Journal of Applied Mathematics
Volume 2008 (2008), Article ID 570825, 23 pages
doi:10.1155/2008/570825
Research Article

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

Kh. S. Mekheimer

Department of Mathematics, Faculty of Science, Al-Azhar University, Nasr City 11884, Cairo, Egypt

Received 2 June 2008; Accepted 29 September 2008

Academic Editor: Jacek Rokicki

Copyright © 2008 Kh. S. Mekheimer. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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.

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 with 𝑋 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 𝐻 0 , which will give rise to an induced magnetic field 𝐻 ( ℎ 𝑋 ′ ( 𝑋 , 𝑌 , 𝑡 ) , ℎ 𝑌 ′ ( 𝑋 , 𝑌 , 𝑡 ) , 0 ) and the total magnetic field will be 𝐻 + ( ℎ 𝑋 ′ ( 𝑋 , 𝑌 , 𝑡 ) , 𝐻 0 + ℎ 𝑌 ′ ( 𝑋 , 𝑌 , 𝑡 ) , 0 ) . The plates of the channel are assumed to be nonconductive, and the geometry of the wall surface is defined as ℎ ( 𝑋 , 𝑡 ) = 𝑎 + 𝑏 c o s 2 𝜋 𝜆 ( 𝑋 − 𝑐 𝑡 ) , ( 2 . 1 ) where 𝑎 0 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 → ∇ ⋅ → 𝑉 𝜌 𝜕 = 0 , → 𝑉 𝜕 𝑡 + → 𝑉 ⋅ → ∇ → 𝑉 = − → ∇ 𝑝 + 𝑘 → ∇ × → 𝑤 + ( 𝜇 + 𝑘 ) → ∇ 2 → 𝑉 + 𝜌 → 𝑓 , 𝜕 𝜌 𝑗 → 𝑉 𝜕 𝑡 + → 𝑉 ⋅ → ∇ → 𝑤 = − 2 𝑘 → 𝑤 + 𝑘 → ∇ × → 𝑉 − 𝛾 → ∇ × → ∇ × → 𝑤 + ( 𝛼 + 𝛽 + 𝛾 ) ∇ → ∇ ⋅ → 𝑤 , ( 2 . 2 ) 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]): | | 𝛽 | | 2 𝜇 + 𝑘 ≥ 0 , 𝑘 ≥ 0 , 3 𝛼 + 𝛽 + 𝛾 ≥ 0 , 𝛾 ≥ . ( 2 . 3 )

The governing equations for a magneto-micropolar fluid are as follows:

Maxwell's equations ∇ ⋅ → 𝐻 = 0 , ∇ ⋅ → 𝐸 ∇ = 0 , ( 2 . 4 ) → 𝐻 = → 𝐽 , w i t h → 𝐽 = 𝜎 → 𝐸 + 𝜇 𝑒 → 𝑉 → 𝐻 + ∇ , ( 2 . 5 ) → 𝐸 = − 𝜇 𝑒 𝜕 → 𝐻 𝜕 𝑡 , ( 2 . 6 ) ∇ ⋅ → 𝑉 = 0 , ( 2 . 7 ) 𝜌 𝜕 → 𝑉 𝜕 𝑡 + → 𝑉 ⋅ → ∇ → 𝑉 𝑝 = − ∇ + 1 2 𝜇 𝑒 𝐻 + 2 + ( 𝜇 + 𝑘 ) ∇ 2 → 𝑉 + 𝑘 → ∇ × → 𝑤 − 𝜇 𝑒 → 𝐻 + ⋅ → ∇ → 𝐻 + , 𝜕 𝜌 𝑗 → 𝑉 𝜕 𝑡 + → 𝑉 ⋅ → ∇ → 𝑤 = − 2 𝑘 → 𝑤 + 𝑘 → ∇ × → 𝑉 − 𝛾 → ∇ × → ∇ × → 𝑤 + ( 𝛼 + 𝛽 + 𝛾 ) → ∇ → ∇ ⋅ → 𝑤 , ∇ 2 = 𝜕 2 𝜕 𝑋 2 + 𝜕 2 𝜕 𝑌 2 , ( 2 . 8 )

the continuity equation → 𝐸

the equations of motion → 𝐽 where 𝜇 𝑒 is an induced electric field, 𝜎 is the electric current density, 𝜕 → 𝐻 + 𝜕 𝑡 = → ∇ → 𝑉 → 𝐻 + + 1 𝜁 ∇ 2 → 𝐻 + , ( 2 . 9 ) is the magnetic permeability, and 𝜁 = 1 / 𝜎 𝜇 𝑒 is the electrical conductivity.

