The Flow Separation of Peristaltic Transport for Maxwell Fluid between Two Coaxial Tubes

and Applied Analysis 3 of strain-rate tensor, and Γ is the relaxation time. The system of nonlinear differential equations has to be solved with the following boundary conditions: w 󸀠 = −c, u 󸀠 = 0, at r󸀠 = r󸀠 1 , w 󸀠 = −c, u 󸀠 = −c dr 󸀠 2 dz , at r󸀠 = r󸀠 2 . (8) Consider the following nondimensional variables and parameters:


Introduction
Peristalsis is produced by sequential waves of contractions in an elastic channel (tube), which push their fluid forward.In the urinary system, peristaltic movement is due to spontaneous muscular contractions of the ureteral wall which drives urine from the kidneys to the bladder through the ureters.Peristalsis is an important property of many biological systems which transports biofluids by its propulsive movements.It is responsible for the movement of chyme in the gastrointestinal tract, intrauterine fluid motion, vasomotion of the small blood vessels and in many glandular ducts.A number of analytical studies of peristaltic flows of different fluids have been reported [1][2][3][4][5].
Non-Newtonian fluids have wide-range applications, so they receive a great attention from scientists.The flows of non-Newtonian fluids are important because of their technological significance.Moreover, physiological flows indicate that non-Newtonian viscoelastic rheology is the correct way of properly describing the peristaltic flow through channels and tubes.Among many models, which have been employed to describe the non-Newtonian behavior exhibited by certain real fluids, Oldroyd fluid has obtained a special status; it includes elastic and memory effects exhibited by dilute solutions.Oldroyd fluid has been extensively used in many applications, and also results of simulations fit experimental data quite well [6,7].
Endoscope and catheters are very important tools for medical diagnosis and they have many clinical applications.The endoscope now is a very important tool used for determining the real causes responsible for many problems in the human organs through which the fluids are transported by peristaltic pumping such as, stomach and small intestine.Also from the fluid dynamic point of view, there is no difference between an endoscope and catheter.In medicine, a catheter is a tube that can be inserted into a body cavity, duct, or vessel.Catheters thereby allow drainage or injection of fluids.The process of inserting a catheter is called catheterization.Also, the insertion of an endoscope (catheter)

Basic Equations
The basic equations governing the motion of Oldroyd fluid take the form where  is the density, V  is the velocity vector, / denotes the material derivative, and T  is the Cauchy stress tensor.The Cauchy stress T  for an incompressible Oldroyd fluid can be expressed as where   is the pressure,  is the coefficient of viscosity, S  is the extra stress, Γ is the relaxation time, and ] is the retardation time.The Rivlin-Ericksen tensors are defined as It is worth mentioning that this model includes the Maxwell model for ] = 0 and the classical linear case for Γ = ] = 0.

Formulation of the Problem and Mathematical Model
In this paper, the flow of the Maxwell fluid is considered through the gap between coaxial tubes.The inner tube is rigid and the outer has a sinusoidal wave traveling down its walls.The geometry of the walls surface (Figure 1) is where  1 is the radius of the inner tube,  2 is the radius of the outer tube at any axial distance   , and  is the amplitude of the wave.Introducing a wave frame (  ,   ) moving with velocity  away from the fixed frame (  ,   ) by the transformation where (  ,   ) and (  ,   ) are the velocity components in the wave and fixed frame, respectively.After using this transformation then the mathematical formulation of this problem results in the following differential equations: The constitutive equations of Maxwell fluid are where  is the density,   is the pressure,    are the components of the extra stress tensor and γ  are the components Abstract and Applied Analysis 3 of strain-rate tensor, and Γ is the relaxation time.The system of nonlinear differential equations has to be solved with the following boundary conditions: Consider the following nondimensional variables and parameters: where  is the amplitude ratio, Re is the Reynolds number,  is the dimensionless wave number and Wi is the Weissenberg number, and  is the radius ratio (the ratio between the radius of the inner tube and the radius of the outer).To proceed, we nondimensionalize ((4), ( 6)-( 7)); this yields The non-dimensional boundary conditions will be

Rate of Volume Flow in the Annulus
The instantaneous volume flow rate in the fixed frame is given by where   2 is a function of   ,   .The rate of volume flow in the wave frame is given by where   2 is a function of   only.If we substitute (5) into (20) and make use of (21), we find that the two rates of volume flow are related through The time mean flow over a period  at a fixed position   is defined as Substituting ( 22) into (23) and integrating, we get On defining the dimensionless time-mean flows  and , respectively, in the fixed and wave frame as one finds that (24) may be written as where

