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

Authors: E. Abo-Eldahab; E. Barakat; Kh. Nowar
Journal: Mathematical Problems in Engineering
 (2012)

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## Abstract

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.


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## Body

## 1. 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 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 [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.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 Section2. 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.


## 2. Problem Formulation

Consider the peristaltic flow of an incompressible viscous fluid in a two-dimensional vertical infinite asymmetric channel of widthd1+d2 through a porous medium. Asymmetry in the channel is produced by choosing the peristaltic wave trains propagating with constant speed c along the walls (H1 is the right hand side wall and H2 is the left hand side wall) to have different amplitudes and phases as shown in Figure 1.Figure 1
The geometry of the problem.The shapes of the channel walls are represented as(2.1)Y=H1(X,t)=d1+a1cos⁡(2πλ[X-ct]),rightwall,Y=H2(X,t)=-d2-b1cos⁡(2πλ[X-ct]+ϕ),leftwall,
where a1, b1 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, d1, d2, a1, b1, and ϕ satisfy the following inequality:(2.2)a12+b12+2a1b1cos⁡ϕ≤(d1+d2)2,
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 T0 and T1, respectively. A uniform magnetic field with magnetic flux density vector B=(0,0,B0) 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(2.3)∇.V=0,-∇P+μ∇2V+ρgα(T-T0)-μk1V+J∧B=ρdVdt,J=σ[V∧B-1eneJ∧B],ρζdTdt=κ∇2T+Q0,
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, ne 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, Q0 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
(2.4)∂U∂X+∂V∂Y=0,ρ(∂U∂t+U∂U∂X+V∂U∂Y)=-∂P∂X+μ(∂2U∂X2+∂2U∂Y2)-μk1U+ρgα(T-T0)+σB021+m2(mV-U),ρ(∂V∂t+U∂V∂X+V∂V∂Y)=-∂P∂Y+μ(∂2V∂X2+∂2V∂Y2)-μk1V-σB021+m2(mU+V),ρζ(∂T∂t+U∂T∂X+V∂T∂Y)=κ(∂2T∂X2+∂2T∂Y2)+Q0,
where U and V are the velocity components in the laboratory frame (X,Y) and(2.5)m=σB0eneistheHallparameter.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(2.6)x=X-ct,y=Y,u=U-c,v=V,p(x)=P(x,t),
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:(2.7)x*=xλ,y*=yd1,u*=uc,v*=vcδ,δ=d1λ,p*=pd12cλμ,t*=ctλ,h1=H1d1,h2=H2d1,d=d2d1,a=a1d1,b=b1d1,R=ρd1cμ,k=k1d12,θ=T-T0T1-T0,Pr=μζκ,β=Q0d12κ(T1-T0),Gr=αg(T1-T0)d13ν2,M=σB02d12μ,
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:(2.8)∂u∂x+∂v∂y=0,(2.9)δR(u∂u∂x+v∂u∂y)=-∂p∂x+δ2∂2u∂x2+∂2u∂y2-(u+1)k+GrθR+δmMv(m2+1)-M(u+1)m2+1,(2.10)δ3R(u∂v∂x+v∂v∂y)=-∂p∂y+δ4∂2v∂x2+δ2∂2v∂y2-δ2vk-δmM(u+1)(m2+1)-δ2Mv(m2+1),(2.11)δRPr(u∂θ∂x+v∂θ∂y)=δ2∂2θ∂x2+∂2θ∂y2+β.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)0=-∂p∂x+∂2u∂y2-(u+1)k+Grθ-M(u+1)m2+1,0=-∂p∂y,0=∂2θ∂y2+β.
The corresponding boundary conditions are(2.13)u=-1,θ=0aty=h1,(2.14)u=-1,θ=1aty=h2,
where(2.15)h1=1+acos⁡2πx,(2.16)h2=-d-bcos⁡(2πx+ϕ),
and a, b, d, and ϕ satisfy the relation(2.17)a2+b2+2abcos⁡ϕ≤(1+d)2.


## 3. Rate of Volume Flow

The volume flow rate in wave frame of reference is given by(3.1)q=∫h2(x)h1(x)u(x,y)dy,
where h1, h2 are functions of x alone.The instantaneous volume flow rate in the fixed frame is given by(3.2)Q=∫H2(x,t)H1(x,t)[U(x,y,t)]dy=∫h2(x)h1(x)[u(x,y)+c]dy=q+ch1-ch2.
The time-mean flow over time period T at a fixed position x is given by(3.3)Q¯(x,t)=1T∫0TQ(x,y)dt.
Using (3.2) and (3.3) we get(3.4)Q¯(x,t)=q+c1d1+c2d2.
On defining the dimensionless mean flow Θ in laboratory frame and F in the wave frame,(3.5)Θ=Q¯cd1,F=qcd1.
Using (3.4) and (3.5) we obtain(3.6)Θ=F+1+d,
in which(3.7)F=∫h2(x)h1(x)u(x,y)dy.


