An Analytical Solution for Effect of Magnetic Field and Initial Stress on an Infinite Generalized Thermoelastic Rotating Nonhomogeneous Diffusion Medium

Abstract

The problem of generalized magneto-thermoelastic diffusion in an infinite rotating nonhomogeneity medium subjected to certain boundary conditions is studied. The chemical potential is also assumed to be a known function of time at the boundary of the cavity. The analytical expressions for the displacements, stresses, temperature, concentration, and chemical potential are obtained. Comparison was made between the results obtained in the presence and absence of diffusion. The results indicate that the effect of nonhomogeneity, rotation, magnetic field, relaxation time, and diffusion is very pronounced.

1. Introduction

Diffusion can be defined as the spontaneous migration of substances from regions of high concentration to regions of low concentration. There is now a great deal of interest in the study of this phenomenon due to its many applications in geophysics and industrial applications. Thermodiffusion in the solids is one of the transport processes which has great practical importance. Thermodiffusion in an elastic solid is due to the coupling of the fields of temperature, mass diffusion, and that of strain. This mater has attracted the attention of many researchers such as [1–5]. Wave propagation in rotating and nonhomogeneous media was studied by Abd-Alla et al. [6–8]. The extended thermoelasticity theory, introducing one relaxation time in the thermoelastic process, was proposed by Lord and Shulman [9]. In this theory, a modified law of heat conduction including both the heat flux and its time derivative replaces conventional Fourier’s law. The heat equation associated with this is a hyperbolic one and hence automatically eliminates the paradox of infinite speeds of propagation inherent in the coupled theory of thermoelasticity. This theory was extended by Dhaliwal and Sherief [10] to include the anisotropic case. Abd-Alla and Mahmoud [11] investigated the magneto-thermoelastic problem in rotating nonhomogeneous orthotropic hollow cylinder under the hyperbolic heat conduction model. Mahmoud [12] investigated wave propagation in cylindrical poroelastic dry bones.

Kumar and Devi [13] studied deformation in porous thermoelastic material with temperature dependent properties. Othman et al. [14] presented the study of the two-dimensional problems of generalized thermoelasticity with one relaxation time with the modulus of elasticity being dependent on the reference temperature for nonrotating and rotating medium, respectively. Kumar and Gupta [15] investigated deformation due to inclined load in an orthotropic micropolar thermoelastic medium with two relaxation times. The temperature-rate dependent theory of thermoelasticity, which takes into account two relaxation times, was developed by Green and Lindsay [16]. Abd-Alla et al. [17, 18] investigated radial vibrations in a nonhomogeneous orthotropic elastic medium subjected to rotation and gravity field. Sherief et al. [19] developed the generalized theory of thermoelastic diffusion with one relaxation time, which allows the finite speed of propagation waves. Sherief and Saleh [20] investigated the problem of a thermoelastic half-space in the context of the theory of generalized thermoelastic diffusion with one relaxation time. The reflection of SV waves from the free surface of an elastic solid in generalized thermoelastic diffusion was discussed by Singh [21]. Kumar and Kansal [22] discussed the propagation of Lamb waves in transversely isotropic thermoelastic diffusive plates. Thermomechanical response of generalized thermoelastic diffusion with one relaxation time due to time harmonic sources was discussed by Ram et al. [23]. Aouadi [24] examined the thermoelastic diffusion problem for an infinite elastic body with spherical cavity. Abd-Alla and Mahmoud [25] investigated analytical solution of wave propagation in nonhomogeneous orthotropic rotating elastic media. Othman et al. [26] discussed the effect of diffusion in a two-dimensional problem of generalized thermoelasticity with Green-Naghdi theory. Xia et al. [27] studied the influence of diffusion on generalized thermoelastic problems of infinite body with a cylindrical cavity. Deswal and Kalkal [28] studied the two-dimensional generalized electromagneto-thermoviscoelastic problem for a half-space with diffusion. Abd-Alla and Abo-Dahab [29] found the time-harmonic sources in a generalized magneto-thermo-viscoelastic continuum with and without energy dissipation. Mahmoud [30] discussed influence of rotation and generalized magnetothermoelastic on Rayleigh waves in a granular medium under effect of initial stress and gravity field. Abd-Alla et al. [31, 32] studied the generalized magneto-thermoelastic Rayleigh waves in a granular medium under the influence of a gravity field and initial stress.

In the present investigation, the temperature, displacements, stresses, diffusion, and concentration as well as chemical potential are obtained in the physical domain using the harmonic vibrations. Also, study of the interaction between the processes of elasticity, nonhomogeneity, rotation, magnetic field, initial stress, heat, and diffusion in an infinite elastic solid with a spherical cavity in the context of the theory of generalized thermoelastic diffusion is presented.

