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

Abstract

We study the peristaltic mechanism of an incompressible non-Newtonian biofluid (namely, Maxwell model) in the annular region between two coaxial tubes. The inner tube represents the endoscope tube. The system of the governing nonlinear PDE is solved by using the perturbation method to the first order in dimensionless wavenumber. The modified Newton-Raphson method is used to predict the flow separation points along the peristaltic wall and the endoscope tube. The results show that the presence of the endoscope (catheter) tube in the artery increases the pressure gradient and shear stress. Such a result seems too reasonable from the physical and medical point of view.

1. 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 [1–5].

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].

Recently, peristaltic motion of non-Newtonian fluids has been an important subject in the field of chemical, biomedical, and environmental engineering and science. A considerable amount of the literature has been reported [8–17].

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) in an artery will alter the flow field and modify the pressure distribution. There are several investigations that studied the effect of an endoscope on peristaltic transport for different fluids [18–22].

So far, no attempt has been made to understand the effect of endoscope on the peristaltic motion for the Maxwell model which is a subclass of Oldroyd model. Therefore the main purpose of the present paper is to study the mathematical modeling of peristaltic transport of flow in a gap between two coaxial tubes, filled with an incompressible non-Newtonian fluid (Maxwell model). The inner tube is rigid (endoscope) and the outer one has wave trains moving independently. In this paper, we are concerned with viscoelastic flows governed by the Maxwell model. The mathematical modeling of this problem is explained below in brief.

2. Basic Equations

The basic equations governing the motion of Oldroyd fluid take the form ∇⋅𝐕^′𝜌=0,𝐷𝐕^′𝐷𝑡=∇⋅𝐓^′, (1) where 𝜌 is the density, 𝐕^′ is the velocity vector, 𝐷/𝐷𝑡 denotes the material derivative, and 𝐓^′ is the Cauchy stress tensor. The Cauchy stress 𝐓^′ for an incompressible Oldroyd fluid can be expressed as 𝐓^′=−𝑝^′𝐈+𝐒^′,𝐒^′+Γ𝑑𝐒^′𝑑𝑡−𝐋^′𝐒^′−𝐒^′𝐋^′𝑇=𝜇𝐀^′₁+𝜈𝑑𝐀^′₁𝑑𝑡−𝐋^′𝐀^′₁−𝐀^′₁𝐋^′𝑇, (2) where 𝑝^′ is the pressure, 𝜇 is the coefficient of viscosity, 𝐒^′ is the extra stress, Γ is the relaxation time, and 𝜈 is the retardation time. The Rivlin-Ericksen tensors are defined as 𝐀^′₁=∇𝐕^′+∇𝐕^′𝑇,𝐋^′=∇𝐕^′. (3)

It is worth mentioning that this model includes the Maxwell model for 𝜈=0 and the classical linear case for Γ=𝜈=0.

3. 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 𝑅^′₁=𝑎₁,𝑅^′₂=𝑎₂+𝑏sin2𝜋𝜆𝑍^′−𝑐𝑡^′, (4) where 𝑎₁ is the radius of the inner tube, 𝑎₂ is the radius of the outer tube at any axial distance 𝑍^′, and 𝑏 is the amplitude of the wave. Introducing a wave frame (𝑟^′,𝑧^′) moving with velocity 𝑐 away from the fixed frame (𝑅^′,𝑍^′) by the transformation 𝑧^′=𝑍^′−𝑐𝑡^′,𝑟^′=𝑅^′,𝑤^′=𝑊^′−𝑐,𝑢^′=𝑈^′, (5) where (𝑢^′,𝑤^′) and (𝑈^′,𝑊^′) 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: 𝜕𝑢^′𝜕𝑟^′+𝜕𝑤^′𝜕𝑧^′+𝑢^′𝑟^′𝜌𝑢=0,_′𝜕𝑢^′𝜕𝑟^′+𝑤^′𝜕𝑢^′𝜕𝑧^′=−𝜕𝑝^′𝜕𝑟^′−1𝑟^′𝜕𝜕𝑟^′𝑟^′𝑆^′₁₁−𝑆^′₂₂𝑟^′+𝜕𝑆^′₁₃𝜕𝑧^′,𝜌𝑢^′𝜕𝑤^′𝜕𝑟^′+𝑤^′𝜕𝑤^′𝜕𝑧^′=−𝜕𝑝^′𝜕𝑧^′−1𝑟^′𝜕𝜕𝑟^′𝑟^′𝑆^′₁₃+𝜕𝑆^′₃₃𝜕𝑧^′. (6)

Figure 1 

Geometry of the problem.

