# Peristaltic Flow of a Magneto-Micropolar Fluid: Effect of Induced Magnetic Field

Authors: Kh. S. Mekheimer
Journal: Journal of Applied Mathematics
 (2008)

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## Abstract

We carry out the effect of the induced magnetic field on peristaltic transport of an incompressible conducting micropolar fluid in a symmetric channel. The flow analysis has been developed for low Reynolds number and long wavelength approximation. Exact solutions have been established for the axial velocity, microrotation component, stream function, magnetic-force function, axial-induced magnetic field, and current distribution across the channel. Expressions for the shear stresses are also obtained. The effects of pertinent parameters on the pressure rise per wavelength are investigated by means of numerical integrations, also we study the effect of these parameters on the axial pressure gradient, axial-induced magnetic field, as well as current distribution across the channel
and the nonsymmetric shear stresses. The phenomena of trapping and magnetic-force lines are further discussed.


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## Body

## 1. Introduction

It is well known that many physiological fluids behave
in general like suspensions of deformable or rigid particles in a Newtonian
fluid. Blood, for example, is a suspension of red cells, white cells, and
platelets in plasma. Another example is cervical mucus, which is a suspension
of macromolecules in a  water-like liquid. In view
of this, some researchers have tried to account for the suspension behavior of
biofluids by considering them to be non-Newtonian [1–6].Eringen [7] introduced the concept of simple
microfluids to characterise concentrated suspensions of neutrally buoyant
deformable particles in a viscous fluid where the individuality of
substructures affects the physical outcome of the flow. Such fluid models can be
used to rheologically describe polymeric suspensions, normal human blood, and
so forth, and have found applications in physiological and engineering problems
[8–10]. A subclass of these microfluids is known as micropolar fluids where the
fluid microelements are considered to be rigid [11, 12]. Basically, these fluids
can support couple stresses and body couples and exhibit microrotational and
microinertial effects.The phenomenon of peristalsis is defined as expansion
and contraction of an extensible tube in a fluid generate progressive waves
which propagate along the length of the tube, mixing and transporting the fluid
in the direction of wave propagation. It is an inherent property of many
tubular organs of the human body. In some biomedical instruments, such as
heart-lung  machines, peristaltic motion is used to
pump blood and other biological fluids. It plays an indispensable role in
transporting many physiological fluids in the body in various situations such
as urine transport from the kidney to the bladder through the ureter, transport
of spermatozoa in the ductus efferentes of the male reproductive tract,
movement of ovum in the fallopian tubes, vasomotion of small blood vessels,
as well as  mixing and transporting the contents of
the gastrointestinal passage.Peristaltic pumping mechanisms have been utilized for
the transport of slurries, sensitive or corrosive fluids, sanitary fluid,
noxious fluids in the nuclear industry, and many others. In some cases, the
transport of fluids is possible without moving internal mechanical components
as  in  the case with peristaltically operated
microelectromechanical system devices [13].The study of peristalsis in the context of fluid
mechanics has received considerable attention in the last three decades, mainly
because of its relevance to biological systems and industrial applications.
Several studies have been made, especially for the peristalsis in non-Newtonian
fluids which have promising applications in physiology [14–23]. The main
advantage of using a micropolar fluid model to study the peristaltic flow of
suspensions in comparison with other classes of non-Newtonian fluids is that it
takes care of the rotation of fluid particles by means of an independent
kinematic vector called the microrotation vector.Magnetohydrodynamic (MHD) is the science which deals
with the motion of a highly conducting fluids in the presence of a magnetic
field. The motion of the conducting fluid across the magnetic field generates
electric currents which change the magnetic field, and the action of the
magnetic field on these currents gives rise to mechanical forces which modify
the flow of the fluid [24]. MHD flow of a fluid in a channel with elastic,
rhythmically contracting walls (peristaltic flow) is of interest in connection
with certain problems of the movement of conductive physiological fluids (e.g.,
the blood and blood pump machines) and with the need for theoretical research
on the operation of a peristaltic MHD compressor. Effect of a moving magnetic
field on blood flow was studied by Stud et al. [25],
Srivastava and  Agrawal
[26] considered the blood as an electrically conducting fluid and
it  constitutes a suspension of red
cells  in plasma. Also Agrawal and Anwaruddin [27]
studied the effect of magnetic field on blood flow by taking a simple
mathematical model for blood through an equally branched channel with flexible
walls executing peristaltic waves using long wavelength approximation method.Some recent studies [28–41] have considered the effect
of a magnetic field on peristaltic flow of a Newtonian and non-Newtonian
fluids, and in all of these studies the effect of the induced magnetic field
have been neglected.The first investigation of the effect of the induced
magnetic field on peristaltic flow was studied by Vishnyakov and Pavlov [42]
where they considers the peristaltic MHD flow of a conductive Newtonian fluid;
they used the asymptotic narrow-band method to solve the problem and only
obtained the velocity profiles in certain channel cross-sections for definite
parameter values. Currently, there is only two attempts [43, 44] for a study of
the effect of induced magnetic field, one for a couple-stress fluid and the
other for a non-Newtonian fluid (biviscosity fluid). To the best of our
knowledge, the influence of a magnetic field on
peristaltic flow of a conductive micropolar fluid has not been investigate with
or without the induced magnetic field.With  keeping  the above
discussion in mind, the goal of this investigation is to study the effect of
the induced magnetic field on peristaltic flow of a micropolar fluid (as a
blood model). The flow analysis is developed in a wave frame of reference
moving with the velocity of the wave. The problem is first modeled and then
solved analytically for the stream function, magnetic-force function, and the
axial pressure gradient. The results for the pressure rise , shear stresses,
the axial induced magnetic field, and the distribution of the current density
across the channel have been discussed for various values of the problem
parameters. Also, the contour plots for the magnetic force and stream functions
are presented, the pumping characteristics and the trapping phenomena are discussed
in detail. Finally, The main conclusions are summarized in the last section.