2. Formulation of the Problem

Consider a perfect electric conductor and linearized Maxwell equations governing the electromagnetic field in the absence of the displacement current (SI) in the form as in Kraus [33]. Applying an initial magnetic field vector 𝐻(0,0,𝐻₀) in spherical coordinates (𝑟,𝜃,𝜙), 𝑢=(𝑢(𝑟,𝑡),0,0). One will consider a nonhomogeneous, isotropic medium, occupying the region 𝑎≤𝑟≤𝑏, where a is the radius of the spherical cavity. The strain tensor has the following components:𝑒_𝑖𝑗=12𝑢_𝑖,𝑗+𝑢_𝑗,𝑖,𝜔_𝑖𝑗=12𝑢_𝑗,𝑖−𝑢_𝑖,𝑗,𝑒_𝑘𝑘=1𝑟²𝜕𝑟𝜕𝑟²𝑢𝑒,𝑖,𝑗,𝑘=1,2,3,_𝑟𝑟=𝜕𝑢𝜕𝑟,𝑒_𝜃𝜃=𝑒_𝜙𝜙=𝑢𝑟. (1a)(1b)(1c)(1d)

The cubical dilatation is given by (1/𝑟²)(𝜕/𝜕𝑟)(𝑟²𝑢), where the nonvanishing displacement component is the radial one 𝑢_𝑟=𝑢(𝑟,𝑡). The elastic medium is rotating uniformly with an angular velocity Ω=Ω𝑛, where 𝑛 is a unit vector representing the direction of the axis of rotation. The displacement equation of motion in the rotating frame has two additional terms: Ω×(Ω×𝑢) which is the centripetal acceleration due to time varying motion only, and 2Ω×𝑢̇ is the Coriolis acceleration, where Ω=(0,Ω,0). Following Sherief’s theory of generalized thermoelastic diffusion [19] and Sherief and Saleh [20], one is going to study an isotropic nonhomogeneous elastic medium which suffers thermal shock. Due to spherical symmetry, the stress-displacement-temperature-diffusion relation or constitutive equations are given by𝜎_𝑟𝑟=2𝜇+𝑃₁𝜕𝑢+𝜕𝑟𝜆+𝑃₁1𝑟²𝜕𝑟𝜕𝑟²𝑢−𝛽₁𝜃−𝜃₀−𝛽₂𝜎𝐶,_𝜙𝜙=𝜎_𝜃𝜃=2𝜇+𝑃₁𝑢𝑟+𝜆+𝑃₁1𝑟²𝜕𝑟𝜕𝑟²𝑢−𝛽₁𝜃−𝜃₀−𝛽₂𝐶. (2a)(2b)

The chemical-displacement-temperature-diffusion relation is given by 𝑃=−𝛽₂1𝑟²𝜕𝑟𝜕𝑟²𝑢+𝑏𝐶−𝑐𝜃−𝜃₀. (3)

The governing equation for an isotropic nonhomogeneous elastic solid with generalized magneto-thermoelastic diffusion under effect of rotation is given by 𝜕1/𝑟²(𝜕/𝜕𝑟)𝑟²𝑢−𝛽𝜕𝑟ℎ²₀𝜕𝐶−𝛽𝜕𝑟ℎ¹₀𝜕𝜃−𝜃₀𝜕𝑟+𝐹_𝑟=𝜌ℎ₀𝜕²𝑢𝜕𝑡²−Ω²𝑢−2Ω𝜕𝑢,𝜕𝑡 (4) where 𝜆 and 𝜇 are Lame’s elastic constants, 𝛿_𝑖𝑗 is Kronecker’s delta, 𝑃₁ is the initial stress, 𝜌 is the density of the medium, and 𝐹 is defined as Lorentz’s force which may be written as 𝐹=𝜇_𝑒𝐽×𝐻=𝜇_𝑒𝐻²₀𝜕𝜕𝑟𝜕𝑢+𝜕𝑟2𝑢𝑟,0,0, (5) where 𝜇_𝑒 is the magnetic permeability, 𝐻 is the magnetic field vector, 𝐽 is the electric current density, 𝑢 is the displacement vector, and 𝑡 is the time.

Equation of heat conduction is given by 𝐾∇²𝜕𝜃=𝜕𝜕𝑡+𝜏²𝜕𝑡²𝜌𝑐_𝑣𝜃+𝜃₀𝛽₁1𝑟²𝜕𝑟𝜕𝑟²𝑢+𝑐𝜃₀𝐶, (6) where the Laplacian operator ∇² is given by ∇²=(𝜕²/𝜕𝑟²)+(2/𝑟)(𝜕/𝜕𝑟) and ℎ₀=2𝜇+𝜆+(𝑃₁/2)+𝜇_𝑒𝐻²₀, ℎ₀ is the coefficient of linear diffusion expansion, 𝐾 is the thermal conductivity, 𝜃 is the absolute temperature, 𝜃₀ is the initial uniform temperature, and |(𝜃−𝜃₀)/𝜃₀|≪1.