The constitutive equations of Maxwell fluid are 𝑆^′₁₁+Γ𝑢^′𝜕𝑆^′₁₁𝜕𝑟^′+𝑤^′𝜕𝑆^′₁₁𝜕𝑧^′−2𝑆^′₁₁𝜕𝑢^′𝜕𝑟^′−2𝑆^′₁₃𝜕𝑢^′𝜕𝑧^′=−𝜇𝛾̇_′11,𝑆^′₁₃+Γ𝑢^′𝜕𝑆^′₁₃𝜕𝑟^′+𝑤^′𝜕𝑆^′₁₃𝜕𝑧^′−𝑆^′₃₃𝜕𝑢^′𝜕𝑧^′−𝑆^′₁₁𝜕𝑤^′𝜕𝑟^′+𝑢^′𝑟^′𝑆^′₁₃=−𝜇𝛾̇_′13,𝑆^′₂₂+Γ𝑢^′𝜕𝑆^′₂₂𝜕𝑟^′+𝑤^′𝜕𝑆^′₂₂𝜕𝑧^′−2𝑢^′𝑟𝑆^′₂₂=−𝜇𝛾̇_′22,𝑆^′₃₃+Γ𝑢^′𝜕𝑆^′₃₃𝜕𝑟^′+𝑤^′𝜕𝑆^′₃₃𝜕𝑧^′−2𝑆^′₃₃𝜕𝑤^′𝜕𝑧^′−2𝑆^′₁₃𝜕𝑤^′𝜕𝑟^′=−𝜇𝛾̇_′33, (7) where 𝜌 is the density, 𝑃^′ is the pressure, 𝑆^′_𝑖𝑗 are the components of the extra stress tensor and 𝛾̇_′𝑖𝑗 are the components 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,at𝑟^′=𝑟^′₁,𝑤^′=−𝑐,𝑢^′=−𝑐𝑑𝑟^′₂𝑑𝑧^′,at𝑟^′=𝑟^′₂. (8)

Consider the following nondimensional variables and parameters: 𝑟𝑟=^′𝑎₂𝑧,𝑧=^′𝜆𝑤,𝑤=^′𝑐,𝑢=𝜆𝑢^′𝑎₂𝑐𝑎,𝑝=²²𝑝𝜆𝜇𝑐^′,𝑟₁=𝑟^′₁𝑎₂𝑏=𝜖,𝜙=𝑎₂,𝑆_𝑖𝑗=𝑎₂𝑆^′_𝑖𝑗𝑐𝜇,𝑟₂=𝑟^′₂𝑎₂=1+𝜙sin(2𝜋𝑧),Re=𝜌𝑐𝑎₂𝜇𝑎,𝛿=²𝜆𝑎,𝜖=¹𝑎₂,Wi=𝑐Γ𝑎₂, (9) where 𝜙 is the amplitude ratio, Re is the Reynolds number, 𝛿 is the dimensionless wave number and Wi is the Weissenberg number, and 𝜖 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 𝑟₁=𝜖,𝑟₂=1+𝜙sin(2𝜋𝑧),1𝑟𝜕()+𝜕𝑟𝑟𝑢𝜕𝑤𝜕𝑧=0,Re𝛿³𝑢𝜕𝑢𝜕𝑟+𝑤𝜕𝑢𝜕𝑧=−𝜕𝑝1𝜕𝑟−𝛿𝑟𝜕𝜕𝑟𝑟𝑆₁₁−𝑆₂₂𝑟+𝛿𝜕𝑆₁₃,𝜕𝑧Re𝛿𝑢𝜕𝑤𝜕𝑟+𝑤𝜕𝑤𝜕𝑧=−𝜕𝑝−1𝜕𝑧𝑟𝜕𝜕𝑟𝑟𝑆₁₃+𝛿𝜕𝑆₃₃,𝜕𝑧𝑆₁₁+Wi𝛿𝑢𝜕𝑆₁₁𝜕𝑟+𝑤𝜕𝑆₁₁𝜕𝑧−2𝑆₁₁𝜕𝑢𝜕𝑟−2𝛿𝑆₁₃𝜕𝑢𝜕𝑧=−2𝛿𝜕𝑢,𝜕𝑟𝑆₁₃+Wi𝛿𝑢𝜕𝑆₁₃𝜕𝑟+𝑤𝜕𝑆₁₃𝜕𝑧−𝛿𝑆₃₃𝜕𝑢+𝑢𝜕𝑧𝑟𝑆₁₃−𝑆₁₁𝜕𝑤𝛿𝜕𝑟=−²𝜕𝑢+𝜕𝑧𝜕𝑤,𝜕𝑟𝑆₂₂+Wi𝛿𝑢𝜕𝑆₂₂𝜕𝑟+𝑤𝜕𝑆₂₂−𝜕𝑧2𝑢𝑟𝑆₂₂𝑢=−2𝛿𝑟,𝑆₃₃+Wi𝛿𝑢𝜕𝑆₃₃𝜕𝑟+𝑤𝜕𝑆₃₃𝜕𝑧−2𝑆₃₃𝜕𝑤𝜕𝑧−2𝑆₁₃𝜕𝑤𝜕𝑟=−2𝛿𝜕𝑤.𝜕𝑧 (10)(11)(12)(13)(14)(15)(16)(17)(18)

The non-dimensional boundary conditions will be 𝑤=−1,𝑢=0,at𝑟=𝜖,𝑤=−1,𝑢=−𝑑𝑟₂𝑑𝑧,at𝑟=𝑟₂. (19)

4. Rate of Volume Flow in the Annulus

The instantaneous volume flow rate in the fixed frame is given by 𝑄=2𝜋^𝑅′2_𝑅′1𝑅^′𝑊^′𝑑𝑅^′, (20) where 𝑅^′₂ is a function of 𝑍^′,𝑡^′. The rate of volume flow in the wave frame is given by 𝑞=2𝜋^𝑟′2_𝑟′1𝑟^′𝑤^′𝑑𝑟^′, (21) where 𝑟^′₂ is a function of 𝑧^′ only. If we substitute (5) into (20) and make use of (21), we find that the two rates of volume flow are related through 𝑄=𝑞+𝜋𝑐𝑟₂^′2−𝑟₁^′2. (22)