## 4. Solution of the Problem

The exact solutions of the set of (2.12) subject to the boundary conditions (2.13) and (2.14) are(4.1)θ=12(h2-h1){-β(h2-h1)y2+[2+β(h22-h12)]y+βh1h2(h1-h2)-2h1},(4.2)u=Acosh⁡Γy+Bsinh⁡Γy+S1y2+S2y+S3-1Γ2dpdx,
where(4.3)Γ=(1+m2)+kMk(1+m2),S1=-βGr2Γ2,S2=GrΓ2(2+β(h22-h12)2(h2-h1)),(4.4)S3=-1Γ2{Gr[βh1h2(h2-h1)+2h1]2(h2-h1)+βGrΓ2+Γ2},(4.5)S4=cosh⁡Γh1sinh⁡Γh2-cosh⁡Γh2sinh⁡Γh1,(4.6)S5=S1(h22sinh⁡Γh1-h12sinh⁡Γh2),(4.7)S6=S2(h2sinh⁡Γh1-h1sinh⁡Γh2),(4.8)S7=sinh⁡Γh1-sinh⁡Γh2,(4.9)S8=S1(h12cosh⁡Γh2-h22cosh⁡Γh1),(4.10)S9=S2(h1cosh⁡Γh2-h2cosh⁡Γh1),(4.11)S10=cosh⁡Γh2-cosh⁡Γh1,(4.12)A=A1+A2dpdx,B=B1+B2dpdx,(4.13)A1=S5+S6+S7+S3S7S4,A2=-S7S4Γ2,(4.14)B1=S8+S9+S10+S3S10S4,B2=-S10S4Γ2,
From (3.7), (4.2), and (4.12) we have(4.15)F=1Γ(A1S7-B1S10)+S11+S12dpdx,
and therefore we can say that(4.16)dpdx=F-(1/Γ)(A1S7-B1S10)-S11S12,
where(4.17)S11=(h1-h2){S13(h12+h1h2+h22)+S22(h1+h2)+S3},(4.18)S12=A2S7Γ-B2S10Γ-(h1-h2)Γ2.
The stream function ψ can be obtained as follows:(4.19)ψ=∫udy.
Using (4.2) and (4.12) we can easily get(4.20)ψ=1Γ(A1sinh⁡Γy+B1cosh⁡Γy)+13S1y3+12S2y2+S3y+[-1Γ2y+A2Γsinh⁡Γy+B2Γcosh⁡Γy]dpdx.
The nondimensional expression for the pressure rise per wavelength is given by(4.21)Δpλ=∫01∂p∂xdx.The coefficient of heat transfer at the wall is given by(4.22)Z=h2xθy.Using (2.16) and (4.1) we get(4.23)Z=2πbsin⁡(2πx+ϕ)[-βy+2+β(h22-h12)2(h2-h1)].The nondimensional shear stress can be calculated as(4.24)τ=τ*(μc/d1)=∂2ψ∂y2.Substituting from (4.20) and (4.12) into (4.24) we get(4.25)τ=Γ(Asinh⁡Γy+Bcosh⁡Γy)+2S1y+S2.The friction forcesFλi at the walls of the channel are given by(4.26)Fλi=∫01hi(-∂p∂x)dx,i=1,2.


