑟𝒬𝑜1+𝜐𝑜𝑠𝑘𝑠𝛽𝜉2𝑘3𝐴1𝑒𝑘1𝑦sinh𝑘1𝑐𝑘1+𝑘3𝑘1𝐴2𝑒𝑘2𝑦sinh𝑘2𝑐𝑘2+𝑘1𝑘2𝐴3𝑒𝑘3𝑦sinh𝑘3𝑐𝑘3𝐺for𝑦𝑐,𝑟𝒬𝑜1+𝜐𝑜𝑠𝑘𝑠𝛽𝜉2𝑘3𝐴1𝑒𝑘1𝑐sinh𝑘1𝑦𝑘1+𝑘3𝑘1𝐴2𝑒𝑘2𝑐sinh𝑘2𝑦𝑘2+𝑘1𝑘2𝐴3𝑒𝑘3𝑐sinh𝑘3𝑦𝑘3for𝑦𝑐.(3.8) Using (3.8) and (3.5), we obtain the velocity distribution𝑢=𝐺𝑟𝒬𝑜1+𝜐𝑜𝑠𝑘𝑠𝛽𝜉2𝑘3𝐴1𝑒𝑘1𝑦sinh𝑘1𝑐+𝑘3𝑘1𝐴2𝑒𝑘2𝑦sinh𝑘2𝑐+𝑘1𝑘2𝐴3𝑒𝑘3𝑦sinh𝑘3𝑐𝐺for𝑦𝑐,𝑟𝒬𝑜1+𝜐𝑜𝑠𝑘𝑠𝛽𝜉2𝑘3𝐴1𝑒𝑘1𝑐cosh𝑘1𝑦+𝑘3𝑘1𝐴2𝑒𝑘2𝑐cosh𝑘2𝑦+𝑘1𝑘2𝐴3𝑒𝑘3𝑐cosh𝑘3𝑦for𝑦𝑐.(3.9) Differentiating (3.9) and using the resulting expressions together with (2.28), we obtain𝐺=𝑟𝜂𝒬𝑜1+𝜐𝑜𝑠𝑘𝑠𝛽2𝑘3𝑘2+𝑘3𝑒𝑤𝑘1𝑦sinh𝑘1𝑐+𝑘3𝑘1𝑘1+𝑘3𝑒𝑤𝑘2𝑦sinh𝑘2𝑐+𝑘1𝑘2𝑘1+𝑘2𝑒𝑤𝑘3𝑦sinh𝑘3𝑐𝐺for𝑦𝑐,𝑟𝜂𝒬𝑜1+𝜐𝑜𝑠𝑘𝑠𝛽2𝑘3𝑘2+𝑘3𝑒𝑤𝑘1𝑐sinh𝑘1𝑦+𝑘3𝑘1𝑘1+𝑘3𝑒𝑤𝑘2𝑐sinh𝑘2𝑦+𝑘1𝑘2𝑘1+𝑘2𝑒𝑤𝑘3𝑐sinh𝑘3𝑦for𝑦𝑐.(3.10) Substituting (3.10) and (2.21), we can get 𝐺𝐸=𝑟𝜂𝒬𝑜1+𝜐𝑜𝑠𝛽𝑘2𝑘3𝑘2+𝑘3𝑒𝑤𝑘1𝑦sinh𝑘1𝑐𝑘1+𝑘3𝑘1𝑘1+𝑘3𝑒𝑤𝑘2𝑦sinh𝑘2𝑐𝑘2+𝑘1𝑘2𝑘1+𝑘2𝑒𝑤𝑘3𝑦sinh𝑘3𝑐𝑘3for𝑦𝑐,𝐺𝑟𝜂𝒬𝑜1+𝜐𝑜𝑠𝛽𝑘2𝑘3𝑘2+𝑘3𝑒𝑤𝑘1𝑐cosh𝑘1𝑦𝑘1+𝑘3𝑘1𝑘1+𝑘3𝑒𝑤𝑘2𝑐cosh𝑘2𝑦𝑘2+𝑘1𝑘2𝑘1+𝑘2𝑒𝑤𝑘3𝑐cosh𝑘3𝑦𝑘3for𝑦𝑐.(3.11) Clearly, 𝜕𝑢(0,𝑠)/𝜕𝑦=(0,𝑠)=0 in agreement with (3.1).