## 2. Mathematical Modelling

Consider the unsteady hydromagnetic flow of a viscous,
incompressible, and electrically conducting micropolar fluid through an
axisymmetric two-dimensional channel of uniform thickness with a sinusoidal
wave traveling down its wall. We choose a rectangular coordinate system for the
channel withX′ along the centerline in the direction of wave
propagation and Y′ transverse to it. The system is stressed by an
external transverse uniform constant magnetic field of strength H0′,
which will give rise to an induced magnetic field H′(hX′′(X′,Y′,t′),hY′′(X′,Y′,t′),0) and the total magnetic field will be H′+(hX′′(X′,Y′,t′),H0′+hY′′(X′,Y′,t′),0).
The plates of the channel are assumed to be nonconductive, and the geometry of
the wall surface is defined as(2.1)h′(X′,t′)=a+bcos⁡2πλ(X′−ct′),where a0 is the half-width at the inlet, b is the wave amplitude, λ is the wavelength, c is the propagation
velocity, and t′ is the time.Neglecting the body couples, the equations of motion
for unsteady flow of an incompressible micropolar fluid are(2.2)∇→⋅V→′=0,ρ{∂V→′∂t′+V→′⋅∇→V→′}=−∇→p′+k∇→×w→′+(μ+k)∇→2V→′+ρf→′,ρj{∂V→′∂t′+V→′⋅∇→w→′}=−2kw→′+k∇→×V→′−γ(∇→×∇→×w→′)+(α+β+γ)∇(∇→⋅w→′),where V→′ is the velocity vector, w→′ is the microrotation vector, p′ is the fluid pressure, f→′ is the body force, and ρ and j are the fluid density and microgyration
parameter. Further, the material constants (new viscosities of the micropolar
fluid) μ,k,α,β,
and γ satisfy the following inequalities (obtained
by Eringen  [11]):(2.3)2μ+k≥0,k≥0,3α+β+γ≥0,γ≥|β|.The governing equations for a magneto-micropolar fluid
are as follows:Maxwell's equations(2.4)∇⋅H→′=0,∇⋅E→′=0,(2.5)∇∧H→′=J→′,withJ→′=σ{E→′+μe(V→′∧H→′+)},(2.6)∇∧E→′=−μe∂H→′∂t′,the continuity equation(2.7)∇⋅V→′=0,the equations of motion(2.8)ρ{∂V→′∂t′+(V→′⋅∇→)V→′}=−∇(p′+12μe(H′+)2)+(μ+k)∇2V→′+k∇→×w→′−μe(H→′+⋅∇→)H→′+,ρj{∂V→′∂t′+(V→′⋅∇→)w→′}=−2kw→′+k∇→×V→′−γ(∇→×∇→×w→′)+(α+β+γ)∇→(∇→⋅w→′),∇2=∂2∂X′2+∂2∂Y′2, where E→′ is an induced electric field, J→′ is the electric current density, μe is the magnetic permeability, and σ is the electrical conductivity.Combining (2.4)
and (2.5)–(2.7), we obtain
the induction equation:(2.9)∂H→′+∂t′=∇→∧{V→′∧H→′+}+1ζ∇2H→′+,where ζ=1/σμe is the magnetic diffusivity.We should carry out this investigation in a coordinate
system moving with the wave speedc, in which the boundary shape is stationary.
The coordinates and velocities in the laboratory frame (X′,Y′) and the wave frame (x′,y′) are related by(2.10)x′=X′−ct′,y′=Y′,u′=U′−c,v′=V′,where U′,V′,
and u′,v′ are the  velocity
components in the corresponding coordinate systems.Using these transformations and introducing the
dimensionless variables(2.11)x=x′λ,y=y′a,u=u′c,v=λv′ac,h=h′(x′)a,p=a2λμcp′(x′),t=ct′λ,j=j′a2,ψ=ψ′ca,ϕ=ϕ′H0a,we find that the equations which
