Thermomechanical Fractional Model of Two Immiscible TEMHD

We introduce a mathematical model of unsteady thermoelectric MHD flow and heat transfer of two immiscible fractional secondgrade fluids, with thermal fractional parameters α i and mechanical fractional parameters β i , i = 1, 2. The Laplace transform with respect to time is used to obtain the solution in the transformed domain. The inversion of Laplace transform is obtained by using numerical method based on a Fourier-series expansion.The numerical results for temperature, velocity, and the stress distributions are represented graphically for different values of α i and β i . The graphs describe the fractional thermomechanical parameters effect on the case of two immiscible fluids and the case of a single fluid.


Introduction
In recent years, the requirement of modern technology has stimulated interest in flow and heat transfer studies which involve interaction of several phenomena such as heat exchangers, transport of heat or cooled fluids, chemical processing equipment, and microelectronic cooling [1].The problem of flow and heat transfer was extensively investigated by many researchers with different hypothesis, in which the mentioned studies pertain to a single-fluid model [2][3][4][5][6].
Many problems relating to plasma physics, aeronautics, and geophysics and in petroleum industry and so forth involve multilayered fluid flow situations [7].There has been some theoretical and experimental work in stratified laminar flow of two immiscible fluids in horizontal pipes.Singh et al. [8] studied the generalized Couette flow of two viscous incompressible immiscible fluids with heat transfer in the presence of heat source through two straight parallel horizontal walls.Umavathi et al. [9] investigated unsteady oscillatory flow and heat transfer in horizontal channel consisting of two viscous immiscible fluids with isothermal permeable walls.Malashetty et al. [10] examined the magnetoconvection of two immiscible fluids in vertical enclosure consisting of two conducting and nonconducting regions.Kamıs ¸lı and Öztop [11] examined the entropy generation in two immiscible incompressible fluid flows under the influence of pressure difference in thin slit of constant wall heat fluxes.Nikodijevic et al. [12] investigate the magnetohydrodynamic (MHD) Couette flow and heat transfer of two immiscible fluids in a parallel-plate channel in the presence of applied electric and inclined magnetic fields.Tigrine et al. [13] studied experimentally the Couette flow of two immiscible fluids between concentric spheres when the outer sphere is fixed and the inner one rotates.
Thermoelectric magnetohydrodynamics (TEMHD) theory was originally developed by Shercliff with direct application to a fusion environment [14].The thermoelectric effect causes a current to develop between a liquid metal and a container wall when a temperature gradient is present along the interface between them [15][16][17].
Recently, the fractional derivatives have been found to be quite flexible in describing the behaviors of viscoelastic fluids and are studied by many mathematicians considering various motions of such fluids [18,19].In their studies, the constitutive equations for generalized non-Newtonian fluids are modified from the well-known fluid models by replacing 2 Advances in Materials Science and Engineering the time derivative of an integer order by the so-called Caputo fractional operator.
El-Shahed [20] studies the effect of transverse magnetic field on the unsteady flow of a generalized second-grade fluid through a porous medium.Jamil et al. [21] obtain an exact solution for the motion of fractionalized second-grade fluid due to oscillations of an infinite circular cylinder.Sherief et al. [22] derive fractional order theory of thermoelasticity using the methodology of fractional calculus.The effects of fractional derivative parameter of two media are discussed in [23].Ezzat [24,25] constructed a mathematical model of the TEMHD in the context of fractional heat conduction equation by using the Taylor series expansion of time fractional order developed by Jumarie [26].Recently, Hamza et al. [27] established a new fractional theory of thermoelasticity associated with two relaxations.
In this work, we introduce a mathematical model of unsteady flow and heat transfer of two immiscible thermoelectric fluids with thermomechanical fractional parameters   and   ,  = 1,2.The thermal parameters   are due to the fractional heat conduction equation of TEMHD introduced by Ezzat [24].The mechanical parameters   are due to the second-grade fluid in fractional form introduced by El-Shahed [20].We will apply our model for the flow of two immiscible electrically conducting, incompressible, fractional viscoelastic second-grade fluids in the presence of a transverse magnetic field.The Laplace transform with respect to time is used.A numerical method based on a Fourier-series expansion is used for the inversion process.The numerical results for temperature, velocity, and the stress distributions are represented graphically for different values of   and   .The graphs describe the fractional thermomechanical parameters effect on the case of two immiscible fluids and the case of a single fluid.