Equation of conservation of mass diffusion may be written as 𝐷𝛽₂∇²1𝑟²𝜕𝑟𝜕𝑟²𝑢+𝐷𝑐∇²𝜕𝜃+𝜕𝜕𝑡+𝜏²𝜕𝑡²𝐶=𝐷𝑏∇²𝐶, (7) where 𝜏 is the diffusion relaxation time, 𝜏₀ is the thermal relaxation time, 𝛼_𝑡 is the coefficient of linear thermal expansion, and 𝜃₀ is constant, where 𝛽₁=(3𝜆+2𝜇)𝛼_𝑡, 𝛽₂=(3𝜆+2𝜇)𝛼_𝑐, 𝜎_𝑖𝑗 are the components of the stress tensor, 𝜏_𝑖𝑗 are the components of stress tensor, 𝑏 and 𝑐 are the measures of thermodiffusion and diffusive effects, 𝐶 is the concentration, 𝐶_𝑣 is the specific heat at constant strain, 𝐷 is the diffusive coefficient, and 𝑒_𝑖𝑗 are the components of the strain tensor. The thermal relaxation time 𝜏₀ will ensure that the heat conduction equation will predict finite speed of heat propagation. The diffusion relaxation time 𝜏, which will ensure the equation satisfied by the concentration 𝐶, will also predict finite speed of propagation of matter from one medium to the other.

3. Dimensionless Quantities

Introduce the following nondimensional parameters: 𝑟^∗=𝑐₁𝜂₀𝑟,𝑢^∗=𝑐₁𝜂₀𝛽𝑢,𝑇=¹𝜃−𝜃₀ℎ₀,𝐶^∗=𝛽₂𝐶ℎ₀,Ω^∗=Ω𝑐²₁𝜂₀,𝜎^∗_𝑖𝑗=𝜎_𝑖𝑗ℎ₀,𝑃^∗=𝑃𝛽₂,𝑡^∗=𝑐²₁𝜂₀𝑡,𝜏^∗₀=𝑐²₁𝜂₀𝜏₀,𝜏^∗=𝑐²₁𝜂₀𝜏,𝜂₀=𝜌𝑐_𝑣𝐾,𝑐²₁=ℎ₀𝜌. (8) The elastic constants 𝜆,𝜇 and the density 𝜌 of nonhomogeneous material in form [32] are as follows: 𝜆=𝑟^2𝑚𝜆₀,𝜇=𝑟^2𝑚𝜇₀,𝜇_ℎ=𝑟^2𝑚𝜇₀,𝜌=𝑟^2𝑚𝜌₀,𝑝^∗=𝑝^∗₀𝑟^2𝑚. (9)

Using the above non-dimensional parameters and (9) in (10)–(14), the non-dimensional system becomes 𝜕1/𝑟²(𝜕/𝜕𝑟)𝑟²𝑢−𝜕𝑟𝜕𝐶−𝜕𝑟𝜕𝑇=𝜕𝜕𝑟²𝑢𝜕𝑡²−Ω²𝑢−2Ω𝜕𝑢,∇𝜕𝑡₂𝜕𝑇=𝜕𝑡+𝜏₀𝜕²𝜕𝑡²𝑇+𝜀₁1𝑟²𝜕𝑟𝜕𝑟²𝑢+𝜀₂𝐶,∇²1𝑟²𝜕𝑟𝜕𝑟²𝑢+ℎ₄∇²𝑇+ℎ₆𝜕𝜕𝜕𝑡+𝜏²𝜕𝑡²𝐶=ℎ₅∇²𝐶, (10)(11)(12)𝜎_𝑟𝑟=ℎ₁𝜕𝑢𝜕𝑟+ℎ₂1𝑟²𝜕𝑟𝜕𝑟²𝑢𝜎−𝑇−𝐶,_𝜃𝜃=ℎ₁𝑢𝑟+ℎ₂1𝑟²𝜕𝑟𝜕𝑟²𝑢𝜎−𝑇−𝐶,_𝜙𝜙=ℎ₁𝑢𝑟+ℎ₂1𝑟²𝜕𝑟𝜕𝑟²𝑢−𝑇−𝐶, (13a)(13b)(13c)1𝑃=−𝑟²𝜕𝑟𝜕𝑟²𝑢+ℎ₃𝐶−ℎ₄𝑇, (14) where 𝜀₁=𝛽²₁𝑇₀𝑚ℎ₀𝜌𝑐_𝑣,𝜀₂=𝛽₁𝑐𝑇₀ℎ₀(𝑚+2)𝛽₂,ℎ₁=2𝜇ℎ₀,ℎ₂=𝜆ℎ₀,ℎ₃=𝑚𝑏ℎ₀𝛽²₂,ℎ₄=𝑐ℎ₀𝛽₁𝛽₂,ℎ₅=𝐷𝑏ℎ₀𝛽₂,ℎ₆=2𝑚+ℎ₀𝛽²₂𝐷𝜂₀. (15)

4. Boundary Conditions

The nonhomogeneous initial conditions are supplemented by the following boundary conditions. The cavity surface is traction free:𝜎_𝑟𝑟(𝑟,𝑡)+𝜏_𝑟𝑟(𝑟,𝑡)=0,𝑟=𝑎. (16a) The cavity surface is subjected to a thermal shock 𝑇(𝑎,𝑡)=𝑇₀𝐻(𝑡), (16b) where 𝐻(𝑡) is the Heaviside unit step function. The chemical potential is also assumed to be a known function of time at the cavity surface: 𝑃(𝑎,𝑡)=𝑃₀𝐻(𝑡),𝑃₀isrealconstant. (16c)

