Influence of Couple Stresses and Thermophoresis on Free Convective Heat and Mass Transfer of Viscoelastic Fluid

A theoretical study has been developed to investigate the influence of thermophoresis and couple stresses on steady flow of non-Newtonian fluid with free convective heat and mass transfer over a channel bounded by two permeable plates. The considered non-Newtonian fluid follows a viscoelastic model. The problem is modulated mathematically by a system of non-linear differential equations which pertaining to describe the continuity, momentum, energy and concentration. These equations involve the effects of viscous dissipation and chemical reaction . The numerical solutions of the dimensionless equations are found as a functions of the physical parameters of this problem. The numerical formulas of the velocity (u), temperature (θ) and concentration (φ) as well as skin friction coefficient τ∗, Nusselt number(Nu) and Sherwood number(Sh) are computed. The physical parameters effects of the problem on these formulas are described and illustrated graphically through some figures and tables. It is observed that any increase in thermophoretic parameter τ leads to reduce in velocity profiles as well as concentration layers. In contrast, the velocity increases with increasing couple stresses inverse parameter.


Introduction
A transport processes of non-Newtonian fluids through a vertical channel is of great interest in diverse fields like ground-water hydrology and petroleum-reservoir. So, process of heat and mass transfer [19] in convective flows has a broad applications which actually exist not only in aeronautics and chemical process industries but also in vast industrial engineering applications and fluid fuel nuclear reactor. Most of scholars and researchers specially mathematicians, physicists, geologists and others have given the attention to these important issues especially under the influences of chemical reaction, couple stresses and thermophoresis. Influence of chemical reaction through heat and mass transfer plays the cornerstone in every chemical process and industries such as food processing and polymer production. In the light of the preceding studies concerned with flow between two vertical plates furthermore, showing important physical phenomenas such as phenomena of thermophoresis and couple stresses on non-Newtonian fluids which affect Journal of Advances in Physics Vol 17 2020 ISSN: 2347-3487 https://rajpub.com/index.php/jap on flow motion; we found that Piazza [1] has presented thermophoresis definition which mean moving particles with thermal gradients, consequently particle thermophoresis is a non-equilibrium cross-flow influence between heat and mass transport. Aside from it is already exploited as a novel tool in macromolecular fractionation, micro-fluidic manipulation and selective tuning of colloidal structures. Uddin et al [2] have investigated effects of hydromagnetic and thermophoresis of unsteady forced convection boundary layer flow over flat plates. The analysis of viscoelastic fluid flow on unsteady two-dimensional in a porous channel with radiative heat transfer and mass transfer has been studied by Das [3]. Sudhakar et al [4] have studied heat and mass transfer on unsteady free convection of a Walters-B viscoelastic fluid flow past a semi infinite plate under thermophoresis effect. Recently, El-Dabe et al [5] have analyzed the effects of thermophoresis and hall currents on non-linear heat and mass transfer of second grade fluid flow over permeable infinite vertical plate with presence of viscous dissipation and chemical reaction.
Couple stresses are consequence of considering that the mechanical action of one part of a body on another, across a surface, is equivalent to a force and a moment distribution; the state of stress is measured by a stress tensor (T ij ) and a couple stress tensor (M ij ). El-Dabe et al [11] have discussed effect of couple stresses on the MHD of a non-Newtonian unsteady flow between two parallel porous plates by using the Eyring -Powell model. The Hall and Ion-slip effects on fully developed electrically conducting couple stress fluid flow between vertical parallel plates in the presence of a temperature dependent heat source are investigated by Srinivasacharya et al [12]. Problem of mixed convection flow of couple stress fluid between parallel vertical plates with Hall and Ion-slip effects has been indicated by Srinivasacharya et al [13]. After that, analysis of Soret and Dufour effects in the presence of chemical reaction on natural convection heat and mass transfer of a couple stress fluid in a vertical channel has been presented by Srinivasacharya et al [14]. Without a doubt to importance of couple stresses effect on viscoelastic second grade model. So to be more closer to the subject of research, El-Dabe et al [15] have studied effect of couple stresses on a pulsatile magnetohydrodynamic viscoelastic flow through a channel bounded by two permeable parallel plates. The flow of hydromagnetic non-Newtonian fluid under couple stresses through a porous channel is investigated using the Eyring-Powell model have been elucidated by Adesanya et al [16]. As can be seen more important related researches to our issue in the final analysis, Kaladhar et al. [17]  This paper focuses to extend the work of [15,17]. So, It is concerned with the influences of thermophoresis and couple stresses on steady flow of second grade fluid with free convective heat and mass transfer over a channel bounded by two permeable plates in the presence of viscous dissipation and chemical reaction.

