Separated Flow Through a Gap Between Two Coaxial Peristaltic Tubes

A peristaltic endoscope is a device that could locomote through curving and tortuous spaces where it has many real-life applications in different disciplines e.g., in the field of medicine, it helps in the process of catheterization in curved tubes which in turn relieve patient’s pain. The aim of this paper is to study a trapping phenomenon at the centerline of a gap between inner peristaltic endoscope and outer peristaltic tube of a fluid with viscosity variation and a novel phenomenon of separated flow at the boundary of these tubes. For understanding these phenomena, we formulate the flow of a fluid with viscosity variation through the gap between two coaxial peristaltic tubes in cylindrical coordinates with neglecting Reynolds number and wave number. Explicit forms for the velocity field, pressure rise and friction forces on inner and outer peristaltic tubes in terms of radius ratio, flow rate, parameter of viscosity and occlusion have been obtained. A new comparison between a rigid endoscope and a peristaltic endoscope through the gastrointestinal tract has been made for the pressure rising and drag (friction) forces results. Also, we identify type of pumping for various physical parameters of interest. In addition, separated flow points are determined numerically by using computer algebra system.

Dimensional wall surfaces of peristaltic endoscope and small intestine at any time respectively. n Radius ratio. a Radius of the small intestine at inlet. b Wave amplitude. c Wave speed. λ Wavelength. t Dimensional time.

R, Z
Dimensional cylindrical fixed coordinates system. (R, Z ) Non-dimensional cylindrical fixed coordinates system.

(r, z)
Dimensional cylindrical moving coordinates system.
The associate editor coordinating the review of this manuscript and approving it for publication was Yeliz Karaca .
(r, z) Non-dimensional cylindrical moving coordinates system. r 1 , r 2 Non-dimensional wall surfaces of peristaltic endoscope and small intestine in the moving coordinates respectively. δ = a λ Wave number. ϕ = b a Amplitude ratio. U , W Dimensional velocity components in the radial and axial directions respectively in fixed coordinates. (U , W ) Non-dimensional velocity components in the radial and axial directions respectively in fixed coordinates. (u, w) Dimensional velocity components in the radial and axial directions respectively in moving coordinates.

(u, w)
Non-dimensional velocity components in the radial and axial directions respectively in moving coordinates. F Non-dimensional volume flow rate in the moving coordinates system. VOLUME

I. INTRODUCTION
Nowadays, there have been several attempts to use biological revelation as the basis for renovation of endoscopes. Several investigators, inspired by the flexible locomotion of snakes, have created new endoscopes that can bend in response to the activation of shape memory alloy wires [1], [2]. A peristaltic endoscope is a device that could locomote through curving and tortuous spaces. It has several applications in different disciplines e.g., in industry and medicine. It could be used to maintain and repair machines with complex internal plumbing. Medically, endoscopy and catheterization within the human body may be done using a peristaltic endoscope, see for example, Mangan et al. [1]. Studies on effects of a concentric and an eccentric catheter on peristaltic motion for Newtonian fluids have shown in literature [3], [4]. Shukla et al. [5] discussed the influence of peripheral-layer viscosity on peristaltic movement of a bio-fluid in uniform tube and channel using the long wavelength approximation as in Shapiro et al. [6]. Shapiro et al. [6] investigated the fluid mechanics of peristaltic pumping in connection with the function of systems such as ureter, gastro-intestinal tract, the small blood vessels, and other glandular ducts. They found that there were two physiologically significant phenomena called ''reflux'' and ''trapping'' in peristaltic flow. From physiological point of view, it is known that the small intestine receives secretions from other organs such as stomach, pancreas, liver and the small intestine itself. This shows that the fluid viscosity on the wall of small intestine is less than that away from the wall. Hence, the viscosity is dependent on the radial distance. Thus, Srivastava et al. [7] studied the influence of viscosity variation and peristaltic transport of physiological fluid flow in nonuniform geometry. They found that the effect of increasing viscosity decreases flow rate. Abd El Naby et al. [8] discussed the hydro-magnetics flow of a fluid with viscosity variation in a uniform tube with peristalsis. Some other studies discussed both the influence of Newtonian and non-Newtonian fluids of varied viscosity and that of a rigid endoscope on peristaltic movement in absence and presence of a magnetic field as given in literature [9]- [16]. Rachid and Ouazzani [17] studied effects of the electro-magneto-hydrodynamics (EMHD) flow of a bi-viscosity fluid through a permeable medium between two deformable coaxial tubes with different phases and amplitudes. McCash et al. [18] studied effects of a Newtonian fluid with constant viscosity and a peristaltic endoscope on trapping by using a curvilinear coordinate system.
To the best of our knowledge, it is clear from previous studies that there is no attempt to study trapping and separated flow phenomena of a fluid with viscosity variation through the gap between two coaxial peristaltic tubes. So, this is the first study to explain these phenomena.
The paper is structured as follows. Section II gives the formulation of the problem in cylindrical coordinates in the non-dimensional form with canceling wave number and Reynolds number. In section III, trapping at the centerline of the gap has been studied. Separated flow (trapping at the boundary of the gap) has been studied in section IV. The peristaltic pumping, augmented pumping, trapping and separated flow have been discussed for various physical parameters of interest in section V. In section VI, concluding remarks are summarized.