The displacement function is as follows: 𝑢(𝑟,𝑡)=0,𝑟=𝑎. (16d)

5. Solution of the Problem

In this section, one obtains the analytical solution of the problem for a spherical region with boundary conditions by taking the harmonic vibrations. One assumes that the solution of (10)–(12) as follows:𝐶(𝑟,𝑡)=𝐶^′(𝑟)𝑒^𝑖𝜔𝑡,𝑇(𝑟,𝑡)=𝑇^′(𝑟)𝑒^𝑖𝜔𝑡,𝑢(𝑟,𝑡)=𝑢^′(𝑟)𝑒^𝑖𝜔𝑡⋅𝑒(𝑟,𝑡)=𝐸^′(𝑟)𝑒^𝑖𝜔𝑡, (17a)(17b)

where 𝑒=(𝜕𝑢/𝜕𝑟)+(2𝑢/𝑟)=(1/𝑟²)(𝜕/𝜕𝑟)(𝑟²𝑢).

Substituting (17a) and (17b) into (10)–(12) yields 𝜕𝐸^′−𝜕𝑟𝜕𝐶^′−𝜕𝑟𝜕𝑇^′𝜕𝑟=𝛽𝑢^′,∇²𝑇^′=𝑘₁𝑇^′+𝜀₁𝐸^′+𝜀₂𝐶^′∇₂,𝐸^′+ℎ₄∇²𝑇^′+ℎ₆𝑘₂𝐺=ℎ₅∇²𝐶^′. (18)(19)(20)

Applying the operator Laplacian operator ∇² to (18), we obtain ∇²−𝛽𝐸^′=∇²𝐶^′+∇²𝑇^′. (21) From (19)–(21), we obtain ∇⁶+𝑏₁∇⁴+𝑏₂∇²+𝑏₃𝐸^′,𝑇^′,𝐶^′=0, (22) where 𝑏₁=−1ℎ₅×𝑘−1₂ℎ₆+𝑘₁ℎ₅+𝜀₂ℎ₄+𝛽ℎ₅+𝑘₁𝜀₁ℎ₅+𝜀₂,𝑏₂=1ℎ₅×𝑘−1₁𝑘₂ℎ₆+𝛽𝑘₂ℎ₆+𝑘₁ℎ₅+𝜀₂ℎ₄+𝑘₁𝑘₂𝜀₁ℎ₆,𝑏₃=−𝛽𝑘₁𝑘₂ℎ₆ℎ₅−1,𝑘₁=𝑖𝜔1+𝑖𝜔𝜏₀,𝑘₂=𝑖𝜔(1+𝑖𝜔𝜏),𝛽=−𝜔²+Ω²+2𝑖𝜔Ω. (23)