Combining (2.4) and (2.5)–(2.7), we obtain the induction equation: 𝑐 where ( 𝑋 , 𝑌 ) is the magnetic diffusivity.

We should carry out this investigation in a coordinate system moving with the wave speed ( 𝑥 , 𝑦 ) , in which the boundary shape is stationary. The coordinates and velocities in the laboratory frame 𝑥 = 𝑋 − 𝑐 𝑡 , 𝑦 = 𝑌 , 𝑢 = 𝑈 − 𝑐 , 𝑣 = 𝑉 , ( 2 . 1 0 ) and the wave frame 𝑈 , 𝑉 are related by 𝑢 , 𝑣 where 𝑥 𝑥 = 𝜆 𝑦 , 𝑦 = 𝑎 𝑢 , 𝑢 = 𝑐 , 𝑣 = 𝜆 𝑣 ℎ 𝑎 𝑐 , ℎ = ( 𝑥 ) 𝑎 , 𝑎 𝑝 = 2 𝑝 𝜆 𝜇 𝑐 ( 𝑥 ) , 𝑡 = 𝑐 𝑡 𝜆 𝑗 , 𝑗 = 𝑎 2 𝜓 , 𝜓 = 𝜙 𝑐 𝑎 , 𝜙 = 𝐻 0 𝑎 , ( 2 . 1 1 ) , and 𝜓 ( 𝑥 , 𝑦 ) are the velocity components in the corresponding coordinate systems.

Using these transformations and introducing the dimensionless variables 𝜙 ( 𝑥 , 𝑦 ) we find that the equations which govern the MHD flow for a micropolar fluid in terms of the stream function 𝜓 R e 𝛿 𝑦 𝜕 𝜕 𝑥 − 𝜓 𝑥 𝜕 𝜓 𝜕 𝑦 𝑦 = − 𝜕 𝑝 𝑚 + 1 𝜕 𝑥 ∇ 1 − 𝑁 2 𝜓 𝑦 + 𝑁 1 − 𝑁 𝜕 𝑤 𝜕 𝑦 + R e 𝑆 2 𝜙 𝑦 𝑦 + R e 𝑆 2 𝛿 𝜙 𝑦 𝜕 𝜕 𝑥 − 𝜙 𝑥 𝜕 𝜙 𝜕 𝑦 𝑦 , ( 2 . 1 2 ) and magnetic-force function R e 𝛿 3 𝜓 𝑥 𝜕 𝜕 𝑦 − 𝜓 𝑦 𝜕 𝜓 𝜕 𝑥 𝑥 = − 𝜕 𝑝 𝑚 − 𝛿 𝜕 𝑦 2 ∇ 1 − 𝑁 2 𝜓 𝑥 − 𝛿 2 𝑁 1 − 𝑁 𝜕 𝑤 𝜕 𝑥 − R e 𝑆 2 𝛿 2 𝜙 𝑥 𝑦 − R e 𝑆 2 𝛿 3 𝜙 𝑦 𝜕 𝜕 𝑥 − 𝜙 𝑥 𝜕 𝜙 𝜕 𝑦 𝑥 , ( 2 . 1 3 ) are R e 𝛿 𝑗 1 − 𝑁 𝑁 𝜓 𝑦 𝜕 𝜕 𝑥 − 𝜓 𝑥 𝜕 𝑤 𝜕 𝑦 = − 2 𝑤 − ∇ 2 𝜓 + 2 − 𝑁 𝑚 2 ∇ 2 𝑤 , ( 2 . 1 4 ) 𝑢 = 𝜕 𝜓 𝜕 𝑦 , 𝑣 = − 𝛿 𝜕 𝜓 𝜕 𝑥 , ℎ 𝑥 = 𝜕 𝜙 𝜕 𝑦 , ℎ 𝑦 = − 𝛿 𝜕 𝜙 , ∇ 𝜕 𝑥 2 = 𝛿 2 𝜕 2 𝜕 𝑥 2 + 𝜕 2 𝜕 𝑦 2 , ( 2 . 1 6 ) R e = 𝑐 𝑎 𝜌 / 𝜇 𝛿 = 𝑎 / 𝜆 where 𝑆 = ( 𝐻 0 √ / 𝑐 ) 𝜇 𝑒 / 𝜌 and the dimensionless parameters as follows:

(i) Reynolds number 𝑅 𝑚 = 𝜎 𝜇 𝑒 𝑎 𝑐 ,(ii)wave number 𝑁 = 𝑘 / ( 𝑘 + 𝜇 ) ( 0 ≤ 𝑁 ≤ 1 ) , 𝑚 2 = 𝑎 2 𝑘 ( 2 𝜇 + 𝑘 ) / ( 𝛾 ( 𝜇 + 𝑘 ) ) ,(iii) Strommer's number (magnetic-force number) 𝑝 𝑚 = 𝑝 + ( 1 / 2 ) R e 𝛿 ( 𝜇 𝑒 ( 𝐻 + ) 2 / 𝜌 𝑐 2 ) ,(iv) the magnetic Reynolds number 𝐸 ( = − 𝐸 / 𝑐 𝐻 0 ) ,(v) the coupling number 𝛼 , 𝛽 is the micropolar parameter,(vi) the total pressure in the fluid, which equals the sum of the ordinary and magnetic pressure, is 𝑘 → 0 , and 𝜓 R e 𝛿 𝑦 𝜕 𝜕 𝑥 − 𝜓 𝑥 𝜕 ∇ 𝜕 𝑦 2 𝜓 = 1 ∇ 1 − 𝑁 4 𝑁 𝜓 + ∇ 1 − 𝑁 2 𝑤 + R e 𝑆 2 ∇ 2 𝜙 𝑦 + R e 𝑆 2 𝛿 𝜙 𝑦 𝜕 𝜕 𝑥 − 𝜙 𝑥 𝜕 ∇ 𝜕 𝑦 2 𝜙 , R e 𝛿 𝑗 1 − 𝑁 𝑁 𝜓 𝑦 𝜕 𝜕 𝑥 − 𝜓 𝑥 𝜕 𝑤 𝜕 𝑦 = − 2 𝑤 − ∇ 2 𝜓 + 2 − 𝑁 𝑚 2 ∇ 2 𝑤 . ( 2 . 1 7 ) is the electric field strength. The parameters 𝑄 = ℎ ′ 0 𝑈 𝑋 , 𝑌 , 𝑡 𝑑 𝑌 , ( 2 . 1 8 ) 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 ℎ .

Excluding the total pressure from (2.12) and (2.13), we obtain 𝑋

The instantaneous volume flow rate in the fixed frame is given by 𝑡 where 𝑞 = ℎ ′ 0 𝑢 𝑥 , 𝑦 𝑑 𝑦 , ( 2 . 1 9 ) is a function of ℎ and 𝑥 .

The rate of volume flow in the wave frame is given by 𝑄 = 𝑞 + 𝑐 ℎ . ( 2 . 2 0 ) where 𝑇 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 1 𝑄 = 𝑇 𝑇 0 𝑄 𝑑 𝑡 . ( 2 . 2 1 ) The time mean flow over a period 𝑄 = 𝑞 + 𝑎 𝑐 . ( 2 . 2 2 ) at a fixed position 𝜃 is defined as: 𝐹 Substituting (2.20) into (2.21), and integrating, we get 𝜃 = 𝑄 𝑞 𝑎 𝑐 , 𝐹 = 𝑎 𝑐 , ( 2 . 2 3 )

On defining the dimensionless time-mean flows 𝜃 = 𝐹 + 1 , ( 2 . 2 4 ) and 𝐹 = ℎ 0 𝜕 𝜓 𝜕 𝑦 𝑑 𝑦 = 𝜓 ( ℎ ) − 𝜓 ( 0 ) . ( 2 . 2 5 ) , respectively, in the fixed and wave frame as ℎ one finds that (2.22) may be written as ℎ ( 𝑥 ) = 1 + 𝛼 c o s ( 2 𝜋 𝑥 ) , ( 2 . 2 6 ) where 𝑏 𝛼 = 𝑎 ( 2 . 2 7 )

We note that ( 𝑦 = 0 ) represents the dimensionless form of the surface of the peristaltic wall: 𝜓 ( 0 ) = 0 , ( 2 . 2 8 ) where ( 𝑦 = ℎ ) is the amplitude ratio or the occlusion.