## 5. 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 withy for various values of m, k, Gr, and β is studied in Figure 2. Figure 2(a) 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).The variation of the velocity distributionu with y for various values of Hall parameter m (a), porosity parameter k (b), Grashof number Gr (c), and width of the channel d (d). The other parameters chosen are: Θ=4, x=0, d=1, M=1.2, a=0.4, b=0.2, β=2, k=1, ϕ=π/4, Gr=10 (a); Θ=4, x=0, d=1, M=1.2, a=0.4, b=0.2, β=2, m=0.5, ϕ=π/4, Gr=10 (b); Θ=4, x=0, d=1, M=1.2, a=0.4, b=0.2, β=2, m=0.5, ϕ=π/4, k=1 (c); Θ=4, x=0, β=2, M=1.2, a=0.4, b=0.2, Gr=10, m=0.5, ϕ=π/4, k=1 (d).
(a)(b)(c)(d)Figure3 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.Variation of velocity distributionu with y for various values of phase angle ϕ (a), amplitude of the right wave a (b), amplitude of the left wave b (c), and nondimensional heat source/sink parameter β (d). The other parameters chosen are Θ=4, x=0, d=1, M=1.2, a=0.4, b=0.2, β=2,k=1, m=0.5, Gr=10 (a); Θ=4, x=0, d=1, M=1.2, k=1, b=0.2, β=2,m=0.5, ϕ=π/4, Gr=310 (b); Θ=4, x=0, d=1, M=1.2, a=0.4, Gr=10, β=2, m=0.5, ϕ=π/4, k=1 (c); Θ=4, x=0, d=1, M=1.2, a=0.4, b=0.2, Gr=10, m=0.5, ϕ=π/4, k=1 (d).
(a)(b)(c)(d)In Figures4 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 Δpλ 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 Δpλ>5.12, the pumping rate decreases by increasing m, further, it  coincides each other between 2.1<Δpλ<5.12, and it increases by increasing m when Δpλ<2.1 (the free pumping Δpλ=0 and copumping Δpλ<0 occur in this region). Figure 8(b) shows that when Δpλ>4.61 the pumping rate decreases with an increase in k, when 1.97<Δpλ<4.61 the curves coincide with each other, and when Δpλ<1.97 the pumping increases by increasing k. The pumping rate increases by increasing each of Gr and β as shown in Figures 8(c) and 8(d).Distribution of pressure gradientdp/dx with x for different values of Hall parameter m (a), amplitudes of right wave a (b), left wave b (c), and the nondimensional heat source/sink parameter β (d). The other parameters chosen are F=-5, y=1, d=1, M=1.2, a=0.4, b=0.2, β=2, k=1, ϕ=π/4, Gr=3 (a); F=-5, y=1, d=1, M=1.2, m=0.5, b=0.2, β=2, k=1, ϕ=π/4, Gr=3 (b); F=-5, y=1, d=1, M=1.2, m=0.5, a=0.4, β=2, k=1, ϕ=π/4, Gr=3 (c); F=-5, y=1, d=1, M=1.2, m=0.5, a=0.4, b=0.2, k=1, ϕ=π/4, Gr=3 (d).
(a)(b)(c)(d)The distribution of the pressure gradientdp/dx with x for different values of porosity parameter k (a), phase angle ϕ (b), width of the channel d (c), and Grashof number Gr (d). The other parameters chosen are F=-5, y=1, d=1, M=1.2, a=0.4, b=0.2, β=2, m=0.5, ϕ=π/4, Gr=3 (a); F=-5, y=1, d=1, M=1.2, a=0.4, b=0.2, β=2, m=0.5, k=1, Gr=3 (b); F=-5, y=1, k=1, M=1.2, a=0.4, b=0.2, β=2, m=0.5, ϕ=π/4, Gr=3 (c); F=-5, y=1, d=1, M=1.2, a=0.4, b=0.2, β=2, m=0.5, ϕ=π/4, k=1 (d).
(a)(b)(c)(d)The shear stress distribution at the right wall withx for different values of Hall parameter m (a), porosity parameter k (b), nondimensional heat source/sink parameter β (c), Grashof number Gr (d), phase angle ϕ (e), and width of the channel d (f). The other parameters chosen are Θ=3, d=0.5, M=1.2, a=0.4, b=0.2, β=2, k=1, ϕ=π/4, Gr=3 (a); Θ=3, d=0.5, M=1.2, a=0.4, b=0.2, β=2, m=0.5, ϕ=π/4, Gr=3 (b); Θ=3, d=0.5, M=1.2, a=0.4, b=0.2, m=0.5, k=1, ϕ=π/4, Gr= (c); Θ=3, d=0.5, M=1.2, a=0.4, b=0.2, β=2, k=1, ϕ=π/4, m=0.5 (d); Θ=3, d=0.5, M=1.2, a=0.4, b=0.2, β=2, k=1, m=0.5, Gr=3 (e); Θ=3, m=0.5, M=1.2, a=0.4, b=0.2, β=2, k=1, ϕ=π/4, Gr=3 (f).