Perturbation Solution
It is evident that (( 12)-( 18)) are highly nonlinear, and, as such, it is not possible to obtain the solutions in a closed form.In order to solve the present problem, we expanded the flow quantities in a power series of the small parameter  as follows: If we substitute (28) into (( 12)-( 19)) and separate the terms of different order in , we obtain the following systems of partial differential equations together with boundary conditions as follows.
System of Order Zero.Consider with the boundary conditions: System of Order One.Consider with the boundary conditions: Solving the above sets of equations with the corresponding boundary conditions, we get the following.
We point out that the zero order solution coincides with the solution obtained by Mekheimer and Abd elmaboud [22] and is still independent of the viscoelastic effects.
First-Order Problem.Substituting the zeroth-order solution ((33) and ( 34)) into the equation of motion obtained for first order, one finds that the solution of the first-order problem will be in the form Abstract and Applied Analysis where (  ) represents differentiation with respect to , and the coefficients Ω  and  6 that appear in the whole paper are listed in the appendix.The total expressions for the velocities and the pressure gradient up to first order can be obtained by substituting ((33)-( 38)) and  0 =  −  1 into (28) and neglecting the terms greater than ().
The pressure rise Δ, in nondimensional form, is given by where Λ = /.The integral in (39) is not integrable in a closed form; it is evaluated numerically with aid of the MATHEMATICA programme.

The Flow Separation Points
In cases of separation from bodies of smooth geometrical form, the surface streamlines are observed to leave (separate) the surface more or less tangentially.In the case where the boundary layer is laminar, in sufficient momentum exchange takes, the flow is unable to adjust to the increasing pressure and separates from the surface.Contemplation of the velocity field shows that it is plausible to identify the point of separation (  ) on the walls with points where the circulation vanishes (the vorticity equal zero) [23].A condition used to prognosticate the separation in boundary-layer theory is to set the vorticity equal to zero on the boundary walls [10], as (the result of ( 40) is omitted because of large output).We can solve the resultant equation (  | = 2 = 0 and   | = = 0) numerically using the modified Newton-Raphson method [24] to get the flow separation points   .The flow separation points   with different values of the amplitude ratio  in the tube ( = 0) and in the annulus ( ̸ = 0) are summarized in Table 1.
The table shows that there are two values of separation points   .An important point here is that there is no separation when  = 0 on the tube wall and on the annulus walls.Moreover, the separation points are different in tube than those in annulus.Also, the separation points are different from the amplitude ratio to the other one and from radius ratio to the other one.The effect of various parameters of interest on the wall vorticity will be discussed in the next section.

Numerical Results and Discussion
This section is divided into three subsections.In the first subsection, the effects of various parameters on the velocity distribution and wall vorticity are investigated.The pumping characteristics and shear stresses are discussed in the second subsection.The trapping phenomenon is illustrated in the last subsection.The emerging parameters of our analysis are the Reynolds number Re, wave number , the Weissenberg number Wi, the radius ratio , the amplitude ratio , and the average flow rate .These have easy physical interpretations:  is a measure of how large the semi depth of the peristaltic motion is, as compared to its wavelength.It is an aspect ratio and thus an expression of shallowness.The Reynolds number Re is formed with the wave speed, the amplitude, and the kinematic viscosity of the Newtonian part of the constitutive behavior.Wi measures the elastic contributions of the stress behavior. gives the ratio between the radius of the inner tube and the radius of the outer tube of the annulus. is a measure of how large the wave amplitude of the peristaltic wave is as compared to the radius of the outer tube.

Axial Velocity, Wall Vorticity, and Flow
Separation.This subsection describes the influences of various emerging parameters of our analysis on the transverse distributions of the longitudinal velocity at the gap of the annulus (, ) and the wall vorticity   on the outer and inner tube walls.Figures 2, 3, 4, 5, 6, 7, 8, and 9 display the effects of Re, , Wi, , , and  on (, ) and   .From Figures 2 and 3, owing to the limit of the no-slip boundary condition in (19), the velocity at wall tubes has the same value  = −1 in the wave frame for all values of the parameters; that is, the fluid velocity at the annulus walls moves with an opposite velocity to that of the wave that propagates on the outer wall of the annulus.As we expected, the magnitudes of forward and backward flows (in the annulus gap) increase when the wave number  (i.e, small wave length for the peristaltic wave, so the amplitude of the wave increases) and Reynolds number Re increase.Moreover, Figure 2, indicates that the location of the reflux phenomenon in the flow depends upon the magnitudes of Re and .Effects of the Weissenberg number Wi and the radius ratio  are illustrated in Figure 3.The curves for Wi match together and a slight difference occurs only for  = 0.01 near the inner wall.
For  = 0.1 the data are identical for Wi values.It is worth mentioning that the shear stress decreases with an increase in the relaxation parameter Γ, Wi = Γ/ 2 , so that the velocity increases.However, an increase of the radius ratio  produces a decrease of the velocity and more back flow occurs.The variation of the flow vorticity at the outer and inner tubes along the annulus is illustrated in Figures 4-9 for different values of the emerging parameters.These figures show that the vorticity increases in the wider part of the annulus,  ∈ (0, 0.5), while it decreases as going to the narrow part, (0.51,1), and separation occurs at the outer tube but at the inner one there are no separation points appearing (Figures 7-9).Moreover, the separation points positions are in the narrow part of the annulus.In solid annulus (no peristaltic  = 0,) we observe that the vorticity is constant along the annulus and, therefore, there are no separation points (this result is compatible with the results in Table 1.).By increasing the value of the amplitude ratio  (peristaltic transport), separation points appear but this depends on the position .