Equation (22) can be factorized as ∇²+𝑄²₁∇²+𝑄²₂∇²+𝑄²₃𝐸^′,𝑇^′,𝐶^′=0, (24) where 𝑄²₁, 𝑄²₂, and 𝑄²₃ are the roots of the characteristic equation 𝑄⁶+𝑏₁𝑄⁴+𝑏₂𝑄²+𝑏₃=0. (25) The solution of (24) which is bounded at infinity is given by 𝑇^′()=1𝑟,𝜔√𝑟³_𝑗=1𝐵_𝑗(𝜔)𝐾_1/2𝑄_𝑗𝑟,𝐸^′()=1𝑟,𝜔√𝑟³_𝑗=1𝐵^′_𝑗(𝜔)𝐾_1/2𝑄_𝑗𝑟,𝐶^′()=1𝑟,𝜔√𝑟³_𝑗=1𝐵^′′_𝑗(𝜔)𝐾_1/2𝑄_𝑗𝑟, (26) where 𝐵_𝑗, 𝐵^′_𝑗, and 𝐵^′′_𝑗 are parameters depending only on 𝜔 and 𝐾_1/2 is the modified spherical Bessel function of the second kind of order 1/2. Compatibility between (26) along with (19) and (20) will give rise to 𝐵^′_𝑗(𝜔)=𝑄²_𝑗−𝑘₁ℎ₅𝑄²_𝑗−ℎ₆𝑘₂−𝜀₂𝑘₁ℎ₄𝑄²_𝑗𝑄²_𝑗𝜀₂𝑘₁+𝑘₁𝜀₁ℎ₅𝑄²_𝑗−ℎ₆𝑘₂𝐵_𝑗(𝜔),𝐵^′′_𝑗(𝜔)=𝜀₁𝑘₁ℎ₄𝑄²_𝑗+𝑄²_𝑗𝑄²_𝑗−𝑘₁𝑄²_𝑗𝜀₂𝑘₁+𝑘₁𝜀₁ℎ₅𝑄²_𝑗−ℎ₆𝑘₂𝐵_𝑗(𝜔). (27) Substituting (26) into (17a) and (17b), we obtain 𝑇()=1𝑟,𝑡√𝑟³_𝑗=1𝐵_𝑗(𝜔)𝐾_1/2𝑄_𝑗𝑟𝑒^𝑖𝜔𝑡,𝑒()=1𝑟,𝑡√𝑟³_𝑗=1𝐵^′_𝑗(𝜔)𝐾_1/2𝑄_𝑗𝑟𝑒^𝑖𝜔𝑡,𝐶()=1𝑟,𝑡√𝑟³_𝑗=1𝐵^′′_𝑗(𝜔)𝐾_1/2𝑄_𝑗𝑟𝑒^𝑖𝜔𝑡. (28)(29)(30) Integrating both sides of (29) from 𝑟 to infinity and assuming that 𝑢(𝑟,𝑡) vanishes at infinity, we obtain 𝑢()=1𝑟,𝑡√𝑟³_𝑗=1𝑄²_𝑗−𝑘₁ℎ₅𝑄²_𝑗−ℎ₆𝑘₂−𝜀₂𝑘₁ℎ₄𝑄²_𝑗𝑄²_𝑗𝜀₂𝑘₁+𝑘₁𝜀₁ℎ₅𝑄²_𝑗−ℎ₆𝑘₂×𝐵_𝑗(𝜔)𝐾_3/2𝑄_𝑗𝑟𝑒^𝑖𝜔𝑡. (31) From (13a), (13b), and (13c)-(14), we get 𝜎_𝑟𝑟=1√𝑟³_𝑗=1𝐵_𝑗(𝜔)𝑄²_𝑗−𝑘₁ℎ₅𝑄²_𝑗−ℎ₆𝑘₂−𝜀₂𝑘₁ℎ₄𝑄²_𝑗𝑄²_𝑗𝜀₂𝑘₁+𝑘₁𝜀₁ℎ₅𝑄²_𝑗−ℎ₆𝑘₂×ℎ₁+ℎ₂−𝑄²_𝑗𝜀₂𝑘₁+𝑘₁𝜀₁ℎ₅𝑄²_𝑗−ℎ₆𝑘₂𝑄²_𝑗−𝑘₁ℎ₅𝑄²_𝑗−ℎ₆𝑘₂−𝜀₂𝑘₁ℎ₄𝑄²_𝑗×𝐾_1/2𝑄_𝑗𝑟−ℎ₁𝑟𝐾_3/2𝑄_𝑗𝑟𝑒^𝑖𝜔𝑡,𝜎_𝜙𝜙=𝜎_𝜃𝜃=1√𝑟³_𝑗=1𝐵_𝑗(𝜔)×ℎ₁𝑟𝑄²_𝑗−𝑘₁ℎ₅𝑄²_𝑗−ℎ₆𝑘₂−𝜀₂𝑘₁ℎ₄𝑄²_𝑗𝑄²_𝑗𝜀₂𝑘₁+𝑘₁𝜀₁ℎ₅𝑄²_𝑗−ℎ₆𝑘₂×𝐾_3/2𝑄_𝑗𝑟+ℎ₂𝑄²_𝑗−𝑘₁ℎ₅𝑄²_𝑗−ℎ₆𝑘₂−𝜀₂𝑘₁ℎ₄𝑄²_𝑗𝑄²_𝑗𝜀₂𝑘₁+𝑘₁𝜀₁ℎ₅𝑄²_𝑗−ℎ₆𝑘₂𝜀−1−¹𝑘₁ℎ₄𝑄²_𝑗+𝑄²_𝑗𝑄²_𝑗−𝑘₁𝑄²_𝑗𝜀₂𝑘₁+𝑘₁𝜀₁ℎ₅𝑄²_𝑗−ℎ₆𝑘₂×𝐾_1/2𝑄_𝑗𝑟𝑒^𝑖𝜔𝑡1𝑃=√𝑟³_𝑗=1𝐵_𝑗(𝜔−𝑄)^2𝑗−𝑘₁ℎ₅𝑄²_𝑗−ℎ₆𝑘₂−𝜀₂𝑘₁ℎ₄𝑄²_𝑗𝑄²_𝑗𝜀₂𝑘₁+𝑘₁𝜀₁ℎ₅𝑄²_𝑗−ℎ₆𝑘₂+ℎ₃𝜀₁𝑘₁ℎ₄𝑄²_𝑗+𝑄²_𝑗𝑄²_𝑗−𝑘₁𝑄²_𝑗𝜀₂𝑘₁+𝑘₁𝜀₁ℎ₅𝑄²_𝑗−ℎ₆𝑘₂−ℎ₄×𝐾_1/2𝑄_𝑗𝑟𝑒^𝑖𝜔𝑡. (32) Using the boundary conditions, we get 𝐵₁(𝜔)=−𝑁₂√𝑎𝜃₀𝑊₃−𝐾_1/2𝑎𝑄₃𝑝₀+𝑁₃√𝑎𝜃₀𝑊₂−𝐾_1/2𝑎𝑄₂𝑝₀×(𝑀)⁻¹,𝐵₂(𝜔)=√𝑎𝑁₁𝜃₀𝑊₃−𝐾_1/2𝑎𝑄₃𝑝₀+√𝑎𝑁₃𝐾_1/2𝑎𝑄₁𝑝₀−𝜃₀𝑊₁×(𝑀)⁻¹,𝐵₃(𝜔)=√𝑎𝑁₁𝐾_1/2𝑎𝑄₂𝑝₀−𝜃₀𝑊₂−√𝑎𝑁₂𝐾_1/2𝑎𝑄₁𝑝₀−𝜃₀𝑊₁×(𝑀)⁻¹,𝑁_𝑗=𝑄²_𝑗−𝑘₁ℎ₅𝑄²_𝑗−ℎ₆𝑘₂−𝜀₂𝑘₁ℎ₄𝑄²_𝑗𝑄²_𝑗𝜀₂𝑘₁+𝑘₁𝜀₁ℎ₅𝑄²_𝑗−ℎ₆𝑘₂×ℎ₁+ℎ₂−𝑄²_𝑗𝜀₂𝑘₁+𝑘₁𝜀₁ℎ₅𝑄²_𝑗−ℎ₆𝑘₂𝑄²_𝑗−𝑘₁ℎ₅𝑄²_𝑗−ℎ₆𝑘₂−𝜀₂𝑘₁ℎ₄𝑄²_𝑗×𝐾_1/2𝑄_𝑗𝑎−ℎ₁𝑎+𝜇_𝑒𝐻²_𝜙3𝑎²−𝑄²_𝑖×𝐾_3/2𝑄_𝑗𝑎,𝑊_𝑗=−𝑄²_𝑗−𝑘₁ℎ₅𝑄²_𝑗−ℎ₆𝑘₂−𝜀₂𝑘₁ℎ₄𝑄²_𝑗𝑄²_𝑗𝜀₂𝑘₁+𝑘₁𝜀₁ℎ₅𝑄²_𝑗−ℎ₆𝑘₂+ℎ₃𝜀₁𝑘₁ℎ₄𝑄²_𝑗+𝑄²_𝑗𝑄²_𝑗−𝑘₁𝑄²_𝑗𝜀₂𝑘₁+𝑘₁𝜀₁ℎ₅𝑄²_𝑗−ℎ₆𝑘₂−ℎ₄×𝐾_1/2𝑄_𝑗𝑎,𝑀=𝑁₁𝑊₃𝐾_1/2𝑎𝑄₂−𝑊₂𝐾_1/2𝑎𝑄₃−𝑁₂𝑊₃𝐾_1/2𝑎𝑄₁−𝑊₁𝐾_1/2𝑎𝑄₁+𝑁₃𝑊₂𝐾_1/2𝑎𝑄₁−𝑊₁𝐾_1/2𝑎𝑄₂,𝑗=1,2,3. (33)

