Effect of Rotation and Magnetic Field through Porous Medium on Peristaltic Transport of a Jeffrey Fluid in Tube

This paper is concerned with the analysis of peristaltic motion of a Jeffrey fluid in a tube with sinusoidal wave travelling down its wall. The effect of rotation, porous medium, and magnetic field on peristaltic transport of a Jeffrey fluid in tube is studied. The fluid is electrically conducting in the presence of rotation and a uniform magnetic field. An analytic solution is carried out for long wavelength, axial pressure gradient, and low Reynolds number considerations. The results for pressure rise and frictional force per wavelength were obtained, evaluated numerically, and discussed briefly.


Introduction
The dynamics of the fluid transport by peristaltic motion of the confining walls has received a careful study in the literature.The need for peristaltic pumping may arise in circumstances where it is desirable to avoid using any internal moving parts such as pistons in a pumping process.The peristalsis is also well known to the physiologists to be one of the major mechanisms of fluid transport in a biological system and appears in urine transport from kidney to bladder through the ureter, movement of chyme in the gastrointestinal tract, the movement of spermatozoa in the ductus efferentes of the male reproductive tract and the ovum in the female fallopian tube, the locomotion of some worms, transport of lymph in the lymphatic vessels, and vasomotion of small blood vessels such as arterioles, venules, and capillaries.Technical roller and finger pumps also operate according to this rule.The behavior of most of the physiological fluids is known to be non-Newtonian.Several models have been proposed to explain the non-Newtonian behavior of fluids.

Formulation of the Problem
Consider the axisymmetric flow of a Jeffrey fluid in a uniform circular tube with a sinusoidal peristaltic wave of small amplitude travelling down its wall see Figure 1 .The geometry of wall surface is therefore described as Here a is amplitudes of the waves, λ is the wavelength, d is average radius of the undisturbed tube.The constitutive equations for an incompressible Jeffrey fluid are: where I 0 and S are Cauchy stress tensor and extra stress tensor, respectively, p is the pressure, I is the identity tensor, μ is dynamic viscosity, λ 1 is the ratio of relaxation to retardation times, λ 2 is the retardation time, γ is the shear rate, and dots over the quantities indicate differentiation with respect to time.In laboratory frame, the equations governing twodimensional motion of an incompressible MHD Jeffrey fluid through a porous medium 24 are as follows: where R, W are the velocity components in the laboratory frame R, Z , ρ is the density, p is the pressure, σ is the electrical conductivity of the fluid, B 0 is a constant of magnetic field, μ is Mathematical Problems in Engineering the kinematic viscosity, Ω is the rotation component, and k 0 is the permeability of the porous medium, and we get 25 2.4 We will carry out this investigation in a coordinate system moving with the wave speed in which the boundary shape is stationary.The coordinates and velocities in the laboratory frame R, Z and the wave frame x, y , are related by where u, w are the velocity components in the wave frame r, z .We introduce the following nondimensional variables and parameters for the flow: where Re is the Reynolds number, δ is the dimensionless wave number, and H is the magnetic parameter Hartman number .Using nondimensional variables and parameters in 2.3 , we get the following: 2.8 We can write 2.7 as follows:

2.10
Re 2.11 Eliminating pressure from 2.9 , 2.11 by cross-differentiation, using the long wavelength δ 1 and low Reynolds number in 2.9 -2.11 , and neglecting δ and higher power, we obtain ∂p ∂r 0, 2.12 ∂p ∂z where 2.14 From 2.12 we show that p / p r .Differentiating 2.13 with respect to r we get where

Rate of Volume Flow
The instantaneous volume flow rate in fixed coordinate system is given by where h is a function of Z and t.On substituting 2.5 into 3.1 and then integrating, one obtains where is the volume flow rate in the moving coordinate system and is independent of time.Here, h is a function of z alone.Using the dimensionless variables, we find The time-mean flow over a period T λ/c at a fixed Z-position is defined as Using 3.2 into 3.5 , 0 < < 1, we obtain Using dimensionless variables we write: where β and F are, respectively, the flow rates in the fixed and wave frames.
We note that h represents the dimensionless form of the surface of the peristaltic wall: 3.9 Choosing the zero value of the streamline along the central line w 0 , we have ψ 0 0. Then the shape of the wave at the boundary is the streamline with value ψ h F in wave frame, the boundary conditions in terms of stream , at r h. 3.10