If we select the zero value of the streamline at the streamline 𝜓 ( ℎ ) = 𝐹 . ( 2 . 2 9 ) 𝜓 ( 𝑥 , 𝑦 ) then the wall 𝜙 ( 𝑥 , 𝑦 ) is a streamline of value 𝜕 𝜓 = 0 , 2 𝜓 𝜕 𝑦 2 = 0 , 𝑤 = 0 , 𝜕 𝜙 𝜕 𝑦 = 0 o n 𝑦 = 0 , 𝜕 𝜓 𝜕 𝑦 = − 1 , 𝜓 = 𝐹 , 𝑤 = 0 , 𝜙 = 0 , 𝜕 𝜙 𝜕 𝑦 = 0 o n 𝑦 = ℎ ( 𝑥 ) . ( 2 . 3 0 )

For a non-conductive elastic channel wall, the boundary conditions for the dimensionless stream function 𝜕 4 𝜓 𝜕 𝑦 4 𝜕 + 𝑁 2 𝑤 𝜕 𝑦 2 + R e 𝑆 2 𝜕 ( 1 − 𝑁 ) 3 𝜙 𝜕 𝑦 3 = 0 , 2 − 𝑁 𝑚 2 𝜕 2 𝑤 𝜕 𝑦 2 𝜕 = 2 𝑤 + 2 𝜓 𝜕 𝑦 2 𝜕 , ( 2 . 3 1 ) 2 𝜙 𝜕 𝑦 2 = 𝑅 𝑚 𝐸 − 𝜕 𝜓 𝜕 𝑦 . ( 2 . 3 2 ) and magnetic-force function 1 𝜓 = 𝐻 2 ( 1 − 𝑁 ) 2 − 𝑁 𝑚 2 𝜕 2 𝑤 𝜕 𝑦 2 − 𝑚 2 𝑤 + 𝜂 𝑦 − 𝐶 1 𝑦 − 𝐶 2 , 𝜕 4 𝑤 𝜕 𝑦 4 − 𝑚 2 + 𝐻 2 𝜕 ( 1 − 𝑁 ) 2 𝑤 𝜕 𝑦 2 + 2 𝑚 2 𝐻 2 ( 1 − 𝑁 ) 2 − 𝑁 𝑤 = 0 , ( 2 . 3 3 ) in the wave frame are [42, 44] 𝐻 2 = R e 𝑆 2 𝑅 𝑚

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: 𝐻 = ( 𝜇 𝑒 𝐻 0 √ ) 𝑎 𝜎 / 𝜇 √ 2 Combining these equations gives 𝜂 = 𝐸 𝐻 2 ( 1 − 𝑁 ) where 𝐶 1 , 𝐶 2 , 𝑤 is the Hartmann number (suitably greater than 𝜓 ), 𝑤 = 𝐴 c o s h ( 𝜃 1 𝑦 ) + 𝐵 s i n h ( 𝜃 1 𝑦 ) + 𝐶 c o s h ( 𝜃 2 𝑦 ) + 𝐷 s i n h ( 𝜃 2 1 𝑦 ) , 𝜓 = 𝐻 2 ( 1 − 𝑁 ) 2 − 𝑁 𝑚 2 [ ( 𝜃 2 1 − 𝑚 2 ) ( 𝐴 c o s h ( 𝜃 1 𝑦 ) + 𝐵 s i n h ( 𝜃 1 𝑦 ) ) + ( 𝜃 2 2 − 𝑚 2 ) ( 𝐶 c o s h ( 𝜃 2 𝑦 ) + 𝐷 s i n h ( 𝜃 2 𝑦 ) ) ] + 𝜂 𝑦 − 𝐶 1 𝑦 − 𝐶 2 , ( 3 . 1 ) , and 𝜃 1 = 1 √ 2 ⎷ ( ( 1 − 𝑁 ) 𝐻 2 + 𝑚 2 ) + ( ( 1 − 𝑁 ) 𝐻 2 + 𝑚 2 ) 2 − 4 2 𝑚 2 ( 1 − 𝑁 ) 𝐻 2 , 𝜃 ( 2 − 𝑁 ) 2 = 1 √ 2 ⎷ ( ( 1 − 𝑁 ) 𝐻 2 + 𝑚 2 ) − ( ( 1 − 𝑁 ) 𝐻 2 + 𝑚 2 ) 2 − 4 2 𝑚 2 ( 1 − 𝑁 ) 𝐻 2 . ( 2 − 𝑁 ) ( 3 . 2 ) are an integration constants.