II. FORMULATION OF THE PROBLEM
We investigate separation of creeping flow of a fluid with constant density and viscosity variation in a gap between two coaxial peristaltic tubes. The geometry of the walls surfaces is described in Fig. 1.
23294 VOLUME 10, 2022 where a is the radius of the outer peristaltic tube at entrance, b is the amplitude of the wave, λ is the wavelength, c is the wave speed, t is the time and n is radius ratio. If we select moving coordinates (r, z) which move in the Z direction with the same speed as the wave (wave frame), the tubes lengths are integral multiplied by wavelength, and the pressure difference through the gap between the two tubes is a constant so the flowing must be considered as steady flowing. The stationary and moving frames have been connected as follows: where (u, w) and U , W are the velocity components in the radial and axial directions in the wave and laboratory frames, respectively.
Since the continuity equation in vector form is ∂ ρ ∂ t + ∇ · ρ V = 0, momentum equation in vector form is where ρ is the density, V is the flow velocity in the stationary frame, P is the pressure, τ is Stokes' stress, µ is the dynamic viscosity, F is the body force per unit mass and t is time and boundary conditions in vector form are the tangential component of the fluid velocity on the wall is determined from no-slip boundary condition V ·ê Z = V wall in the stationary frame, whereê Z is the unit vector in the axial direction. Also, the normal component of the fluid velocity on the wall is determined from the boundary surface condition of a fluid DF i Dt = 0 at R = r i Z , t , i = 1, 2 with using the previous condition, where F i R, Z , t = R − r i Z , t = 0. Then following [10], [14], [16] the governing equations of the flow with boundary conditions, under creeping flow (Re 1), long wavelength approximation (δ 1), the density is constant, the viscosity is variable, the flow is steady, the diameter Reynolds number is small ρ wa µ 1 and with negligence of gravitational force, in the wave frame in the non-dimensional form are continuity equation: Navier-Stokes equations: We neglect wave number by assuming that the wavelength is long compared to radius of the tube (a λ), then (δ 1) . Also, since the inertial force components in the dimensionless form for steady flow take the following form Reδ 3 u ∂u ∂r + w ∂u ∂z , Reδ u ∂w ∂r + w ∂w ∂z . Then (Reδ 1), hence we may neglect the inertial terms. From the condition of boundary surfaces of a fluid, where the fluid particles originally on the walls must remain on the walls, and mechanical property of the walls in the stationary coordinates W = 0 at R = r 1 , r 2 . By using the transformations (3) and (4), we obtain the boundary conditions in the moving coordinates in the dimensionless form as follows: Hence, the no-slip condition for a viscous flow is satisfied. The viscosity causes the fluid to stick to the walls and thus the velocity of the fluid at the walls assumes the velocity of the walls.
Where the dimensionless variables are given by where ϕ is the occlusion and µ 0 is the coefficient of viscosity on the surface of inner peristaltic tube. By integrating Eq. (7) and using the dimensionless boundary conditions (8), by using separation of variables method we can get the velocity profile as (10), shown at the bottom of the next page, where The volume flow rate in the stationary and moving coordinates are given by By substitution from Eqs. (3) and (4) in Eq. (13) and making use of Eq. (14) gives The time-mean flowing through a period T = λ c at a stationary position Z is defined as: VOLUME 10, 2022 By substitution from Eq. (15) in Eq. (16) and integrating yields:Q On defining the dimensionless time-mean flow and the volume flow rate F in the wave frame as follows: then Eq. (17) may be written as Substituting from Eq. (10) in Eq. (20) yields where The pressure rising P λ and drag (friction) forces F (i) λ and F (o) λ (at the walls) in the two peristaltic tubes of length λ, in the dimensionless form, are given by The influence of viscosity variation through the gap between two coaxial peristaltic tubes can be shown in Eqs. (23)- (25) to give a function of viscosity µ (r). For the present implementation, we assume that the viscosity variation in non-dimensional form as mentioned by Srivastava et al. [7] as: or Here α is a viscosity parameter.
This assumption was justified physiologically as shown in reference [10].
(34-36) reduce to which are the same results obtained by Shapiro et al. [6] and Srivastava [14] in absence of the viscosity parameter α. Also, these results are derived by Shukla et al. [5] in absence of the peripheral layer.

