Oscillatory Flow MHD of Jeffrey Fluid with Temperature-Dependent Viscosity (TDV) in a Saturated Porous Channel

In this research, we studied the impact of Magnetohydrodynamic (MHD) on Jeffrey fluid with porous channel saturated with temperature-dependent viscosity (TDV). It is obtained on the movement of fluid flow equations by using the method of perturbation technique in terms of number Weissenberg ( ) to get clear formulas for the field of velocity. All the solutions of physical parameters of the Reynolds number , Magnetic parameter , Darcy parameter , Peclet number and are discussed under the different values, as shown in the plots.


Introduction
In many biological streams and engineering requests, with arterial blood flow, Magnetohydrodynamic generators , geothermal liveliness withdrawal, petroleum engineering, atomic reactors and much more, -MHD-flux investigation of viscous liquid in a permeable channel that is full with insignificant permeable media is insignificant. The rheological fluid in the ducts is of fundamental importance for particular natural and scientific presentations of transit flux (or oscillation).
For example, the semi-intermittent flow inside the heart background container been represented by pressure (regularity component) and flow frequency beats. A lot of infections accompanying with 82 the vascular method are due to a trouble in the native flow of blood vessels. In many generalized applications such as photogravure from inkjet imprinters, a quick switch is necessary between "nonflow" and "flow" for a non-Newtonian fluid.
There are few authors extended the investigation to the impact of temperature-dependent viscosity Falade et al. [ 5], and Abbas et al. [ 6] on oscillatory flow of Casson fluid with porous channel. There are same researchers extended the problem by presenting -injection/suction- [ 7] and time-dependent boundary conditions (TDB) [ 8].
This study aims to employ a chain of perturbation method to fix the issue of an elevated medium with variable viscosity for the impact magneto hydrodynamics of temperature-dependent viscosity on Jeffrey fluid in a saturated porous channel.

Preparation of the problem
Consider the incompressible unstable flow for Magnetohydrodynamic of Jeffrey fluid with -TDV-in a saturated permeable channel with variable viscosity and at height , see (Fig.1). We choose the Cartesian coordinates system is velocity vector in which is the -component) of velocity and is perpendicular to -axis).

Figure 1-Construction of the Problem
The Jeffrey fluid model, the constitutive equation below [ 9]: where is temperature-dependent viscosity, ̇ is the shear rate, is the ratio of relaxation to retardation time and is the retardation time. The velocity field and the heat field of the present problem are: [ ] (2) The Jeffrey fluid model equations for flow with -TDV-are given by: Khudair et al.
with associated boundary conditions In which ̅ is the axial velocity, is an accelerating due to importance, k is a penetrability, is a fluid density, is a specific heat at constant pressure, is a measurement of volume intensification due to heat, is a radioactive heat flux, is a conductivity of the fluid, is a attractive field and is a thermal conductivity. Non-dimensional parameters are given by (Wissam et al., 2019) [ 10]: and Eq. (5) becomes } To solve the temperature equation (8), let: (10) Substituting the Eq. (10) into the Eq. (8), we have: (11) The exact solution of Eq. (11) and Eq. (9) is given by: where . Therefore

Results and discussion
In this part of the paper, magneto hydrodynamics oscillatory flow problematic of Jeffrey fluid with -TDV-in a saturated permeable channel is discoursed. The perturbation method is comprised in the direction of solving the motion equations, while the meticulous solution of temperature equation is attained. The numerical calculations were have been performed using (Mathematical ver.11) using with the set of values: . From Figure-2, the cumulative frequency of the oscillation , during the canal decreases is clarified. The significances payable to variation in on are shown in Figure-.3. From this figure, it is initially distinguished that in the absenteeism absence of (injection/suction), the fluid temperature is linearly dispersed during the channel. Conversely, the temperature of the fluid increases in the channel by increasing the injection on the heated bowl.
The linearity inspected at provides an approach to concave circulation. By increasing , the concavity is a important due to the pathway of the flow of temperature which is absorbed each time from the animated superficial to the cold superficial. From Figure-4, it is distinguished that increases by increasing the values of because the temperature is communicated from the animated wall near the fluid. Figure-5 displays that velocity distribution increases, with increasing the thoughts . Figure-6 illustrates the effect of on the velocity field. The figure demonstrates that after increasing the velocity field increased. Figure-7 illustrations the effects of the parameter of . By increasing , velocity distribution is illustrates. Figure-8 shows that the velocity distribution is increased with increasing . Figure-9 shows the influence on the velocity field function . By increasing the velocity distribution is decreased. Figure-10 illustrates that by increasing velocity distribution is increased. Figure-11 displays that increasing , leads to an increase in velocity distribution . Figure-12 it is displays that velocity distribution is decreased with increasing . Figure-13 Shows that increasing results in an increase in the velocity field is increased. Figure-14 Shows that velocity is decreased with increasing .

Conclusion
The oscillatory magneto hydrodynamics flow problem of Jeffrey fluid with -TDVin a saturated porous channel is discussed. It is established that the velocity field and temperature, examined by exhausting the perturbation method are satisfactory to resolve the problem. Different sets of values have been employed to tackle the problem. We conclude that, by increasing the and , the velocity distribution is increased. While increasing the and leads to a decrease in the velocity field. It is also shown that by increasing and causes an increase in the temperature but, however, increasing leads to a decrease in the temperature .