4. 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). The inversion formula of Laplace transforms states that 1𝑔(𝑡)=2𝜋𝑖𝑑+𝑖𝑑𝑖𝑒𝑠𝑡𝑔(𝑠)𝑑𝑠,(4.1) where 𝑑 is an arbitrary positive constant greater than all the real parts of the singularities of 𝑔(𝑠). Taking 𝑠=𝑑+𝑖𝑦, we get 𝑒𝑔(𝑡)=𝑑𝑡2𝜋𝑒𝑖𝑡𝑦𝑔(𝑑+𝑖𝑦)𝑑𝑦.(4.2) This integral can be approximated by𝑒𝑔(𝑡)=𝑑𝑡2𝜋𝑘=𝑒𝑖𝑘𝑡Δ𝑦𝑔(𝑑+𝑖𝑘Δ𝑦)Δ𝑦.(4.3) Taking Δ𝑦=𝜋/𝑡1, we obtain𝑒𝑔(𝑡)=𝑑𝑡𝑡112𝑔(𝑑)+Re𝑘=1𝑒𝑖𝑘𝜋𝑡/𝑡1𝑔𝑑+𝑖𝑘𝜋𝑡1.(4.4) For numerical purposes, this is approximated by the function𝑔𝑁𝑒(𝑡)=𝑑𝑡𝑡112𝑔(𝑑)+Re𝑁𝑘=1𝑒𝑖𝑘𝜋𝑡/𝑡1𝑔𝑑+𝑖𝑘𝜋𝑡1,(4.5) where 𝑁 is a sufficiently large integer chosen such that𝑒𝑑𝑡𝑡1𝑒Re𝑖𝑁𝜋(𝑡/𝑡1)𝑔𝑑+𝑖𝑁𝜋𝑡1𝜂,(4.6) 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 0𝑡2𝑡1 [22]. In particular, we choose 𝑡=𝑡1, getting𝑔𝑁𝑒(𝑡)=𝑑𝑡𝑡12𝑔(𝑑)+Re𝑁𝑘=1(1)𝑘𝑔𝑑+𝑖𝑘𝜋𝑡.(4.7)

5. Numerical Results

The constants of the problem were taken as 𝜀𝑜=0.003, 𝛼=0.3, 𝐺𝑟=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, , and 𝐸 are evaluated. The results are shown in Figures 1, 2, 3, and 4. In these figures, solid lines represent the solution corresponding to using the generalized Fourier equation of heat conduction (𝜐𝑜=0.4), while dashed lines represent the solution corresponding to using the classical Fourier heat equation (𝜐𝑜=0.03).

731876.fig.001
Figure 1: Variation of 𝜃 with 𝑦, (−) for 𝜐0=0.03, (− ·) for 𝜐0=0.4, and 𝑡=0.7,1 at 𝑃𝑟=7, 𝐺𝑟=4, 𝐾=1.2, 𝛼=0.3, and 𝑐=2 representing onset stationary convection.
731876.fig.002
Figure 2: Variation of 𝑢 with 𝑦, (−) for 𝜐0=0.03, (− ·) for 𝜐0=0.4, and 𝑡=0.7,1 at 𝑃𝑟=7, 𝐺𝑟=4, 𝐾=1.2, 𝛼=0.3, and 𝑐=2 representing onset stationary convection.
731876.fig.003
Figure 3: Variation of with 𝑦, (−) for 𝜐0=0.03, (− ·) for 𝜐0=0.4, and 𝑡=0.7,1 at 𝑃𝑟=7, 𝐺𝑟=4, 𝐾=1.2, 𝛼=0.3, and 𝑐=2 representing onset stationary convection.
731876.fig.004
Figure 4: Variation of 𝐸 with 𝑦, (−) for 𝜐0=0.03, (− ·) for 𝜐0=0.4, and 𝑡=0.7,1 at 𝑃𝑟=7, 𝐺𝑟=4, 𝐾=1.2, 𝛼=0.3, and 𝑐=2 representing onset stationary convection.

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 (𝜐𝑜=0.03) and to using the non-Fourier case (𝜐𝑜=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 0𝑦𝑦(𝑡) for the semispace problem and by Min(0,𝑦(t)𝑐)𝑦𝑦+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.