3. Exact Solution

The general solutions of the microrotation component 𝐴 = 0 , 𝐶 = 0 , 𝐶 2 𝐻 = 0 , 𝐷 = 2 𝑚 2 ( 2 − 𝑁 ) ( 1 − 𝑁 ) ( 𝐹 + ℎ ) s i n h ( 𝜃 2 ℎ ) [ 𝜉 2 ( 1 − 𝜃 2 ℎ c o t h ( 𝜃 2 ℎ ) ) − 𝜉 1 ( 1 − 𝜃 1 ℎ c o t h ( 𝜃 1 , ℎ ) ) ] 𝐵 = − 𝐻 2 𝑚 2 2 − 𝑁 ( 1 − 𝑁 ) ( 𝐹 + ℎ ) s i n h ( 𝜃 1 ℎ ) [ 𝜉 2 ( 1 − 𝜃 2 ℎ c o t h ( 𝜃 2 ℎ ) ) − 𝜉 1 ( 1 − 𝜃 1 ℎ c o t h ( 𝜃 1 , 𝐶 ℎ ) ) ] 1 = 𝐻 2 ( 1 − 𝑁 ) 1 + ( 𝐹 + ℎ ) 𝜁 ( 𝜃 2 𝜉 2 c o t h ( 𝜃 2 ℎ ) − 𝜃 1 𝜉 1 c o t h ( 𝜃 1 ℎ ) ) + 𝜂 , 𝜁 = 𝜉 2 ( 1 − 𝜃 2 ℎ c o t h ( 𝜃 2 ℎ ) ) − 𝜉 1 ( 1 − 𝜃 1 ℎ c o t h ( 𝜃 1 ℎ ) ) , 𝜉 𝑖 = 𝜃 2 𝑖 − 𝑚 2 , 𝑖 = 1 , 2 . ( 3 . 3 ) and the stream function 𝑤 are 𝜓 ( 𝑥 , 𝑦 ) = ( 𝐹 + ℎ ) 𝜁 𝜉 2 s i n h ( 𝜃 2 𝑦 ) s i n h ( 𝜃 2 ℎ ) − 𝜉 1 s i n h ( 𝜃 1 𝑦 ) s i n h ( 𝜃 1 − ℎ ) 1 + ( 𝐹 + ℎ ) 𝜁 ( 𝜃 2 𝜉 2 c o t h ( 𝜃 2 ℎ ) − 𝜃 1 𝜉 1 c o t h ( 𝜃 1 , = ℎ ) ) 𝑦 𝑤 ( 𝑥 , 𝑦 ) ( 𝐹 + ℎ ) ( 1 − 𝑁 ) 𝐻 2 𝑚 2 ( 2 − 𝑁 ) 𝜁 s i n h ( 𝜃 2 𝑦 ) s i n h ( 𝜃 2 − ℎ ) s i n h ( 𝜃 1 𝑦 ) s i n h ( 𝜃 1 . ℎ ) ( 3 . 4 ) where 𝜙 ( 𝑥 , 𝑦 ) = 𝑅 𝑚 𝜉 1 ( 𝐹 + ℎ ) 𝜃 1 𝜁 c o s h ( 𝜃 1 𝑦 ) s i n h ( 𝜃 1 − 𝜉 ℎ ) 2 ( 𝐹 + ℎ ) 𝜃 2 𝜁 c o s h ( 𝜃 2 𝑦 ) s i n h ( 𝜃 2 + 𝑦 ℎ ) 2 2 1 + 𝐹 + ℎ 𝜁 ( 𝜃 2 𝜉 2 c o t h ( 𝜃 2 ℎ ) − 𝜃 1 𝜉 1 c o t h ( 𝜃 1 ℎ ) ) + 𝐸 + 𝐶 3 𝑦 + 𝐶 4 , ( 3 . 5 )

Using the corresponding boundary conditions in (2.20), we get 𝐶 3 𝐶 = 0 , 4 = − 𝑅 𝑚 𝜉 1 ( 𝐹 + ℎ ) 𝜃 1 𝜁 c o s h ( 𝜃 1 ℎ ) s i n h ( 𝜃 1 − 𝜉 ℎ ) 2 ( 𝐹 + ℎ ) 𝜃 2 𝜁 c o s h ( 𝜃 2 ℎ ) s i n h ( 𝜃 2 − ℎ ℎ ) 2 2 1 + ( 𝐹 + ℎ ) 𝜁 𝜃 2 𝜉 2 c o t h ( 𝜃 2 ℎ ) − 𝜃 1 𝜉 1 c o t h ( 𝜃 1 . ℎ ) + 𝐸 ( 3 . 6 )