Method of Solution
Integration of 2.15 along with boundary conditions 3.10 gives where c 1 is an arbitrary function of z.Equation 4.1 after using the transformation can be reduced into the following modified Bessel equation: whose solution along with 4.2 and boundary conditions 3.10 is given below: where I 0 , I 1 , and I 2 are the modified Bessel function of order zero, one, and two, respectively.Substitution of 4.4 into 2.8 and 2.13 yields the following expressions for axial velocity w and axial pressure gradient: The expressions for pressure rise ΔP λ and frictional force F λ per wavelength are, respectively, given by 4.6

Results and Discussion
To investigate the effects of rotation Ω , magnetic parameter M , material parameter λ 1 , permeability of the porous medium k , and mean flux F , we plotted Figures 2-6.
The stress distribution S rr , S rz , and S zz in tube for different values of the rotation Ω is presented in Figures 2 a , 2 b , and 2 c , respectively.We notice that the stress is  in oscillatory behaviour, which may be due to peristalsis.The absolute value of stress distribution S rr , S rz , and S zz increases at first with increasing the rotation Ω , and then it decreases with increasing the rotation Ω when large values of r have been taken into account.It is observed that the absolute values of the stress are larger in case of a Jeffrey fluid when compared with Newtonian fluid.The effects of the rotation Ω and magnetic parameter M on the velocity is plotted in Figure 3. Figure 3 shows that influence of the rotation Ω and magnetic parameter M on the velocity increases with the increase of magnetic parameter M , and it decreases with the increase of the rotation Ω .
Figure 4 shows the variation of ΔP λ with flow rate F for values of rotation Ω , magnetic parameter M , and material parameter λ 1 for tube.We observe that the peristaltic pumping rate increases with increase of magnetic parameter M and material parameter λ 1 , and it decreases with the increase of the rotation Ω .The phenomenon of trapping is another interesting topic in peristaltic transport.The formation of an internally circulating bolus of the fluid by closed streamlines is called trapping, and this trapped bolus pushed ahead along the peristaltic wave.Figure 5 shows the variation of F λ with flow rate F for values of rotation Ω , magnetic parameter M , and material parameter λ 1 for tube.Figures 5 a , 5 b , and 5 c display the influence of Ω , magnetic parameter M , and material parameter λ 1 , respectively, for tube on F λ .Figure 5 a refers to the case when F −0.2.Here it is noted that F λ increases with decrease of rotation Ω when −0.6 ≤ F ≤ −0.2 and it increases with the increase of the rotation Ω when −0.2 < F. Figure 5 a refers to the case when Ω 1.2, here, it is noted that F λ is negative and positive when −0.6 ≤ F ≤ 0.6 and 0.6 < F, respectively.When Ω 0.8.Here it is noted that F λ is negative and positive when −0.6 ≤ F ≤ 0.83 and 0.83 < F, respectively.When Ω 0.4, F λ is negative for −0.6 ≤ F ≤ 1.4 and positive for 1.4 < F. Also, when Ω 0.0, F λ is negative for −0.6 ≤ F ≤ 3.0 and positive for 3.0 < F.  Figure 6 shows the distributions of the pressure gradient within a wavelength for various values of the rotation Ω , magnetic parameter M and material parameter λ 1 .The effects of magnetic parameter M , on the pressure gradient dp/dz within a wavelength are plotted in Figure 2 b .It is noticed that magnetic parameter M and material parameter λ 1 increase the maximum amplitude of dp/dz when compared to the case with zero magnetic parameter and zero material parameter λ 1 .

