Hall Currents and Heat Transfer Effects on Peristaltic Transport in a Vertical Asymmetric Channel through a Porous Medium

The influences of Hall currents and heat transfer on peristaltic transport of a Newtonian fluid in a vertical asymmetric channel through a porous medium are investigated theoretically and graphically under assumptions of low Reynolds number and long wavelength. The flow is investigated in a wave frame of reference moving with the velocity of the wave. Analytical solutions have been obtained for temperature, axial velocity, stream function, pressure gradient, and shear stresses. The trapping phenomenon is discussed. Graphical results are sketched for various embedded parameters and interpreted.


Introduction
Peristalsis is a phenomenon found in several physiological and industrial processes.The word peristalsis stems from the Greek word "Peristalikos" which means clasping and compressing.
Peristalsis is an important mechanism for mixing and transporting fluids that occurs when a progressive wave of area contraction or expansion propagates along the wall of the tube.Peristaltic flows occur widely in chemical processes such as in distillation towers and fixed-bed reactors, urine transport from kidney to bladder through the ureter, transport of lymph in the lymphatic vessels, swallowing food through the esophagus, the movement of chyme in the gastrointestinal tract, ovum movement in the fallopian tube, transportation of spermatozoa in the ductus efferentes of the male reproductive tracts, in the vasomotion of small blood vessels, transport of corrosive fluids, in sanitary fluid transport, and blood pumps in heart lung machine.In addition, peristaltic pumping occurs in many practical applications involving biomechanical systems.Several attempts had been made to know Mathematical Problems in Engineering and understand peristaltic action in different situations.Since the first attempt of Latham 1 some interesting investigations in this direction have been given in 2-17 .Flow through a porous medium attracted the attention of many researchers in the last few decades because of its very important practical applications.It occurs in filtration of fluids and seepage of water in river beds, sandstone, limestone, bile duct, wood, the human lung, gall bladder with stones, and in small blood vessels.El Shehawey and Husseny 18 formulated a mathematical model for the peristaltic transport of a viscous incompressible fluid through a porous medium bounded by two porous plates.El Shehawey et al. 19 studied the peristaltic transport in an asymmetric channel through a porous medium.Vajravelu et al. 20 have examined flow through vertical porous tube with peristalsis and heat transfer.Srinvas and Kothandapani 21 have studied the influence of heat and mass transfer on MHD peristaltic flow through a porous space with compliant walls.Srinvas and Gayathri 22 have discussed the peristaltic transport of Newtonian fluid in a vertical asymmetric channel with heat transfer and porous medium.The Hall effect is important when the Hall parameter which is the ratio between the electron-cyclotron frequency and the electron-atom-collision frequency is high; this can occur if the collision frequency is low or when the magnetic field is high.This is a current trend in magnetohydrodynamics because of its important influence of the electromagnetic force.Hence, it is important to study Hall effects and heat transfer effects on the flow to be able to determine the efficiency of some devices such as power generators and heat exchangers.Attia  The aim of this paper is to investigate the effects of Hall current and heat transfer on peristaltic transport of a Newtonian fluid in a vertical asymmetric channel through a porous medium.We introduce the governing equations and boundary conditions in Section 2. Section 3 represents the volume flow rate.The exact solution of the problem is derived in Section 4. Section 5 deals with numerical results and discussion.The trapping phenomenon is discussed in Section 6.The conclusions are summarized in Section 7.

