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

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 𝐻_0→ 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 _→ℎ 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] cucrulrℎ_→l𝐸_→=𝐽_→+𝜀_𝑜𝐸̇,(2.1)divℎ_→=−𝜇_𝑜ℎ̇,(2.2)𝐵=0,div𝐸_→=0,(2.3)_→=𝜇_𝑜𝐻^0→+ℎ_→,𝐷_→=𝜀_𝑜𝐸_→,(2.4) where 𝐽_→ is the electric current density, 𝜇_𝑜 and 𝜀_𝑜 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_→=𝜎_𝑜𝐸_→+𝜇_𝑜𝑣_→×𝐻_0→,(2.5) where _→𝑣=(𝑢,0,0) is velocity vector of the fluid and 𝜎_𝑜 is the electric conductivity.

As mentioned above, the applied field 𝐻_𝑜 and the induced magnetic field has the components 𝐻_0→=0,𝐻_𝑜,0,ℎ_→=(ℎ,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=𝜎_𝑜𝐸+𝜇_𝑜𝐻_𝑜𝑢.(2.8) The vector (2.1) and (2.2) reduce to the following scalar equations ℎ_,𝑦=𝐸−𝐽+𝜀_𝑜𝐸̇,(2.9)_,𝑦=−𝜇_𝑜ℎ̇.(2.10) Eliminating 𝐽 between (2.8) and (2.9), we obtainℎ_,𝑦=−𝜎_𝑜𝐸+𝜀_𝑜𝐸̇−𝜎_𝑜𝜇_𝑜𝐻_𝑜𝑢.(2.11) The pondermotive force 𝐹_→=𝐽_→×𝐵_→ has one nonvanishing component in the 𝑥-direction𝐹_𝑥=𝜇_𝑜𝐻_𝑜ℎ_,𝑦+𝜀_𝑜𝐸̇,(2.12) 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]𝜌𝑢̇=𝜌𝜈𝑢_,𝑦𝑦−𝜌𝜈𝐾𝑢+𝜌𝐻𝛼²_𝑜ℎ_,𝑦ℎ_,𝑦+𝜀_𝑜𝐸̇+𝜌𝑔𝛽𝑇−𝑇_∞,=−𝜎_𝑜𝐸𝐸+𝜀_𝑜𝐸̇−𝜎_𝑜𝜇_𝑜𝐻_𝑜𝑢,_,𝑦𝜌𝑐_𝑝=−𝜇_𝑜ℎ̇,𝑇̇=𝜆𝑇,_𝑦𝑦−𝜌𝑐_𝑝𝜐_𝑜𝑇̈+𝒬𝜐_𝑜𝒬̇.(2.13) And the constitutive equation 𝜌_∞−𝜌=𝜌𝛽𝑇−𝑇_∞.(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, 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 variables𝑦^∗=𝑦𝛼𝜈,𝑡^∗=𝑡𝛼²𝜈,𝑢^∗=𝑢𝛼,𝐸^∗=𝐸𝜇_𝑜𝐻_𝑜𝛼,ℎ^∗=ℎ𝐻_𝑜,𝜐^∗𝑜=𝛼²𝜐_𝑜𝐾_∗𝜈,=𝐾𝛼²𝜈²,𝜃=𝑇−𝑇^∞𝑇_𝑜−𝑇_∞,𝐺_𝑟=𝜈𝛽𝑔𝑇^𝑜−𝑇_∞𝛼³,𝒬^∗=𝑣²𝒬𝜆𝛼²𝑇_𝑜−𝑇_∞.(2.15) In view of these transformations, (2.13)–(2.14) become𝑢̇=𝑢_,𝑦𝑦−1𝐾𝑢+ℎ^,𝑦ℎ+𝑎𝐸̇+𝐺_𝑟𝜃,_,𝑦=−𝐸𝜂(𝐸+𝑢)−𝑎𝐸̇,_𝑦𝜃=−ℎ̇,_,𝑦𝑦−𝑝_𝑟𝜃̇+𝜐_𝑜𝜃̈=−𝒬−𝜐_𝑜𝒬̇,(2.16) where 𝐺_𝑟 is the Grashof number, 𝑐 is the speed of light given by 𝑐²=1/𝜀_𝑜𝜇_𝑜, 𝑎=𝛼²/𝑐², 𝜂=𝑣𝜇_𝑜𝜎_𝑜 is a measure of the magnetic viscosity, and 𝑝_𝑟=𝜌𝑐_𝑝𝜐/𝜆 is the Prandtl number. From now on, we will consider a heat source of the form𝒬=𝒬_𝑜𝛿(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 𝒬_𝑜 is a constant.