Conclusion
The influence of the rotation Ω , magnetic parameter M , and material parameter λ 1 on the peristaltic flow of a Jeffrey fluid in tube has been analyzed.The analytical expressions are constructed for axial velocity, F λ , and pressure gradient.Numerical investigation is plotted and discussed.The main findings can be summarized as follows.
i The axial velocity for the MHD fluid is less when compared with hydrodynamic fluid in the central part of the tube.
ii The magnitude of dp/dz and F λ increases with increase of magnetic parameter M and material parameter λ 1 and it increases with decrease of the rotation Ω .
iii The size of trapped bolus is smaller in Jeffrey fluid when compared with that of Newtonian fluid λ 1 0 .
iv The magnitudes of dp/dz , ΔP λ , and F λ for Newtonian fluid are smaller than that of Jeffrey fluid.
v For large values of magnetic parameter M and material parameter λ 1 , the magnitudes of ΔP λ and ΔP λ increase with decrease of the rotation Ω .

Figure 1 :
Figure 1: Geometry of peristaltic motion on asymmetric channel through porous medium.

5 K 7 cFigure 6 :
Figure 6: Plot showing variation of the pressure gradient dp/dz within wavelength for various values of rotation Ω a magnetic parameter M b , and material parameter λ 1 c .

Figure 5 b
Figure5shows the variation of F λ with flow rate F for values of rotation Ω , magnetic parameter M , and material parameter λ 1 for tube.Figures 5 a , 5 b , and 5 c display the influence of Ω , magnetic parameter M , and material parameter λ 1 , respectively, for tube on F λ .Figure5a refers to the case when F −0.2.Here it is noted that F λ increases with decrease of rotation Ω when −0.6 ≤ F ≤ −0.2 and it increases with the increase of the rotation Ω when −0.2 < F. Figure5a refers to the case when Ω 1.2, here, it is noted that F λ is negative and positive when −0.6 ≤ F ≤ 0.6 and 0.6 < F, respectively.When Ω 0.8.Here it is noted that F λ is negative and positive when −0.6 ≤ F ≤ 0.83 and 0.83 < F, respectively.When Ω 0.4, F λ is negative for −0.6 ≤ F ≤ 1.4 and positive for 1.4 < F. Also, when Ω 0.0, F λ is negative for −0.6 ≤ F ≤ 3.0 and positive for 3.0 < F. Figure 5 b refers to the case when F −0.8.Here it is noted that F λ increases with decrease of magnetic parameter M when −1.5 ≤ F ≤ −0.8, and it increases with the increase of the magnetic parameter M when −0.8 < F.Figure 5 c refers to the case when F −0.8.Here it is noted that F λ increases with decrease of material parameter λ 1 when −1 ≤ F ≤ −0.48, and it increases with the increase of the magnetic parameter M when −0.48 < F.

Figure 5 c
Figure5shows the variation of F λ with flow rate F for values of rotation Ω , magnetic parameter M , and material parameter λ 1 for tube.Figures 5 a , 5 b , and 5 c display the influence of Ω , magnetic parameter M , and material parameter λ 1 , respectively, for tube on F λ .Figure5a refers to the case when F −0.2.Here it is noted that F λ increases with decrease of rotation Ω when −0.6 ≤ F ≤ −0.2 and it increases with the increase of the rotation Ω when −0.2 < F. Figure5a refers to the case when Ω 1.2, here, it is noted that F λ is negative and positive when −0.6 ≤ F ≤ 0.6 and 0.6 < F, respectively.When Ω 0.8.Here it is noted that F λ is negative and positive when −0.6 ≤ F ≤ 0.83 and 0.83 < F, respectively.When Ω 0.4, F λ is negative for −0.6 ≤ F ≤ 1.4 and positive for 1.4 < F. Also, when Ω 0.0, F λ is negative for −0.6 ≤ F ≤ 3.0 and positive for 3.0 < F. Figure 5 b refers to the case when F −0.8.Here it is noted that F λ increases with decrease of magnetic parameter M when −1.5 ≤ F ≤ −0.8, and it increases with the increase of the magnetic parameter M when −0.8 < F.Figure 5 c refers to the case when F −0.8.Here it is noted that F λ increases with decrease of material parameter λ 1 when −1 ≤ F ≤ −0.48, and it increases with the increase of the magnetic parameter M when −0.48 < F.