6. Particular Case

If we neglect the initial stress and diffusion effects by eliminating (3) and (8) and putting 𝑃₁=𝛽₂=𝐶=0 in (4) and (6), we get (𝑇,𝑒), 𝑢(𝑟,𝑡), 𝜎_𝑟𝑟, 𝜎_𝜙𝜙, and 𝜎_𝜃𝜃: (𝑇,𝑒)=𝐴𝑒^{−𝜆𝑟+𝑖𝜔𝑡}+𝐵𝑒^{−𝜆𝑟+𝑖𝜔𝑡}, (34) where 𝜂₁=−ℓ₁+𝛽,𝜂₂=ℓ₁𝛽−𝜀₁,𝜆²₁,𝜆²₂=12𝜂₁±𝜂²₁−4𝜂₂,𝑢()𝑟𝑟,𝑡=−²𝜆²₁+2𝑟𝜆₁+2𝑟²𝜆³₁𝐴(𝜔)𝑒^−𝜆1𝑟+𝑟²𝜆²₂+2𝑟𝜆₂+2𝑟²𝜆³₂𝐵(𝜔)𝑒^−𝜆2𝑟+𝐶(𝜔)𝑒^𝑖𝜔𝑡,𝜎_𝑟𝑟=𝛼₁4+4𝑟𝜆₁+2𝑟²𝜆²₁+𝑟³𝜆³₁𝑟³𝜆³₁+𝛼₂−1𝐴(𝜔)𝑒^−𝜆1𝑟+𝛼₁4+4𝑟𝜆₂+2𝑟²𝜆²₂+𝑟³𝜆³₂𝑟³𝜆³₂+𝛼₂(𝜔)𝑒−1×𝐵^−𝜆2𝑟𝑒^𝑖𝜔𝑡,𝜎_𝜙𝜙=𝜎_𝜃𝜃=−𝛼₁𝑟²𝜆²₁+2𝑟𝜆₁+2𝑟³𝜆³₁−𝛼₂(𝜔)𝑒−1×𝐴^−𝜆1𝑟+𝛼₁𝑟²𝜆²₂+2𝑟𝜆₂+2𝑟³𝜆³₂−𝛼₂(𝜔)𝑒−1×𝐵^−𝜆2𝑟𝑒^𝑖𝜔𝑡. (35) Using the boundary conditions, we obtain 𝐴(𝜔)=−ℎ₂𝑇₀ℎ₁𝑒^−𝜆2𝑎−ℎ₂𝑒^−𝜆1𝑎(𝜔)=ℎ,𝐵¹𝑇₀ℎ₁𝑒^−𝜆2𝑎−ℎ₂𝑒^−𝜆1𝑎,𝐶(𝜔)=ℎ₂ℎ₃−ℎ₁ℎ₄𝜃₀ℎ₁𝑒^−𝜆2𝑎−ℎ₂𝑒^−𝜆1𝑎,ℎ₁=𝛼₁4+4𝑎𝜆₁+2𝑎²𝜆²₁+𝑎³𝜆³₁𝑎³𝜆³₁+𝛼₂−1−𝜇_𝑒𝐻²_𝜙𝜆₁𝑒^−𝜆1𝑎,ℎ₂=𝛼₁4+4𝑎𝜆₂+2𝑎²𝜆²₂+𝑎³𝜆³₂𝑎³𝜆³₂+𝛼₂−1−𝜇_𝑒𝐻²_𝜙𝜆₂𝑒^−𝜆2𝑎,ℎ₃=𝑎²𝜆²₁+2𝑎𝜆₁+2𝑎²𝜆³₁𝑒^−𝜆1𝑎,ℎ₄=𝑎²𝜆²₂+2𝑎𝜆₂+2𝑎²𝜆³₂𝑒^−𝜆2𝑎. (36)

