Thermomechanical Fractional Model of Two Immiscible TEMHD

Abstract

We introduce a mathematical model of unsteady thermoelectric MHD flow and heat transfer of two immiscible fractional second-grade fluids, with thermal fractional parametersαiand mechanical fractional parametersβi,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 ofαiandβi. The graphs describe the fractional thermomechanical parameters effect on the case of two immiscible fluids and the case of a single fluid.

Thermomechanical Fractional Model of TEMHD Rotational Flow

“…To clarify the physical applications of the graph, the assumptions [41] are required in cylindrical coordinate as follow: The fluid between the cylinders moves gradually with velocity V = (0, v ( r , t ), 0) while the constitutive equation of generalized second grade fluid, corresponding to this motion is given by [7] where τ ( r , t ) = T rϕ is the non-zero shear stress, μ is the viscosity, α 1 is the first normal material modulus.The modified Fourier law defined by Shercliff [18] for thermoelectric medium is extended using the Taylor series expansion of time fractional order would take the form [38] where κ is thermal conductivity, τ 0 is thermal relaxation time, q is heat conduction vector, T is the temperature and J is the current density vector given by the modified Ohm’s law as follows; π 0 is the Peltier coefficient, k 0 is the Seebeck coefficient at reference temperature T 0 and σ is electrical conductivity. The electric intensity vector and the magnetic induction vector are E , B respectively, where B has one constant non-vanishing component B z which has the form; …”

Application of Generalized Fractional Thermoelasticity Theory with Two Relaxation Times to an Electromagnetothermoelastic Thick Plate

“…Hamza et al established a new mathematical model of Maxwell's equations in an electromagnetic field in [15] and derived a fractional model for thermoelasticity associated with two relaxation times in [16]. A model for unsteady thermoelectric magnetohydrodynamics (TEMHD) flow and heat transfer of two immiscible second-grade fluids with two fractional parameters was introduced in [17].…”