govern the MHD flow for a micropolar fluid in terms of the stream function ψ(x,y) and magnetic-force function ϕ(x,y) are(2.12)Reδ{(ψy∂∂x−ψx∂∂y)ψy}=−∂pm∂x+11−N∇2ψy+N1−N∂w∂y+ReS2ϕyy+ReS2δ(ϕy∂∂x−ϕx∂∂y)ϕy,(2.13)Reδ3{(ψx∂∂y−ψy∂∂x)ψx}=−∂pm∂y−δ21−N∇2ψx−δ2N1−N∂w∂x−ReS2δ2ϕxy−ReS2δ3(ϕy∂∂x−ϕx∂∂y)ϕx,(2.14)Reδj(1−NN){(ψy∂∂x−ψx∂∂y)w}=−2w−∇2ψ+(2−Nm2)∇2w,(2.15)ψy−δ(ψyϕx−ψxϕy)+1Rm∇2ϕ=E,
where(2.16)u=∂ψ∂y,v=−δ∂ψ∂x,hx=∂ϕ∂y,hy=−δ∂ϕ∂x,∇2=δ2∂2∂x2+∂2∂y2,and the dimensionless parameters
as follows:(i)
Reynolds
numberRe=caρ/μ,(ii)
wave
numberδ=a/λ,(iii)
Strommer's number (magnetic-force number)S=(H0/c)μe/ρ,(iv)
the
magnetic Reynolds numberRm=σμeac,(v)
the
coupling numberN=k/(k+μ)(0≤N≤1),m2=a2k(2μ+k)/(γ(μ+k))  is the micropolar parameter,(vi)
the total
pressure in the fluid, which equals the sum of the ordinary and magnetic
pressure, ispm=p+(1/2)Reδ(μe(H+)2/ρc2),
and E(=−E/cH0) is the electric field strength. The parameters α,β do not appear in the governing equations as
the microrotation vector is solenoidal. However,
(2.12)–(2.15) reduce to the
classical MHD Navier-Stokes equations as k→0.Excluding the total pressure from (2.12)
and (2.13), we obtain(2.17)Reδ{(ψy∂∂x−ψx∂∂y)∇2ψ}=11−N∇4ψ+N1−N∇2w+ReS2∇2ϕy+ReS2δ(ϕy∂∂x−ϕx∂∂y)∇2ϕ,Reδj(1−NN){(ψy∂∂x−ψx∂∂y)w}=−2w−∇2ψ+(2−Nm2)∇2w.The instantaneous volume flow rate in the fixed frame
is given by(2.18)Q=∫0h′U′(X′,Y′,t)dY′,where h′ is a function of X′ and t.The rate of volume flow in the wave frame is given
by(2.19)q=∫0h′u′(x′,y′)dy′,where h′ is a function of x′ alone. If we
substitute (2.10)
into (2.18) and make use of (2.19), we find that the two rates of volume flow
are related through(2.20)Q=q+ch′.The time mean flow over a period
T at a fixed position X′ is defined as:(2.21)Q¯=1T∫0TQ⁢dt.Substituting (2.20) into
(2.21),
and integrating, we get(2.22)Q¯=q+ac.On defining the dimensionless time-mean flowsθ and F,
respectively, in the fixed and wave frame as(2.23)θ=Q¯ac,F=qac,one finds that (2.22) may be
written as(2.24)θ=F+1,where(2.25)F=∫0h∂ψ∂ydy=ψ(h)−ψ(0).We note thath represents the dimensionless form of the
surface of the peristaltic wall:(2.26)h(x)=1+α⁢cos⁡(2πx),where(2.27)α=bais the amplitude ratio or the
occlusion.If we select
the zero value of the streamline at the streamline(y=0)(2.28)ψ(0)=0,then the wall (y=h) is a streamline of value(2.29)ψ(h)=F.For a non-conductive elastic channel wall, the boundary
conditions for the dimensionless stream functionψ(x,y) and magnetic-force function ϕ(x,y) in the wave frame are
[42, 44](2.30)ψ=0,∂2ψ∂y2=0,w=0,∂ϕ∂y=0ony=0,∂ψ∂y=−1,ψ=F,w=0,ϕ=0,∂ϕ∂y=0ony=h(x).Under the long wavelength and low Reynolds number
consideration [4, 5, 6, 34–36], the dimensionless equations of the problem are
expressed in the following form:(2.31)∂4ψ∂y4+N∂2w∂y2+ReS2(1−N)∂3ϕ∂y3=0,2−Nm2∂2w∂y2=2w+∂2ψ∂y2,(2.32)∂2ϕ∂y2=Rm(E−∂ψ∂y).Combining these equations
gives(2.33)ψ=1H2(1−N){2−Nm2(∂2w∂y2−m2w)+ηy−C1y−C2},∂4w∂y4−{m2+H2(1−N)}∂2w∂y2+2m2H2(1−N)2−Nw=0, where H2=ReS2Rm, H=(μeH0)aσ/μ is the Hartmann number (suitably greater than 2), η=EH2(1−N),
and C1,C2 are an integration constants.


