Effect of rotation on peristaltic flow of a micropolar fluid through a porous medium with an external magnetic field

https://doi.org/10.1016/j.jmmm.2013.06.030Get rights and content

Highlights

  • The effects of induced magnetic field and rotation in peristaltic motion of a two dimensional of a micropolar fluid through a porous medium

  • The exact and closed form solutions are presented

  • Different wave shapes are considered to observe the behavior of the axial velocity, pressure rise, mechanical efficiency, spin velocity, stream function and pressure gradient.

Abstract

In this paper, the effects of both rotation and magnetic field of a micropolar fluid through a porous medium induced by sinusoidal peristaltic waves traveling down the channel walls are studied analytically and computed numerically. Closed-form solutions under the consideration of long wavelength and low-Reynolds number is presented. The analytical expressions for axial velocity, pressure rise per wavelength, mechanical efficiency, spin velocity, stream function and pressure gradient are obtained in the physical domain. The effect of the rotation, density, Hartmann number, permeability, coupling number, micropolar parameter and the non-dimensional wave amplitude in the wave frame is analyzed theoretically and computed numerically. Numerical results are given and illustrated graphically in each case considered. Comparison was made with the results obtained in the presence and absence of rotation and magnetic field. The results indicate that the effect of rotation, density, Hartmann number, permeability, coupling number, micropolar parameter and the non-dimensional wave amplitude are very pronounced in the phenomena.

Introduction

Peristaltic pumping has been the object of scientific and engineering research in recent years. The word peristaltic comes from a Greek word Peristaltikos which means clasping and compressing. The peristaltic transport is traveling contraction wave along a tube-like structure, and it results physiologically from neuron-muscular properties of any tubular smooth muscle. Peristaltic motion of blood (or other fluid) in animal or human bodies has been considered by many authors. It is an important mechanism for transporting blood, where the cross-section of the artery is contracted or expanded periodically by the propagation of progressive wave. 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 spermatoza in the ductus efferentes of the male reproductive tract and the movement of ovum in the fallopin tubes.

The study of the peristaltic transport of a fluid in the presence of an external magnetic field and rotation is of great importance with regard to certain problems involving the movement of conductive physiological fluids, e.g. blood and saline water. Pandey and Chaube [1] investigated an analytical study of the MHD flow of a micropolar fluid through a porous medium induced by sinusoidal peristaltic waves traveling down the channel wall. The magnetohydrodynamic flow of a micropolar fluid in a circular cylindrical tube has been investigated by Wang et al. [2]. Nadeem and Akram [3] studied the analytical and numerical treatment of peristaltic flows in viscous and non-Newtonian fluids. Vajravelu et al. [4] studied the influence of heat transfer on peristaltic transport of a Jeffrey fluid in a vertical porous stratum. Hayat et al. [5] discussed the influence of compliant wall properties and heat transfer on the peristaltic flow of an incompressible viscous fluid in a curved channel. Bhargava et al. [6] have studied finite element study of nonlinear two-dimensional deoxygenated biomagnetic micropolar flow. Ali et al [7] discussed the peristaltic motion of a non-Newtonian fluid in a channel having compliant boundaries. Abd-Alla et al. [8] studied the effects of rotation and magnetic field on nonlinear peristaltic flow of second-order fluid in an asymmetric channel through a porous medium. Hayat et al. [9] analyzed the effect of an induced magnetic field on the peristaltic flow of an incompressible Carreau fluid in an asymmetric channel. Pandey and Chaube [10] concerned with the theoretical study of two-dimensional peristaltic flow of power-law fluids in three layers with different viscosities. Jiménez-Lozano and Sen [11] investigated the streamline patterns and their local and global bifurcations in a two-dimensional planar and axisymmetric peristaltic flow for an incompressible Newtonian fluid. Hayat et al. [12] analyzed the effect of an induced magnetic field on the peristaltic flow of an incompressible Carreau fluid in an asymmetric channel. Srinivas and Kothandapani [13] investigated the effects of heat and mass transfer on peristaltic transport in a porous space with compliant walls. Gad [14] discussed the effect of hall currents on the interaction of pulsatile and peristaltic transport induced flows of a particle fluid suspension. Abd-Alla et al. [15] investigated the peristaltic flow in a tube with an endoscope subjected to magnetic field. Hayat and Noreen [16] discussed the influence of an induced magnetic field on the peristaltic flow of an incompressible fourth grade fluid in a symmetric channel with heat transfer. Nadeem and Akbar [17] investigated the peristaltic flow of an incompressible MHD Newtonian fluid in a vertical annulus. Nadeem and Akbar [18] investigated the deals with the peristaltic motion of an incompressible non-Newtonian fluid in a non-uniform tube for long wavelength. Nadeem et al. [19] studied the concentrates on the heat transfer characteristics and endoscope effects for the peristaltic flow of a third order fluid. Srinivas et al. [20] studied the effects of both wall slip conditions and heat transfer on peristaltic flow of MHD Newtonian fluid in a porous channel with elastic wall properties. Abd-Alla et al. [21] investigated the effect of the rotation, magnetic field and initial stress on peristaltic motion of micropolar fluid. Mahmoud et al. [22] discussed the effect of the rotation on wave motion through cylindrical bore in a micropolar porous medium. The dynamic behavior of a wet long bone that has been modeled as a piezoelectric hollow cylinder of crystal class 6 is investigated by Abd-Alla et al. [23]. Akram and Nadeem [27] discussed the peristaltic motion of a two dimensional Jeffrey fluid in an asymmetric channel under the effects of induced magnetic field and heat transfer.

