Heat Transfer Analysis and Magnetohydrodynamics Effect on Peristaltic Transport of Ree–Eyring Fluid in Rotating Frame

Abstract This paper discusses Ree–Eyring fluid’s peristaltic transport in a rotating frame and examines the impacts of Magnetohydrodynamics (MHD). The results deal with systematically (analytically) applying each of the governing equations of Ree–Eyring fluid, the axial and secondary velocities, flow rate due to auxiliary stream, and bolus. The effects of some distinctive variables, such as Hartman number, heat source/sink, and amplitude ratio, are taken under consideration and illustrated through graphs.


Introduction
A non-Newtonian fluid is a fluid that is not subject to Newton's laws of viscosity. It has become important in many researches. The viscosity (the gradual deformation by shear or tensile stresses) of non-Newtonian fluids is dependent on shear rate or shear rate history. In a Newtonian fluid, the relation between the shear stress and the shear rate is linear but, in a non-Newtonian fluid, this relation is different. For example, many salt solutions and molten polymers are non-Newtonian fluids. Here we studied the Ree-Eyring model [1,2] because it is one of the important fluids that can be converted into a Newtonian fluid model for high and low shear rates. Ellahi et al. [3] studied the mathematical analysis of peristaltic transport of an Eyring-Powell fluid through a porous rectangular duct. MHD is the study of the behavior of electrically conductive fluids when exposed to a magnetic field, that is, the study of magnetic properties and the behavior of those fluids along with the changes that occur. Examples of this fluid include plasma, liquid minerals, and others. The set of equations describing MHD is a combination of the Navier-Stokes equations of fluid dynamics and Maxwell's equations of electromagnetism. Solving these equations is achieved either analytically or numerically.

ISSN: 0067-2904
Almusawi and Abdulhadi Iraqi Journal of Science, 2021, Vol. 62, No. 8, pp: 2714-2725 2715 The rotation phenomenon has vast applications in cosmic and geophysical flows and helps to better comprehend galaxy formation and ocean circulation. Rotational diffusion accounts for nanoparticle orientation in fluids. The following is a review of studies that discuss the effects of rotation. Hayat and Zahir [4] presented heat transfer analysis on peristaltic transport of Ree-Eyring fluid in rotating frame. Safa and Abdulhadi [5] performed unsteady heat transfer analysis on MHD flow of a second grade fluid in a channel with porous medium. Farah and Abdulhadi [6] analyzed the effect of inclined magnetic field on peristaltic flow of Bingham plastic fluid in an inclined symmetric channel with slip conditions. Khan et al. [7] studied the entropy generation in radiative flow of Ree-Eyring fluid due to due rotating disks. Wissam and Dheia [8] presented the influence of heat transfer on magneto hydrodynamics oscillatory flow for Williamson fluid through a porous medium. There have also been other attempts to study MHD [9][10][11].
The aim of the present investigation is to examine the peristaltic transport of Ree-Eyring liquid in a rotating frame, taking into consideration the convective condition. It is worth specifying that such examination for Ree-Eyring liquid is critical for different applications in biomedicine and designing. For further clarification, this article is organized into Section 2 that presents the physical modeling statement for our problem. The axial velocity, secondary velocity, temperature, and stream function are interpreted for relevant parameters and analyzed through graphs in Section 4. Finally, conclusions are given in Section 5.