We will assume that the initial state of the medium is quiescent. Taking the Laplace transform, defined by the relation𝑓(𝑠)=^∞0𝑒^−𝑠𝑡𝑓(𝑡)𝑑𝑡,(2.18) of both sides of (2.16), we obtain𝜕²𝑢𝜕𝑦²−1𝐾+𝑠𝑢=−𝐺_𝑟𝜃−𝜕ℎ𝜕𝑦−𝑎𝑠𝜕𝐸,(2.19)ℎ𝜕𝑦=−(𝜂+𝑎𝑠)𝜕𝐸−𝜂𝑢,(2.20)𝐸𝜕𝑦=−𝑠𝜕ℎ,(2.21)₂𝜕𝑦²−𝑝_𝑟𝑠1+𝜏_𝑜𝑠𝜃=−𝒬_𝑜1+𝜏_𝑜𝑠𝑠𝛿(𝑦).(2.22) Eliminating 𝐸 between (2.19)-(2.20), we get 𝜕𝜕𝑦²²−𝐹𝑢=−𝐺_𝑟𝜃−𝑛𝜕ℎ𝜕𝜕𝑦₂,𝜕𝑦²−𝜉ℎ=−𝜂𝜕𝑢𝜕𝑦,(2.23) where 𝐹=𝑠+1𝐾+𝜂𝑎+𝑠𝜂𝑎𝑠,𝑛=𝜂+𝜂𝑎𝑠,𝜉=𝑠(𝜂+𝑎𝑠).(2.24)

We will choose as state variables the temperature 𝜃, the velocity component 𝑢, and induced magnetic field ℎ and their gradients. Equations (2.22)–(2.23) can be written as 𝜕𝜃𝜕𝑦=𝜃^,𝜕𝑢𝜕𝑦=𝑢^,𝜕ℎ𝜕𝑦=𝜕ℎ^,𝜃^𝜕𝑦=𝑝𝑠𝜃−𝒬_𝑜𝛿(𝑦)1+𝜐^𝑜𝑠𝜕𝑠,𝑢^𝜕𝑦=−𝐺^𝑟𝜕𝜃+𝐹𝑢−𝑛ℎ^,ℎ^𝜕𝑦=−𝜂𝑢^+𝜉ℎ,(2.25) where 𝑝=𝑝_𝑟(1+𝜐_𝑜𝑠). 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)=⎡⎢⎢⎢⎢⎢⎢⎣−𝑝000𝐺𝑠_𝑟000000000100001000010𝐹00𝜉00−0𝜂−0𝑛⎤⎥⎥⎥⎥⎥⎥⎦,𝑣(𝑦,𝑠)=⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣𝜃(𝑦,𝑠)𝑢(𝑦,𝑠)ℎ(𝑦,𝑠)𝜃^(𝑦,𝑠)𝑢^(𝑦,𝑠)𝐵(𝑦,𝑠)=−𝒬_𝑜ℎ^(𝑦,𝑠)⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦,𝛿(𝑦)1+𝜐^𝑜𝑠𝑠⎡⎢⎢⎢⎢⎢⎢⎣000100⎤⎥⎥⎥⎥⎥⎥⎦.(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 form𝑘⁶−𝑍₁𝑘⁴+𝑍₂𝑘²−𝑍₃=0,(2.28) where𝑍𝑍₁=𝐹+𝜉+𝑝𝑠+𝜂𝑛,₂=𝐹𝜉𝑍+𝑝𝑠(𝐹+𝜉)+𝜂𝑛𝑝𝑠,₃=𝑝𝑠𝐹𝜉.(2.29) The roots ±𝑘₁, ±𝑘₂, and ±𝑘₃ of (2.28) satisfy the relations𝑍₂𝑍₁=𝑘²¹+𝑘²²+𝑘²³,=𝑘𝑍²¹₃𝑘²²+𝑘²²𝑘²³+𝑘²¹𝑘²³,=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+𝑘𝑘²³=𝐹+𝜉+𝜂𝑛,₂₂𝑘²³=𝐹𝜉.(2.32) The Taylor series expansion of the matrix exponential has the formexp[𝐴(𝑠)𝑦]=^∞𝑛=01𝑛![𝐴(𝑠)𝑦]^𝑛.(2.33)