Problem Formulation
Consider the peristaltic flow of an incompressible viscous fluid in a two-dimensional vertical infinite asymmetric channel of width d 1 d 2 through a porous medium.Asymmetry in the channel is produced by choosing the peristaltic wave trains propagating with constant speed c along the walls H 1 is the right hand side wall and H 2 is the left hand side wall to have different amplitudes and phases as shown in Figure 1.
The shapes of the channel walls are represented as where a 1 , b 1 are the amplitudes of the right and left waves, λ is the wave length, the phase difference φ varies in the range 0 ≤ φ ≤ π, φ 0 corresponds to symmetric channel with waves out of phase, and φ π describes the case where waves are in phase.Further, d 1 , d 2 , a 1 , b 1 , and φ satisfy the following inequality: so that the walls will not intersect with each other.The heat transfer in the channel is taken into account by maintaining the right and left walls at temperatures T 0 and T 1 , respectively.A uniform magnetic field with magnetic flux density vector B 0, 0, B 0 is applied, and the induced magnetic field is neglected by taking a very small magnetic Reynolds number.The fundamental equations governing this model together with the generalized Ohm's law taking the effects of Hall currents and Maxwell's equations into account are where V is the velocity vector, P is the pressure, μ is the dynamic viscosity, ∇ 2 is the Laplacian operator, ρ is the density of the fluid, d/dt is the material derivative, t is the time, J is the current density, B is the total magnetic field, σ is the electric conductivity, e is the electric charge, n e is the number density of electrons, ζ is the specific heat at constant pressure, α is the coefficient of linear thermal expansion of the fluid, κ is the thermal conductivity of the fluid, Q 0 is the constant heat addition/absorption, and T is the temperature of the fluid.The equations governing the two-dimensional motion of this model are where U and V are the velocity components in the laboratory frame X, Y and m σB 0 en e is the Hall parameter.

2.5
Following Shapiro et al. 3 we introduce a wave frame of reference x, y moving with velocity c in which the motion becomes independent of time when the channel is an integral multiple of the wavelength and the pressure difference at the ends of the channel is a constant.The transformation from the laboratory frame of reference X, Y to the wave frame of reference x, y is given by where u, v , p and U, V , P are the velocity components and pressure in the wave and laboratory frames of reference, respectively.We introduce the following nondimensional variables: where R is the Reynolds number, δ is the dimensionless wave number, k is the porosity parameter, Pr is the Prandtl number, Gr is the Grashof number, ν is the kinematic viscosity of the fluid, M is the magnetic parameter, and β is the nondimensional heat source/sink parameter.Using 2.6 and 2.7 in 2.4 , we can get the following nondimensional form in a wave frame of reference after dropping the stars: Assuming that the wave length is long and since the Reynolds number is low, then under this assumption 2.9 -2.11 will take the following forms after dropping terms of order δ and higher:

2.12
The corresponding boundary conditions are where and a, b, d, and φ satisfy the relation 2.17

Rate of Volume Flow
The volume flow rate in wave frame of reference is given by where h 1 , h 2 are functions of x alone.
The instantaneous volume flow rate in the fixed frame is given by The time-mean flow over time period T at a fixed position x is given by Using 3.2 and 3.3 we get On defining the dimensionless mean flow Θ in laboratory frame and F in the wave frame, Using 3.4 and 3.5 we obtain

Solution of the Problem
The exact solutions of the set of 2.12 subject to the boundary conditions 2.13 and 2.14 are where  where The stream function ψ can be obtained as follows: Using 4.2 and 4.12 we can easily get

4.20
The nondimensional expression for the pressure rise per wavelength is given by Δp λ The coefficient of heat transfer at the wall is given by Z h 2x θ y .

4.22
Using 2.16 and 4.1 we get The nondimensional shear stress can be calculated as 4.24 Substituting from 4.20 and 4.12 into 4.24 we get τ Γ A sinh Γy B cosh Γy 2S 1 y S 2 .