The objective of the present analysis is to analyze the effects of rotation and magnetic field on the peristaltic flow of a micropolar fluid through a porous medium. The two-dimensional equations of motion are simplified using the assumptions of long wavelength and low Reynolds number. The exact and numerical solutions of the reduced equations are found of wave shape. Numerical calculations are carried out and illustrated graphically in each case considered. The magnetic field and rotation may be utilized as transport of bio-fluid in the ureters, intestines and arterioles. The results indicate that the effects of rotation, density, Hartmann number, permeability, coupling number, micropolar parameter and the non-dimensional wave amplitude are very pronounced and effective in the peristaltic phenomena. Comparison is made with existing results.

Section snippets

Formulation of the problem

Let us consider the peristaltic flow of an incompressible micropolar fluid through a porous channel of width 2a in the presence of a magnetic field. Let Y=±H be the upper and lower boundaries of the channel. The motion is considered to be induced by sinusoidal wave trains propagating along the channel walls with a constant speed c, such thatH=a+bsin2πλ(Xct)where b is the wave amplitude, λ is the wavelength and t is the time.

We choose a wave frame (x,y) moving with speed c. The coordinates

Rate of volume flow and boundary conditions

In the laboratory frame, the instantaneous volume rate of flow is given asQ(X,t)=0HU(X,Y,t)dY,where H=H(X,t) which in the wave frame may be expressed asq(x)=0Hu(x,y)dy,where, H=H(x).

Substituting from Eqs. (2), (15) into Eq. (14), and then integrating, one obtainsQ(X,t)=q(x)+cH(X,t).The time-mean flow rate over a period T at a fixed X-position is defined asQ(X)=1T0TQ(X,t)dt.Using Eq. (16) into Eq. (17) and integrating, we getQ(X)=q(x)+acThe dimensionless mean flow

Solution of the problem

In order to solve the present problem, we are differentiating Eq. (13) with respect to y and then adding to Eq. (11), we getu=K(1N)(1+KM2)KD{(2Nm2)3wy3(2N)wy(1N)px(1N)(1+KM2)K}Then using this value of Eq. (13), we get4wy4{(1N)(1+KM2)KDK+m2}2wy2+2m2(1N)(1+KM2)KDK(2N)w=0,whose general solution isw=A(x)cosh(θ1y)+B(x)sinh(θ1y)+C(x)cosh(θ2y)+D(x)sinh(θ2y)whereθ1=12{(1N)(1+KM2)KDK+m2}+{(1N)(1+KM2)KDK+m2}24{2m2(1N)(1+KM2)KDK(2N)},θ2=12{(1N)(1+KM2)KDK+m2}++{(1N)(1+KM

Numerical results and discussion

In order to gain physical insight into the axial velocity u, pressure rise Δp, mechanical efficiency E, spin velocity W, stream function ψ and pressure gradient dPdx have been discussed by assigning numerical values to the parameter encountered in the problem in which the numerical results are displayed with the graphical illustration. The variations are shown in Fig. 1, Fig. 2, Fig. 3, Fig. 4, Fig. 5, Fig. 6, respectively, with the constant values N=0.2, m=2, M=1, ϕ=0.6, x=0.2 and K=0.1.

Fig. 1

Conclusion

Due to the complicated nature of the governing equations of the pertinent field equations governing the peristaltic flow, the work done in this field is unfortunately limited in number. The method used in this study provides a quite successful in dealing with such problems. This method gives exact and numerical solutions of the peristaltic flow with long wavelength approximation and neglecting the wave number of the actual physical quantities that appear in the governing equations of the

References (27)

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