Using the well-known Cayley-Hamilton theorem, the infinite series can be truncated to the following form:exp[𝐴(𝑠)𝑦]=𝐿(𝑠,𝑦)=𝑏_𝑜𝐼+𝑏₁𝐴+𝑏₂𝐴²+𝑏₃𝐴³+𝑏₄𝐴⁴+𝑏₅𝐴⁵,(2.34) where 𝐼 is the unit matrix of order 6 and 𝑏_𝑜–𝑏₅ are some parameters depending on 𝑠 and 𝑦. The characteristic roots ±𝑘_𝑖, 𝑖=1,2,3 of the matrix 𝐴 must satisfy the equationseexxpp±±𝑘𝑘₁₂𝑦=𝑏_𝑜±𝑏₁𝑘₁+𝑏₂𝑘²¹±𝑏₃𝑘³¹+𝑏₄𝑘⁴¹±𝑏₅𝑘⁵¹,exp±𝑘₃𝑦=𝑏_𝑜±𝑏₁𝑘₂+𝑏₂𝑘²²±𝑏₃𝑘³²+𝑏₄𝑘⁴²±𝑏₅𝑘⁵²,𝑦=𝑏_𝑜±𝑏₁𝑘₃+𝑏𝑎₂𝑘²³±𝑏₃𝑘³³+𝑏₄𝑘⁴³±𝑏₅𝑘⁵³.(2.35) The solution of this system of linear equations is given by 𝑏_𝑜𝑏₁=−𝑅𝑘²²𝑘²³𝐶₁+𝑘²¹𝑘²³𝐶₂+𝑘²²𝑘²¹𝐶₃,𝑏₂=−𝑅𝑘²²𝑘²³𝑆₁+𝑘²³𝑘²¹𝑆₂+𝑘²¹𝑘²²,𝑏₃=𝑅𝑘²²+𝑘²³𝐶₁+𝑘²³+𝑘²¹𝐶₂+𝑘²¹+𝑘²²𝐶₃,=𝑅𝑘²²+𝑏𝑘²³₄𝑆₁+𝑘²³+𝑘²¹𝑆₂+𝑘²¹+𝑘²²𝑆₃,𝑏₅=−𝑅𝐶₁+𝐶₂+𝐶₃,=−𝑅𝑆₁+𝑆₂+𝑆₃,(2.36) where 𝑅=1𝐶₁𝑘²¹−𝑘²²𝑘²²−𝑘²³𝑘²³−𝑘²¹,=𝑘²²−𝑘²³cosh𝑘₁𝑦,𝑆₁=𝑘²²−𝑘²³𝑘₁𝐶₂sinh𝑘₁𝑦,=𝑘²³−𝑘²¹cosh𝑘₂𝑦,𝑆₂=𝑘²³−𝑘²¹𝑘₂𝐶₃sinh𝑘₂𝑦,=𝑘²¹−𝑘²²cosh𝑘₃𝑦,𝑆₃=𝑘²¹−𝑘²²𝑘₃sinh𝑘₃𝑦.(2.37) Substituting for the parameters 𝑏_𝑜–𝑏₅ from (2.36) into (2.34) and computing 𝐴², 𝐴³, 𝐴⁴, and 𝐴⁵, we get the elements (𝐿_𝑖𝑗=1,2,3,4,5,6) of the matrix 𝐿(𝑠,𝑦) to