Thus the stream function and the microrotation component ℎ 𝑥 ( 𝑥 , 𝑦 ) = 𝑅 𝑚 𝜉 1 ( 𝐹 + ℎ ) 𝜁 s i n h ( 𝜃 1 𝑦 ) s i n h ( 𝜃 1 − 𝜉 𝑦 ) 2 ( 𝐹 + ℎ ) 𝜁 s i n h ( 𝜃 2 𝑦 ) s i n h ( 𝜃 2 ℎ ) + 𝑦 1 + 𝐹 + ℎ 𝜁 𝜃 2 𝜉 2 c o t h ( 𝜃 2 ℎ ) − 𝜃 1 𝜉 1 c o t h ( 𝜃 1 , 𝐽 ℎ ) + 𝐸 𝑧 ( 𝑥 , 𝑦 ) = 𝑅 𝑚 𝜃 1 𝜉 1 ( 𝐹 + ℎ ) 𝜁 c o s h ( 𝜃 1 𝑦 ) s i n h ( 𝜃 1 − 𝜃 ℎ ) 2 𝜉 2 ( 𝐹 + ℎ ) 𝜁 c o s h ( 𝜃 2 𝑦 ) s i n h ( 𝜃 2 + ℎ ) 1 + 𝐹 + ℎ 𝜁 𝜃 2 𝜉 2 c o t h ( 𝜃 2 ℎ ) − 𝜃 1 𝜉 1 c o t h ( 𝜃 1 . ℎ ) + 𝐸 ( 3 . 7 ) will take the forms 𝐸

Now solving (2.32) with the corresponding boundary conditions in (2.30), we get the magnetic force function in the form 𝜙 where 𝜓

Also, the axial-induced magnetic field and the current density distribution across the channel will take the forms 𝐸 = 𝐹 / ℎ

In the formulation under consideration, the field strength 𝜕 𝑝 = 1 𝜕 𝑥 𝜕 ( 1 − 𝑁 3 𝜓 𝜕 𝑦 3 + 𝑁 1 − 𝑁 𝜕 𝑤 𝜕 𝑦 + 𝐻 2 𝐸 − 𝜕 𝜓 𝜕 𝑦 . ( 3 . 8 ) 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 𝜕 𝑝 = 𝜕 𝑥 ( 𝐹 + ℎ ) 𝜃 2 𝜁 𝜉 2 𝜃 2 2 1 − 𝑁 − 𝐻 2 + 𝑁 𝐻 2 𝑚 2 2 − 𝑁 c o s h ( 𝜃 2 𝑦 ) s i n h ( 𝜃 2 − ℎ ) ( 𝐹 + ℎ ) 𝜃 1 𝜁 𝜉 1 𝜃 2 1 1 − 𝑁 − 𝐻 2 + 𝑁 𝐻 2 𝑚 2 2 − 𝑁 c o s h ( 𝜃 1 𝑦 ) s i n h ( 𝜃 1 ℎ ) + 𝐻 2 ( 𝐸 + 1 ) − ( 𝐹 + ℎ ) 𝜁 ( 𝜃 1 𝜉 1 c o t h ( 𝜃 1 ℎ ) − 𝜃 2 𝜉 2 c o t h ( 𝜃 2 . ℎ ) ) ( 3 . 9 ) and Δ 𝑝 𝜆 in (2.30). In this case, we obtain the dimensionless 𝐿 .

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 Δ 𝑝 𝜆 = 1 0 𝜕 𝑝 𝜕 𝑥 𝑑 𝑥 . ( 3 . 1 0 )

Using (3.4), the axial pressure gradient will take the form 𝜏 𝑥 𝑦 = 𝜕 2 𝜓 𝜕 𝑦 2 − 𝑁 𝜏 1 − 𝑁 𝑤 , 𝑦 𝑥 = 1 𝜕 1 − 𝑁 2 𝜓 𝜕 𝑦 2 + 𝑁 1 − 𝑁 𝑤 . ( 3 . 1 1 )

The pressure rise 𝜏 𝑥 𝑦 for a channel of length 𝜏 𝑦 𝑥 in its nondimensional forms is given by 𝜕 𝑝 / 𝜕 𝑥 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 Δ 𝑝 𝜆

The shear stresses 𝜏 𝑥 𝑦 , 𝜏 𝑦 𝑥 and 𝑁 → 0 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 𝛼 = 0 . 3 , 𝜃 = − 1 . 2 , 𝐻 = 2 , the pressure rise per wavelength 𝑚 , and the shear stresses 𝑁 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 𝛼 = 0 . 4 , 𝑁 = 0 . 4 , 𝐻 = 2 corresponds to that of Newtonian fluid.