The time mean flow over a period 𝑇 at a fixed position 𝑍^′ is defined as 𝑄^′=1𝑇^𝑇₀𝑄𝑑𝑡. (23) Substituting (22) into (23) and integrating, we get 𝑄^′=𝑞+𝜋𝑐𝑎²₂𝜙1+²2−𝜋𝑐𝑟₁^′2. (24)

On defining the dimensionless time-mean flows 𝑄 and 𝐹, respectively, in the fixed and wave frame as 𝑄𝑄=^′𝜋𝑐𝑎²₂𝑞,𝐹=𝜋𝑐𝑎²₂, (25) one finds that (24) may be written as 𝜙𝑄=𝐹+1+²2−𝜖², (26) where 𝐹=2^𝑟2_𝑟1𝑟𝑤𝑑𝑟. (27)

5. 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 𝛿 as follows: 𝑢=𝑢₀+𝛿𝑢₁+𝛿²𝑢₂+⋯,𝑤=𝑤₀+𝛿𝑤₁+𝛿²𝑤₂+⋯,𝑝=𝑝₀+𝛿𝑝₁+𝛿²𝑝₂𝜏+⋯,_𝑖𝑗=𝑆⁽⁰⁾_𝑖𝑗+𝛿𝑆⁽¹⁾_𝑖𝑗+𝛿²𝑆⁽²⁾_𝑖𝑗+⋯where𝑖,𝑗=1,2,3,𝐹=𝐹₀+𝛿𝐹₁+𝛿²𝐹₂+⋯. (28)

If we substitute (28) into ((12)–(19)) and separate the terms of different order in 𝛿, we obtain the following systems of partial differential equations together with boundary conditions as follows.

System of Order Zero. Consider 0=−𝜕𝑝₀,𝜕𝑟0=−𝜕𝑝₀−1𝜕𝑧𝑟𝜕𝜕𝑟𝑟𝑆⁰₁₃,𝑆⁰₁₁𝑆=0,₀₁₃+Wi−𝑆⁰₁₁𝜕𝑤₀𝜕𝑟=−𝜕𝑤₀,𝑆𝜕𝑟₀₂₂𝑆=0,₀₃₃+Wi−2𝑆⁰₁₃𝜕𝑤₀𝜕𝑟=0, (29) with the boundary conditions: 𝑤₀=−1,𝑢₀=0,at𝑟=𝑟₁𝑤=𝜖,₀=−1,𝑢₀=−𝑑𝑟₂𝑑𝑧,at𝑟=𝑟₂. (30)

System of Order One. Consider 0=−𝜕𝑝₁−1𝜕𝑟𝑟𝜕𝜕𝑟𝑟𝑆⁰₁₁−1𝑟𝑆⁰₂₂,Re𝑢₀𝜕𝑤₀𝜕𝑟+𝑤₀𝜕𝑤₀𝜕𝑧=−𝜕𝑝₁−1𝜕𝑧𝑟𝜕𝜕𝑟𝑟𝑆¹₁₃+𝜕𝑆⁰₃₃,𝑆𝜕𝑧₁₁₁+Wi𝑢₀𝜕𝑆⁰₁₁𝜕𝑟+𝑤₀𝜕𝑆⁰₁₁𝜕𝑧−2𝑆⁰₁₁𝜕𝑢₀𝜕𝑟=−2𝜕𝑢₀,𝑆𝜕𝑟₁₁₃+Wi𝑢₀𝜕𝑆⁰₁₃𝜕𝑟+𝑤₀𝜕𝑆⁰₁₃+𝑢𝜕𝑧𝑟⁰𝑆⁰₁₃−𝑆⁰₁₁𝜕𝑤₁𝜕𝑟−𝑆¹₁₁𝜕𝑤₀𝜕𝑟=−𝜕𝑤₁,𝑆𝜕𝑟₁₂₂+Wi𝑢₀𝜕𝑆⁰₂₂𝜕𝑟+𝑤₀𝜕𝑆⁰₂₂−𝜕𝑧2𝑢₀𝑟𝑆⁰₂₂𝑢=−2⁰𝑟,𝑆¹₃₃+Wi𝑢₀𝜕𝑆⁰₃₃𝜕𝑟+𝑤₀𝜕𝑆⁰₃₃𝜕𝑧−2𝑆⁰₃₃𝜕𝑤₀𝜕𝑧−2𝑆⁰₁₃𝜕𝑤₁𝜕𝑟−2𝑆¹₁₃𝜕𝑤₀𝜕𝑟=−2𝜕𝑤₀,𝜕𝑧 (31) with the boundary conditions: 𝑤₁=0,𝑢₁=0,at𝑟=𝑟₁=𝜖,𝑤₁=0,𝑢₁=0,at𝑟=𝑟₂. (32)

Solving the above sets of equations with the corresponding boundary conditions, we get the following.