be𝐿𝐿₁₁₁₂=𝑅𝑘²¹−𝑘²²𝑘²³−𝑘²¹𝐶₁,𝐿=𝐿₁₃=0,₁₄𝐿₁₅=𝑅𝑘²¹−𝑘²²𝑘²³−𝑘²¹𝑆₁,𝐿=𝐿₁₆=0,₂₁𝐿₂₂=𝐺_𝑟𝑅𝑘²¹−𝜉𝐶₁+𝑘²²−𝜉𝐶₂+𝑘²³−𝜉𝐶₃,𝐿₂₃=𝑅𝑘²¹−𝑘²²𝐹−𝑘²³𝐶₂+𝑘²¹−𝑘²³𝐹−𝑘²²𝐶₃,𝐿₂₄=𝑛𝜉𝑅𝑘²²−𝑘²¹𝑆₂+𝑘²³−𝑘²¹𝑆₃,𝐿₂₅=𝐺_𝑟𝑅𝑘²¹−𝜉𝑆₁+𝑘²²−𝜉𝑆₂+𝑘²³−𝜉𝑆₃,𝐿₂₆=𝑅𝑘²²−𝜉𝑘²¹−𝑘²²𝑆₂+𝑘²³−𝜉𝑘²¹−𝑘²³𝑆₃,𝐿₃₁=𝑛𝑅𝑘²²−𝑘²¹𝐶₂+𝑘²³−𝑘²¹𝐶₃,𝐿₃₂=−𝐺_𝑟𝜂𝑅𝑘²¹𝑆₁+𝑘²²𝑆₂+𝑘²³𝑆₃,𝐿₃₃=𝜂𝐹𝑅𝑘²²−𝑘²¹𝑆₂+𝑘²³−𝑘²¹𝑆₃,𝐿₃₄=𝑅𝑛−𝑘²³𝑘²¹−𝑘²²𝐶₂+𝑛−𝑘²²𝑘²¹−𝑘²³𝐶₃,𝐿₃₅=−𝐺_𝑟𝜂𝑅𝐶₁+𝐶₂+𝐶₃,𝐿₃₆=𝜂𝑅𝑘²²−𝑘²¹𝐶₂+𝑘²³−𝑘²¹𝐶₃,𝐿₄₁=𝑅𝑘²²−𝐹𝑘²¹−𝑘²²𝑆₂+𝑘²³−𝐹𝑘²¹−𝑘²³𝑆₃,𝐿₄₂=−𝑅𝑘²¹𝑘²¹−𝑘²²𝑘²¹−𝑘²³𝑆₁,𝐿=𝐿₄₃=0,₄₄𝐿₄₅=−𝑅𝑘²¹−𝑘²²𝑘²¹−𝑘²³𝐶₁,𝐿=𝐿₄₆=0,₅₁𝐿₅₂=𝐺𝑅𝑘²¹𝑘²¹−𝜉𝑆₁+𝑘²²𝑘²²−𝜉𝑆₂+𝑘²³𝑘²³−𝜉𝑆₃,𝐿₅₃=𝐹𝑅𝑘²¹−𝑘²²𝑘²¹−𝜉𝑆₁+𝑘²¹−𝑘²³𝑘²³−𝜉𝑆₃,=𝑛𝜉𝑅𝜂𝐿³⁵𝐿₅₄,𝐿₅₅=𝐿₂₁,𝐿₅₆=𝑅𝑘²¹−𝑘²²𝑘²²−𝑛𝐶₂+𝑘²¹−𝑘²³𝑘²³−𝑛𝐶₃,𝐿₆₁=−𝑛𝑅𝑘²²𝑘²¹−𝑘²²𝑆₂+𝑘²³𝑘²¹−𝑘²³𝑆₃,𝐿₆₂=−𝜂𝐺_𝑟𝑅𝑘²¹𝐶₁+𝑘²²𝐶₂+𝑘²³𝐶₃,=𝜂𝐹𝑛𝐿²⁶𝐿₆₃,𝐿₆₄=𝜉𝐿₃₆,𝐿₆₅=𝐿₃₁,=𝜂𝑛𝐿⁵⁶𝐿₆₆,=𝑅𝑘²²−𝐹𝑘²¹−𝑘²²𝐶₂+𝑘²³−𝐹𝑘²¹−𝑘²³𝐶₃.(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 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)=−𝒬⁰1+𝜐_𝑜𝑠2𝑠⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣12𝑘₁𝐺_𝑟𝑘₁𝑘₂𝑘₃+𝜉𝑘₁+𝑘₂+𝑘₃2𝑘₁𝑘₂𝑘₃𝑘₁+𝑘₂01𝑘₁+𝑘₃𝑘₂+𝑘₃−𝜂20𝐺_𝑟2𝑘₁+𝑘₂𝑘₁+𝑘₃𝑘₂+𝑘₃⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦.(2.41) Equation (2.40) expresses the solution of the problem in the Laplace transform domain for 𝑦≥0 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 ℎ𝑢((00,,𝑡𝑡))==00,,oorrℎ(0,𝑠)=0,𝑢(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 𝑞(0,𝑡)=𝒬^𝑜2𝐻(𝑡),or𝑞(0,𝑠)=𝒬^𝑜2𝑠.