## 3. Exact Solution

The general  solutions  of
the microrotation componentw and the stream function ψ are(3.1)w=A⁢cosh⁡(θ1y)+B⁢sinh⁡(θ1y)+C⁢cosh(θ2y)+D⁢sinh(θ2y),ψ=1H2(1−N){2−Nm2[(θ12−m2)(A⁢cosh(θ1y)+B⁢sinh(θ1y))+(θ22−m2)(C⁢cosh(θ2y)+D⁢sinh(θ2y))]+ηy−C1y−C2}, where (3.2)θ1=12((1−N)H2+m2)+((1−N)H2+m2)2−4(2m2(1−N)H2(2−N)),θ2=12((1−N)H2+m2)−((1−N)H2+m2)2−4(2m2(1−N)H2(2−N)).Using the corresponding boundary
conditions in (2.20), we get(3.3)A=0,C=0,C2=0,D=H2m2(2−N)(1−N)(F+h)sinh(θ2h)[ξ2(1−θ2hcoth(θ2h))−ξ1(1−θ1hcoth(θ1h))],B=−H2m22−N(1−N)(F+h)sinh(θ1h)[ξ2(1−θ2hcoth(θ2h))−ξ1(1−θ1hcoth(θ1h))],C1=H2(1−N){1+(F+h)ζ(θ2ξ2coth(θ2h)−θ1ξ1coth(θ1h))}+η,ζ=ξ2(1−θ2hcoth(θ2h))−ξ1(1−θ1hcoth(θ1h)),ξi=θi2−m2,i=1,2.Thus the stream function and the microrotation componentw will take the forms(3.4)ψ(x,y)=(F+h)ζ{ξ2sinh(θ2y)sinh(θ2h)−ξ1sinh(θ1y)sinh(θ1h)}−{1+(F+h)ζ(θ2ξ2coth(θ2h)−θ1ξ1coth(θ1h))y},w(x,y)=(F+h)(1−N)H2m2(2−N)ζ{sinh(θ2y)sinh(θ2h)−sinh(θ1y)sinh(θ1h)}.Now solving (2.32) with the corresponding boundary
conditions in (2.30), we get the magnetic force function in the
form(3.5)ϕ(x,y)=Rm{ξ1(F+h)θ1ζcosh(θ1y)sinh(θ1h)−ξ2(F+h)θ2ζcosh(θ2y)sinh(θ2h)+y22(1+F+hζ(θ2ξ2coth(θ2h)−θ1ξ1coth(θ1h))+E)}+C3y+C4,where(3.6)C3=0,C4=−Rm{ξ1(F+h)θ1ζcosh(θ1h)sinh(θ1h)−ξ2(F+h)θ2ζcosh(θ2h)sinh(θ2h)−h22(1+(F+h)ζ(θ2ξ2coth(θ2h)−θ1ξ1coth(θ1h))+E)}.Also, the  axial-induced
magnetic field and the current density distribution across the channel will
take  the forms(3.7)hx(x,y)=Rm{ξ1(F+h)ζsinh(θ1y)sinh(θ1y)−ξ2(F+h)ζsinh(θ2y)sinh(θ2h)+y(1+F+hζ(θ2ξ2coth(θ2h)−θ1ξ1coth(θ1h))+E)},Jz(x,y)=Rm{θ1ξ1(F+h)ζcosh(θ1y)sinh(θ1h)−θ2ξ2(F+h)ζcosh(θ2y)sinh(θ2h)+(1+F+hζ(θ2ξ2coth(θ2h)−θ1ξ1coth(θ1h))+E)}.In the formulation under consideration, the field
strengthE is the determining factor and its value can be found by integrating
(2.32), which represents Ohm's law in differential form, across the channel,
taking into account the boundary conditions for ϕ and ψ in  (2.30). In this case, we obtain the dimensionless E=F/h.When the flow is steady in the wave frame, one can
characterize the pumping performance by means of the the pressure rise per
wavelength. So, the axial pressure gradient can be  obtained from the
equation(3.8)∂p∂x=1(1−N∂3ψ∂y3+N1−N∂w∂y+H2(E−∂ψ∂y).Using (3.4), the axial pressure gradient
will take the form(3.9)∂p∂x=(F+h)θ2ζ{ξ2(θ221−N−H2)+NH2m22−N}cosh(θ2y)sinh(θ2h)−(F+h)θ1ζ{ξ1(θ121−N−H2)+NH2m22−N}cosh(θ1y)sinh(θ1h)+H2{(E+1)−(F+h)ζ(θ1ξ1coth(θ1h)−θ2ξ2coth(θ2h))}.The pressure riseΔpλ for a channel of length L in its nondimensional forms is given by(3.10)Δpλ=∫01∂p∂xdx.The integral in (3.10) is not
integrable in closed form, it is evaluated numerically using a digital
computer.An interesting property of the micropolar fluid is
that the stress tensor is not symmetric. The nondimensional shear stresses in
the problem under consideration are given by(3.11)τxy=∂2ψ∂y2−N1−Nw,τyx=(11−N)∂2ψ∂y2+N1−Nw.The shear stressesτxy and τyx are calculated at both the lower and upper
walls and graphical results are shown in
Figures 4–6.


## 4. Numerical Results and Discussion

This section is divided into three subsections. In the
first subsection, the effects of various parameters on the pumping
characteristics of a magneto-miropolar fluid are investigated. The magnetic
field characteristics are discussed in the second subsection. The trapping
phenomenon and the magnetic-force lines are illustrated in the last subsection.


