PERISTALTIC VISCOELASTIC FLUID MOTION IN A TUBE

Peristaltic motion of viscoelastic incompressible fluid in an axisymmetric tube with a sinusoidal wave is studied theoretically in the case that the radius of the tube is small relative to the wavelength. Oldroyd flow has been considered in this study and the problem is formulated and analyzed using a perturbation expansion in terms of the variation of the wave number. This analysis can model the chyme movement in the small intestine by considering the chyme as an Oldroyd fluid. We found out that the pumping rate of Oldroyd fluid is less than that for a Newtonian fluid. Further, the effects of Reynolds number, Weissenberg number, amplitude ratio and wave number on the pressure rise and friction force have been discussed. It is found that the pressure rise does not depend on Weissenberg number at a certain value of flow rate. The results are studied for various values of the physical parameters of interest. 2000 Mathematics Subject Classification. 76Z05.

number using long-wavelength approximation and also the same problem was studied by Elshehawey et al. [3] in the case of nonuniform channel.
The purpose of this paper is to study the peristaltic motion of Oldroyd fluid in a tube.This problem can model the movement of the chyme, which may be considered as Oldroyd fluid, through small intestine.In our analysis, we assumed that the velocity components, the pressure, the shearing stress, and the flow rate may be expanded in a regular perturbation series in the wave number.Expressions for pressure rise, velocity components and friction force were obtained in terms of the flow rate, the occlusion, the Reynolds number, the Weissenberg number, and the wave number.

Formulation and analysis.
Consider the flow of an incompressible Oldroyd fluid in a circular tube of radius a.We assume an infinite wave train traveling with velocity c along the wall.Taking R and Z as cylindrical coordinates, the geometry of the wall surface is where b is the wave amplitude, λ is the wave length, and Z is the same direction of the wave propagation.
Choosing moving coordinates (r , z) (wave frame) which travel in the Z-direction with the same speed as the wave, the unsteady flow in the laboratory frame ( R, Z) can be treated as steady [8].The coordinates frame are related through where Ū, W and ū, w are, respectively, the radial and the axial velocity components in the corresponding coordinate systems.Equations of motion in the moving coordinates are The constitutive equations of Oldroyd fluid are, [7], where p is the pressure, τij , i, j = 1, 2, 3, are the components of the extra stress tensor, Γ is the relaxation time, ρ is the viscosity, and γij , i, j = 1, 2, 3, are the components of strain-rate tensor and given by The boundary conditions are Introducing the nondimensional variables and parameters where Wi is the Weissenberg number, δ is the wave number, Re is the Reynolds number, and h = h/a = 1 + (b/a) sin 2πz = 1 + ϕ sin 2πz, ϕ = b/a < 1, is the amplitude ratio, equations (2.3), (2.4), after using (2.5), become where h is a function of Z and t.The rate of volume flow in the moving frame (wave frame) is given by where h is a function of z.Using (3.2), one finds that the two rates of volume flow are related by The time-mean flow over a period T = λ/c at a fixed position Z is defined as which can be written, using (2.1) and (3.3), as Defining the dimensionless time-mean flows θ and F in the fixed and wave frame, respectively, as then making use of (3.6), equation (3.5) can be rewritten as where

Perturbation solution.
Beginning by expanding the following quantities in a power series of the small parameter δ as follows: 13 + δ 2 τ (2) ) and collecting terms of like powers of δ, we obtain three sets of coupled linear differential equations with their corresponding boundary conditions in u 0 ,w 0 ,u 1 ,w 1 , and u 2 ,w 2 for the first three powers of δ.The first set of differential equations in u 0 ,w 0 , subject to the corresponding boundary conditions, yields the following classical Poiseuille flow: ) where On substituting the zeroth-order solution (4.2) in the second set of differential equations and using its corresponding boundary conditions, the first-order solution can be obtained in the form ) where We now solve the second-order system.Using the zeroth-order and the first-order solutions in the third set of differential equations and using the boundary conditions, we obtain where (4.7) At this order, the perturbation solution for the axial velocity can be, using (4.2a), (4.4a), and (4.6), written as A close look at (4.8) reveals that the axial velocity is affected by the wave number, the Reynolds number and the viscoelastic parameter (Weissenberg number).
5. Pressure gradient.An expression for the pressure gradient, ∂p/∂z, can be obtained by substituting (4.1) into the dimensionless equation of motion (2.10) and equating the coefficients of like powers of δ, we obtain three sets of partial differential equations for ∂p 0 /∂z, ∂p 1 /∂z, and ∂p 2 /∂z.Using this form of ∂p/∂z, the pressure rise and the friction force per wavelength can be obtained.
The nondimensional pressure rise and the nondimensional friction force per wavelength are defined, respectively, as Since ∂p/∂z is periodic in z, the pressure rise and the friction force per wavelength in the longitudinal direction are independent of r , [5].Accordingly, the integrals in (5.1) can be evaluated on the axis at r = 0. Further, the pressure rise and friction force can be expanded as a power series in δ as where (5.3) We now use the zeroth-, first-, and second-order terms for the pressure gradient in (5.3), integrating from 0 to 1, then substituting in (5.2) we obtain where Here, we used the relation and ∆p (2) λ ,F (2) λ , and F (2) are, respectively, the pressure rise, the friction force and the flow rate in the wave frame to the second order in δ, and where (5.8) The substitution of (5.8) into (5.4) and (5.5) yields and where (5.11)