(2.43) We will use the generalized Fourier’s law of heat conduction in the nondimensional form [31], namely,𝑞+𝜐_𝑜𝜕𝑞𝜕𝑡=−𝜕𝜕𝜃𝑦.(2.44) Taking the Laplace transform of both sides of this equation and using (2.43), we obtain 𝜃^(0,𝑠)=−𝒬^𝑜1+𝜐_𝑜𝑠2𝑠.(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)=−𝒬^𝑜1+𝜐_𝑜𝑠2𝑠𝑘₁,𝑢^(0,𝑠)=𝑤𝐺^𝑟𝒬₀1+𝜐_𝑜𝑠2𝑠𝑘₁𝑘₁+𝑘₂𝑘₁+𝑘₃𝑘₂𝑘₃+𝜉,ℎ^(0,𝑠)=𝜂𝐺^𝑟𝒬_𝑜1+𝜐_𝑜𝑠2𝑠𝑘₁𝑘₁+𝑘₂𝑘₁+𝑘₃𝑘₂𝑘₃+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 obtain𝜃(𝑦,𝑠)=𝒬^𝑜1+𝜐_𝑜𝑠2𝑠𝑘₁𝑒^−𝑘1𝑦,(2.47)𝑢(𝑦,𝑠)=−𝐺^𝑟𝒬_𝑜1+𝜐_𝑜𝑠2𝑠𝛽𝜉𝑘²−𝑘₃𝐴₁𝑒^−𝑘1𝑦+𝑘₃−𝑘₁𝐴₂𝑒^−𝑘2𝑦+𝑘₁−𝑘₂𝐴₃𝑒^−𝑘3𝑦,(2.48)ℎ(𝑦,𝑠)=−𝐺^𝑟𝜂𝒬_𝑜1+𝜐_𝑜𝑠2𝑠𝛽𝑘²−𝑘₃𝑘₂+𝑘₃−𝑤𝑒^−𝑘1𝑦+𝑘₃−𝑘₁𝑘₁+𝑘₃−𝑤𝑒^−𝑘2𝑦+𝑘₁−𝑘₂𝑘₁+𝑘₂−𝑤𝑒^−𝑘3𝑦,(2.49) where𝛽=𝑘𝑤₁=𝑘₁𝑘₂𝑘₃+𝜉𝑘₁+𝑘₂+𝑘₃,𝐴₁𝑘²¹−𝑘²²𝑘²³−𝑘²¹𝑘₂𝑎₃−𝑘₃𝑎₂,=𝑤𝐴𝜉𝑘₁−𝑘₂−𝑎₂𝑎₃−𝜉𝑘₃𝑘₁+𝑘₃₂,𝐴=−𝑎₂𝑎₃−𝜉𝑘₃𝑘₁+𝑘₃₃,=−𝑤𝜉𝑎𝑘₂−𝑘₃−𝑎₂𝑎₃−𝜉𝑘₃𝑘₁+𝑘₃₂,=𝑘²²−𝜉,𝑎₃=𝑘²³−𝜉.(2.50) Also, substituting from (2.49) into (2.21) the induced electric field is given by 𝐸(𝑦,𝑠)=𝐺^𝑟𝜂𝒬_𝑜1+𝜏_𝑜𝑠2𝛽𝑘²−𝑘₃𝑘₂+𝑘₃−𝑤𝑒^−𝑘1𝑦𝑘₁+𝑘₃−𝑘₁𝑘₁+𝑘₃−𝑤𝑒^2𝑦−𝑘𝑘₂+𝑘₁−𝑘₂𝑘₁+𝑘₂−𝑤𝑒^−𝑘3𝑦𝑘₃.(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 have𝜃(𝑦,𝑠)=𝒬^𝑜1+𝜐_𝑜𝑠2𝑠𝑘₁𝑒^{±𝑘1(𝑦−𝑐)},(2.52)𝑢(𝑦,𝑠)=−𝐺^𝑟𝒬_𝑜1+𝜐_𝑜𝑠2𝑠𝛽𝜉𝑘²−𝑘₃𝐴₁𝑒^{±𝑘1(𝑦−𝑐)}+𝑘₃−𝑘₁𝐴₂𝑒^{±𝑘2(𝑦−𝑐)}+𝑘₁−𝑘₂𝐴₃𝑒^{±𝑘3(𝑦−𝑐)},(2.53)ℎ(𝑦,𝑠)=𝐺^𝑟𝜂𝒬_𝑜1+𝜐_𝑜𝑠2𝑠𝛽𝑘²−𝑘₃𝑘₂+𝑘₃−𝑤𝑒^{±𝑘1(𝑦−𝑐)}+𝑘₃−𝑘₁𝑘₁+𝑘₃−𝑤𝑒^{±𝑘2(𝑦−𝑐)}+𝑘₁−𝑘₂𝑘₁+𝑘₂−𝑤𝑒^{±𝑘3(𝑦−𝑐)},(2.54)𝐸(𝑦,𝑠)=𝐺^𝑟𝜂𝒬_𝑜1+𝜐_𝑜𝑠2𝛽𝑘²−𝑘₃𝑘₂+𝑘₃−𝑤𝑒^{±𝑘1(𝑦−𝑐)}𝑘₁+𝑘₃−𝑘₁𝑘₁+𝑘₃−𝑤𝑒^{±𝑘2(𝑦−𝑐)}𝑘₂+𝑘₁−𝑘₂𝑘₁+𝑘₂−𝑤𝑒^{±𝑘3(𝑦−𝑐)}𝑘₃,(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.

