Couple Stresses and Thermophoresis Influences on Free Convective Heat and Mass Transfer of Viscoelastic Fluid with Hall and Ion-Slip Effects.

An analysis is presented to investigate the Hall and Ion-slip effects on non-Newtonian couple stress fluid flow between vertical channel in the presence of thermophoresis phenomena. The considered fluid obeys to the viscoelastic second-grade model. The problem is modulated mathematically to describe continuity, momentum, temperature, and concentration equations. The influences of physical parameters of dimensionless equations of velocity components ( 𝑢 ) and ( 𝑤 ), temperature ( 𝜃 ) and concentration ( 𝜙 ) have been shown in some of figuthe res, as well as skin friction ( 𝜏 ∗ ), Nusselt number( 𝑁𝑢 ) and Sherwood number( 𝑆ℎ ) have been computed. It is found that the velocity 𝑢(𝑦) decreases with increasing in Hall parameter and Ion slip parameters.


Introduction
During recent years the study of convection heat and mass transfer in non-Newtonian fluids has received much attention, and this is because the traditional Newtonian fluids cannot precisely describe the characteristics of the real fluids. The spin couple stresses the theory of fluids showed size-dependent due to microrotation of freely suspended particles set up an antisymmetric stress, known as couple-stress, and thus forming couplestress fluid. These fluids such as lubricants, blood, suspension fluids, etc. and their applications industry such as the extrusion of polymer fluids, solidification of liquid crystals, cooling of the metallic plate in a bath, and colloidal solutions, etc. Stokes [1] discussed the hydromagnetic steady flow of a fluid with a couple stress effects. In this paper, we have investigated the Hall and Ion-slip effects on steady free convective heat transfer flow between two vertical parallel plates in a couple of stress fluid. Piazza [2] has displayed thermophoresis which mean moving particles with thermal gradients, consequently, particle thermophoresis is a non-equilibrium crossflow effect between mass and heat transport. In addition, it is already exploited as a novel tool in macromolecular fractionation, micro-fluidic manipulation, and selective tuning of colloidal structures. Analysis of thermophoretic MHD slip flow over a permeable surface has been presented by Das et al [3]. Chamkha et al [4] study the Hall current, thermal radiation, and chemical reaction effects on heat and mass transfer. Hayat and Nawaz [5] have analyzed second-grade fluid flow in three dimensional under the Hall and ion slip effects. Chaudhary and Jain [6] have elucidated the effect of Hall current on MHD mixed convection flow of a viscoelastic fluid past an infinite vertical porous plate with mass transfer and radiation. Similarly, Kumar and Chand [7] have investigated the effect of slip conditions and Hall current on unsteady MHD flow of a viscoelastic fluid past an infinite porous plate embedded in a porous medium. Sahoo [8] has studied heat and mass transfer effect on MHD flow of a viscoelastic fluid through a porous medium bounded by an oscillating porous plate in slip flow regime. Furthermore, Jha et al. [9] have examined the effect of Soret and Hall current on MHD mixed convection flow of visco-elastic fluid past a vertical surface. Recently (in 2016), K.Das et al [10] have studied influences of thermophoresis and thermal radiation on heat and mass transfer of second grade MHD fluid flow past a semiinfinite stretching sheet with convective surface heat flux. Also, K. Das [11] has investigated the combined effects of thermophoresis and thermal radiation on MHD mixed convective heat and mass transfer flow of a secondgrade fluid past a semi-infinite stretching sheet in the presence of viscous dissipation and Joule heating. M.VeeraKrishna and B.V.Swarnalathamma [12] have studied Hall effects on unsteady MHD free convection flow of an incompressible electrically conducting Second-grade fluid through a porous medium over an infinite rotating vertical plate fluctuating with Heat Source/Sink and chemical reaction. Nayak and Panda [13] have presented an interesting result on mixed convective MHD flow of second-grade fluid past an infinite vertical plate with mass transfer, Joule heating, and viscous dissipation. Recently, El-Dabe et al. [14] have analyzed the effects of thermophoresis and hall currents on non-linear heat and mass transfer of second-grade fluid ow over a permeable infinite vertical plate with the presence of viscous dissipation and chemical reaction. The focus of this study extends the work of Srinivasacharya and Kaladhar [15], in addition, to study a non-Newtonian model which obeyed to viscoelastic second-grade fluid and Thermophoresis phenomena.