The Pumping Characteristics and Shear
Stresses.This subsection describes the influences of various emerging parameters of our analysis on the axial pressure gradient (/), the pressure rise per wavelength (Δ), and the shear stresses at the annulus walls.The effects of these parameters are shown in Figures 10,11  ∈ [0, 0.6] and [0.9, 1.0] the pressure gradient is relatively small; that is, the flow can easily pass without imposition of large pressure gradient.However, in a narrow part of the channel  ∈ [0.6, 0.9] a much larger pressure gradient is required to maintain the same flux to pass it, especially for the narrowest position near  = 0.8.This is in good agreement with the physical situation.Also from these two figures we observe the effect of  and  on the pressure gradient, for fixed values of the other parameters, where the amplitude of / increases with increasing  and . Figure 11 shows that the amplitude of the axial pressure gradient decreases as Reynolds number increases.Reynolds number is the ratio of inertial to viscous forces and with the increase of Reynolds number the viscous forces decrease so the velocity increases and so the pressure gradient decreases.Also, as the Wi increases (stresses decreases) the pressure gradient increases.In Figure 12, the pressure rise Δ is illustrated in terms of the wave amplitude  of the wall disturbance with various values of the radius ratio .Firstly, it is obvious that with increasing occlusion  the pressure drop increases; that is, an increase of the pressure gradient is needed to push the same flux to pass the gap annulus.Secondly, the pressure rise for an annulus is higher than that for a tube.Figure 12 is sectored so that the upper right-hand quadrant denotes the region of peristaltic pumping where  > 0 (positive pumping) and Δ > 0 (adverse pressure gradient).The lower righthand quadrant denotes the region of augmented pumping where  > 0 (positive pumping) and Δ < 0 (favorable pressure gradient) and the upper left-hand quadrant denotes the region of retrograde pumping (or backward pumping) where  < 0 and Δ > 0 (adverse pressure gradient); in this region the flows is opposite to the direction of the peristaltic motion.It is clear that the peristaltic pumping region becomes wider as the radius ratio , amplitude ratio  (the ratio between wave amplitude and the radius of the outer tube), increases.
The influences of various parameters of our analysis on shear stress  13 at the outer and inner walls are shown in Figures 13-16.The shear stress  13 in the outer tube varies along the annulus from small value in the wider part to large value in the narrow part but the shear stress in the inner tube has an opposite behavior.The maximum amplitude of the shear stress  13 increases with an increase in both  and  as displayed in Figure 13. Figure 14 shows that there is no variation in the shear stress  13 along the annulus in the case of solid wall annulus ( = 0), but an apparent variation is noticed at  > 0 (peristaltic movement).The figure also shows that, in the peristaltic movement case ( > 0), there is an apparent variation in the shear stress  13 with various values  With further increase in Re, we find the presence of a second eddy as seen in Figure 17(c).We can see that there is no substantial difference to be seen on trapping by changing Weissenberg parameter Wi as shown in Figure 18.In order to see the effect of the curvature  on trapping we have plotted Figure 19.
We see that the shape of the trapped bolus is deformed by increasing the values of the dimensionless wave number .
The streamlines graph for different values of radius ratio  is plotted in Figure 19.The "trapping" phenomena occur for  = 0.2 (Figure 19(a)).But for a further decrease in ( = 0) (tube), the trapped bolus is seen with large size and the trapping range increases gradually as in Figure 19(c).

Concluding Results
The mathematical model of peristaltic transport of flow in a gap between two coaxial tubes filled with Maxwell fluid is studied analytically.The main findings can be summarized as follows.
(i) There exist two values of a separation points at the outer wall of the annulus.An important result is that there is no separation when  = 0 (solid wall) on both the tube wall and the annulus walls.Moreover, the separation points are different in the tube than those in the annulus.
(ii) In the narrow part of the annulus, the maximum amplitude of the pressure gradient for Maxwell fluid is different from the Newtonian fluid.
(iii) Inertial force effect on the peristaltic flow is stronger for the case of larger amplitude ratio.
(iv) The shear stress  13 in the outer tube along the annulus varies from small values in the wider part to large values in the narrow part, but the shear stress in the inner tube behaves oppositely.
(v) The presence of the endoscope tube ( ̸ = 0) in an artery increases the pressure gradient and shear stress.Such a result seems reasonable from the physical and medical point of view.
(vi) The presence of a second eddy is seen by increasing the values of the Reynolds number. Appendix

Figure 1 :
Figure 1: Geometry of the problem.

Table 1 :
Computed values of the flow separation points   for different values of amplitude ratio  at  = 1, Re = 10, Wi = 0.01, and  = 0.01.