III. TRAPPING AT THE CENTERLINE OF THE GAP
Trapping phenomenon was studied by several investigators as shown in literature [4], [6], [19]- [21]. Following Siddiqui and Schwarz [22], the trapping limits have been determined as min max is a function of occlusion ϕ, where min is the minimum flow rate and max is the maximum flow rate. The minimum flow rate is obtained by Eqs. (10), (19) and (28) when w = 0 at r = k 4 r 2 2 + α k 5 r 3 2 1/2 , where k 4 = −k 1 2 log (n) and and solving it with respect to , then we get where k 6 , k 7 , k 8 , as shown at the bottom of the page. The maximum flow rate has been obtained by setting P λ = 0 in Eq. (34) and solving it with respect to , then we get and hence the flow rate ratio min max , neglecting the terms of order α 2 , reduces to (41), as shown at the bottom of the page, where By taking the limit of minimum and maximum flow rate ratio min max as n tends to zero, one obtains the following form (42), as shown at the bottom of the next page. This result is the same as given by Shapiro et al. [6] as α = 0.
Physically, minimum and maximum flow rate must be real and satisfy the inequality 0 ≤ min ≤ max . So, trapping takes place such that 0 ≤ min max ≤ 1 for all values of radius ratio n and amplitude ratio ϕ. The boundary of the trapping region has been obtained by calculating minimum and maximum flow rate ratio min max for different values of the parameter ϕ at n = 0, 0.32.

IV. SEPARATED FLOW (TRAPPING AT THE BOUNDARY OF THE GAP)
To predict separated flow at the boundary, the magnitude of the vorticity vector equals to zero at the boundary, i.e., By substituting from Eqs. (11), (12) and (28) in Eq. (10) and making use of Eq. (27), we get the axial velocity of the fluid as: where the prime means differentiation with respect to z.
To obtain the radial velocity of the fluid, we substitute from Eq. (44) into Eq. (5) using the boundary condition (8b), we have the radial velocity of the fluid in the following form: Substituting from Eq. (9) in Eqs. (46) and (47), we have (48) and (49), as shown at the bottom of the next page. Since separated flow occurs on the boundaries that are functions of z, then we must find z as a function of parameters that are including parameters of both the peristaltic motion of the walls and viscous flow in the thin annuli. Thus by substituting from Eq. (19) in Eqs. (48) and (49) and then solve them numerically. Hence, we obtain the axial component of separated flow points z s on the outer and inner peristaltic tubes respectively as a function of parameters α, ϕ, n and .