(a)(b)(c)(d)(e)(f)The shear stress distribution at the left wall withx for different values of Hall parameter m (a), porosity parameter k (b), nondimensional heat source/sink parameter β (c), Grashof number Gr (d), phase angle ϕ (e), and width of the channel d (f). The other parameters chosen are Θ=3, d=0.5, M=1.2, a=0.4, b=0.2, β=2, k=1, ϕ=π/4, Gr=3 (a); Θ=3, d=0.5, M=1.2, a=0.4, b=0.2, β=2, m=0.5, ϕ=π/4, Gr=3 (b); Θ=3, d=0.5, M=1.2, a=0.4, b=0.2, m=0.5, k=1, ϕ=π/4, Gr=3 (c); Θ=3, d=0.5, M=1.2, a=0.4, b=0.2, β=2, k=1,ϕ=π/4, m=0.5 (d); Θ=3, d=0.5, M=1.2, a=0.4, b=0.2, β=2, k=1, m=0.5, Gr=3 (e); Θ=3, m=0.5, M=1.2, a=0.4, b=0.2, β=2, k=1, ϕ=π/4, Gr=3 (f).
(a)(b)(c)(d)(e)(f)The variation of pressure riseΔpλ with Θ for different values of Hall parameter m (a), porosity parameter k (b), Grashof number Gr (c), and nondimensional heat source/sink parameter β (d). The other parameters chosen are d=1, M=1.2, a=0.1, b=0.2, ϕ=π/2, k=1, β=2, Gr=3 (a); d=1, M=1.2, a=0.1, b=0.2, ϕ=π/2, m=0.5, β=2, Gr=3 (b); d=1, M=1.2, a=0.1, b=0.2, ϕ=π/2, k=1, m=0.5, Gr=3 (c); d=1, M=1.2, a=0.1, b=0.2, ϕ=π/2, k=1, β=2, m=0.5 (d).
(a)(b)(c)(d)Figures9 and 10 study the variation of friction forces Fλ1 and Fλ2 with Θ for different values of m, k, Gr, and β. In Figure 9(a) we notice that when Fλ1>-1.66 the pumping rate decreases by increasing m, also when -5.74<Fλ1<-1.66 the curves coincide with each other, and when Fλ1<-5.74 the pumping rate increases by increasing m. Nearly the same situation occurs in Figure 9(b); when Fλ1>-1.85 the pumping rate decreases by increasing k, when -4.29<Fλ1<-1.85 the curves coincide with each other; and when Fλ1<-4.29 the pumping rate increases by increasing k. Figures 9(c) and 9(d) shows that pumping rate decreases by increasing each of Gr and β for all values of Fλ1. Figure 10(a) shows that when Fλ2>5.25 the pumping rate decreases by increasing m and curves coincide with each other for 1.97<Fλ2<5.25 and when Fλ2<-1.97 the pumping rate increases by increasing m. From Figure 10(b) we find that when Fλ2>4.73 the pumping rate decreases with an increase in k while it coincides each other between 1.88<Fλ2<4.73, also the pumping rate increases by increasing k when Fλ2<1.88. The pumping rate increases by increasing each of Gr and β as shown in Figures 10(c) and 10(d).The variation of friction force at the right wallFλ1 with Θ for different values of Hall parameter m (a), porosity parameter k (b), Grashof number Gr (c), and nondimensional heat source/sink parameter β (d). The other parameters chosen are d=1, M=1.2, a=0.1, b=0.2, ϕ=π/2, k=1, β=2, Gr=3 (a); d=1, M=1.2, a=0.1, b=0.2, ϕ=π/2, m=0.5, β=2, Gr=3 (b); d=1,M=1.2, a=0.1, b=0.2, ϕ=π/2, k=1, m=0.5, Gr=3 (c); d=1, M=1.2, a=0.1, b=0.2, ϕ=π/2, k=1, β=2, m=0.5 (d).
(a)(b)(c)(d)The variation of friction force at the left wallFλ2 with Θ for different values of Hall parameter m (a), porosity parameter k (b), Grashof number Gr (c), and nondimensional heat source/sink parameter β (d). The other parameters chosen are d=1, M=1.2, a=0.1, b=0.2, ϕ=π/2, k=1, β=2, Gr=3 (a); d=1, M=1.2, a=0.1, b=0.2, ϕ=π/2, m=0.5, β=2, Gr=3 (b); d=1, M=1.2, a=0.1, b=0.2, ϕ=π/2, k=1, m=0.5, Gr=3 (c); d=1, M=1.2, a=0.1, b=0.2, ϕ=π/2, k=1, β=2, m=0.5 (d).
(a)(b)(c)(d)The variation of heat transfer coefficientZ 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.The variation of heat transfer coefficientZ with x for different values of amplitude of the left wave b (a), channel width d (b), phase angle ϕ (c), and nondimensional heat source/sink parameter β (d). The other parameters chosen are d=1, a=0.4, β=-4, ϕ=π/4 (a); a=0.4, b=0.3, β=-4, ϕ=π/4 (b); b=0.3, a=0.4, β=-4, d=1 (c); d=1, a=0.4, ϕ=π/4, b=0.2 (d).
(a)(b)(c)(d)