Problem Formulation:
Consider an incompressible couple stress fluid flow between two vertical parallel plates distance 2d apart. Choose the coordinate system such that the x-axis is taken along vertically upward direction through the central line of the channel, y is perpendicular to the plates, and the two plates are infinitely extended in the direction of x and z. The plates of the channel are at y = ± . The effect of Hall and Ion-slip current gives rise to force in the z-direction, which induces a cross flow in that direction and hence the flow becomes three dimensional. Assume that the flow is steady and the magnetic Reynolds number is very small so that the induced magnetic field can be neglected in comparison with the applied magnetic field. Further, assume that all the fluid properties are constant except the density in the buoyancy term of the balance of momentum equation.
Such that is the rate of deformation tensor, is the Kronecker delta, and are the material constants of viscosity, is the curvature-twist tensor, , and ′ are the material constants of momentum and is the couple stress tensor and is the pressure. The equation of motion as obtained by Stokes in Gibbs notation as: where, = + . ∇ , For incompressible fluids = 0 so, if the body force and the body moment are absent, the equations of motion reduce to where the fourth derivative in the above equation gives the effect of couple stresses.
We used a second-grade viscoelastic model For these according to Truesdell and Noll (1965): the coefficients , 1 and 2 describe the viscosity, visco-elasticity, and cross-viscosity, respectively and must satisfy: material constants and 1 and 2 are the kinematic Rivlin-Ericksen tensors, Wu and Greif have determined the thermophoretic velocity as A thermophoretic parameter can be defined as follows; where, is some reference temperature, represents the thermophoretic diffusivity while, is the coefficient of thermophoresis which ranges in value from 0.2 to 1.2.
the typical values of are 0.01, 0.05 and 0.1 corresponding to approximate values of ( 2 − 1 ) equal to 3.15 and 30 for a reference temperature of = 300 .
the generalized Ohm's law taking the effects of Hall and ion-slip currents, and Maxwell's equations into account are: where is the velocity vector, is the density of the fluid, is the time, is the pressure, is the dynamic viscosity, 1 is the couple stress parameter, ∇ 2 is the Laplacian operator, is the electric current, is the magnetic induction vector, is the intensity of the electric field, is the electric conductivity, is the magnetic field strength, and = 1 + , is the ion-slip parameter, and is the Hall parameter. By using the relation ∇. = 0 for the magnetic field = ( , , ) we obtain = (imposed magnetic field) everywhere in the field. This assumption is valid only when the magnetic Reynolds number is very small. For the current density = ( , , ), we obtain from the relation ∇. = 0 that = . Hence we consider that the channel is non-conducting and therefore = 0 at the channel and hence is zero everywhere.
Within the above framework, the governing partial differential equations of the flow under the usual Boussinesq approximation are: momentum conservation in x-direction: heat equation: The appropriate boundary conditions for the problem are: such that all differentiations are respected to * and the nomenclature in the above equations are: is the kinematic viscosity, is the fluid density, is the gravitational acceleration, and are thermal expansion volumetric coefficient and volumetric coefficient of concentration expansion, respectively. * is denote the dimensional temperature of the fluid, * is the dimensional concentration of the solute [?], is the specific heat at constant pressure, is the thermal conductivity of the medium, and is chemical molecular diffusivity. From eq.(12), it can be seen that * is either a constant or a function of time. Thus, assuming that the suction velocity to be constant. where, 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 (18), the system of equations (14), (15) and (16) with boundary conditions (17) can be written in dimensionless form as: with the boundary conditions : where primes denote differentiation with respect to y; the dimensionless numbers are = 1 2 2 is the viscoelastic parameter, is the Eckert number, the Schmidt number = that represents the ratio of momentum to mass diffusivity, = 2 is the nondimensional chemical reaction parameter, = ( 2 * − 1 * ) It is the thermophoretic parameter.