Zero-Order Problem. Consider 𝑤₀(𝑟,𝑧)=−1+log[𝑟]Ω₇+Ω₈+𝑟²Ω₉,𝑢₀(𝑟,𝑧)=Ω₃−𝑟Ω₄+𝑟log[𝑟]Ω₅−𝑟Ω₆,𝑑𝑝₀=8𝐹𝑑𝑧⁰−𝜖²+𝑟²₂log[𝜖]−log𝑟₂×𝜖²−𝑟²₂−𝜖²+𝑟²₂+𝜖²log[𝜖]+𝑟²₂log[𝜖]−𝜖²log𝑟₂−𝑟²₂log𝑟₂⁻¹. (33)(34)(35)

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 𝑤₁(𝑟,𝑧)=𝑑𝑝₁𝑟𝑑𝑧4²+𝜖²/4−𝑟²₂/4log[𝑟]log𝑟₂[𝜖]+𝜖−log₂log𝑟₂−𝑟²₂log[𝜖]4log[𝜖]−log𝑟₂+𝑟²Ω₁₇+𝑟²log[𝑟]Ω₁₈+𝑟⁴Ω₁₉+log[𝑟]Ω₂₀+Ω₂₁+log[𝑟]Ω₂₂+Ω₁₅+2Wilog[𝑟]Ω₅Ω₇𝑟+29𝑟³ReΩ₃Ω₉+𝑟Ω₁₆+4Wilog[𝑟]Ω₅Ω₉+14𝑟²Relog[𝑟]²Ω₇Ω^′₇+12Wilog[𝑟]²Ω₇Ω^′₇+1𝑟36⁶ReΩ₉Ω^′₉,𝑢₁(𝑟,𝑧)=𝑐₆+1𝑟−1𝑟36⁶6Ω^′₁₉−Ω₂₀−𝑟Ω^′₁₅−2WiΩ₇Ω^′₅+Ω₅Ω^′₇−2𝑟45⁵ReΩ₉Ω^′₃+Ω₃Ω^′₉−19𝑟³3Ω^′₁₆−4WiΩ₉Ω^′₅+Ω₅Ω^′₉−1𝑟16²𝑟²Re+4Wilog[𝑟]²×Ω₇^′2+Ω₇Ω₇^′′−1𝑟128⁴32Ω^′₁₇−8Ω^′₁₈+ReΩ₇^′2+Ω₇Ω₇^′′−𝑟²16𝑟₂log[𝜖]−log𝑟₂²×𝑑𝑝₁𝑟𝑑𝑧^′2[𝜖]×𝜖1+2log₂+𝑟²₂2log𝑟₂−1−2log[𝜖]+2𝑟₂log[𝜖]−log𝑟₂²×4Ω^′₂₁−2Ω^′₂₂+WiΩ₇^′2+Ω₇Ω₇^′′[𝑟]1−log6𝑟⁶Ω^′₂₀+2𝑟WiΩ₇Ω^′₅+Ω₅Ω^′₇+43𝑟³WiΩ₉Ω^′₅+Ω₅Ω^′₉+1𝑟32⁴8Ω^′₁₈−ReΩ₇^′2+Ω₇Ω₇^′′+𝑟²8𝑟₂log[𝜖]−log𝑟₂²×𝑑𝑝₁𝑟𝑑𝑧^′2−𝜖²𝑘𝑖+𝑟²₂[𝜖]𝑟1+2log−2log₂+2𝑟₂log[𝜖]−log𝑟₂²×2Ω^′₂₂−WiΩ₇^′2+Ω₇Ω₇^′′−𝑟⁸ReΩ288^9′2+Ω₉Ω₉^′′,𝑑𝑝₁−𝜖𝑑𝑧=1⁴8+𝜖⁴8log[𝜖]−log𝑟₂−𝜖²𝑟²₂4log[𝜖]−log𝑟₂+𝑟⁴₂1+log[𝜖]−log𝑟₂8log[𝜖]−log𝑟₂⁻¹×−𝐹₁−2𝜖Ω₁₅−23𝜖³Ω₁₆−12𝜖⁴Ω₁₇+18𝜖⁴Ω₁₈−12𝜖⁴log[𝜖]Ω₁₈−13𝜖⁶Ω₁₉+1𝜖18⁶Ω₂₀−13𝜖⁶log[𝜖]Ω₂₀+1𝑟18⁶²6Ω₁₉+−1+6log𝑟₂Ω₂₀−𝜖²Ω₂₁+12𝜖²Ω₂₂−𝜖²log[𝜖]Ω₂₂+4Wi𝜖Ω₅Ω₇−4Wi𝜖log[𝜖]Ω₅Ω₇+2𝑟₂Ω₁₅+2Wi−1+log𝑟₂Ω₅Ω₇−445Re𝜖⁵Ω₃Ω₉+445Re𝑟⁵₂Ω₃Ω₉+89Wi𝜖³Ω₅Ω₉−83Wi𝜖³log[𝜖]Ω₅Ω₉+29𝑟³₂3Ω₁₆+4Wi−1+3log𝑟₂Ω₅Ω₉−14Wi𝜖²Ω₇Ω^′₇−164Re𝜖⁴Ω₇Ω^′₇+12Wi𝜖²log[𝜖]Ω₇Ω^′₇+116Re𝜖⁴log[𝜖]Ω₇Ω^′₇−12Wi𝜖²log[𝜖]²Ω₇Ω^′₇−18Re𝜖⁴log[𝜖]²Ω₇Ω^′₇+14𝑟²₂4Ω₂₁+−2+4log𝑟₂Ω₂₂+Wi1+2−1+log𝑟₂×log𝑟₂Ω₇Ω^′₇+1𝑟64⁴²32Ω₁₇+8−1+4log𝑟₂Ω₁₈+Re1−4log𝑟₂+8log𝑟₂²Ω₇Ω^′₇−1144Re𝜖⁸Ω₉Ω^′₉+1144Re𝑟⁸₂Ω₉Ω^′₉, (36)(37)(38) where (^′) represents differentiation with respect to 𝑧, and the coefficients Ω_𝑖𝑗 and 𝑐₆ 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 𝐹₀=𝐹−𝛿𝐹₁ into (28) and neglecting the terms greater than 𝑂(𝛿).