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Figure 1: The axial pressure gradient versus the wavelength for 𝐻 = 8 and different values of 𝑚 and 𝛼 = 0 . 4 , 𝑚 = 2 , 𝐻 = 2 .
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Figure 2: The pressure rise versus flow rate for 𝐻 = 8 , and 𝑁 at different values of 𝜏 𝑥 𝑦 .
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Figure 3: The pressure rise versus flow rate for 𝛼 = 0 . 5 , 𝜃 = 1 . 2 , 𝑚 = 3 , 𝑁 = 0 . 6 , and 𝐻 at different values of 𝜏 𝑦 𝑥 .
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Figure 4: The shear stresses 𝛼 = 0 . 5 , 𝜃 = 1 . 2 , 𝑚 = 3 , 𝑁 = 0 . 6 for 𝐻 , and different values of 𝜏 𝑦 𝑥 .
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Figure 5: The shear stresses 𝛼 = 0 . 5 , 𝜃 = 1 . 2 , 𝐻 = 3 for 𝑚 , and different values of 𝑁 .
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Figure 6: The shear stresses 𝑥 for 𝑁 , and different values of 𝑥 𝜖 [ 0 , 0 . 2 ] and 𝑥 𝜖 [ 0 . 2 , 0 . 8 ] .

Figure 1 illustrates the variation of the axial pressure gradient with 𝑥 = 0 . 5 for different values of the microrotation parameter m and the coupling number 𝑚 . We can see that in the wider part of the channel 𝑁 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 𝑑 𝑝 / 𝑑 𝑥 , a much larger pressure gradient is required to maintain the same flux to pass it, especially for the narrowest position near 𝑁 . This is in well agreement with the physical situation. Also from this figure, we observe the effect of 𝑁 → 0 and 𝐻 on the pressure gradient for fixed values of the other parameters, where the amplitude of 𝑑 𝑝 / 𝑑 𝑥 decrease as m increases and increases with increasing 𝑑 𝑝 / 𝑑 𝑥 , and the smallest value of such amplitude corresponds to the case 𝐻 (Newtonian fluid). The effect of the Hartmann number Δ 𝑝 𝜆 on 𝜃 is not included, where it is illustrated in a previous paper [44], where the amplitude of 𝑚 increases as 𝑁 = 0 . 4 , 𝛼 = 0 . 4 increases.

Figures 2 and 3 illustrate the change of the pressure rise 𝐻 versus the time-averaged mean flow rate 𝑁 ( = 0 . 2 , 0 . 4 , 0 . 6 for various values of the parameters 𝑚 = 2 , 𝛼 = 0 . 4 (= 0.001, 10,100 with 𝐻 and different values of 𝜃 > 0 ) and Δ 𝑝 𝜆 > 0 with Δ 𝑝 𝜆 < 0 and different values of 𝜃 > 0 ).

The graph is sectored so that the upper right-hand quadrant (I) denotes the region of peristaltic pumping, where Δ 𝑝 𝜆 > 0 (positive pumping) and 𝜃 < 0 (adverse pressure gradient). Quadrant (II), where Δ 𝑝 𝜆 (favorable pressure gradient) and 𝜃 (positive pumping), is designated as augmented flow (copumping region). Quadrant (IV), such that Δ 𝑝 𝜆 > 0 (adverse pressure gradient) and 𝑚 , 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 Δ 𝑝 𝜆 < 0 and 𝑚 , 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 ( 𝑁 ), but as Δ 𝑝 𝜆 increases, the curves tend to coincide. In copumping region ( 𝑁 ), the pumping increases with an increase in 𝑁 . Figure 3 shows the effects of the coupling number 𝜏 𝑥 𝑦 on 𝜏 𝑦 𝑥 , where the pumping increases with an increase in 𝜏 𝑥 𝑦 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 𝜏 𝑦 𝑥 .

