Effects of Magnetic Field and an Endoscope on Peristaltic Motion

Abstract

The Problem of peristaltic transport of a magnetic fluid with variable viscosity through the gap between coaxial tubes where the outer tube is nonuniform with sinusoidal wave traveling down its wall and the inner tube is rigid. The relation between the pressure gradient and friction force on the inner and outer tubes is obtained in terms of magnetic and viscosity parameter. The numerical solutions of pressure gradient, outer friction and inner friction force, and flow rate are shown graphically.

1. Introduction

The purpose of this paper is an attempt to understand the fluid mechanics in a physiological situation with the presence of an endoscope placed concentrically. The pressure rise, peristaltic pumping, augmented pumping, and friction force on the inner tube (endoscope) and outer tube are discussed by Srivastava et al. [1] and Siddiqui and Schwarz [2]. Latham [3] investigated the fluid mechanics of peristaltic pump and since then, other work on the same subject has been followed by Burns and Parkes [4]. Barton and Raynor [5] have been studied the case of a vanishingly small Reynolds number. Lykoudis and Roos [6] studied the fluid mechanics of the ureter from a lubrication theory point of view. Zien and Ostrach [7] have investigated a long wave approximation to peristaltic motion, and the analysis is aimed at the possible application to urine flow in human ureters. Roos and Lykoudis [8] studied the effect of the presence of a catheter upon the pressure distribution inside the ureter. Ramchandra and Usha [9] studied the influence of an eccentrically inserted catheter on the peristaltic pumping in a tube under long wavelength and low Reynolds numbers approximations. Abd El Naby and El Misery [10] studied the effect of an endoscope and generalized Newtonian fluid on peristaltic motion. Gupta and Sheshadri [11] studied peristaltic transport of a Newtonian fluid in nonuniform geometries. L. M. Srivastava and V. P. Srivastava [12] have investigated the effect of power law fluid in uniform and nonuniform tube and channel under zero Reynolds number and long wavelength approximation. Provost and Schwarz [13] have investigated a theoretical study of viscous effect in peristaltic pumping and assumed that the flow is free of inertial effect and that non-Newtonian normal stresses are negligible. Boehme and Friedrich [14] have investigated peristaltic flow of viscoelastic liquids and assumed that the relevant Reynolds number is small enough to neglect inertia forces, and that the ratio of the wavelength and channel height is large, which implies that the pressure is constant over the cross section. El Misery et al. [15] have investigated the effect of a Carreau fluid in peristaltic transport for uniform channel. Elshehaway et al. [16] studied peristaltic motion of generalized Newtonian fluid in a nonuniform channel under zero Reynolds number with long wavelength approximation. Most of studies on peristaltic motion, that assume physiological fluids behave like a Newtonian fluid with constant viscosity, fail to give a better understanding when peristaltic mechanics is involved in small blood vessel, lymphatic vessel, intestine, ducts efferent of the male reproductive tracts, and in transport of spermatozoa in the cervical canal. According to Haynes [17], Bugliarllo and Sevilla [18] and Goldsmith and Skalak [19], it is clear that in prementioned body organs, viscosity of the fluid varies across the thickness of the duct. Cotton and Williams [20] study the practical gastrointestinal endoscope. Rathod and Asha [21] studied the peristaltic transport of a couple stress fluids in uniform and nonuniform annulus moving with a constant velocity. Rathod and Asha [22] studied the effect of couple stress fluid and an endoscope on peristaltic motion.

In the view of above discussion, the effect of magnetic fluid with variable viscosity through the gap between inner and outer tubes where the inner tube is an endoscope and the outer tube has a sinusoidal wave traveling down its wall is the aim of present investigation.

2. Formulation and Analysis

Consider the two-dimensional flow of an incompressible Newtonian fluid with variable viscosity through the gap between inner and outer tubes where the inner tube is an endoscope and the outer tube has a sinusoidal wave traveling down its wall. The geometry of the two wall surface is given by the equation: 𝑟₁=𝑎₁,𝑟₁=𝑎₂₀+𝑏sin2𝜋𝜆𝑧−𝑐𝑡, (2.1)(2.2) where 𝑎₁ is the radius of endoscope, 𝑎₂₀ is the radius of the small intestine at inlet, 𝑏 is the amplitude of the wave, 𝜆 is the wavelength, 𝑡 is time, and 𝑐 is the wave speed.