The pressure rise Δ𝑝, in nondimensional form, is given by Δ𝑝=^Λ₀𝑑𝑝𝑑𝑧𝑑𝑧, (39) where Λ=𝐿/𝜆.

The integral in (39) is not integrable in a closed form; it is evaluated numerically with aid of the MATHEMATICA programme.

6. 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 (𝑧_𝑠) 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 𝜂_𝑧=𝜕𝑢−𝜕𝑧𝜕𝑤𝜕𝑟,at𝑟=𝜖,𝑟₂ (40) (the result of (40) is omitted because of large output). We can solve the resultant equation (𝜂_𝑧|_𝑟=𝑟2=0and𝜂_𝑧|_𝑟=𝜖=0) numerically using the modified Newton-Raphson method [24] to get the flow separation points 𝑧_𝑠. The flow separation points 𝑧_𝑠 with different values of the amplitude ratio 𝜙 in the tube (𝜖=0) and in the annulus (𝜖≠0) are summarized in Table 1.

Table 1 

Computed values of the flow separation points 𝑧_𝑠 for different values of amplitude ratio 𝜙 at 𝑄=1, Re=10, Wi=0.01, and 𝛿=0.01.

The table shows that there are two values of separation points 𝑧_𝑠. An important point here is that there is no separation when 𝜙=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.

7. 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 𝛿, the Weissenberg number Wi, the radius ratio 𝜖, the amplitude ratio 𝜙, and the average flow rate 𝑄. These have easy physical interpretations: 𝛿 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. 𝜖 gives the ratio between the radius of the inner tube and the radius of the outer tube of the annulus. 𝜙 is a measure of how large the wave amplitude of the peristaltic wave is as compared to the radius of the outer tube.

7.1. 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 𝑤(𝑟,𝑧) and the wall vorticity 𝜂_𝑧 on the outer and inner tube walls. Figures 2, 3, 4, 5, 6, 7, 8, and 9 display the effects of Re, 𝛿, Wi, 𝜖, 𝜙, and 𝑄 on 𝑤(𝑟,𝑧) and 𝜂_𝑧. 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 𝑤=−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 𝛿 (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 𝛿. Effects of the Weissenberg number Wi and the radius ratio 𝜖 are illustrated in Figure 3. The curves for Wi match together and a slight difference occurs only for 𝜖=0.01 near the inner wall. For 𝜖=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=𝑐Γ/𝑎₂, so that the velocity increases. However, an increase of the radius ratio 𝜖 produces a decrease of the velocity and more back flow occurs.

Figure 2 

The velocity distribution 𝑤, across the annulus with different values of Re and 𝛿 at 𝜖=0.01, Wi=0.01, 𝑄=1.5, 𝜙=0.4, and 𝑧=0.1, where 𝑟∈[𝜖,𝑟2].

Figure 3 

The velocity distribution 𝑤, across the annulus with different values of 𝜖 and Wi at Re=3, 𝛿=0.1, 𝑄=1.5, 𝜙=0.4, and 𝑧=0.1, where 𝑟∈[𝜖,𝑟2].

Figure 4 

Variation of wall vorticity 𝜂𝑟2𝑧 versus 𝑧 with different values of 𝛿 and Wi at Re=5, 𝜖=0.01, 𝑄=1.5, and  𝜙=0.4.

Figure 5 

Variation of wall vorticity 𝜂𝑟2𝑧 versus 𝑧 with different values of 𝜖 and Re at Wi=0.01, 𝛿=0.1, 𝑄=1, and 𝜙=0.4.

Figure 6 

Variation of wall vorticity 𝜂𝑟2𝑧 versus 𝑧 with different values of 𝑄 and 𝜙 at Wi=0.02, 𝛿=0.1, 𝜖=0.01, and Re=1.

Figure 7 

Variation of wall vorticity 𝜂𝜖𝑧 versus 𝑧 with different values of 𝛿 and Wi at Re=5, 𝜖=0.01, 𝑄=1.5, and 𝜙=0.4.

Figure 8 

Variation of wall vorticity 𝜂𝜖𝑧 versus 𝑧 with different values of 𝜖 and Re at Wi=0.01, 𝛿=0.1, 𝑄=1, and 𝜙=0.4.

Figure 9 

Variation of wall vorticity 𝜂𝜖𝑧 versus 𝑧 with different values of  𝑄 and 𝜙 at Wi=0.02, 𝛿=0.1, 𝜖=0.01, and Re=1.

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, 𝑧∈(0,0.5), while it decreases as going to the narrow part, 𝑧𝜖(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 𝜙=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 𝜙 (peristaltic transport), separation points appear but this depends on the position 𝑧.