It is known that the stress tensor is not symmetric in micropolar fluid, that is why the expressions for 𝐻 and 𝑥 = 0 are different. In Figures 4 and 5, we have plotted the shear stresses 𝐻 and 𝜏 𝑦 𝑥 at the upper and lower walls for various values of the Hartmann number 𝑚 . It can be seen that both shear stress are symmetric about the line 𝑁 . However, its magnitude increases as ℎ 𝑥 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 𝑥 = 0 decreases with an increase in the microrotation parameter 𝐽 𝑧 , while it increases as the coupling number 𝑚 , 𝑁 , 𝐻 , 𝑅 𝑚 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 field 𝛼 across the channel at 𝛼 = 0 . 4 , 𝜃 = 1 . 2 , 𝑁 = 0 . 9 , 𝑅 𝑚 = 1 and the current density distribution 𝑚 across the channel for various values of 𝐻 , and 𝛼 = 0 . 4 , 𝜃 = 1 . 2 , 𝑚 = 2 , 𝑅 𝑚 = 1 are displayed in Figures 7–12.

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Figure 7: Variation of the axial-induced magnetic field across the channel for 𝑁 , and different values of 𝐻 and 𝑚 = 3 , 𝑁 = 0 . 6 , 𝐻 = 4 , 𝑅 𝑚 = 2 , 𝜃 = − 1 . 2 .
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Figure 8: Variation of the axial-induced magnetic field across the channel for 𝛼 , and different values of 𝛼 = 0 . 5 and 𝑁 .
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Figure 9: Variation of the axial-induced magnetic field across the channel for 𝑚 = 2 , 𝐻 = 4 , 𝑅 𝑚 = 2 and different values of 𝑚 .
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Figure 10: Axial-induced magnetic field versus flow rate for 𝑁 = 0 . 7 , 𝐻 = 7 , 𝑅 𝑚 = 1 for different values of 𝐻 at 𝑚 = 4 , 𝑁 = 0 . 6 , 𝑅 𝑚 = 2 for different values of 𝛼 = 0 . 5 , 𝜃 = 1 . 6 , 𝑚 = 3 , 𝑁 = 0 . 2 , 𝑅 𝑚 = 1 at 𝐻 , and for different values of 𝛼 = 0 . 5 , 𝜃 = 1 . 6 , 𝐻 = 2 , 𝑅 𝑚 = 1 at 𝑚 .
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Figure 11: Variation of the current density distribution across the channel for 𝑁 , and different values of 𝑚 .
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Figure 12: Variation of the current density distribution across the channel for 𝑁 = 0 . 9 , 𝑅 𝑚 = 1 , 𝛼 = 0 . 4 , and different values of 𝜃 = 1 . 2 and 𝐻 .

In Figures 7-8, 𝑁 (= 5, 20, 100 with 𝑚 = 2 , 𝑅 𝑚 = 1 , 𝛼 = 0 . 4 and 𝜃 = 1 . 2 and different values of 𝐻 ), ℎ 𝑥 (= 0.2, 0.4, 0.6 with 𝑚 and 𝐻 and different values of 𝑁 ).

These figures indicate that the magnitude of the axial-induced magnetic field 𝑦 = 0 decreases as the microrotation parameter ℎ 𝑥 and the Hartmann number 𝛼 increases while it increases as the coupling number 𝑥 = 0 . 5 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 𝛼 . Figure 9 illustrates the variation of 𝑚 = 3 , 𝑁 = 0 . 6 , 𝐻 = 4 , 𝑅 𝑚 = 2 across the channel for different values of the amplitude ratio 𝜃 = − 1 . 2 at ℎ 𝑥 , where 𝛼 = 0 (= 0, 0.3, 0.6, 0.9 with 𝛼 , and ℎ 𝑥 ). It is clear that the magnitude of the axial-induced magnetic field 𝜃 at 𝑥 = 0 . 2 5 , 𝑦 = 0 . 5 (no peristalsis) is larger, and it decreases with increasing ℎ 𝑥 . The distributions of 𝑁 within the time-averaged mean flow rate 𝑚 = 2 , 𝐻 = 4 , 𝑅 𝑚 = 2 , 𝛼 = 0 . 5 are exhibited in Figure 10 at 𝑚 , where 𝑁 = 0 . 7 , 𝐻 = 7 , 𝑅 𝑚 = 1 , 𝛼 = 0 . 5 is plotted for various values of the parameters 𝐻 (= 0.2, 0.8 with 𝑚 = 4 , 𝑁 = 0 . 6 , 𝑅 𝑚 = 2 , 𝛼 = 0 . 5 ),