Hall and Joule's heating Influences on Peristaltic Transport of Bingham plastic Fluid with Variable Viscosity in an Inclined Tapered Asymmetric Channel

This paper presents an investigation of peristaltic flow of Bingham plastic fluid in an inclined tapered asymmetric channel with variable viscosity. Taking into consideration Hall current, velocity, thermal slip conditions, Energy equation is modeled by taking Joule heating effect into consideration and by holding assumption of long wavelength and low Reynolds number approximation, these equations are simplified into a couple of non-linear ordinary differential equations that are solved by using perturbation technique. Graphical analysis has been involved for various flow parameters emerging in the problem. We observed two opposite behaviors for Hall parameter and Hartman number on velocity axial and temperature curves.


Introduction
Peristaltic transport is a successive sinusoidal waves movement of fluids along a flexible channel walls. It is naturally found in human living body such as urine movement from kidney to bladder, food swallowing process and blood flow in the small vessels [1][2][3]. Moreover, the peristaltic transport of non-Newtonian fluid gained much attention in various modern industrial and biomedical phenomena like polymer industry and artificial hearts that their devices designed in a manner where the fluid flows without internal moving parts [4][5][6]. Inspired by this fact and since the modern industrial fluids are characterized by their variable viscosity, a few researchers indicate studies regarding the peristaltic transport of fluids having variable viscosity. Adnan and Abdulhadi [7] analyzed the effect of an inclined magnetic field on peristaltic flow of Bingham plastic fluid in an inclined symmetric channel with slip conditions. In the same year Adnan and Abdulhadi [8] investigated the peristaltic flow of the Bingham plastic fluid in a curved channel. Hayat et al. [9] studied the effect of soret and dufour on the peristaltic transport of Bingham plastic fluid considering magnetic field. While Lakshminarayana et al. [10] investigated the heat transfer and the effect of slip condition and wall properties on the peristaltic transport of Bingham fluid. Ara et al. [11] explored the Jeffery-Hamel flow of Bingham plastic fluid in converging channel in the presence of external magnetic field. However, Salih [12] illustrated the influence of varying temperature and concentration on (MHD) peristaltic transport of Jeffery fluid with variable viscosity through porous channel. For more information see [7,13].
In this paper, the influence of Hall and Joule's heating on the peristaltic flow of Bingham plastic fluid passing through an inclined tapered asymmetric channel with variable viscosity is studied. A long wave number and low Reynolds number are taken into consideration to simplify the problem. Perturbation technique is used to solve and find the last shape of stream function.
Finally, the effects of various parameters on axial velocity, temperature, stream function and heat transfer coefficients are discussed graphically.

Mathematical Modeling
The peristaltic transport of an incompressible Bingham plastic fluid in a tapered inclined channel at an angle which is an asymmetric channel with a total width(2 ) is considered. Characterizing the flow by the existence of a strong transverse magnetic field = (0, 0, 0 ). A magnetic Reynolds number is taken small and the induced magnetic field is prescribed neglected. The flow is achieved by the peristaltic waves of length with different amplitude and phases moving with a constant speed along the channels walls.
The geometry of the walls surfaces is described by Where 1 , 2 are the upper and lower wall respectively, 1 is the non-uniform parameter, 1 , 2 are the wave amplitudes, is the time and ( ̅ , ̅ ) the rectangular coordinates in a fixed frame. Ø Is the phase different and Ø ∈ [0, ] such that when Ø = 0 corresponds to asymmetric channel with waves out of phase, and when Ø = , the waves are in phase. Further , , ∅ satisfy the necessary condition By applying the generalized Ohm's law [2], we include the Hall current as follows Such that Hence, In which ⃑ is the magnetic force, ⃑ assigns to the current density vector, ⃑⃑ = ( , , 0) the velocity field, the electrical conductivity, the number density of electron, the electric charge, 0 the magnetic field strength and ( = 0 ) the Hall parameter.
The fluid satisfies Bingham plastic model and its extra stress tensor is given as follows [11]: Such that And ̅ represents the extra stress tensor, = ( / ̅ , / ̅ , 0) the gradient vector, the yield stress ̅ 1 is the first Rivlin-Erickson, and ( ̅ ) is the dynamic variable viscosity.
The fundamental equations of the flow can be written as below: -component of momentum equation -component of momentum equation And Energy equation with Joule heating effect is In which , , , ̅ , , , , are the electrical conductivity, the thermal conductivity, the porosity parameter, the dynamic viscosity, the density, the specific heat, and the gravity respectively.
The corresponding boundary slip conditions are And the wall flexibility condition is Where 0 , , 2 , ′ , , 1 are the temperature at the upper and lower walls, the elastic tension, the mass per unit area and the coefficient of viscous damping, velocity slip coefficient, and temperature slip coefficient respectively.
Dimensional analysis is used for normalizing the flow equations Eqs. (7) -(15) by using the following as bellows: Where is the wave number, 1 the wall elastance parameter, 2 the mass per unit area parameter, 3 17), the continuity equation (11) vanishs identically, other flow equations take the following form Adopting the assumptions of peristaltic long wavelength and low Reynolds number, Eqs. (17)-(19) will be reduced into following form And the dimensionless boundary conditions are Where Furthermore, heat transfer coefficient at lower wall is derived as Through Eqs. (20) and (21), we obtain We toke the dimensionless approximate expression for ( )as is non-dimensional viscosity parameter.