Also, we observe from Figure 5. that the effect of heating by free convention currents when 𝐺𝑟0. In this case, it is noticed that as Alfven velocity 𝛼 increases, the velocity is found to decrease. This is mainly due to the fact that the effect of the magnetic field corresponds to a term signifying a positive force that tends to decelerate the fluid particles. Also, it is noticed that the velocity increases as 𝐺𝑟 increases, while it decreases when the Prandtl number 𝑝𝑟 increases.

731876.fig.005
Figure 5: Velocity distribution 𝑢 with 𝑦, for different values of 𝛼, 𝑃𝑟, and 𝐺𝑟, with 𝑡=0.7, 𝜐0=0.4, 𝐾=1, and 𝑐=2 representing onset stationary convection.

We notice that results for the temperature distribution when the relaxation time is appears in the heat equation are distinctly different from those when the relaxation time is not mentioned in the heat equation. This is due to the fact that thermal waves in the Fourier theory of heat equation travel with an infinite speed of propagation as opposed to finite speed in the non-Fourier case. We also notice that for small values of time, the solution is localized in a finite region near the plane surface. This region grows with increasing time until it fills the whole boundary-layer region. As time 𝑡 increases, results for both theories of heat equation almost coincide which is expected, since magnetohydrodynamic free convection flow effects are short lived. At all values of time, the velocity distributions for both theories coincide.

The effect of the relaxation time on the variation temperature and velocity distribution for this problem is shown in Figures 1 and 2, respectively.

The values of Grashof number 𝐺𝑟 have been chosen as they are interesting from physical point of view.

6. Concluding Remarks

Many metallic materials are manufactured after they have been redefined sufficiently in the molten state. Therefore, it is a central problem in metallurgical chemistry to study the free convection effects on conducting fluid metal. For instance, liquid sodium Na (100°C) exhibit very small electrical resistivity.

The effects of Grashof number, Alfven velocity, Prandtl number, and relaxation time on the oneset temperature and velocity distribution are discussed. A discussion is provided for the effects of heating on a viscous conducting fluid is given.

The importance of state space analysis is recognized in fields where the time behavior of any physical process is of interest.

The state-space approach is more general than the classical Laplace and Fourier transform techniques. Consequently, state space 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 [32].

Owing to the complicated nature of the governing equations for the unsteady magnetohydrodynamic with gradient pressure flow, few attempts have been made to solve problems in this field. These attempts utilized approximate methods valid for only a specific range of some parameters.

In this work, the method of direct integration by means of the matrix exponential, which is a standard approach in modern control theory and developed in detail in many texts such as Ogata [33], and Ezzat et al. [34, 35], is introduced in the field of magnetohydrodynamics and applied to two specific problems in which the temperature and velocity are coupled. This method gives exact solutions in the Laplace transform domain without any assumed restrictions on either the applied magnetic field or the velocity, temperature distributions, and viscoelastic parameter.

The method used in the present work is applicable to a wide range of problems. It can be applied to problems which are described by the linearized Navier-Stokes equations. The same approach was used quite successfully in dealing with problems in thermoelasticity theory [14, 15, 19, 34, 35].

Nomenclature

𝑡:Time
𝑥, 𝑦:Coordinates system
𝑣:=(𝑢,0,0) velocity vector of the fluid
𝑇:Temperature distribution
𝑇𝑜:Temperature of the plate
𝑇:Temperature of the fluid away from the plate
𝑝𝑟:Prandtl number
𝐺𝑟:Grashof number
𝑐𝑝:Specific heat at constant pressure
𝑔:Acceleration due to gravity
𝜌:Density
𝜆:Thermal diffusivity
:Induced magnetic field
𝒬𝑜:Constant
𝒬:The strength of the applied heat source
𝐸:Electric field
𝑐:Spead of light
𝐹:Magnetic viscosity
𝐽:Electric current density
𝜇𝑜:Magnetic permeability
𝛽:Coefficient of volume expansion
𝜀𝑜:Electric permeability
𝐻𝑜:Magnetic intensity vector
𝜐0:Relaxation time
𝛿(𝑦):Dirac delta function
𝐻(𝑡):Heaviside unit step function
𝐾:Permeability of the porous medium
𝛼:Alfven velocity
𝜎𝑜:Electric conductivity.

Acknowledgment

The author is very grateful to the referees for their helpful suggestions which improved the paper.

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