Studying the Magnetohydrodynamics for Williamson Fluid with Varying Temperature and Concentration in an Inclined Channel with Variable Viscosity

In this paper, the Magnetohydrodynamic (MHD) for Williamson fluid with varying temperature and concentration in an inclined channel with variable viscosity has been examined. The perturbation technique in terms of the Weissenberg number (We ≪ 1) to obtain explicit forms for the velocity field has been used. All the solutions of physical parameters of the Darcy parameter(Da), Reynolds number(Re), Peclet number (Pe) and Magnetic parameter (M) are discussed under the different values as shown in plots.


Introduction:
In physiology, the Non-Newtonian fluids that modify their consistency in line with the forces they managed. The molecule mixture traps them in situ, nevertheless, the opposite section assumes that hanging these substances expels fluids from the colloids and pushes them nearer along and hardens. However, a recent analysis has tested each assumption. The thickness of this liquid depends primarily on the friction between its molecules and hydraulics forces that play a role for the chemical process once the mixture becomes less dense.
This classic model includes an accurate benefit over non-Newtonian fluids, because it is derived from the kinetic theory of gases and not from the interactions of inquiry. Formulating flows of newly established materials such as black lead residue and glycerol may be common.
Viscosity is a fluid's inner asset that has flow stability. Consistency could also be a fundamental fluid property that is essential in some respects as used in fossil fuel, industrial chemistry, packaging and writing, meat and drink, engineering, energy, and environment, etc. Consistency is regarded as a performance of either temperature or strain in natural science.
There are several studies in the scientific works on fluid movement in the channel, for example; the movement of Williamson fluid for two types (Poiseuille flow and Couette flow) in an inclined channel dependent on the viscosity studied by Nadeem et al. ( 3) in (2015).
In (2016) the effect of temperature on (MHD) for Jeffrey liquid with flexible viscosity model in to porous channel was considered by Al-Khafajy ( 4).
In (2017) Jassim ( 5) studied the effects of Williamson fluid of wall tapered and magnetic field on peristaltic movement in an inclined channel. In (2014) the flexible viscosity flows in channel with great temperature generation was studied by Ting ( 6).
In (2017) Immaculate et al. ( 7) discussed the Williamson Nano-fluid with unsteady flow MHD in a porous channel and oscillating wall temperature to solve the momentum equations by using the homotopy analysis method.
This study aims to employ a series of perturbation method to fix the issue of an elevated medium with variable viscosity for the impact of (MHD) of Williamson fluid with varying temperature and concentration. ( Fig.1). Consider the Cartesian Coordinates system is considered such that ( ( , ), 0,0) is velocity vector in which is the -component of velocity and is perpendicular to -axis.

Figure 1. Geometry of the problem
The Williamson fluid governing equations are ( 8): (2) where ∞ is the infinite shear rate viscosity, 0 = ( ) , Γ is the time constant, is the unit tensor, ̅ is the pressure and ̅ is the extra stress tensor,. Then ̇ is given by: The Williamson fluid extra stress tensor as follows: (4) The momentum, temperature and concentration equations are given by: The stress variable viscosity for Williamson fluid is: where (̅) is the axial velocity, ( ) is the acceleration due to gravity, ( ) is the electrical conductivity, ( ) is the density of the fluid, ( 0 ) is the strength of the magnetic field, ( , ) are temperature and is the concentration, ( ) is thermal conductivity, ( ) is heat generation, ( ) is the coefficient of mass diffusivity ( ) is specific heat at constant pressure and ( ) is the thermal diffusion ratio. The corresponding boundary conditions are given below: where ( ) is the radiation absorption and ( ) is the radioactive heat flux. The non-dimensional conditions are as follows (8) The '' perturbation technique'' was used to solve the equation (28), as follows ( 10) Figure 2 shows that the velocity distribution decreases with the increasing of . Figure 3 displays the effect on the velocity distribution function. By increasing the velocity distribution decreases. Figure 4 illustrates that velocity distribution increases with increasing the considerations . Figure 5 and 6 illustrate the effect and , on the velocity distribution. It has been observed that by the increasing and the velocity distribution function increases. Figure 7 shows that the velocity distribution u is rising up by increasing the effect of the consideration . Figure  8 illustrates that velocity distribution increases with increasing the considerations . 534 Figure 9 shows the velocity distribution decreases with increasing . Figure 10 shows the effect of the parameter of . By increasing , velocity distribution rises up. Figure 11 displays the velocity distribution increases with increasing . Figure 12 shows the velocity distribution increases with the increasing . Figure 13 shows that by increasing the then velocity distribution increases. Figure 14 displays that with increasing , velocity distribution decreases. Figure 15 shows that velocity distribution decreases with increasing . Figure 16 displays that increasing , velocity distribution decreases. Figure 17 illustrates that the temperature increases with increasing . Figure 18 shows that the effect on temperature , that increases by increasing . Figure 19 shows us that temperature decreases with increasing .
The concentration field is shown in Figs. 20 -25. Figure 20 shows that with increasing , the concentration field Φ decreases. By Fig.21, it is observed that the effect frequency of the oscillation on concentration field Φ by increasing , which leads to Φ decreases. From Fig.22 and 23 it is noted that by increasing each of parameters and then Φ decreases. With increasing , then concentration field Φ decreases in Fig.24. In Fig.25, it is clear that by increasing , the concentration field Φ decreases.

Conclusion:
The (MHD) oscillatory flow for Williamson fluid with varying temperature and concentration in an inclined channel with variable viscosity are investigated. It is confirmed that the velocity field, concentration profile and temperature analyzed by using the perturbation method is adequate to solve the problem. Different sets of values have been employed to tackle the problem. The conclusions that can be drawn are:  By increasing the , , , , , , , and the velocity distribution increases.  When increasing the , , , , and the velocity distribution decreases.  The concentration profile decreases by increasing , , , , and .  It is observed that by increasing and the temperature increases, while, by increasing the temperature decreases .