### 4.1. Pumping Characteristics

This subsection
describes the influences of various emerging parameters of our analysis on the
axial pressure gradient∂p/∂x,
the pressure rise per wavelength Δpλ,
and the shear stresses τxy,τyx on the lower and upper walls. The effects of
these parameters are shown in
Figures 1–6, and in most of the
figures, the
case of N→0
corresponds  to
that of Newtonian fluid.Figure 1
The axial pressure gradient versus the
wavelength forα=0.3,θ=−1.2,H=2 and different values of m and N.Figure 2
The pressure rise versus flow rate forα=0.4,N=0.4,H=2, and H=8 at different values of m.Figure 3
The pressure rise versus flow rate forα=0.4,m=2,H=2, and H=8 at different values of N.Figure 4
The shear stressesτxy for α=0.5,θ=1.2,m=3,N=0.6, and different values of H.Figure 5
The shear stressesτyx for α=0.5,θ=1.2,m=3,N=0.6, and different values of H.Figure 6
The shear stressesτyx for α=0.5,θ=1.2,H=3, and different values of m and N.Figure1
illustrates  the
variation of the axial pressure gradient with x for different values of the
microrotation parameter m and the coupling number N. We can see that in the
wider part of the channel xϵ[0,0.2] and [0.8,1.0], the pressure gradient
is relatively small, that is, the flow can easily pass without imposition of
large pressure gradient. Where, in a narrow part of the channel xϵ[0.2,0.8], a much larger pressure gradient is
required to maintain the same flux to pass it, especially for the narrowest
position near x=0.5. This is in well agreement with the physical situation.
Also from this figure, we observe the effect of m and N on the pressure
gradient for fixed values of the other parameters, where the amplitude of dp/dx decrease as m increases and increases with
increasing N, and the smallest value of such amplitude corresponds to the case N→0 (Newtonian fluid). The effect of the Hartmann
number H on dp/dx is not included, where it is illustrated in a
previous paper [44], where the amplitude of dp/dx increases as H increases.Figures2 and 3 illustrate the change of the
pressure rise Δpλ versus the time-averaged mean flow rate θ for various values of the parameters m  (= 0.001, 10,100 with N=0.4,α=0.4 and different values of H) and  N(=0.2,0.4,0.6
with m=2,α=0.4 and different values of H).The graph is sectored so that the upper right-hand
quadrant (I) denotes the region of peristaltic pumping, whereθ>0 (positive pumping) and Δpλ>0 (adverse pressure gradient). Quadrant (II),
where Δpλ<0 (favorable pressure gradient) and θ>0 (positive pumping), is designated as augmented
flow (copumping region). Quadrant (IV), such that Δpλ>0 (adverse pressure gradient) and θ<0,
is called retrograde or backward pumping. The flow is opposite to the direction
of the peristaltic motion, and there is no flows in the last
quadrant  (Quadrant (III)). It is shown in both
Figures 2 and 3, that there is an inversely linear relation between Δpλ and θ, that is, the pressure rise decreases with
increasing the flow rate and the pumping curves are linear both for Newtonian
and micropolar fluid. Moreover, the pumping curves for micropolar fluid
lie above the Newtonian fluid in pumping region (Δpλ>0), but as m increases, the curves tend to
coincide. In copumping region (Δpλ<0), the pumping increases with an increase in
m. Figure 3
shows  the effects of the coupling
number N on Δpλ, where the pumping increases with an increase
in N and the pumping curve for the Newtonian fluid lies below the curves for
micropolar fluid in the pumping region, and in the copumping region, the
pumping decreases with an increase in N.It is known that the stress tensor is not symmetric in
micropolar fluid, that is why the expressions forτxy and τyx are different. In
Figures
4 and 5, we have plotted the shear
stresses τxy and τyx at the upper and lower walls for various
values of the Hartmann number H. It can be seen that both shear stress are
symmetric about the line x=0. However, its magnitude increases as H increases.
Moreover, both shear stresses have directions opposite to the upper wave
velocity, while the directions of these shear stresses are along the direction
of the lower wave velocity. Figure 6 indicates that the shear stress τyx decreases with an increase in the
microrotation parameter m, while it increases as the coupling number N
increases, and so, the magnitude value of shear stress for a Newtonian fluid is
less than that for a micropolar fluid.