Solution Methodology
By using the perturbation method for a small non-dimensional viscosity parameter and expanding the flow quantities in a power series of , we obtain  27) and then comparing the coefficients of same power of up to the first order, we obtain the following two systems

Zeroth order system
The general form of zeroth-order system is: - Now with the respect to the boundary conditions, we have: -

Result and Discussions
In this section we visualize graphically the influence of different inclusive parameters on velocity profile, temperature distribution, heat transfer coefficient and trapping phenomenon.  Fig.1 (a) is plotted to describe the effect of wall elasticity parameters on velocity profile. One can conclude a significant increase upon enhancement of wall rigidity and tension parameters respectively and Bingham number are shown in Figure 2 (b) and Figure 3 (a). From Figure 3 (b), we notice that the velocity profile enhances for higher values of phase difference parameter .

1.
(b)   ( , , , 1 , , ). Figure 4(a) illustrates the impact of Brinkman number on ( ) . It is seen that react directly on temperature profile. However, in Figure 4(b) Froude number record quite opposite behavior compared to Brinkman number. The effect of Hartman number on temperature profile testified in Fig. 5(a). It is evident that the rise in Lorentz force produces a resistance for a larger value of Hartman number and consequently reduces the temperature profile. Figures.  5(b), 6(a), and 6(b) clarify an increment in temperature slip parameter 1 Hall number and Bingham number causing rise in the temperature distribution curve.

Trapping phenomenon
A phenomenon in which an amount of fluid trapped in closed streamlines is called bolus. In this part of work, some results of the phenomenon of trapping are portrayed. Figures 9-15 highlight the impact of wall elasticity parameters 1 , 2 , 3 and for , , , , , values. Graphical results show two asymmetric regions, the first region begins from (0.2 ≤ ≤ 0.7)while the second region (0.7 ≤ ≤ 1.2). One can observe an increment in size and number of trapped bolus, whereas the second region witnesses a less number of generated bolus. The effect of wall rigidity and tension parameters respectively 1 , 2 and mass characterization parameter 3 on trapping phenomenon are shown in Figure 9. However, it is important to note that both parameters 1 , 2 increase the trap bolus in magnitude and number while a higher value of 3 parameter that reduces the size of bolus but in the right side, its number reduces. Figure 10 reveals that ascending value of Hartman number is due to increases in Lorentz force which resists the fluid flow as a result decreases the size of trapping bolus. Opposite to this result, permeability parameter directly acts on trapping bolus in size and number; see Figure 11. The variation of Hall number and dimensionless viscosity parameter on trapped bolus are reflected in Figures 12 and 13. One can observe the increasing function for them on trapped bolus size. In Figure 14, we demonstrate that a larger value of Froude number reduces both the size and circulation of bolus.

Conclusions
The peristaltic transport of non-Newtonian Bingham plastic fluid with variable viscosity in an inclined tapered asymmetric channel is performed, taking into account Hall and Joule's heating influences. Adopting assumptions of long wavelength and low Reynolds number the problem is modeled and reduced into a pair of nonlinear differential equations which are solved approximately by using a perturbation method. A parametric analysis is permitted through various graphs that made us outcome with some following important observations