Effects of Hall Current and Ohmic Heating on Non-Newtonian Fluid Flow in a Channel due to Peristaltic Wave

In this paper, the study of non-Newtonian fluid flow with heat transfer in a porous asymmetric channel due to peristaltic wave was taken out. Hall current and Ohmic heating effects were introduced in the present study. A Casson non-Newtonian constitutive model was employed for the transport of fluid. Analytical solutions were obtained for stream function, temperature and heat transfer coefficient. The coupled nonlinear equations have also been solved numerically using MATLAB software (by bvp4c function). The influences of many evolving parameters on the flow characteristic have been explained with the help of 2D and 3D plots. Again the obtained results were compared with the results available in the literature and were found in good agreement.


Introduction
At present, investigation regarding the peristaltic flow of non-Newtonian fluid cannot be ignored due to their vast applications in physiology and industry. This mechanism is associated with a spontaneous relaxing and compressing movement along the length of the fluid filled channel or tube. Digestive track, urine transport form kidney to bladder, blood flow in small vessels, egg movement in fallopian tube are few examples that can be seen in our body [1]. Latham [2] studied the initial pioneer work on peristaltic mechanism of viscous fluid and later we saw many studies [3] [4] [5] [6]. Again the study of heat transfer in pe-How to cite this paper: Hasan, M.M., Samad, M.A. and Hossain, M.M. (2020) Effects of Hall Current and Ohmic Heating on Non-Newtonian Fluid Flow in a Chan-ristaltic flow has some vital roles in biomedical processes such as hemodialysis and blood oxygenation. This property helps in crude oil refinement, food processing, sanitary fluid transport, and noxious fluid transport in nuclear industries [7].
It is well known the industrial and physiological fluids are non-Newtonian in nature. Due to various rheological properties of non-Newtonian fluids, several constitutive relations have been suggested. Casson fluid is one of the non-Newtonian fluids which was introduced by Casson [8]. The behaviors of Human blood, honey and jelly can be represented by Casson's model [9]. Most of the time Hall effect is neglected in the Ohm's law as they have no remarkable effect for a weak magnetic field. It is mentioned that this effect is important when the magnetic field is high [10]. On the other hand, Ohmic heating (Joule heating) effect, which is the transformation between electric energy to thermal energy, arises from the applied electric field and fluid electrical resistivity. This effect is useful for a variety of applications in food industry [11]. Existing literatures witness that no proper attention is given to peristaltic motion of Casson fluid in an asymmetric channel with Hall current and Ohmic heating effects. The aim of the study is to fulfill this void. Mathematical modeling is carried out for low Reynolds number and long wave length approximations. The transformed equations have been solved analytically and numerically. The effects of various important parameters on velocity and temperature are displayed graphically and discussed. Trapping phenomena are also explained.

Mathematical Formulation
Consider the peristaltic flow of non-Newtonian fluid in a two dimensional porous asymmetric channel. We choose a stationary frame of reference ( ) , X Y such that X measured along the axis of the channel and Y perpendicular to it.
is applied and the Hall and Joule heating effects are taken into consideration. The induced magnetic field is neglected for small magnetic Reynolds number and also the externally applied electric field is assumed to be zero. The geometry (in Figure 1) where 1 2 , a a denote the waves amplitudes, 1 2 d d + is the channel width, λ is the wave length, t is the time, c is the velocity of propagation and φ is the phase difference ( ) 0 φ π ≤ ≤ , in which 0 φ = corresponds to symmetric channel with waves out of phase and φ π = corresponds to waves in phase. Here The rheological relation for non-Newtonian Casson [12] fluid is where ij e is the ( ) Generalized Ohm's law is Ohmic heating (Joule heating) is ( ) where J is the current density, q is the fluid velocity, P is the pressure, 1 e en γ = is the Hall factor/Hall Current, e is the charge electron, e n is the mass of the electron, E is the electric field. The corresponding boundary conditions are where , , , u v p T are the velocity components, pressure and temperature in the wave frame respectively. Now we introduce the following dimensionless quantities The governing Equations (4)- (7) under the assumptions of long wave length and low Reynolds number in terms of stream function ψ (dropping the das symbols) become The dimensionless boundary conditions become is the magnetic field para-

Exact Solution
Equation (15) gives that ( ) p p y ≠ . Eliminating the pressure terms from (14) we get Solving Equations (18) and (16) where the constants involved in the solutions are given in the appendix.
The dimensionless mean flow rate Q′ in the laboratory frame is related to the dimensionless mean flow rate F in the wave frame by in which Again the heat transfer coefficient at the upper wall ( )

Numerical Solution
The dimensionless equations have again been solved numerically for different values of model parameters with another code written in MATLAB (bvp4c function).
It is a finite difference code that implements a collocation formula. Numerical

Results and Discussions
The effect of magnetic field parameter M on velocity field is sketched in Figure 2.
Here we see that the velocity decreases with an increase in M. This is due to the fact that the applied magnetic field produces a Lorentz force to the flow and this force reduces the velocity. Figure 3 is plotted to see the velocity profiles for different permeability parameter K. . There is a grow in the velocity of fluid near the central part of the channel when we increase in K. This is due to the fact that large K provides less resistance to the flow. From Figure 4, it is clear that β has an increasing effect on velocity field. Large β accelerates the fluid flow. The magnitude of the velocity u increases with increasing Hall parameter m as seen in Figure 5. The fact is that the effective conductivity       In order to see the job of Ohmic (Joule) heating effect we have resolved the heat equation after neglecting the Ohmic heating term and then investigative the magnetic field parameter M and Hall parameter m effects on temperature field through graphs. Figure 12 and Figure 13 show that temperature decreases for increasing M and increases for large m. From the above discussion it is clear that when we neglect the Ohmic heating term, the behaviors of M and m on temperature field are quite opposite.
Trapping, the procedure of contours of streamlines, is an attractive characte-  Figure 16 indicates the impact of phase difference φ on streamlines. It is clear that the bolus travels upwards and decreases in size as φ gets large. Also the trapping vanishes when φ reaches to π .       To verify the accuracy of numerical results, the present study is compared with the previous study of Hayat et al. [13]. We have considered the same parameter values (Newtonian case and in absence of Ohmic heating) to do it. These comparisons are given in Figure 17, which are found in very good agreement. Again Table 1 gives a comparison of heat transfer coefficient 1 Z at upper wall for different values of K (Newtonian case and in absence of Ohmic heating).

Conclusions
This study is presented on the peristaltic motion of non-Newtonian fluid in a porous asymmetric channel with Hall current and Ohmic heating effects. A 2) The fluid becomes less viscous when we increase β .

3) Temperature decreases when m increases and increases when M and Br in-
crease.
4) The activities of M and m on temperature field are quite reverse when we neglect the Ohmic heating term.
5) The size of trapped bolus decreases for large values of phase difference φ and trapping vanishes when φ π = . E + =