Formulation of the Problem
Consider unsteady laminar stratified two immiscible fluid flows through nonconducting half space  ≥ 0 in contact with an infinite plate that is rigidly fixed.Take the positive -axis in the direction of the flow and the positive -axis vertically downward perpendicular to the flow.
The regions 0 ≤  ≤ ℎ and  ≥ ℎ are denoted as Region-1 and Region-2, respectively, and occupied by two immiscible electrically conducting incompressible thermoelectric generalized second-grade viscoelastic fluids with different viscosity   , density   , specific heat at constant pressure   , thermal conductivity   , electrical conductivity   , first normal stress modulus  1 , thermal relaxation time   , Seebeck coefficient   , and Peltier coefficient   , where the subscript  = 1, 2 represents the values of the parameters of the first fluid and second fluid, respectively.
A constant magnetic field of strength   is applied in the -direction.The magnetic Reynolds number is assumed to be small so that the induced magnetic field is neglected.Due to the formulation of the problem all variables depend on  and  only.
The following assumptions are required.
(1) The stress tensor which is different from zero for generalized second-grade viscoelastic fluid is given by [20] where   is the stress tensor Τ  of the two fluids and    is Caputo fractional time derivative operator defined as [28]     () where Γ(⋅) is the gamma function.
(2) The modified Ohm and Fourier laws defined by Shercliff [14] for the two thermoelectric media are given by where q  , J  , and V  are heat conduction vector, conduction current density vector, and velocity vector of the two fluids and E and B are the electric density and magnetic induction vectors, respectively.
(3) A mathematical model of fractional heat conduction equation by using the Taylor series expansion of time fractional order developed by Jumarie in the context of thermoelectric MHD [24] is where is the Caputo fractional derivative operator of order   where 0 <   ≤ 1,  = 1, 2.
(4) The magnetic induction has one constant nonvanishing component: (5) The Lorentz force F = J × B has only component in the -direction for each fluid and is given by Under these assumptions the governing equations for such flow will take the following form.The energy equation: equation of motion: the modified fractional Fourier law of heat conduction: the constitutive equation: where   are the components of the fluids velocity,   are the temperature of the regions of the two fluids, and are thermal and mechanical Caputo fractional time derivative operator of orders (0 <   ≤ 1), (0 <   ≤ 1),  = 1, 2.
The thermal boundary and interface condition for the two fluids are written as where () is the Heaviside unit step function.
The hydrodynamic boundary and interface condition for the two fluids are considered as Let us introduce dimensionless variables: where   is reference temperature defined as Therefore, ( 7)-( 10) are reduced to the nondimensional equations where where   ,  2 ,   , and   are Prandtl number, Hartmann number, viscoelastic parameter, and dimensionless thermoelectric figure-of-merit [29], respectively,   is the first Thomson relation in thermoelectric medium [30], and , , , ,   ,   , and  are the ratios of the materials parameter.
The thermal boundary and interface condition in nondimensional form become The hydrodynamic boundary and interface condition in nondimensional form become Applying the Laplace transform for both sides of ( 14), defined by we get the following system of equations: where   ,   ,   ,   , ℓ  ,   , and   are given in the Appendix.In obtaining the above equations we shall use the homogenous initial condition and the following relation [28]: The thermal boundary and interface conditions become The hydrodynamic boundary and interface conditions become Now, we can rewrite (19) and (20) in the form of Eliminating   ,  = 1, 2, between ( 26) and ( 27), we obtain the following fourth-order differential equation satisfied by   : Equation ( 28) can be factorized as where  2 1 ,  2 2 and  2 1 ,  2 2 are the roots of the characteristic equations The general solution of (28) which is bounded at infinity is given by where the characteristic roots of Region-1 are ± 1 , ± 2 while  1 and  2 are the roots with positive real parts of the characteristic equation of Region-2.These roots are given in the Appendix.The other roots in Region-2 are neglected in order to make the functions field bounded in this region. 1 - 6 are parameters depending on  only.By the same manner, we get where  1 - 6 are parameters depending on  only.From ( 31) and ( 32) into (26), we get ( We thus have the following. Region-1 Region-2 In order to determine the unknown   ,  = 1, 2, . . ., 6, we shall use (34) and (35) with boundary and interface conditions equations ( 24) and ( 25); we get Solving the above system of equations we get  1 - 6 ; thus we have the solution in the Laplace transformed domain.
To obtain the solution in the time domain, we can use

Numerical Results and Discussion
Two viscoelastic fluids were chosen for the purposes of numerical evaluations, namely, high density polyethylene (HDPE) and polypropylene (PP) [32][33][34][35].The constants of the problem are shown in Table 1.
The computations were carried out for the case of two immiscible fluids (HDPE/PP) and the case of a single fluid (HDPE) on the whole region.
The temperature, the velocity, and the stress were calculated by using the numerical method of the inversion of the Laplace transform outlined above.The FORTRAN programming language was used on a personal computer.The accuracy maintained was 6 digits for the numerical program.
Figure 1 shows the temperature, the velocity, and the stress distributions for different values of time, namely,  = 0.02, 0.03, 0.06, 0.15, and 0.3, when all fractional parameters   =   ,  = 1, 2, are equal to one.Figures 2-5 show the temperature, the velocity, and the stress profiles for one value of time, namely,  = 0.09, and different values of the fractional parameters   ,   ,  = 1, 2. Figure 6 shows the thermal and mechanical effects of the fractional parameters on the two fluids.
In Figure 1, we observe that for the first two smaller values of time the temperature and stress have sharp jump (discontinuity) at the locations of the wave front while it disappears in the velocity profile.This is consistent with the physical observation that the effect of the second fluid does not appear for the small value of time while it appears for the larger time.For the larger two values of time the temperature vanishes for the case of two fluids before the case of one fluid; this is convenient with the physical phenomena of the incident waves and the reflected waves during the interface between the two fluids while the velocity and the stress have a larger value for the case of two fluids than the case of a single fluid.
Figures 2-3 show the temperature, the velocity, and the stress distributions for different values of thermal fractional parameters  1 and  2 .These figures show that the fractional parameter  1 has a noticeable effect on the temperature profile more than  2 , while the mechanical effects of  1 are small on the velocity and the stress profiles and do not appear for  2 .
Figures 4-5 show the temperature, the velocity, and the stress distributions for different values of mechanical fractional parameters  1 and  2 .These figures show that the fractional parameter  1 has noticeable effects on the velocity and the stress profiles more than  2 , while the thermal effects of  1 and  2 do not appear.
Figure 6 describes the thermal effects of the fractional parameters  1 and  2 as shown in case (a) and the mechanical effects of  1 and  2 as shown in case (b).
Tables 2-7 show the temperature, the velocity, and the stress values for two values of time, namely,  = 0.02 and  = 0.09.For the smaller value of , the function fields have a very small change for all values of the fractional parameters.This means that the effect of the presence of the second fluid disappears for small values of time, which is in agreement with the behavior of the function fields as in [23].For the larger value of time the interaction between the two fluids is obvious in comparison with the absence of the second fluid.

Appendix
Consider

Table 1 :
The constants of the problem.