Nusselt Number
The rate of heat transfer between plates and the fluid can be expressed in terms of non-dimensional Nusselt number as = − = where the Nusselt number is given by:

Sherwood Number
The rate of mass transfer between plates and the fluid can be expressed in terms of non-dimensional Sherwood number as = − = ℎ where the Sherwood number is given by:

Results and discussions
A parametric ND solve package was designed in Mathematica 10 software program to simulate the numerical computations of differential equations system which describe our problem. The purpose of these numerical solutions to illustrate the effect of different governing physical parameters such as viscoelastic second-grade parameter , Hall parameter , Ion-slip parameter , inverse of couple stress parameter , Grashof number , modified Grashof number , Prandtl number , Eckert number , Schmidt number , thermophoretic parameter and chemical reaction parameter on the velocity ( ) , the temperature ( ) and the concentration ( ) fields which they have been done at the following values: = 0.5, = 2 = 5, = 0.1, = 0.2, = 5, = 0.71, = 0.01, = 0.22, ℎ = 1, = 2, = 0.5. As well as Skin friction * , Nusselt number and Sherwood number ℎ have been computed and represented in tabulated forms at the same values of parameters. In order to get a clear insight into the physics of the problem, a representative set of numerical results are shown graphically in figures (2 − 21). In fig. 2, by increasing the viscoelastic second-grade parameter , the velocity profile ( ) decreases. It is observed from fig. 3 that an increase in the couple stress inverse parameter leads to decreasing in the values of velocity ( ). The velocity ( ) curves decrease when the thermophoretic parameter increases, as seen in fig. 4. The values of Grashof number are taken to be positive and negative as they respectively represent symmetric heating of the plates when < 0 and symmetric cooling of the plates when > 0. The velocities profiles ( ) and ( ) increase by increasing the Grashof number as happened in modified Grashof number see fig. 5 and fig. 6, respectively. This is due to the buoyancy forces, which is acting on the fluid particles due to the gravitational force that enhances the fluid velocity. The increase in the Prandtl number results in the decrease in ( ) 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 is to enhance ( ) as shown in fig. 8. Higher Eckert number values lead to greater viscous dissipative heat. On the contrary, the reversed effect has been observed in fig. 9 where it indicates that velocity curves decreases with increasing of Schmidt number . Physically it is true as the concentration increase the density of the fluid increases which results a decrease in fluid particles. The combination effect for hall and ion slip parameter has been shown in fig.10 fig. 11 depicts that ( ) decreases as the chemical reaction parameter increases because of the presence of viscous dissipation. As a general description, velocity curves of this fluid are increased from injected plate to reach the top of the curves (the highest value for the speed) which locate next to the center of channel and then they decrease until it reaches zero at the sucked plate. This mean that the velocity profiles of the fluid is maximum at the center of the channel and zero at the plates. Fig. 12 depicts that ( ) decreases as ion-slip parameter increases, this also happen in fig.13but with hall parameter .

Concentration profiles
Influence of Schmidt number on concentration ( ) is shown in the figure 18, from this figure it is noticed that concentration decreases with an increase in . Because, Schmidt number is a dimensionless number determined by the ratio of momentum diffusivity and mass diffusivity, and is used to characterize fluid flows in which there are simultaneous momentum and mass diffusion convection processes. Therefore, concentration boundary layer decreases with an increase in Schmidt number. The increase in chemical reaction parameter at values = 0.1, 0.3, 0.5, 0.8 leads to a decrease in the concentration distributions ( ) as shown in fig. 19. Effect of thermophoresis on concentration profiles ( ) is displayed in fig.20 at values of = 0.1, 0.6, 0.9 and 1.2 . It is noticed that the concentration profiles ( ) decreases as increases. Specially, the effect of increasing the thermophoretic parameter 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 ℎ is large as chemical reaction effect increases and compared to the convection effect. So, the thermophoretic parameter is expected to alter the concentration boundary layer significantly. The concentration profiles diverge at values of couple stress inverse parameter that equal to = 8. As a result of the effect of the previous parameters on the concentration layers, we found that the fluid be more concentrated in the adjacent side of the absorption process of th plate which located at = 1 as shown in 21.       Table 3 shows numerical values of mass transfer coefficient in terms of sherwood number ℎ. It is observed that Sherwood number increases with increasing in chemical reaction , Schmidt number and Hall currents ℎ . likewise, the increase of Eckert number and thermophoretic parameter lead to decrease the Sherwood number ℎ .

Conclusions
The effects of Couple stresses and thermophoresis on free convective heat and mass transfer of viscoelastic fluid with Hall and ion-slip effects have been studied. The behavior of velocities, temperature and concentrations has been noticed at physical values of parameters furthermore, Skin friction, Nusselt and Sherwood numbers have been calculated. It is found that The velocity ( ) decrease as the Hall parameter increases. the velocity ( ) increases and induced flow velocity ( ) in the z-direction decreases with an increase in the Ion slip parameter.