### 4.2. Magnetic Field Characteristics

The variations
of the  axial-induced  magnetic fieldhx across the channel at x=0 and the current
density distribution Jz across the channel for various values of m,N,H,Rm,
and α are displayed in
Figures 7–12.Figure 7
Variation of the axial-induced magnetic field
across the channel forα=0.4,θ=1.2,N=0.9,Rm=1,
and different values of m and H.Figure 8
Variation of the
axial-induced  magnetic field across the channel forα=0.4,θ=1.2,m=2,Rm=1, and different values of N and H.Figure 9
Variation of the
axial-induced  magnetic field across the channel form=3,N=0.6,H=4,Rm=2,θ=−1.2 and different values of α.Figure 10
Axial-induced
magnetic field versus flow rate forα=0.5 for different values of N at m=2,H=4,Rm=2
for different values of m at N=0.7,H=7,Rm=1, and for different values of H at
m=4,N=0.6,Rm=2.Figure 11
Variation of the current density distribution
across the channel forα=0.5,θ=1.6,m=3,N=0.2,Rm=1, and different values of H.Figure 12
Variation of the current density distribution
across the channel forα=0.5,θ=1.6,H=2,Rm=1, and different values of m and N.In Figures7-8, m (= 5,  20, 100 with N=0.9,Rm=1,α=0.4
and
θ=1.2 and different values of H), N (= 0.2,
0.4, 0.6 with m=2,Rm=1,α=0.4  and θ=1.2 and different values of H).These figures  indicate
that the magnitude of the axial-induced magnetic fieldhx decreases as the microrotation parameter m and
the Hartmann number H increases while it increases as the coupling number N
increases. Further in the half region, the induced magnetic field is one
direction, and in the other half, it is in the opposite direction and it is
zero at y=0. Figure 9 illustrates the variation of hx across the channel for different values of the
amplitude ratio α at x=0.5, where α (= 0, 0.3, 0.6, 0.9 with m=3,N=0.6,H=4,Rm=2, and θ=−1.2).  It  is
clear that the magnitude of the  axial-induced
magnetic field hx at α=0 (no peristalsis) is larger, and it
decreases with increasing α.
The  distributions  of hx within the time-averaged mean flow rate θ are exhibited in Figure 10 at x=0.25,y=0.5,
where hx is plotted for various  values of the
parameters N (= 0.2, 0.8 with m=2,H=4,Rm=2,α=0.5 ), m (= 0.01, 40 with N=0.7,H=7,Rm=1,α=0.5) and
H (= 5, 10 with m=4,N=0.6,Rm=2,α=0.5). From this figure, we observe that there
is an inversely linear relation between hx and θ for any value of the above-mentioned
parameters, that is, hx decreases with increasing the flow rate θ and that the obtained curves will
intersect at the point θ=0. It is found also that for any value of θ≤0,
the effect of increasing each of M and H is to decrease hx values whereas the effect of increasing N is
to increase the value of hx.
On the other hand, for any value of θ≥0,
all the obtained lines will behave in an opposite manner to  this
behavior when θ≤0.Figures11–13 describe the distribution of the
current density Jz within y for different values of m,N,H, and
Rm at the central line of the channel (x=0), where H (= 2,  4,  6, 8, 10 with m=3,N=0.2,Rm=1,α=0.5,θ=1.6), m (= 3,  6,  9 with N=0.8,H=2,Rm=1,α=0.5andθ=1.6), N  (= 0.1, 0.3, 0.5 with m=3,H=2,Rm=1,α=0.5andθ=1.6), and Rm (= 1, 2, 3, with m=3,N=0.2,H=4,α=0.5,θ=1.6). The graphical results of these figures
indicate  that the dimensionless current density Jz decreases as H and M
increase  in the region near the center of the
channel while it increases for the same values of H and M in the region near to
the lower and upper walls, that is, the net current flow through the channel is
zero and this  corresponds  to the case of open
circuit. However, an opposite behavior is noticed as N and Rm increases. Also,
the current value for a micropolar fluid is higher than that for a Newtonian
fluid.Figure 13
Variation of the current density distribution
across the channel forα=0.5,θ=1.6,H=4,m=3,N=0.2, and different values of Rm.


### 4.3. Trapping Phenomena and Magnetic-Force Lines

Another interesting phenomenon in peristaltic motion
is trapping. In the wave frame, streamlines under certain conditions split to
trap a bolus which moves as a whole with the speed of the wave. To see the
effects of microrotation parameterm and the coupling number N on the trapping,
we prepared Figures 14 and 15 for various values of the parameters m (= 3,
6, 9 with N=0.5,H=2,θ=0.7,α=0.5) and N = (0.2, 0.4, 0.8 with m=7,H=2,θ=0.7,α=0.5). Figures 14 and 15
reveal  that the trapping is about the center line
and the trapped bolus decrease in size as the microrotation parameter m
increases, while the size of the bolus increases as N increases. Also the
effects of m and N on the magnetic-force function ϕ are illustrated in Figures 16 and
17 for
various values of the parameters m (= 0.01, 3, 10, with N=0.5,H=2,θ=0.7,Rm=1,α=0.4) and N = (0.1, 0.4, 0.8, with m=3,H=2,θ=0.7,Rm=1,α=0.4). It is observed that as m increases, the
size of the magnetic-force lines will decrease, and
for large values of m, these lines will  vanish  at
the center of the channel. An opposite behavior  will
occur  as the coupling number N increase, where the width of the
magnetic-force lines  will increase  and more
magnetic-force lines will be created at the center of the channel as the
coupling number N increases.Stream lines for different values ofm.
(a)(b)(c)Stream lines for different values ofN.
(a)(b)(c)Magnetic-force  lines   for different
values ofm.
(a)(b)(c)Magnetic-force  lines  for different
values ofN.
(a)(b)(c)


