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

Authors: S. R. Mahmoud
Journal: Mathematical Problems in Engineering
 (2011)

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## Abstract

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.


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## Body

## 1. 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.Mahmoud et al. [1–3] investigated effect of the rotation on the radial vibrations in a nonhomogeneous orthotropic hollow cylinder and effect of the rotation on wave motion through cylindrical bore in a micropolar porous cubic crystal, and he investigated effect of the rotation on the radial vibrations in a nonhomogeneous orthotropic hollow cylinder. Abd-Alla et al. [4–7] investigated  effect of the rotation on a nonhomogeneous infinite cylinder of orthotropic material, influences of rotation, magnetic field, initial stress and gravity on rayleigh waves in a homogeneous orthotropic elastic half space, and magneto-thermoelastic problem in rotating nonhomogeneous orthotropic hollow cylindrical under the hyperbolic heat conduction model, and they studied effect of the rotation on propagation of thermoelastic waves in a nonhomogeneous infinite cylinder of isotropic material. Mahmoud [8] studied effect of rotation on generalized magneto-thermoelastic Rayleigh waves in a granular medium under influence of gravity field and initial stress. Afifi et al. [9–11] investigated  effect of magnetic field and wall properties on peristaltic motion of micropolar fluid in circular cylindrical tubes and interaction  of  peristaltic  flow  with pulsatile magnetofluid through a porous medium, and they studied aspects  of  a magnetofluid  with  suspended particles. Various attempts [12–14] are made to solve the extremely complex equations of motion of non-Newtonian fluids. The good number of recent investigations [15–23] on the peristalsis of non-Newtonian  fluids has been presented with various perspectives, in channels or tubes. Most of the analytic studies are asymptotic expansions with small Reynolds number, wave number, and amplitude ratio as a perturbation parameter. Siddiqui et al. [24] examined the peristaltic motion of a magnetohydrodynamic Newtonian fluid in a tube by taking long wavelength approximation.  More recently Hayat and Ali [22] studied the peristaltic motion of a third-order fluid in a tube under long wavelength and small Deborah number approximation. However, no attempt has been made to discuss the peristaltic motion of a magnetohydrodynamic (MHD) non-Newtonian  fluid in a tube which holds for all values of non-Newtonian parameters. In the present analysis, such an attempt has been made. The liquid considered is of the Jeffrey type and is electrically conducting. This shows worthwhile the first attempt for MHD non-Newtonian flow in a tube for all values of the rheological parameters. The Jeffrey model is relatively simpler linear model using time derivatives instead of convected derivatives, for example,  what the Oldroyd-B model does;  it represents a rheology different from the Newtonian. Although more sophisticated viscoelastic models than the Jeffrey model exist, in a first study of the MHD peristaltic motion of a non-Newtonian  fluid in circular cylindrical tube, the choice of Jeffrey fluid model is motivated by the following.In spite of its relative simplicity, the Jeffrey model can indicate the changes of the rheology on the peristaltic flow even under the assumption of large wavelength, low Reynolds number, and small or large amplitude ratio. In Newtonian fluid, Mekheimer [25] studied the MHD peristaltic flow in a channel under the assumption of small wave number. Therefore, at least in an initial study, this motivates an analytic study of MHD peristaltic non-Newtonian tube flow that holds for all non-Newtonian parameters. By choosing the Jeffrey fluid modele it became possible to treat both the MHD Newtonian and non-Newtonian problems analytically under long wavelength and low Reynolds number consideration. Considering the blood as an MHD fluid, it may be possible to control blood pressure and its flow behavior by using an appropriate magnetic field. The influence of magnetic field may also be utilized as a blood pump for cardiac operations for blood flow in arterial stenosis or arteriosclerosis.