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,or𝜕𝑢(0,𝑠)ℎ(0,𝑡)=0,or𝜕𝑦=0,ℎ(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+𝑐≻0. Thus, 𝜃₂ is given by 𝜃₂(𝑦,𝑠)=𝒬^𝑜1+𝜐_𝑜𝑠2𝑠𝑘₁𝑒^{−𝑘1(𝑦+𝑐)}.(3.3) Combining (2.52) and (3.2), we obtain 𝜃(𝑦,𝑠)=𝒬^𝑜1+𝜐_𝑜𝑠2𝑠𝑘₁𝑒^−𝑘1𝑦sinh𝑘₁𝑐for𝑦≥𝑐,𝜃(𝑦,𝑠)=𝒬^𝑜1+𝜐_𝑜𝑠2𝑠𝑘₁𝑒^−𝑘1𝑐sinh𝑘₁𝑦for𝑦≺𝑐.(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 as𝜓=𝐺^𝑟𝒬_𝑜1+𝜐_𝑜𝑠2𝑠𝛽𝜉𝑘²−𝑘₃𝐴₁𝑒^{±𝑘1(𝑦−𝑐)}𝑘₁+𝑘₃−𝑘₁𝐴₂𝑒^{±𝑘2(𝑦−𝑐)}𝑘₂+𝑘₁−𝑘₂𝐴₃𝑒^{±𝑘3(𝑦−𝑐)}𝑘₃,(3.6) 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 𝜓=𝐺^𝑟𝒬_𝑜1+𝜐_𝑜𝑠2𝑠𝛽𝜉𝑘²−𝑘₃𝐴₁𝑒^{−𝑘1(𝑦+𝑐)}𝑘₁+𝑘₃−𝑘₁𝐴₂𝑒^{−𝑘2(𝑦+𝑐)}𝑘₂+𝑘₁−𝑘₂𝐴₃𝑒^{−𝑘3(𝑦+𝑐)}𝑘₃.(3.7) Since 𝜓 is a scalar field, we can use superposition to obtain the stream function for the semispace problem as𝜓=⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩𝐺^𝑟𝒬_𝑜1+𝜐_𝑜𝑠𝑠𝛽𝜉𝑘²−𝑘₃𝐴₁𝑒^−𝑘1𝑦sinh𝑘₁𝑐𝑘₁+𝑘₃−𝑘₁𝐴₂𝑒^−𝑘2𝑦sinh𝑘₂𝑐𝑘₂+𝑘₁−𝑘₂𝐴₃𝑒^−𝑘3𝑦sinh𝑘₃𝑐𝐺𝑘₃for𝑦≥𝑐,_𝑟𝒬_𝑜1+𝜐_𝑜𝑠𝑠𝛽𝜉𝑘²−𝑘₃𝐴₁𝑒^−𝑘1𝑐sinh𝑘₁𝑦𝑘₁+𝑘₃−𝑘₁𝐴₂𝑒^−𝑘2𝑐sinh𝑘₂𝑦𝑘₂+𝑘₁−𝑘₂𝐴₃𝑒^−𝑘3𝑐sinh𝑘₃𝑦𝑘₃for𝑦≺𝑐.(3.8) Using (3.8) and (3.5), we obtain the velocity distribution𝑢=⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩−𝐺^𝑟𝒬_𝑜1+𝜐_𝑜𝑠𝑠𝛽𝜉𝑘²+𝑘₃−𝑘₃𝐴₁𝑒^−𝑘1𝑦sinh𝑘₁𝑐+𝑘₁−𝑘₁𝐴₂𝑒^−𝑘2𝑦sinh𝑘₂𝑐𝐺−𝑘₂𝐴₃𝑒^−𝑘3𝑦sinh𝑘₃𝑐for𝑦≥𝑐,_𝑟𝒬_𝑜1+𝜐_𝑜𝑠𝑠𝛽𝜉𝑘²+𝑘₃−𝑘₃𝐴₁𝑒^−𝑘1𝑐cosh𝑘₁𝑦+𝑘₁−𝑘₁𝐴₂𝑒^−𝑘2𝑐cosh𝑘₂𝑦−𝑘₂𝐴₃𝑒^−𝑘3𝑐cosh𝑘₃𝑦for𝑦≺𝑐.