Mathematical formulation
Let us consider an incompressible Ree-Eyring fluid material through a channel of width ( ). The fluid is electrically conducting by an external magnetic field, B = (0, 0, ), that is applied to it. Both the channel and fluid rotate with a uniform angular velocity ̅ about the -axis, as shown in Figure-1. Mathematical expression for wave propagation along the walls of the channel at ̆ ̆ can be written as: in which ̆ ̆ are the upper and lower wall, respectively, are the amplitudes of the waves along the lower and upper walls, respectively, the phase difference, and is the non-uniform parameter.
The governing equations of continuity and energy and the momentum equation for incompressible, irrotational, laminar flow can be written as [12] ̆ ̆ where , c , ̆ , , , , , ̆, , and ̆ represent the wavelength, constant speed, time of the wave, fluid density, specific heat, thermal conductivity, heat source/sink, modified pressure, electric conductivity, and material time derivative, respectively.
The stress tensor of Ree-Eyring fluid model is defined as [13] where µ is the fluid dynamic viscosity, and are material constants, and ̆ ̆ ( ̆ ̆ ̆) and ̆ ̆ ( ̆ ̆ ) Hence, the conditions are: indicate the heat transfer coefficient and temperature at walls, respectively. Then, by using the wave frame transformations and conditions (9-12), Eqs. (3)-(7) become: It is worth noting the definitions of the non-dimensional quantities [14], which are: where the terms , , , , , , , , , , and represent the Reynolds number, kinematic viscosity, Taylor number, fluid parameter, Brinkman number, Prandtl number, heat source/sink parameter, amplitude ratio parameters, Biot number, and Hartman number, respectively. Now, if we use Eq. (18) and the facts of into Eqs. (13)-(17), the following dimensionless equations are obtained:

The dimensionless form of the boundary and boundary conditions are
The pressure and the flow rate are: Eqs. (19) and (20) and the secondary flow pressure due to the rotational effect were neglected, and then we get:

Solutions of the problem
This section reviews the results of velocities, temperature, and stream function analytically. Eqs.
Where, , are constants which are found by using the boundary conditions below.

Results and Discussion
We demonstarte Figures-2-21 to monitor the resulting effects when changing the parameter values on the axial and secondary velocities as well as against Hartmann parameter ( ), fluid parameter ( ), Taylor number ( ), the amplitude parameters ( , ), phase differences ( ), Biot number ( ), heat source/sink parameter ( ) and Brinkman number ( ). Figure-2 shows that is enhanced in the left side of the channel, but it is decayed in its right side when increases. The same outcomes are observed for the fluid parameter , as noted in Figure-3. One can observe from Figure-4 that when increased, this will lead to the emergence of a force called "Lorentz force" due to the influence of the magnetic field exerted on the flow of the fluid. Thus, a resistance against the flow will be created, resulting in a diminishing in the left side, which has an increasing value in the right side. In Figure-5, we can notice that we get the same results as in the previous figure for .     Figure-6 shows that if is increased, it leads to a decrease in velocity at the fluid flow center. To know the effect of on the secondary velocity , we notice that increasing the values of leads to an increase in . Figure-7 illustrates this effect, for the same reasons of the effect of on axial velocity . Figure-8 shows the effect of on . One can notice that increases slowly by increasing the values of . The influences of and parameters on the velocity are portrayed respectively in Figure-9 and Figure-10. It is seen that decreases with the increase of and . In Fig. 11, one can observe the effect of on the secondary velocity which increases in the channel.

Temperature profile
Now, temperature ( ) for different variables is demonstrated through Figures-16-21. In Figure-16, we observe, for fluid parameter against that temperature distribution increases when the fluid parameter increases. In Figure-17, we observe that if increases then also increases. If both of and increase, respectively, we notice no effect on temperature ( Figures-18 and 19). Figure- Figure-25 shows that there is an increase in trapping when increases. Decreasing behavior in the size of the trapped bolus is seen upon rising up the amplitude (Figure-26). Figure-27 shows that as increases, the bolus decreases. Figure-28 shows that as increase the bolus tends to rise. The impact of is illustrated in figure 29, which shows that as increases, the bolus increases. While Figure-

Conclusions
In this paper, the heat transfer and the magnetic field were discussed and the equations that describe the motion of flow for Ree-Eyring fluid were simplified under assumptions of low Reynolds number and long wavelength. These equations were solved analytically; the main results of the flow problem are summarized as follows.
When increasing , the velocity increases in one side of the fluid course, while it decreases at the other; we also obtain the same results when increasing the amount of , but when increases the velocity increases, while it decreases with the increase in . As for the temperature it decreases due to the increase in the size of the magnetic field, but there is no effect on it profile for both and . For and , when increases, increases in one side and decreases in the other. and increase with increasing , but decreases with increasing However, opposite results are obtained with respect to . The temperature increases with increasing and . As for the phenomenon of ingestion that occurs in the fluid stream, it increases with increasing , , , and , while it decreases with increasing , , and .