7.2. The Pumping Characteristics and Shear Stresses

This subsection describes the influences of various emerging parameters of our analysis on the axial pressure gradient (𝑑𝑝/𝑑𝑧), the pressure rise per wavelength (Δ𝑝), and the shear stresses at the annulus walls. The effects of these parameters are shown in Figures 10, 11, 12, 13, 14, 15, and 16. Figures 10 and 11 illustrate that in the wider part of the channel 𝑧∈[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 𝑧∈[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 𝑧=0.8. This is in good agreement with the physical situation. Also from these two figures we observe the effect of 𝜖 and 𝛿 on the pressure gradient, for fixed values of the other parameters, where the amplitude of 𝑑𝑝/𝑑𝑥 increases with increasing 𝜖 and 𝛿. 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 Δ𝑝 is illustrated in terms of the wave amplitude 𝜙 of the wall disturbance with various values of the radius ratio 𝜖. Firstly, it is obvious that with increasing occlusion 𝜙 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 𝑄>0 (positive pumping) and Δ𝑝>0 (adverse pressure gradient). The lower right-hand quadrant denotes the region of augmented pumping where 𝑄>0 (positive pumping) and Δ𝑝<0 (favorable pressure gradient) and the upper left-hand quadrant denotes the region of retrograde pumping (or backward pumping) where 𝑄<0 and Δ𝑝>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 𝜖, amplitude ratio 𝜙 (the ratio between wave amplitude and the radius of the outer tube), increases.

Figure 10 

Variation of pressure gradient versus 𝑧 with different values of 𝛿 and 𝜖 at Wi=0.05, 𝑄=0.5, 𝜙=0.4, and Re=2.

Figure 11 

Variation of pressure gradient versus 𝑧 with different values of Wi and Re at 𝛿=0.1, 𝑄=0.1, 𝜙=0.4, and 𝜖=0.1.

Figure 12 

The variation of pressure rise over the length Δ𝑝 with 𝑄 for different values of 𝜙 and 𝜖 at Wi=0.01, Re=10, and 𝛿=0.08.

Figure 13 

Wall shear stress 𝜏13 at the outer wall versus 𝑧 with different values of 𝛿 and 𝜖 at Wi=0.01, 𝑄=1.5, 𝜙=0.4, and Re=10.

Figure 14 

Wall shear stress 𝜏13 at the outer wall versus 𝑧 with different values of 𝜙 and Wi at 𝜖=0.1, 𝑄=1, 𝛿=0.08, and Re=10.

Figure 15 

Wall shear stress 𝜏13 at the inner wall versus 𝑧 with different values of 𝛿 and 𝜖 at Wi=0.01, 𝑄=1.5, 𝜙=0.4, and Re=10.

Figure 16 

Wall shear stress 𝜏13 at the inner wall versus 𝑧 with different values of 𝜙 and Wi at 𝜖=0.1, 𝑄=1, 𝛿=0.08, and Re=10.

The influences of various parameters of our analysis on shear stress 𝜏₁₃ at the outer and inner walls are shown in Figures 13–16. The shear stress 𝜏₁₃ 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 𝜏₁₃ increases with an increase in both 𝜖 and 𝛿 as displayed in Figure 13. Figure 14 shows that there is no variation in the shear stress 𝜏₁₃ along the annulus in the case of solid wall annulus (𝜙=0), but an apparent variation is noticed at 𝜙>0 (peristaltic movement). The figure also shows that, in the peristaltic movement case (𝜙>0), there is an apparent variation in the shear stress 𝜏₁₃ with various values of the Weissenberg number Wi along the annulus. The inverse results are seen through Figures 15 and 16 for the shear stress at the inner wall.

7.3. Trapping Phenomenon

An important phenomenon in peristaltic motion is trapping. In a wave frame, the streamline under particular conditions split to trap a bolus which moves as a whole with the speed of the peristaltic wave. The effects of the Reynolds number Re, Weissenberg parameter Wi, the wave number 𝛿, and radius ratio 𝜖 on trapping can be observed through Figures 17, 18, 19, and 20. The effect of inertia on trapping is shown in Figure 17. The figure displays that the streamlines corresponding to Re=0 in Figure 17(a) are in the form of trapped bolus and by elevating the values of Re more trapped boluses appear as in Figure 17(b). 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 𝛿 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 𝛿. The streamlines graph for different values of radius ratio 𝜖 is plotted in Figure 19. The “trapping” phenomena occur for 𝜖=0.2 (Figure 19(a)). But for a further decrease in (𝜖=0) (tube), the trapped bolus is seen with large size and the trapping range increases gradually as in Figure 19(c).

(a)

(b)

(c)

Figure 17 

The effect of Reynolds number on the streamline patterns in the wave frame for different three values of Re: (a) Re=0, (b) Re=10, (c) Re=50 at Wi=0.02, 𝜖=0.1, 𝑄=0.7, 𝛿=0.156, and 𝜙=0.3, 𝑟∈[𝜖,𝑟2].

(a)

(b)

(c)

Figure 18 

Graph of streamlines for different three values of Wi: (a) Wi=0, (b) Wi=0.04, (c) Wi=0.08 at 𝜖=0.1, 𝑄=0.7, 𝛿=0.156, Re=0.1, and 𝜙=0.3, 𝑟∈[𝜖,𝑟2].