V. NUMERICAL RESULTS AND DISCUSSION
For discussion our results, we calculate the non-dimensional pressure rising P λ , and drag (friction) forces (at the walls) in the two peristaltic tubes for various given values of the non-dimensional time-mean flow , occlusion φ, and radius ratio n. As shown in Srivastava and Srivastava [23], one uses the values of various parameters as: a 2 = 1.25 cm.  [7]. Since the radius of the small intestine, a 2 = 1.25 cm. , is small compared with the wavelength λ = 8.01 cm., then the theory of long wavelength and creeping flow of the present implementing remains usable. Also, Lew et al. [24] observed that Reynolds number in the small intestine was very small. Cotton and Williams [25] reported diameter of gastrointestinal rigid endoscopes are among 8 and 11 mm. Besides, Srivastava and Srivastava [23] reported the radius of the small intestine is 1.25 cm. , then the maximum value of the radius ratio n for rigid endoscopes becomes 0.44, but the radius ratio n for peristaltic endoscopes takes the values greater than zero or less than one. Equations (34), (35) and (36) are drawn in Figs. 2-7 respectively. Whereas, Eq (41) is drawn in Fig. 8.
Experimental results in vivo pig had been obtained by Grundfest et al. [26] showed that the intestinal diameter increases with increasing intraluminal intestinal pressure in presence of a robotic endoscope and this agrees with our theoretical results which state that the pressure rising increases when radius ratio increases. Physically, increasing the radius ratio implies increasing of the intestinal diameter.
Figs. 4-7 represent the variation of drag (friction) forces on a peristaltic endoscope and on a peristaltic tube F   parameter α = 0, 0.1 and ϕ = 0.2, 0.4. Furthermore, drag (friction) force on the peristaltic endoscope is smaller than that on the outer peristaltic tube. This happens because amplitude ratio φ on the outer peristaltic tube is greater than that on the peristaltic endoscope.
The next results in tables 1 and 2 for the pressure rise P λ and friction forces F It is obvious from tables 1 and 2 that the absolute values of pressure rising and friction forces in presence of rigid endoscope are smaller than the corresponding values in presence of a peristaltic endoscope at 0.1 ≤ ϕ ≤ 0.2, but they are greater at 0.3 ≤ ϕ ≤ 0.6 for the fluid with constant and variable viscosity. Fig. 8 shows that the trapping limit, for fluids of a constant viscosity (α = 0) and that of a variable viscosity (α = 0.1), takes place at 0.26 ≤ ϕ < 0.5 and 0.27 ≤ ϕ ≤ 0.52 for α = 0, 0.1 respectively when n = 0 (absence of the peristaltic endoscope), but it takes place at 0.36 ≤ ϕ ≤ 0.6 when n = 0.32 (presence of the peristaltic endoscope). Also, it is clear that the trapping limit increases when viscosity parameter α increases in absence of the inner peristaltic tube, but it is independent of viscosity parameter in presence of  the peristaltic endoscope. Moreover, the trapping limit, which was obtained by Shapiro et al. [6], agrees with our results when n = 0 and α = 0.
To discuss the phenomenon of separated flow (trapping) at walls bounded the gap between the peristaltic endoscope and the small intestine. We calculate numerically the axial component of separated flow points on the outer peristaltic tube wall and on the inner peristaltic tube wall from Eqs.

VI. CONCLUSION
As a result of previous analyses, we conclude that the flow field of a fluid with constant density and viscosity variation through a gap between two coaxial peristaltic tubes is remarkable.
More exactly: • The peristaltic endoscope is more appropriate than the rigid endoscope for peristaltic organs whose amplitude ratios are 0.3, 0.4, 0.5 and 0.6, but the rigid endoscope is more appropriate than the peristaltic endoscope for peristaltic organs whose amplitude ratios are 0.1 and 0.2.
• Increasing radius ratio increases the pressure rise in the peristaltic pumping and augmented pumping regions.
• The friction forces increase with increasing radius ratio in peristaltic pumping and reflux regions.
• The trapping limit on the centerline of the gap between coaxial peristaltic tubes is independent of viscosity parameter in presence of the peristaltic endoscope (n = 0.32), but it increments with increasing viscosity parameter in absence of the peristaltic endoscope (n = 0) .
• The separated limit on the boundary increments with increasing radius ratio and volume flow rate, but it declines when viscosity parameter increases.
• Separated flow occurs at the surfaces of peristaltic movement.
• The present mathematical model of a peristaltic endoscope is considered as benchmark research for endoscopes' development.