In the fixed coordinates (𝑟,𝑧), the flow in the gap between inner and outer tubes is unsteady but if we choose moving coordinates (𝑟,𝑧) which travel in the 𝑧-direction with the same speed as the wave, then the flow can be treated as steady. The coordinate’s frames are related through𝑧=𝑍−𝑐𝑡,𝑟=𝑅,𝑤=𝑊−𝑐,𝑢=𝑈, (2.3) where 𝑈, 𝑊 and 𝑢, 𝑤 are the velocity components in the radial and axial direction in the fixed and moving coordinates, respectively.

Equations of boundary condition in the moving coordinates are continuity equation:1𝑟𝛿(𝑟𝑢)𝛿𝑟+𝛿𝑤𝛿𝑧=0 (2.4) and the Navier Stokes equation:𝜌𝑢𝜕𝑢𝜕𝑟+𝑤𝜕𝑢𝜕𝑧𝜕=−𝑝𝜕𝑟+𝜕𝜕𝑟2𝜇(𝑟)𝜕𝑢𝜕𝑟+2𝜇(𝑟)𝑟𝜕𝑢𝜕𝑟−𝑢𝑟+𝜕𝜕𝑧𝜇(𝑟)𝜕𝑢𝜕𝑧+𝜕𝑤𝜕𝑟−𝜎𝐵²₀(𝑢),𝜌𝑢𝜕𝑤𝜕𝑟+𝑤𝜕𝑤𝜕𝑧𝜕=−𝑝𝜕𝑧+𝜕𝜕𝑧2𝜇(𝑟)𝜕𝑤𝜕𝑧+1𝑟𝜕𝜕𝑟𝜇(𝑟)𝑟𝜕𝑢𝜕𝑧+𝜕𝑤𝜕𝑟−𝜎𝐵²₀(𝑤). (2.5)𝑝 is the pressure, 𝜇(𝑟) is the viscosity function, 𝜎 is Electric conductivity, and 𝐵₀ is applied magnetic field. The boundary conditions are written as follows:𝑤=−𝑐at𝑢=0,𝑟=𝑟₁,𝑟=𝑟₂,𝑢=0at𝑟=𝑟₁. (2.6) We introduce the nondimensional variable and the Reynolds number (Re) and the wave number (𝛿) introduced.𝑟=𝑟𝑎₂₀,𝑅=𝑅𝑎₂₀,𝑟₁=𝑟₁𝑎₂₀=𝑎₁₀𝑎₂₀=𝜀<1,𝑧=𝑧𝜆,𝑍=𝑍𝜆,𝜇(𝑟)=𝜇(𝑟)𝜇₀𝜆,𝑢=𝑢𝑎₂₀𝑐𝜆,𝑈=𝑈𝑎₂₀𝑐,𝑤=𝑤𝑐,𝑊=𝑊𝑐𝑎,𝛿=²⁰𝜆≪1,Re=𝑐𝑎₂₀𝜌𝜇₀𝑎,𝑝=²²⁰𝑝𝑐𝜆𝜇₀𝑐,𝑡=𝑡𝜆𝑏,𝜙=𝑎₂₀𝑟<1,₂=𝑟₂𝑎₂₀=1+𝜙sin2𝜋𝑧, (2.7) where 𝜀 is the radius ratio, 𝜙 is the amplitude ratio, and 𝜇₀ is the viscosity on the endoscope. Equation of motion and boundary conditions in the dimensionless form become1𝑟𝜕(𝑟𝑢)+𝜕𝑟𝜕𝑤𝜕𝑧=0,Re𝛿³𝑢𝜕𝑢𝜕𝑟+𝑤𝜕𝑢𝜕𝑧=−𝜕𝑝𝜕𝑟+𝛿²𝜕(𝑟)𝜕𝑟2𝜇𝜕𝑢𝜕𝑟+𝛿²𝜕𝜇(𝑟)𝛿𝜕𝑧²𝜕𝑢+𝜕𝑧𝜕𝑤+𝜕𝑟2𝛿²𝜇(𝑟)𝑟𝜕𝑢−𝑢𝜕𝑟𝑟−𝛿²𝑀²(𝑢),Re𝛿𝑢𝜕𝑤𝜕𝑟+𝑤𝜕𝑤𝜕𝑧=−𝜕𝑝+1𝜕𝑧𝑟𝜕𝜇(𝑟)𝑟𝛿𝜕𝑟²𝜕𝑢+𝜕𝑧𝜕𝑤𝜕𝑟+𝛿²𝜕(𝑟)𝜕𝑧2𝜇𝜕𝑤𝜕𝑧−𝑀²(𝑤). (2.8)𝑀=𝜎/𝜇𝐵₀𝑎₂₀ is Hartmann number and 𝜎 is Electric conductivity