## 6. 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 parameterm 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 each of Gr and k. On the other hand the size of trapped bolus decreases by increasing β as shown in Figure 16. The effects of a, b, and d on trapping are studied in Figures 17, 18, and 19, respectively. It is clear that the number of trapped bolus increases by increasing a and decreases by increasing b. Furthermore, increasing the channel width led to an increase in size of trapped bolus. Figure 20 studies the case of symmetric channel (a=b,d=1,andϕ=0). It is observed that the size of trapped bolus increases by increasing each of m, k, and Θ and decreases by increasing each of Gr and β.Streamlines for three different values ofm: (a) m=1, (b) m=2, (c) m=3. The other parameters chosen are Θ=4, d=1, M=1.2, a=0.4, b=0.2, β=-2, k=0.5, ϕ=π/4,  Gr=2.
(a)(b)(c)Streamlines for three different values ofΘ: (a) Θ=2, (b) Θ=2.5, (c) Θ=5. The other parameters chosen are m=1.5, d=1, M=1.2, a=0.4, b=0.2, β=-2, k=0.5, ϕ=π/4, Gr=2.
(a)(b)(c)Streamlines for three different values of Gr: (a)Gr=0, (b) Gr=2, (c) Gr=5. The other parameters chosen are Θ=4, d=1, M=1.2, a=0.4, b=0.2, β=-2, k=0.5, ϕ=π/4, m=1.5.
(a)(b)(c)Streamlines for three different values ofk: (a) k=0.2, (b) k=0.5, (c) k=0.8. The other parameters chosen are Θ=4, d=1, M=1.2, a=0.4, b=0.2, β=-2, m=1.5, ϕ=π/4, Gr=2.
(a)(b)(c)Streamlines for three different values ofβ: (a) β=-4, (b) β=0, (c) β=4. The other parameters chosen are Θ=4, d=1, M=1.2, a=0.4, b=0.2, m=1.5, k=0.5, ϕ=π/4, Gr=2.
(a)(b)(c)Streamlines for three different values ofa: (a) a=0.2, (b) a=0.4, (c) a=0.6. The other parameters chosen are Θ=4, d=1, M=1.2, m=1.5, b=0.2, β=-2, k=0.5, ϕ=π/4, Gr=2.
(a)(b)(c)Streamlines for three different values ofb: (a) b=0.1, (b) b=0.3, (c) b=0.5. The other parameters chosen are Θ=4,d=1,M=1.2,a=0.4,m=1.5,β=-2,k=0.5,ϕ=π/4,Gr=2.
(a)(b)(c)Streamlines for three different values ofd: (a) d=1, (b) d=1.2, (c) d=1.6. The other parameters chosen are Θ=4, m=1.5, M=1.2, a=0.4, b=0.2, β=-2, k=0.5, ϕ=π/4, Gr=2.
(a)(b)(c)Streamlines for symmetric channel (a=b=0.4,d=1,ϕ=0). The other parameters chosen are (a) Θ=4, β=-2, m=0.5, k=0.5, M=1.5, Gr=2; (b) Θ=4, β=-2,m=0.5, k=0.5, M=1.5, Gr=5; (c) Θ=4, β=-2, m=1.5, k=0.5, M=1.5, Gr=5; (d) Θ=4, β=-2, m=1.5, k=0.8, M=1.5, Gr=5; (e) Θ=4, β=2, m=1.5, k=0.8, M=1.5, Gr=5; (f) Θ=5, β=2, m=1.5, k=0.8, M=1.5, Gr=5.
(a)(b)(c)(d)(e)(f)


## 7. 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.(i)
The velocity increases by increasing each ofm, k, Gr, ϕ, a, and β and decreases by increasing d and b.(ii)
The pressure gradient decreases with the increase ofm, k, ϕ, and d while it increases by increasing Gr, a, b, and β.(iii)
Increasing each ofm, 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 ofm, k, β, and Gr and decreases by increasing d.(v)
An increase inGr and θ led to an increase in the pressure rise Δpλ.(vi)
The friction force at the right wall decreases by increasingGr and β.(vii)
The friction force at the left wall increases by increasingGr and β.(viii)
Increasingm 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 increasingd and ϕ and increases by increasing β.(x)
The size of trapped bolus increases by increasingm, Θ, Gr, k, a, and d and decreases by increasing β and b.(xi)
In the case of symmetric channel the size of trapped bolus increases by increasing each ofm, k, and Θ and decreases by increasing each of Gr and β.



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*Source: 840203-2012-05-21.xml*