Results and conclusion.
It is clear that our results calculate the velocity, the pressure rise and the friction force without restrictions on the amplitude ratio, the Reynolds number and the Weissenberg number but we used a small wave number.Further, the results extend the work of Shapiro et al. [8] as well as it include the effect of Weissenberg number Wi.
In Figure 6.1, the dimensionless pressure rise (∆p λ ) (2) is graphed versus the dimensionless flow rate θ (2) for different values of Weissenberg number (Wi = 0, 0.04, 0.08) at wave number δ = 0.156 and Reynolds number Re = 0.1, for both cases (ϕ = 0.35 and ϕ = 0.6).As shown, for ϕ = 0.35, the effect of Weissenberg number is very small and the three curves coincide.But for ϕ = 0.6, the effect of Weissenberg number is very clear and show that the pumping rate of Oldroyd fluid is less than that for a Newtonian having a shear viscosity the same as Oldroyd fluid and the pressure rise decreases with increasing Weissenberg number.Further, it is clear that the pressure rise is independent on Weissenberg number at a certain value of flow rate and the peristaltic pumping, where (∆p λ ) (2) > 0 and θ (2) > 0, occur at 0 ≤ θ (2) ≤ 0.9 and the augmented pumping, where (∆p λ ) (2) < 0 and θ (2) > 0, occur at 0.9 ≤ θ (2) ≤ 1.5, for ϕ = 0.6.The linear relation for a Newtonian fluid is obvious in (5.9), with Wi = 0 and δ = 0. Figures 6.2 and 6.3 show the effect of the wave number δ on the pressure rise at Re = 0.1, Wi = 0.08, and ϕ = 0.35, 0.6, respectively.Figure 6.2 reveals that, for ϕ = 0.35, an increase in the wave number yields a slight increase in the magnitude of the pressure rise but for ϕ = 0.6, the effect of the wave number δ is very clear and the pressure rise decreases as wave number increases as shown in Figure 6.3.Shown in Figure 6.4 the dimensionless pressure rise versus flow rate at Wi = 0, δ = 0.02, ϕ = 0.6, and Re = 0, 50, 100.The results reveal that the magnitude of the pressure rise increases with increasing Reynolds number.The dimensionless friction force is plotted versus flow rate in Figures 6.5, 6.6, 6.7, and 6.8. Figure 6.5 shows the friction force versus flow rate at δ = 0.156, Re = 0.1, and Wi = 0, 0.04, 0.08 in both cases ϕ = 0.35 and ϕ = 0.6.It is shown that the friction force is independent on Weissenberg number at ϕ = 0.35 but its magnitude decreases with increasing Weissenberg number at ϕ = 0.6 and it does not depend on Weissenberg number at a certain value of flow rate in this case.Shown in Figures 6.6 and 6.7 the effect of wave number δ on the friction force at ϕ = 0.35 and ϕ = 0.6, respectively.We notice from Figure 6.6 that the magnitude of the friction force increases with increasing the wave number and it is independent on wave number at a certain value of flow rate.This result is different at ϕ = 0.6 as shown in Figure 6.7.Finally, the friction force is displayed versus flow rate in Figure 6.8 at Wi = 0, δ = 0.02, and ϕ = 0.6, for various values of Reynolds number (Re = 0, 50, 100).It is clear that the magnitude of friction force decreases with increasing Reynolds number and it does not depend on Reynolds number at a certain value of flow rate.