7. Numerical Results and Discussion

For the purposes of numerical evaluations. The copper material was chosen. The constants of the problem given by Aouadi [24], Sokolnikoff [34] and Thomas [35] are 𝜇=3.86×10¹⁰kg/ms³,𝜆=7.76×10¹⁰kg/ms³,𝜌=8954kg/m³,𝑐_𝑣𝛼=383.1J/kg⋅K,_𝑡=1.78×10⁻⁵K⁻¹,𝛼_𝑐=1.98×10⁻⁴m³/kg,𝑘=386W/mK,𝐷=0.85×10⁸kg⋅s/m³,𝑇₀=293K,𝑐=1.2×10⁴m²/s²K,𝑏=0.9×10⁶m⁵/s²kg,𝜂₀=8886.73s/m². (37) Using the above values, we get 𝜇_𝑒=1, 𝜃₀=1, 𝑃₀=1, 𝑎=2, 𝜔=9.5, 𝐻=0.7×10⁻⁵, 𝜏₀=0.1, and 𝜏=0.2. The values of radial displacement 𝑢, temperature distribution 𝜃, concentration 𝐶, stresses 𝜎_𝑟𝑟, 𝜎_𝜙𝜙, and chemical potential distribution 𝑃 for thermoelastic diffusion and thermoelasticity are studied for force thermal source and chemical potential source. The output is plotted in Figures 1–10. Figure 1 shows that the values of chemical potential distribution 𝑃 have oscillatory behavior with diffusion in the whole range of radius 𝑟. The effects of nonhomogeneity 𝑚, rotation Ω, time 𝑡 and relaxation time 𝜏₀ on chemical potential distribution is shifting from the positive into the negative gradually with the radius 𝑟. Figure 2 shows that the value of concentration distribution 𝐶 has oscillatory behavior for diffusion in the whole range of radius 𝑟 under the effects of nonhomogeneity, rotation, and relaxation time, while it is decreasing with an increase of nonhomogeneity 𝑚. In these figures, it is clear that the distribution has a nonzero value only in the bounded region of space for 𝑡=0.15 where the infinite speed of propagation is inherent. The effects of nonhomogeneity, rotation Ω, time 𝑡, and relaxation time 𝜏₀ on concentration distribution is shifting from the positive into the negative gradually. This indicates that the equations are satisfied by the concentration 𝐶 which predict a finite speed of propagation of matter from first medium to another one. Figure 3 shows that the value of temperature distribution 𝜃 has an oscillatory behavior for thermoelastic diffusion in the whole range of the radius 𝑟, while the solution is notably different inside the sphere. This is due to the fact that, the thermal waves in the coupled theory travel with an infinite speed of propagation as opposed to finite speed in the generalized case. The effects of nonhomogeneity, rotation Ω, time 𝑡 and relaxation time 𝜏₀ on temperature distribution shift from the positive into the negative gradually. This indicates that the heat propagates as a wave with finite velocity. Figure 4 shows that the value of radial displacement 𝑢 has oscillatory behavior with diffusion in the whole range of radius 𝑟. These figures indicate that the medium along 𝑟 undergoes expansion deformation due to the thermal shock, while the other one shows the compressive deformation. The effect of nonhomogeneity, rotation Ω, and relaxation time 𝜏₀ on radial displacement becomes large. Increasing the nonhomogeneity, the radial displacement is shifted upward from negative values to positive values. At a given instant, the radial displacement is finite which is due to the effect of nonhomogeneity, rotation, time, and relaxation time. Figures 5 and 6 show the variations of the radial stress 𝜎_𝑟𝑟 and tangential stress 𝜎_𝜙𝜙 with respect to the radius 𝑟, respectively. The values of radial stress and tangential stress are increased and decreased due to the diffusion in a nonuniform behavior for all values of the radius 𝑟. For the values of 𝜎_𝑟𝑟 and 𝜎_𝜙𝜙, depicting the effect of nonhomogeneity, diffusion, rotation, and relaxation time, it is shown that the radial stress is compressive in its nature.