4.25
The friction forces F λi at the walls of the channel are given by

Results and Discussion
This section represents the graphical results in order to be able to discuss the quantitative effects of the sundry parameters involved in the analysis.The variation of velocity distribution with y for various values of m, k, Gr, and β is studied in Figure 2.  studies the effect of Hall parameter m on the velocity u and it is noticed that the velocity increases by increasing m.Also, it is noticed that increasing each of the porosity parameter k and Grashof number Gr led to an increase in the velocity as shown in Figures 2 b to 2 c , but the velocity decreases by increasing the width of the channel d as shown in Figure 2 d .Figure 3 illustrates the effects of phase angle φ, amplitudes of the right and left waves a, b, respectively, and nondimensional heat source/sink parameter β on velocity distribution.It reveals that the velocity increases by increasing each of phase angle φ, amplitude of the right wave a, and nondimensional heat source/sink parameter β.On the other hand the velocity decreases by increasing the amplitude of the left wave b.
In Figures 4 and 5 we considered variation of the pressure gradient with x for different values of m, a, b, and β Figure 4 and k, φ, d, and Gr Figure 5 .It is clear from Figure 4 that the pressure gradient decreases by increasing the Hall parameter m while it increases by increasing each of a, b, and β.Further it is observed from Figure 5 that the pressure gradient decreases by increasing each of k, φ, and d, while increasing the Grashof number Gr led to increasing the pressure gradient.Figures 6 and 7 show the stress distribution at the right and left walls, respectively.It is obvious from Figure 6 that the amplitude of the shear stress decreases by increasing each of m, k, β, and Gr. Figure 6 e indicates that the amplitude of the shear stress increases by increasing the phase angle φ and after a certain value of x the situation is reversed.Figure 6 f shows that increasing the width of the channel led to an increase in the shear stress at the right wall.Situation is in sharp contrast in Figure 7 we note that the shear stress at the left wall increases by increasing each of m, k, β, and Gr, also increasing φ led to a decrease in the shear stress, and after a certain value of x the situation is reversed.Figure 7 f reveals that the shear stress at the left wall decreases by increasing the width of the channel.Variation of the pressure rise per wavelength Δp λ with the mean flow Θ for different values m, k, Gr, and β is studied in Figure 8. From Figure 8 a it is observed that, when Δp λ > 5.12, the pumping rate decreases by increasing m, further, it coincides each other between 2.1 < Δp λ < 5.12, and it increases by increasing m when Δp λ < 2.1 the free pumping Δp λ 0 and copumping Δp λ < 0 occur in this region .The variation of heat transfer coefficient Z with x for different values b, a, d, and φ is studied in Figure 11.It is obvious from Figure 11 a that increasing the amplitude of the left wave b led to increasing the heat transfer coefficient and after a certain value of x the situation is reversed; also from Figure 11 b we notice that increasing the channel width decreases the heat transfer coefficient.The maximum value of heat transfer coefficient decreases by increasing φ as shown in Figure 11 c . Figure 11 d shows that increasing β produces an increase in heat transfer coefficient parameter.

Trapping Phenomenon
The formation of an internally circulating bolus of fluid is called trapping.The trapped bolus moves along with the wave.The effect of Hall parameter m on trapping can be observed through Figure 12 where the size of bolus increases by increasing m. Figure 13 depicts that increasing Θ led to an increase in trapped bolus size and also increases the number of trapped bolus.In Figures 14 and 15 we notice that the size of trapped bolus increases by increasing

Conclusions
In the present paper, the effects of Hall current and heat transfer on the peristaltic transport of a Newtonian fluid in an asymmetric channel through a porous medium under assumptions of a constant external magnetic field, low Reynolds number, and long wavelength are investigated.The governing equations are first modeled and then solved analytically.The effects of various emerging parameters on axial velocity, stream function, pressure gradient, friction forces, and shear stresses are observed from the graphs.The results are summarized as follows.ii The pressure gradient decreases with the increase of m, k, φ, and d while it increases by increasing Gr, a, b, and β.
iii Increasing each of m, k, β, and Gr led to decreasing the shear stress at the right wall but it increases by increasing d.
iv The shear stress at the left wall increases by increasing each of m, k, β, and Gr and decreases by increasing d.
v An increase in Gr and θ led to an increase in the pressure rise Δp λ .
vi The friction force at the right wall decreases by increasing Gr and β.
vii The friction force at the left wall increases by increasing Gr and β.
viii Increasing m and k increases the friction forces in some regions and decreases the friction forces in some other regions.
ix The heat transfer coefficient decreases by increasing d and φ and increases by increasing β.
23 had examined unsteady Hartmann flow with heat transfer of a viscoelastic fluid taking the Hall effect into account.Asghar et al. 24 studied the effects of Hall current and heat transfer on flow due to a pull of eccentric rotating disk.Hayat et al. 25 studied the Hall effects on peristaltic flow of a Maxwell fluid in a porous medium.Abo-Eldahab et al. 26, 27 investigated the effects of Hall and ion-slip currents on magnetohydrodynamic peristaltic transport and couple stress fluid.For the benefit of the readers in this direction you can take these additional papers 28-35 into account.This paper may be considered as an extension of the paper 22 by Srinvas and Gayathri, if we neglected the effect of Hall current.

Figure 1 :
Figure 1: The geometry of the problem.