## 4.1. Pumping Characteristics

This subsection
describes the influences of various emerging parameters of our analysis on the
axial pressure gradient∂p/∂x,
the pressure rise per wavelength Δpλ,
and the shear stresses τxy,τyx on the lower and upper walls. The effects of
these parameters are shown in
Figures 1–6, and in most of the
figures, the
case of N→0
corresponds  to
that of Newtonian fluid.Figure 1
The axial pressure gradient versus the
wavelength forα=0.3,θ=−1.2,H=2 and different values of m and N.Figure 2
The pressure rise versus flow rate forα=0.4,N=0.4,H=2, and H=8 at different values of m.Figure 3
The pressure rise versus flow rate forα=0.4,m=2,H=2, and H=8 at different values of N.Figure 4
The shear stressesτxy for α=0.5,θ=1.2,m=3,N=0.6, and different values of H.Figure 5
The shear stressesτyx for α=0.5,θ=1.2,m=3,N=0.6, and different values of H.Figure 6
The shear stressesτyx for α=0.5,θ=1.2,H=3, and different values of m and N.Figure1
illustrates  the
variation of the axial pressure gradient with x for different values of the
microrotation parameter m and the coupling number N. We can see that in the
wider part of the channel xϵ[0,0.2] and [0.8,1.0], the pressure gradient
is relatively small, that is, the flow can easily pass without imposition of
large pressure gradient. Where, in a narrow part of the channel xϵ[0.2,0.8], a much larger pressure gradient is
required to maintain the same flux to pass it, especially for the narrowest
position near x=0.5. This is in well agreement with the physical situation.
Also from this figure, we observe the effect of m and N on the pressure
gradient for fixed values of the other parameters, where the amplitude of dp/dx decrease as m increases and increases with
increasing N, and the smallest value of such amplitude corresponds to the case N→0 (Newtonian fluid). The effect of the Hartmann
number H on dp/dx is not included, where it is illustrated in a
previous paper [44], where the amplitude of dp/dx increases as H increases.Figures2 and 3 illustrate the change of the
pressure rise Δpλ versus the time-averaged mean flow rate θ for various values of the parameters m  (= 0.001, 10,100 with N=0.4,α=0.4 and different values of H) and  N(=0.2,0.4,0.6
with m=2,α=0.4 and different values of H).The graph is sectored so that the upper right-hand
quadrant (I) denotes the region of peristaltic pumping, whereθ>0 (positive pumping) and Δpλ>0 (adverse pressure gradient). Quadrant (II),
where Δpλ<0 (favorable pressure gradient) and θ>0 (positive pumping), is designated as augmented
flow (copumping region). Quadrant (IV), such that Δpλ>0 (adverse pressure gradient) and θ<0,
is called retrograde or backward pumping. The flow is opposite to the direction
of the peristaltic motion, and there is no flows in the last
quadrant  (Quadrant (III)). It is shown in both
Figures 2 and 3, that there is an inversely linear relation between Δpλ and θ, that is, the pressure rise decreases with
increasing the flow rate and the pumping curves are linear both for Newtonian
and micropolar fluid. Moreover, the pumping curves for micropolar fluid
lie above the Newtonian fluid in pumping region (Δpλ>0), but as m increases, the curves tend to
coincide. In copumping region (Δpλ<0), the pumping increases with an increase in
m. Figure 3
shows  the effects of the coupling
number N on Δpλ, where the pumping increases with an increase
in N and the pumping curve for the Newtonian fluid lies below the curves for
micropolar fluid in the pumping region, and in the copumping region, the
pumping decreases with an increase in N.It is known that the stress tensor is not symmetric in
micropolar fluid, that is why the expressions forτxy and τyx are different. In
Figures
4 and 5, we have plotted the shear
stresses τxy and τyx at the upper and lower walls for various
values of the Hartmann number H. It can be seen that both shear stress are
symmetric about the line x=0. However, its magnitude increases as H increases.
Moreover, both shear stresses have directions opposite to the upper wave
velocity, while the directions of these shear stresses are along the direction
of the lower wave velocity. Figure 6 indicates that the shear stress τyx decreases with an increase in the
microrotation parameter m, while it increases as the coupling number N
increases, and so, the magnitude value of shear stress for a Newtonian fluid is
less than that for a micropolar fluid.


## 4.2. Magnetic Field Characteristics

The variations
of the  axial-induced  magnetic fieldhx across the channel at x=0 and the current
density distribution Jz across the channel for various values of m,N,H,Rm,
and α are displayed in
Figures 7–12.Figure 7
Variation of the axial-induced magnetic field
across the channel forα=0.4,θ=1.2,N=0.9,Rm=1,
and different values of m and H.Figure 8
Variation of the
axial-induced  magnetic field across the channel forα=0.4,θ=1.2,m=2,Rm=1, and different values of N and H.Figure 9
Variation of the
axial-induced  magnetic field across the channel form=3,N=0.6,H=4,Rm=2,θ=−1.2 and different values of α.Figure 10
Axial-induced
magnetic field versus flow rate forα=0.5 for different values of N at m=2,H=4,Rm=2
for different values of m at N=0.7,H=7,Rm=1, and for different values of H at
m=4,N=0.6,Rm=2.Figure 11
Variation of the current density distribution
across the channel forα=0.5,θ=1.6,m=3,N=0.2,Rm=1, and different values of H.Figure 12
Variation of the current density distribution
across the channel forα=0.5,θ=1.6,H=2,Rm=1, and different values of m and N.In Figures7-8, m (= 5,  20, 100 with N=0.9,Rm=1,α=0.4
and
θ=1.2 and different values of H), N (= 0.2,
0.4, 0.6 with m=2,Rm=1,α=0.4  and θ=1.2 and different values of H).These figures  indicate
that the magnitude of the axial-induced magnetic fieldhx decreases as the microrotation parameter m and
the Hartmann number H increases while it increases as the coupling number N
increases. Further in the half region, the induced magnetic field is one
direction, and in the other half, it is in the opposite direction and it is
zero at y=0. Figure 9 illustrates the variation of hx across the channel for different values of the
amplitude ratio α at x=0.5, where α (= 0, 0.3, 0.6, 0.9 with m=3,N=0.6,H=4,Rm=2, and θ=−1.2).  It  is
clear that the magnitude of the  axial-induced
magnetic field hx at α=0 (no peristalsis) is larger, and it
decreases with increasing α.
The  distributions  of hx within the time-averaged mean flow rate θ are exhibited in Figure 10 at x=0.25,y=0.5,
where hx is plotted for various  values of the
parameters N (= 0.2, 0.8 with m=2,H=4,Rm=2,α=0.5 ), m (= 0.01, 40 with N=0.7,H=7,Rm=1,α=0.5) and
H (= 5, 10 with m=4,N=0.6,Rm=2,α=0.5). From this figure, we observe that there
is an inversely linear relation between hx and θ for any value of the above-mentioned
parameters, that is, hx decreases with increasing the flow rate θ and that the obtained curves will
intersect at the point θ=0. It is found also that for any value of θ≤0,
the effect of increasing each of M and H is to decrease hx values whereas the effect of increasing N is
to increase the value of hx.
On the other hand, for any value of θ≥0,
all the obtained lines will behave in an opposite manner to  this
behavior when θ≤0.Figures11–13 describe the distribution of the
current density Jz within y for different values of m,N,H, and
Rm at the central line of the channel (x=0), where H (= 2,  4,  6, 8, 10 with m=3,N=0.2,Rm=1,α=0.5,θ=1.6), m (= 3,  6,  9 with N=0.8,H=2,Rm=1,α=0.5andθ=1.6), N  (= 0.1, 0.3, 0.5 with m=3,H=2,Rm=1,α=0.5andθ=1.6), and Rm (= 1, 2, 3, with m=3,N=0.2,H=4,α=0.5,θ=1.6). The graphical results of these figures
indicate  that the dimensionless current density Jz decreases as H and M
increase  in the region near the center of the
channel while it increases for the same values of H and M in the region near to
the lower and upper walls, that is, the net current flow through the channel is
zero and this  corresponds  to the case of open
circuit. However, an opposite behavior is noticed as N and Rm increases. Also,
the current value for a micropolar fluid is higher than that for a Newtonian
fluid.Figure 13
Variation of the current density distribution
across the channel forα=0.5,θ=1.6,H=4,m=3,N=0.2, and different values of Rm.