(3.9) Differentiating (3.9) and using the resulting expressions together with (2.28), we obtainℎ=⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩𝐺^𝑟𝜂𝒬_𝑜1+𝜐_𝑜𝑠𝑠𝛽𝑘²+𝑘₃−𝑘₃𝑘₂+𝑘₃−𝑤𝑒^−𝑘1𝑦sinh𝑘₁𝑐+𝑘₁−𝑘₁𝑘₁+𝑘₃−𝑤𝑒^−𝑘2𝑦sinh𝑘₂𝑐𝐺−𝑘₂𝑘₁+𝑘₂−𝑤𝑒^−𝑘3𝑦sinh𝑘₃𝑐for𝑦≥𝑐,_𝑟𝜂𝒬_𝑜1+𝜐_𝑜𝑠𝑠𝛽𝑘²+𝑘₃−𝑘₃𝑘₂+𝑘₃−𝑤𝑒^−𝑘1𝑐sinh𝑘₁𝑦+𝑘₁−𝑘₁𝑘₁+𝑘₃−𝑤𝑒^−𝑘2𝑐sinh𝑘₂𝑦−𝑘₂𝑘₁+𝑘₂−𝑤𝑒^−𝑘3𝑐sinh𝑘₃𝑦for𝑦≺𝑐.(3.10) Substituting (3.10) and (2.21), we can get 𝐸=⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩𝐺^𝑟𝜂𝒬_𝑜1+𝜐_𝑜𝑠𝛽𝑘²−𝑘₃𝑘₂+𝑘₃−𝑤𝑒^−𝑘1𝑦sinh𝑘₁𝑐𝑘₁+𝑘₃−𝑘₁𝑘₁+𝑘₃−𝑤𝑒^−𝑘2𝑦sinh𝑘₂𝑐𝑘₂+𝑘₁−𝑘₂𝑘₁+𝑘₂−𝑤𝑒^−𝑘3𝑦sinh𝑘₃𝑐−𝐺_𝑟𝑘₃for𝑦≥𝑐,𝜂𝒬_𝑜1+𝜐_𝑜𝑠𝛽𝑘²−𝑘₃𝑘₂+𝑘₃−𝑤𝑒^−𝑘1𝑐cosh𝑘₁𝑦𝑘₁+𝑘₃−𝑘₁𝑘₁+𝑘₃−𝑤𝑒^−𝑘2𝑐cosh𝑘₂𝑦𝑘₂+𝑘₁−𝑘₂𝑘₁+𝑘₂−𝑤𝑒^−𝑘3𝑐cosh𝑘₃𝑦𝑘₃for𝑦≺𝑐.(3.11) Clearly, 𝜕𝑢(0,𝑠)/𝜕𝑦=ℎ(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 𝑔(𝑡)=12𝜋𝑖^{𝑑+𝑖∞𝑑−𝑖∞}𝑒^𝑠𝑡𝑔(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 𝑔(𝑡)=𝑒^𝑑𝑡2𝜋^∞−∞𝑒^𝑖𝑡𝑦𝑔(𝑑+𝑖𝑦)𝑑𝑦.(4.2) This integral can be approximated by𝑔(𝑡)=𝑒^𝑑𝑡2𝜋^{∞𝑘=−∞}𝑒^{𝑖𝑘𝑡Δ𝑦}𝑔(𝑑+𝑖𝑘Δ𝑦)Δ𝑦.(4.3) Taking Δ𝑦=𝜋/𝑡₁, we obtain𝑔(𝑡)=𝑒^𝑑𝑡𝑡₁21𝑔(𝑑)+Re^∞𝑘=1𝑒^{𝑖𝑘𝜋𝑡/𝑡1}𝑔𝑑+𝑖𝑘𝜋𝑡₁.