Materials and Methods
Let us consider a steady free convection flow of a couple stress viscoelastic fluid [18] between two permeable vertical plates distance 2d apart. By choosing the coordinate system such that x -axis be taken along vertically upward direction through the central line of the channel, y-axis is perpendicular to the plates. The plate y = d has given the constant temperature T 1 and concentration C 1 , while the plate y = d is subjected to a uniform temperature  The main influence of couple stresses will be to introduce a length-dependent effect that is not present in the classical non-polar theories. The governing equations of motion are [9]: • Continuity equation • Cauchy's first law of motion • Cauchy's second law of motion where, V i is the velocity vector, ρ is the density of the fluid, a i is the acceleration, T ji is the stress tensor, f i is the body force and i is the body moment per unit mass. Stokes, Mindlin and Tiersten [9,10] are obtained the constitutive equations as: where, P is the pressure, D ij is the rate of deformation tensor, δ ij is the Kronecker delta, α and µ are the material constants of viscosity, K ij is the curvature-twist tensor, η and η are the material constants of momentum and M ij is the couple stress tensor. The equation of motion as obtained by Stokes is: which, in the notation of Gibbs, can be written as: where, a = ∂V ∂t + V.∇V , For incompressible fluids divV = 0 so, if the body force f and the body moment are absent, the equations of motion reduce to The where, −P I is the indeterminate spherical stress due to the constraint of incompressibility while, P is the hydrostatic pressure, µ is is the dynamic viscosity, α 1 and α 2 are material constants and A 1 and A 2 are the kinematic Rivlin-Ericksen tensors, here, d/dt is the material time derivative. If the fluid of second grade modeled by eq.9 is to be compatible with thermodynamics and is to satisfy the inequality of Clausius-Duhem for all motions and the assumption that the specific Helmholtz free energy of the fluid is a minimum when it is locally at rest, then the coefficients µ, α 1 and α 2 describe the viscosity, visco-elasticity and cross-viscosity respectively and must satisfy: It is necessary to be noted that when α 1 = α 2 = 0 then, the constitutive equation of second grade fluid is respond to that of viscous fluid.
Wu and Greif [8] have determined the thermophoretic velocity V T as where, T r is some reference temperature, k t v represents the thermophoretic diffusivity while, k t is coefficient of thermophoresis which ranges in value from 0.2 to 1.2 as indicated by Batchelor and Shen [7], Also by Talbot et al. [6]: where, C s , C t , C 1 , C 2 , C 3 , C m are constants and λ g , λ p , K n are the thermal conductivities of the fluid and diffused particles, Knudsen number respectively. A thermophoretic parameter τ can be defined as follows; typical values of τ are 0.01, 0.05 and 0.1 corresponding to approximate values of (T 2 − T 1 ) equal to 3.15 and 30k for a reference temperature of T r = 300k. Within the above framework, the governing partial differential equations of the flow under the usual Boussinesq approximation are: Continuity equation: Momentum conservation: Energy equation:

Species concentration equation
The appropriate boundary conditions for the problem are: where, nomenclature in the above equations are: ν is the kinematic viscosity, ρ is the fluid density, g is the gravitational acceleration, β T and β c are thermal expansion volumetric coefficient and volumetric coefficient of concentration expansion respectively. T * is denote the dimensional temperature of the fluid, C * is the dimensional concentration of the solute [20], c p is the specific heat at constant pressure, κ is the thermal conductivity of the medium and D is chemical molecular diffusivity. From eq.(15), it can be seen that v * is either a constant or a function of time. Thus, assuming that the suction velocity to be constant. where, v o is a scale of mean suction velocity , Let us obtain the dimensionless equations by using the following non-dimensional variables: Using the above dimensionless quantities (20), the system of equations (16), (17) and (18) with boundary conditions (19) can be written in dimensionless form as: with the boundary conditions : The dimensionless numbers are: α = α 1 v 2 o ρv 2 is the viscoelastic parameter, ρν 3 is the couple stresses parameter, is the modified Grashof number, the Prandtl number P r = µc p κ that represents the ratio of momentum to thermal diffusivity, is the Eckert number, the Schmidt number Sc = ν D that represents the ratio of momentum to mass diffusivity, is the thermophoretic parameter .

Skin friction
The skin friction coefficient at the surface y = 0 is given by: Table 1 shows numerical values of the Skin friction coefficient τ * for various values of viscoelastic parameter α, couple stress inverse parameter a, Grashof number Gr, modified Grashof number Gm, Prandtl number P r, Schmidt number Sc and thermophoresis parameter τ . From that table, we noticed that an increase in P r, Sc and τ increases the value of the skin friction coefficient τ * , while an increase in α, a, Gr and Gm decreases the value of the skin friction coefficient τ * .

Nusselt Number
From the temperature field; the rate of heat transfer between plates and the fluid can be expressed in terms of non-dimensional Nusselt number as q w = −κ dθ dy = κN u where the Nusselt number is given by: Nusselt number N u is given in table2. It can be clearly seen that an increase in the Prandtl number P r result in an increase in Nusselt number N u. Because of the increase in viscoelastic parameter α, couple stress inverse parameter a, Grashof number Gr and Eckert number Ec then Nusselt number N u decreases.

Sherwood Number
From the concentration field; the rate of mass transfer between plates and the fluid can be expressed in terms of non-dimensional Sherwood number as j m = −ρD dφ dy = ρDSh where the Sherwood number is given by:

Results and Discussion
In order to investigate the physical significance of the problem, a program was designed by using Mathematica 10 software including using of parametric ND solve package to simulate the numerical solutions of the system of the  19). In fig. 2, by increasing the viscoelastic second grade parameter α, the velocity profile u(y) decreases.
Since the influence of α on the temperature θ(y) and concentration φ(y) fields is very small; then the results are not shown here. It is observed from fig. 3 that an increase in couple stress inverse parameter a leads to a large arise in the values of velocity u(y). The velocity u(y) curves decrease when the thermophoretic parameter τ increases as seen in fig. 4. The values of Grashof number Gr are taken to be positive and negative as they respectively represent symmetric heating of the plates when Gr < 0 and symmetric cooling of the plates when Gr > 0. The velocity profiles u(y) increases not only by increasing the Grashof number Gr but also by increasing the modified Grashof number Gm as shown in fig. 5 and fig. 6 respectively. This is due to the buoyancy forces which is acting on the fluid particles due to gravitational force that enhances the fluid velocity. The increase in the Prandtl number P r results in the decrease in u(y) as displayed in fig. 7. This is physically true because, the Prandtl number is a non-dimensional number which is the ratio of momentum diffusivity (kinematic viscosity) to thermal diffusivity. The effect of increasing Eckert number                decreases as τ increases. Specially, the effect of increasing the thermophoretic parameter τ is limited to increasing the wall slope of the concentration profiles, but decreasing the concentration. This is true only for small values of Schmidt number for which the Sherwood number Sh is large as chemical reaction effect increases and compared to the convection effect. So, the thermophoretic parameter τ is expected to alter the concentration boundary layer significantly. The concentration profiles diverge at values of couple stress inverse parameter that equal to a = 8. As a result of the effect