## 2. 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 Figure1). The geometry of wall surface is therefore described as(2.1)h¯(z¯,t¯)=d+acos⁡[2πλ(z¯-ct¯)].
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:(2.2)I̅0=-p̅I̅+S̅,S̅=μ1+λ1(γ̇¯+λ2γ̈¯),
where I̅0 and S̅ are Cauchy  stress  tensor and extra stress tensor, respectively, p̅ is the pressure, I̅ is  the identity tensor, μ  is dynamic  viscosity, λ1 is  the  ratio  of relaxation  to  retardation  times, λ2 is the retardation time, γ̇̅ is the shear rate, and dots over the quantities indicate differentiation with  respect to time.  In  laboratory  frame, the  equations  governing  two-dimensional  motion of  an incompressible MHD Jeffrey fluid  through a porous medium [24] are as follows:(2.3)∂U¯∂R¯+∂W¯∂Z¯+U¯R¯=0,ρ∂U¯∂t¯+U¯∂U¯∂R¯+W¯∂U¯∂Z¯+Ω¯2U¯=-∂p¯∂R¯+1R¯∂∂R¯(R¯∂S¯R¯R¯∂R¯)-S¯R¯R¯R¯2+∂2S¯R¯Z¯∂Z¯2-μκ¯0U¯,ρ(∂W¯∂t¯+U¯∂W¯∂R¯+W¯∂W¯∂Z¯)=-∂p¯∂Z¯+1R¯∂∂R¯(R¯∂S¯R¯Z¯∂R¯)+∂2S¯Z¯Z¯∂Z¯2-σB02W¯-μk0W¯+ρΩ¯2W¯,
where R̅, W̅ are the velocity components in  the laboratory  frame (R̅, Z̅), ρ is the density, p̅  is the pressure, σ  is  the electrical  conductivity of  the fluid, B0  is a constant of magnetic field, μ  is the kinematic viscosity, Ω  is the rotation component, and k0 is the permeability of the porous medium, and we get [25](2.4)S¯=μ1+λ1(1+λ2[U¯∂∂R¯+W¯∂∂Z¯])γ̇¯,S¯R¯R¯=2μ1+λ1(1+λ2[U¯∂∂R¯+W¯∂∂Z¯])∂U¯∂R¯,S¯Z¯Z¯=2μ1+λ1(1+λ2[U¯∂∂R¯+W¯∂∂Z¯])∂U¯∂Z¯,S¯R¯Z¯=μ1+λ1(1+λ2[U¯∂∂R¯+W¯∂∂Z¯])(∂U¯∂Z¯-∂W¯∂R¯).
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 (R̅, Z̅) and the wave frame (x̅, y̅), are related by(2.5)r̅=R¯-ct¯,z̅=Z¯,u̅=U¯-c,w̅=W¯,p¯=p¯(R¯,t),
where   u̅, w̅   are the velocity components in the wave frame  (r̅,z̅). We  introduce  the following  nondimensional  variables  and  parameters for the flow:(2.6)r=R¯d1,z=2πZ¯λ,u=U¯cδ,w=W¯c,δ=2πd1λ,t=2πct¯λ,p=2πd12p¯μcλ,H=σμBο2d12,Re=ρcd1μ,S=d1μcS¯,Ω2=Ω¯2νd12,k¯=k0d12,h1=h¯1d1,a=a1d1,
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: (2.7)δ∂u∂r+δ∂W∂Z+δur=0,Reδ(u∂∂r+w∂∂z)u=-∂p∂r+δ2[1r∂∂r(r∂Srr∂r)-Srrr2+δ2∂2Srz∂z2]-1k¯u-Ω2u,Reδ3(u∂∂r+w∂∂z)w=-∂p∂z+1r∂∂r(r∂Srz∂r)+δ2∂2Szz∂z2-H2w-1k¯w+Ω2w,
introducing the stream function ψ as (2.8)u(r,z)=δr∂ψ∂z,w(r,z)=-1r∂ψ∂r.
We can write  (2.7) as follows:(2.9)δ2∂∂r(1r∂ψ∂z)-δ∂∂z(1r∂ψ∂r)+δ2∂r2(1r∂ψ∂z)=0,(2.10)Reδ(δr∂ψ∂z∂∂r-1r∂ψ∂r∂∂z)1r∂ψ∂z=-∂p∂r+δ2[1r∂∂r(r∂Srr∂r)-Srrr2+δ2∂2Srz∂z2]-δk¯(1r∂ψ∂z)-δΩ2(1r∂ψ∂z),(2.11)Reδ3(δr∂ψ∂z∂∂r+1r∂ψ∂r∂∂z)1r∂ψ∂r=-∂p∂z+1r∂∂r(r∂Srz∂r)+δ2∂2Szz∂z2+H2(1r∂ψ∂r)+1k¯(1r∂ψ∂r)-Ω2(1r∂ψ∂r).
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 (2.12)∂p∂r=0,(2.13)∂p∂z=1r∂∂r(r(1+λ1)(1r∂ψ∂r))-ω2(1r∂ψ∂r),
where   ω2=1/k¯-Ω2+H2,(2.14)Srr=2δ1+λ1[1+δcλ2d1(∂ψ∂z∂∂r-∂ψ∂r∂∂z)]∂∂r(1r∂ψ∂z),Szz=2δ1+λ1[1+δcλ2d1(∂ψ∂z∂∂r-∂ψ∂r∂∂z)]∂∂z(1r∂ψ∂r).
From (2.12)  we show that p≠p(r).  Differentiating (2.13)  with respect   to r we get(2.15)∂∂r=(1r∂∂r(r∂∂r(1r∂ψ∂r)))-χ2(1r∂ψ∂r)=0,
where  χ2=(1+λ1)ω2.Figure 1
Geometry of peristaltic motion on asymmetric channel through porous medium.