(a)

(b)

Figure 1 

Variation of chemical potential 𝑃 with radius 𝑟 (thermoelastic diffusion nonhomogeneity medium).

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Figure 2 

Variation of concentration 𝐶 with radius 𝑟 (thermoelastic diffusion nonhomogeneity medium).

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(b)

Figure 3 

Variation of temperature 𝜃 with radius 𝑟 (thermoelastic diffusion nonhomogeneity medium).

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(b)

Figure 4 

Variation of displacement 𝑢 with radius 𝑟 (thermoelastic diffusion nonhomogeneity medium).

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Figure 5 

Variation of radial stress 𝜎𝑟𝑟 with radius 𝑟 (thermoelastic diffusion nonhomogeneity medium).

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Figure 6 

Variation of tangential stress 𝜎𝜙𝜙 with radius 𝑟 (thermoelastic diffusion nonhomogeneity medium).

Figure 7 

Variation of displacement 𝑢 with radius 𝑟 (thermoelastic nonhomogeneity medium).

Figure 8 

Variation of temperature 𝜃 with radius 𝑟 (thermoelastic nonhomogeneity medium).

Figure 9 

Variation of radial stress 𝜎𝑟𝑟 with radius 𝑟 (thermoelastic nonhomogeneity medium).

Figure 10 

Variation of tangential stress 𝜎𝜙𝜙 with radius 𝑟 (thermoelastic nonhomogeneity medium).

Figure 7 shows the values of radial displacement 𝑢 in thermoelastic medium without diffusion. This figure indicates clearly that the radial displacement at the cavity surface tends to zero which agrees with the boundary conditions prescribed. This coincides with the mechanical boundary condition of the cavity, in case of fixed surface. Figure 8 shows the values of temperature distribution 𝜃 without diffusion in the whole range of radius 𝑟. It was found that the values of 𝜃 under effect of nonhomogeneity and rotation Ω are increase with an increase of nonhomogeneity and rotation Ω but are decreasing with the increase of the values of 𝑚. Figures 9 and 10 show the values of radial stress 𝜎_𝑟𝑟 and the tangential stress 𝜎_𝜙𝜙 without diffusion in the whole range of radius 𝑟, respectively. It was found that the values of 𝜎_𝑟𝑟 under the effects of nonhomogeneity 𝑚 and rotation Ω are increasing with an increase of the values of nonhomogeneity 𝑚 and rotation Ω, while the values of 𝜎_𝑟𝑟 are decreasing with an increase of nonhomogeneity 𝑚, while the tangential stress 𝜎_𝜙𝜙 is decreasing with the increase of the values of nonhomogeneity 𝑚 and rotation Ω, but the values of 𝜎_𝜙𝜙 are increasing with an increase of 𝑚. Due to the complicated nature of the governing equations of the generalized magneto-thermoelastic diffusion theory, the done works in this field are unfortunately limited. The method used in this study provides quite a success in dealing with such problems. This method gives exact solutions in the elastic medium without any restrictions on the actual physical quantities that appear in the governing equations of the considered problem.

8. Conclusions

The results presented in this paper will be very helpful for researchers concerned with material science, designers of new materials, and low-temperature physicists, as well as for those working on the development of a theory of hyperbolic propagation of hyperbolic thermodiffusion. Study of the phenomenon of nonhomogeneity, rotation, magnetic field, and diffusion is also used to improve the conditions of oil extractions. It was found that, for values of rotation and nonhomogeneity, the coupled theory and the generalization give close results. The case is quite different when we consider small value of rotation and nonhomogeneity. Comparing Figures 1–6 in case of thermoelastic diffusion medium with the Figures 7–10 in case of thermoelastic medium, it was found that 𝑢, 𝜎_𝑟𝑟, 𝜎_𝜙𝜙, 𝐶, and 𝑃 have the same behavior in both media. But with the passage of nonhomogeneity and rotation, the numerical values of 𝑢, 𝜎_𝑟𝑟, 𝜎_𝜙𝜙, 𝐶, and 𝑃 in thermoelastic diffusion medium are large in comparison with those in thermoelastic medium due to the influences of nonhomogeneity, magnetic field, rotation, and mass diffusion.

Acknowledgment

This article was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The author, therefore, acknowledge with thanks DSR technical and financial support.