(a)

(b)

(c)

Figure 19 

Graph of streamlines for different three values of 𝛿: (a) 𝛿=0, (b) 𝛿=0.08, (c) 𝛿=0.156 at 𝜖=0.13, 𝑄=0.7, Wi=0.05, Re=10, and 𝜙=0.3, 𝑟∈[𝜖,𝑟2].

(a)

(b)

(c)

Figure 20 

Graph of streamlines for different three values of 𝜖: (a) 𝜖=0.2, (b) 𝜖=0.1, (c) 𝜖=0.0 at 𝛿=0.156, 𝑄=0.7, Wi=0.05, Re=15, and 𝜙=0.4, 𝑟∈[𝜖,𝑟2].

8. 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 𝜙=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 𝜏₁₃ 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 (𝜖≠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

Ω₁=2𝑟₂𝜖²−𝑟²₂𝜖²−1+log[𝜖]−log𝑟₂+𝑟²₂1+log[𝜖]−log𝑟₂×log[𝜖]−log𝑟₂,Ω₂=2𝜖²𝑟²₂+𝜖⁴−1+log[𝜖]−log𝑟₂+𝑟⁴₂−1−log[𝜖]+log𝑟₂,Ω₃=𝜖𝐹₀−𝜖²+𝑟²₂×𝜖²+𝑟²₂−1−2log[𝜖]+2log𝑟₂𝑟^′₂×Ω₁⁻¹,Ω₄=𝐹₀−𝜖²+𝑟²₂×𝜖²+𝑟²₂−1−2log[𝜖]+2log𝑟₂𝑟^′₂×Ω₁⁻¹,Ω₅=2𝐹₀−𝜖²+𝑟²₂×𝜖²+𝑟²₂−1−2log[𝜖]+2log𝑟₂𝑟^′₂×Ω₁⁻¹,Ω₆=2𝐹₀−𝜖²+𝑟²₂[𝜖]×𝜖log₂+𝑟²₂−1−2log[𝜖]+2log𝑟₂𝑟^′₂×Ω₁⁻¹,Ω₇=−2𝜖²𝐹₀−𝜖²+𝑟²₂Ω₂+2𝑟²₂𝐹₀−𝜖²+𝑟²₂Ω₂,Ω₈=2𝜖²𝐹₀−𝜖²+𝑟²₂log𝑟₂Ω₂−2𝑟²₂𝐹₀−𝜖²+𝑟²₂log[𝜖]Ω₂,Ω₉=2𝐹₀−𝜖²+𝑟²₂log[𝜖]−log𝑟₂Ω₂,Ω₁₀=2WiΩ₄Ω₇+2WiΩ₆Ω₇,Ω₁₁=4WiΩ₃Ω₉−4WiΩ₄Ω₉+4WiΩ₅Ω₉−4WiΩ₆Ω₉,Ω₁₂=WiΩ₈Ω^′₇−WiΩ^′₇,Ω₁₃=−4WiΩ₄Ω₉−4WiΩ₆Ω₉+WiΩ₉Ω^′₇−2WiΩ^′₉+2WiΩ₈Ω^′₉,Ω₁₄=2Ω₅Ω₉+Ω₇Ω^′₉,Ω₁₅=−Ω₁₀+4WiΩ₅Ω₇,Ω₁₆=Ω₁₁+ReΩ₃Ω₇−4WiΩ₅Ω₉,Ω₁₇=Ω₁₃2−12WiΩ₁₄−14ReΩ₄Ω₇−14ReΩ₅Ω₇−14ReΩ₆Ω₇+14ReΩ^′₇+38ReΩ₇Ω^′₇−14ReΩ₈Ω^′₇−14ReΩ^′₈−14ReΩ₇Ω^′₈+14ReΩ₈Ω^′₈,Ω₁₈=WiΩ₁₄+14ReΩ₅Ω₇−14ReΩ^′₇−12ReΩ₇Ω^′₇+14ReΩ₈Ω^′₇+14ReΩ₇Ω^′₈,Ω₁₉1=−8ReΩ₄Ω₉−116ReΩ₅Ω₉−18ReΩ₆Ω₉−132ReΩ₉Ω^′₇+116ReΩ₉Ω^′₈−116ReΩ^′₉−132ReΩ₇Ω^′₉+116ReΩ₈Ω^′₉+12WiΩ₉Ω^′₉,Ω₂₀=18ReΩ₅Ω₉+116ReΩ₉Ω^′₇+116ReΩ₇Ω^′₉,Ω₂₁=−𝜖²Ω₁₇−𝜖²log[𝜖]Ω₁₇−log[𝜖]+log𝑟₂+𝑟²₂log[𝜖]Ω₁₇−log[𝜖]+log𝑟₂−𝜖²log[𝜖]Ω₁₈−𝜖²log[𝜖]²Ω₁₈−log[𝜖]+log𝑟₂+𝑟²₂log[𝜖]log𝑟₂Ω₁₈−log[𝜖]+log𝑟₂−𝜖⁴Ω₁₉+log[𝜖]Ω₂₀−𝜖⁴log[𝜖]Ω₁₉+log[𝜖]Ω₂₀−log[𝜖]+log𝑟₂+𝑟⁴₂log