(4.4) For numerical purposes, this is approximated by the function𝑔_𝑁(𝑡)=𝑒^𝑑𝑡𝑡₁21𝑔(𝑑)+Re^𝑁𝑘=1𝑒^{𝑖𝑘𝜋𝑡/𝑡1}𝑔𝑑+𝑖𝑘𝜋𝑡₁,(4.5) where 𝑁 is a sufficiently large integer chosen such that𝑒^𝑑𝑡𝑡₁Re𝑒^{𝑖𝑁𝜋(𝑡/𝑡1)}𝑔𝑑+𝑖𝑁𝜋𝑡₁≺𝜂,(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 0≤𝑡≤2𝑡₁ [22]. In particular, we choose 𝑡=𝑡₁, getting𝑔_𝑁(𝑡)=𝑒^𝑑𝑡𝑡12𝑔(𝑑)+Re^𝑁𝑘=1(−1)^𝑘𝑔𝑑+𝑖𝑘𝜋𝑡.(4.7)

5. 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, 3, and 4. In these figures, solid lines represent the solution corresponding to using the generalized Fourier equation of heat conduction (𝜐_𝑜=0.4), while dashed lines represent the solution corresponding to using the classical Fourier heat equation (𝜐_𝑜=0.03).

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.

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.

Figure 3 

Variation of ℎ 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.

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 (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.

Also, we observe from Figure 5. that the effect of heating by free convention currents when 𝐺_𝑟≻0. 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 𝐺_𝑟 increases, while it decreases when the Prandtl number 𝑝_𝑟 increases.

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 𝐺_𝑟 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
ℎ:	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.