## 3. Rate of Volume Flow

The  instantaneous  volume  flow  rate  in  fixed  coordinate  system  is given by(3.1)Q¯(z¯,t¯)=2π∫0h̅w̅(Z¯,t¯)R¯dR¯,
where h̅ is a function of  Z̅  and t̅.  On  substituting (2.5) into (3.1) and then integrating, one obtains  (3.2)Q¯=q¯+πch¯2,
where(3.3)q¯=2π∫0h¯w¯r¯dr¯
is  the  volume flow  rate in the moving  coordinate system and is independent of time. Here, h̅ is a function of z alone. Using the dimensionless variables, we find(3.4)F=q¯2πca12=∫0hwrdr.
The  time-mean  flow  over a period  T=λ/c at a fixed Z-position is defined  as(3.5)Q=1T∫0TQ¯dt̅.
Using  (3.2)  into  (3.5), 0<ϵ<1, we obtain(3.6)Q=q¯+πca2(1+ϵ22).
Using dimensionless variables we write:(3.7)Q2πca2=q¯2πca2+12(1+ϵ22).
Equation (3.6) becomes
(3.8)β=F+12(1+ϵ22),β=Q2πc∂2,F=∫0h∂ψ∂rdr=ψ(h)-ψ(0),
where β  and  F  are,  respectively,  the  flow  rates  in the fixed and wave frames.We note thath  represents the dimensionless form of the surface of the peristaltic wall:(3.9)h(z)=1+ϵcos⁡2πz,ϵ=a1d1.
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 (3.10)ψ=0,∂∂r(1r∂ψ∂r)=0,atr=0,ψ=F,1r∂ψ∂r=1,atr=h.


## 4. Method of  Solution

Integration of (2.15) along with boundary conditions (3.10) gives(4.1)∂∂r(1r∂ψ∂r)-χ2rψ=r2c1,
where c1 is an arbitrary function of z. Equation (4.1) after using the transformation(4.2)ϕ1=ψr+r2χ2c1
can be reduced into the following modified Bessel equation:(4.3)r2∂2ϕ1∂r2+r∂ϕ1∂r-(χ2r2+1)ϕ1=0,
whose solution along with (4.2) and boundary conditions (3.10) is given below:(4.4)ψ=r[rFhI0(χh)+hrI1(χh)-(2F+h2)I1(χr)]h2χI2(χh),
where I0,I1,and I2  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:(4.5)w=2FχI0(χh)+2hχI1(χh)-2χ(2f+h2)I0(χr)h2χI2(χh),dpdz=-χ2(2F+h2)I0(χr)h2I2(χh).
The expressions for pressure rise (ΔPλ) and frictional force (Fλ) per wavelength are, respectively, given by (4.6)Δpλ=∫02dpdzdz,Fλ=∫02πh2(-dpdz)dz.


## 5. Results and Discussion

To investigate the effects of rotation (Ω), magnetic parameter (M), material parameter (λ1), permeability of the porous medium (k),  and mean flux (F), we  plotted Figures 2–6.Show the stress distributions for tubea=0.3, b=0.4, and d=1.1.
(a)(b)(c)Show the velocity distributions fora=0.3, b=0.4, d=1.1, and λ=0.5.
(a)(b)Show the Graph of pressure rise versusF in symmetric channels for a=0.3, b=0.4, and d=1.1.
(a)(b)(c)Plot showingFλ  for F∈[1,1] for changing rotation (Ω),   (a), Hartman number M (b), and material parameter λ1   (c).
(a)(b)(c)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 (λ1) (c).
(a)(b)(c)The stress distribution (Srr), (Srz), and (Szz) 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 (Srr), (Srz), and (Szz) 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).Figure4 shows the variation of ΔPλ with flow rate F for values of rotation (Ω), magnetic parameter (M), and material parameter (λ1) for tube. We observe that the peristaltic pumping rate increases with increase of magnetic parameter (M) and material parameter (λ1), 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.Figure5 shows the variation of Fλ with flow rate F for values of rotation (Ω), magnetic parameter (M), and material parameter (λ1) for tube.  Figures 5(a), 5(b), and 5(c) display the influence of (Ω), magnetic parameter (M), and material parameter (λ1), respectively, for tube on Fλ.  Figure 5(a) refers to the case when F=-0.2. Here  it is noted that Fλ  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 Fλ  is negative and positive when -0.6≤F≤0.6   and 0.6<F, respectively. When Ω=0.8. Here it is noted that Fλ  is negative and positive when  -0.6≤F≤0.83  and 0.83<F, respectively. When Ω=0.4, Fλ  is negative for -0.6≤F≤1.4 and  positive for 1.4<F. Also, when Ω=0.0,  Fλ 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 Fλ 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 Fλ  increases with decrease of material parameter (λ1) when  -1≤F≤-0.48, and it increases with the increase of the magnetic parameter (M) when -0.48<F.Figure6 shows the distributions of the pressure gradient within a wavelength for various values of the rotation (Ω), magnetic parameter (M) and material parameter (λ1). 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 (λ1) increase the maximum amplitude of (dp/dz) when compared to the case with zero magnetic parameter and  zero material parameter (λ1).


## 6. Conclusion

The influence of the rotation (Ω), magnetic parameter (M), and material parameter (λ1) on the peristaltic flow of a Jeffrey fluid in tube has been analyzed. The analytical expressions are constructed for  axial velocity, Fλ,  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 Fλ   increases with increase of magnetic parameter (M) and material parameter (λ1) 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 (λ1=0).(iv)
The magnitudes of (dp/dz), ΔPλ,  and  Fλ  for Newtonian fluid are smaller than that of Jeffrey fluid.(v)
For large values of magnetic parameter (M) and material parameter (λ1), the magnitudes of ΔPλ and ΔPλ increase with decrease of the rotation (Ω).



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