## 4.3. Trapping Phenomena and Magnetic-Force Lines

Another interesting phenomenon in peristaltic motion
is trapping. In the wave frame, streamlines under certain conditions split to
trap a bolus which moves as a whole with the speed of the wave. To see the
effects of microrotation parameterm and the coupling number N on the trapping,
we prepared Figures 14 and 15 for various values of the parameters m (= 3,
6, 9 with N=0.5,H=2,θ=0.7,α=0.5) and N = (0.2, 0.4, 0.8 with m=7,H=2,θ=0.7,α=0.5). Figures 14 and 15
reveal  that the trapping is about the center line
and the trapped bolus decrease in size as the microrotation parameter m
increases, while the size of the bolus increases as N increases. Also the
effects of m and N on the magnetic-force function ϕ are illustrated in Figures 16 and
17 for
various values of the parameters m (= 0.01, 3, 10, with N=0.5,H=2,θ=0.7,Rm=1,α=0.4) and N = (0.1, 0.4, 0.8, with m=3,H=2,θ=0.7,Rm=1,α=0.4). It is observed that as m increases, the
size of the magnetic-force lines will decrease, and
for large values of m, these lines will  vanish  at
the center of the channel. An opposite behavior  will
occur  as the coupling number N increase, where the width of the
magnetic-force lines  will increase  and more
magnetic-force lines will be created at the center of the channel as the
coupling number N increases.Stream lines for different values ofm.
(a)(b)(c)Stream lines for different values ofN.
(a)(b)(c)Magnetic-force  lines   for different
values ofm.
(a)(b)(c)Magnetic-force  lines  for different
values ofN.
(a)(b)(c)


## 5. Concluding Remarks

The  effect of the
induced magnetic field on peristaltic flow of a magneto-micropolar fluid is
studied. The exact expressions for stream function, magnetic-force function,
axial pressure gradient, axial-induced magnetic field, and current density are
obtained analytically. Graphical results are presented for the pressure rise
per wave length, shear stresses on the lower and upper walls, axial-induced
magnetic field,  and  current density and trapping.
The main findings can be summarized as follows.(1)
The value of the axial pressure gradientdp/dx is higher for a magneto-micropolar
fluid  than that for a Newtonian fluid.(2)
The magnitude of pressure rise per wavelength
for a magneto-micropolar  fluid  is greater than that
of  a  Newtonian  fluid  in
the pumping region, while in the  copumping  region,
the situation is reversed.(3)
Shear stresses at the lower wall are quite
similar to those for the upper wall except that at the upper wall has its
direction opposite to the upper wave velocity while at the lower wall has its
direction along the wave velocity.(4)
The shear stresses
decrease  with an
increase  ofm and increases with increasing N.(5)
The  axial-induced
magnetic field is higher for a Newtonian fluid than that for a micropolar fluid
and smaller as the transverse magnetic field increases.(6)
There is an inversely linear relation between
the  axial-induced  magnetic field and the flow rate.(7)
The current densityJz at the center of the channel is higher for a
micropolar fluid than that for a Newtonian fluid, and it will
decrease as the microrotation parameter and
transverse magnetic field increases, while it increases as the coupling number
increases.(8)
As we move from the Newtonian fluid to a
micropolar fluid, more trapped bolus created at the center line and the size of
these bolus increases.(9)
More magnetic force lines created at the
center of the channel as the fluid moves from Newtonian
fluid to a micropolar fluid.



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*Source: 570825-2008-12-22.xml*