[𝜖]Ω₁₉+log𝑟₂Ω₂₀−log[𝜖]+log𝑟₂−Ω₁₅+2Wilog[𝜖]Ω₅Ω₇𝜖−log[𝜖]Ω₁₅+2Wilog[𝜖]Ω₅Ω₇𝜖−log[𝜖]+log𝑟₂+log[𝜖]Ω₁₅+2Wilog𝑟₂Ω₅Ω₇𝑟₂−log[𝜖]+log𝑟₂−29Re𝜖³Ω₃Ω₉−2Re𝜖³log[𝜖]Ω₃Ω₉9−log[𝜖]+log𝑟₂+2Re𝑟³₂log[𝜖]Ω₃Ω₉9−log[𝜖]+log𝑟₂−𝜖Ω₁₆+4Wilog[𝜖]Ω₅Ω₉−𝜖log[𝜖]Ω₁₆+4Wilog[𝜖]Ω₅Ω₉−log[𝜖]+log𝑟₂+𝑟₂log[𝜖]Ω₁₆+4Wilog𝑟₂Ω₅Ω₉−log[𝜖]+log𝑟₂−12Wilog[𝜖]²Ω₇Ω^′₇−14Re𝜖²log[𝜖]²Ω₇Ω^′₇−Wilog[𝜖]³Ω₇Ω^′₇2−log[𝜖]+log𝑟₂−Re𝜖²log[𝜖]³Ω₇Ω^′₇4−log[𝜖]+log𝑟₂+Wilog[𝜖]log𝑟₂²Ω₇Ω^′₇2−log[𝜖]+log𝑟₂+Re𝑟²₂log[𝜖]log𝑟₂²Ω₇Ω^′₇4−log[𝜖]+log𝑟₂−136Re𝜖⁶Ω₉Ω^′₉−Re𝜖⁶log[𝜖]Ω₉Ω^′₉36−log[𝜖]+log𝑟₂+Re𝑟⁶₂log[𝜖]Ω₉Ω^′₉36−log[𝜖]+log𝑟₂,Ω₂₂1=−−log[𝜖]+log𝑟₂×−𝜖²Ω₁₇+𝑟²₂Ω₁₇−𝜖²log[𝜖]Ω₁₈+𝑟²₂log𝑟₂Ω₁₈−𝜖⁴Ω₁₉+log[𝜖]Ω₂₀+𝑟⁴₂Ω₁₉+log𝑟₂Ω₂₀−Ω₁₅+2Wilog[𝜖]Ω₅Ω₇𝜖+Ω₁₅+2Wilog𝑟₂Ω₅Ω₇𝑟₂−29Re𝜖³Ω₃Ω₉+29Re𝑟³₂Ω₃Ω₉−𝜖Ω₁₆+4Wilog[𝜖]Ω₅Ω₉+𝑟₂Ω₁₆+4Wilog𝑟₂Ω₅Ω₉−12Wilog[𝜖]²Ω₇Ω^′₇−14Re𝜖²log[𝜖]²Ω₇Ω^′₇+12Wilog𝑟₂²Ω₇Ω^′₇+14Re𝑟²₂log𝑟₂²Ω₇Ω^′₇−136Re𝜖⁶Ω₉Ω^′₉+136Re𝑟⁶₂Ω₉Ω^′₉,𝑐₆1=−𝜖−1𝜖36⁶6Ω^′₁₉−Ω^′₂₀−𝜖Ω^′₁₅−2WiΩ₇Ω^′₅+Ω₅Ω^′₇−245Re𝜖⁵Ω₉Ω^′₃+Ω₃Ω^′₉−19𝜖³3Ω^′₁₆−4WiΩ₉Ω^′₅+Ω₅Ω^′₉−1𝜖16²4Wi+Re𝜖²log[𝜖]²Ω₇^′2+Ω₇Ω₇^′′−1𝜖128⁴32Ω^′₁₇−8Ω^′₁₈+ReΩ₇^′2+Ω₇Ω₇^′′−𝜖²16𝑟₂log𝜖/𝑟₂²×𝑑𝑝₁[𝜖]×𝜖𝑑𝑧1+2log₂+𝑟²₂𝜖−1−2log𝑟₂𝑟^′₂+2𝑟₂𝜖log𝑟₂²×4Ω^′₂₁−2Ω^′₂₂+WiΩ₇^′2+Ω₇Ω₇^′′−log[𝜖]8𝑟₂log𝜖/𝑟₂²×16𝜖⁶Ω^′₂₀+2Wi𝜖Ω₇Ω^′₅+Ω₅Ω^′₇+43Wi𝜖³Ω₉Ω^′₅+Ω₅Ω^′₉+𝜖⁴𝜖32⁴8Ω^′₁₈−ReΩ₇^′2+Ω₇Ω₇^′′+𝜖²𝑑𝑝₁𝑑𝑧−𝜖²+𝑟²₂𝜖1+2log𝑟₂𝑟^′₂+2𝑟₂𝜖log𝑟₂²×2Ω^′₂₂−WiΩ₇^′2+Ω₇Ω₇^′′−1288Re𝜖⁸Ω₉^′2+Ω₉Ω₉^′′. (A.1)

Conflict of Interests

The authors declare that they have no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors are grateful to the referees and editors for their constructive suggestions. This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah, under Grant no. 66-130-D1432. The authors, therefore, acknowledge with thanks DSR technical and financial support.