With the dimensionless boundary condition, 𝑤=−1,𝑢=0,at𝑟=𝑟₁,𝑟=𝑟₂,𝑢=0,at𝑟=𝑟₁. (2.9)

Using the long wavelength approximation and neglecting the wave number (𝛿=0), one can reduce Navier-Stokes equation𝜕𝑝𝜕𝑟=0,𝜕𝑝=1𝜕𝑧𝑟𝜕𝜇(𝑟)𝑟𝜕𝑟𝜕𝑤𝜕𝑟−𝑀²(𝑤). (2.10)

The instantaneous volume flow rate in the fixed coordinate system is given by𝑄=2𝜋^𝑟2_𝑟1𝑊𝑅𝑑𝑅, (2.11) where 𝑟₁ is a constant and 𝑟₂ is a function of 𝑍 and 𝑡. On substituting (2.3) into (2.11) and the integrating, one obtains𝑄=𝑞+𝜋𝑐𝑟²₂−𝑟²₁, (2.12) where 𝑞=2𝜋^𝑟2_𝑟1𝑤𝑟𝑑𝑟. (2.13) Is the volume flow rate in the moving coordinate system and is it independent of time? Here, 𝑟₂ is a function of 𝑧 alone and is defined through (2.2). Using the dimensionless variable, we find that (2.13) becomes𝐹=𝑞2𝜋𝑎²₂₀𝑐=^𝑟2_𝑟1𝑤𝑟𝑑𝑟. (2.14) The time-mean flow over a period 𝑇=𝜆/𝑐 at a fixed Z position is defined as1𝑄=𝑇^𝑇₀𝑄𝑑𝑡. (2.15) Using (2.12) and (2.13) in (2.15) and integrating, we get𝑄=𝑞+𝜋𝑐𝑎²₂−𝑎²₁+𝑏²2, (2.16) which may be written as𝑄2𝜋𝑎²₂₀𝑐=𝑞2𝜋𝑎²₂₀𝑐+121−𝜀²+𝜙²2. (2.17) On defining the dimensionless time-mean flow asΘ=𝑄2𝜋𝑎²₂₀𝑐, (2.18) writing (2.17) as1Θ=𝐹+21−𝜀²+𝜙²2, (2.19) and solving (2.9)-(2.10), we obtain1𝑊=2𝑑𝑝𝐼𝑑𝑧¹(𝑟)−𝐼₂(𝑟)𝐼₁𝑟₂−𝐼₁𝑟₁−𝐼₁𝑟₁𝐼₂𝑟₂−𝐼₂𝑟₁𝐼₁𝑟₂𝐼₂𝑟₂−𝐼₂𝑟₁−12𝑀²(𝐷)−1, (2.20) where 𝐷=(𝐼₂(𝑟){𝐼₁(𝑟₂)−𝐼₁(𝑟₁)}−𝐼₁(𝑟₁)𝐼₂(𝑟₂)−𝐼₂(𝑟₁)𝐼₁(𝑟₂))/(𝐼₂(𝑟₂)−𝐼₂(𝑟₁)),𝐼₁(𝑟)=𝑟𝜇(𝑟)𝐼𝑑𝑟,₂(𝑟)=𝑑𝑟(𝑟).𝑟𝜇 (2.21) Using (2.14) we obtain the relationship between 𝑑𝑝/𝑑𝑧 and 𝐹 as follows:1𝐹=4𝑑𝑝𝐼𝑑𝑧¹𝑟₂−𝐼₁𝑟₁²𝐼₂𝑟₂−𝐼₂𝑟₁−𝐼₃+14𝑀²𝐼₁𝑟₂−𝐼₁𝑟₁²𝐼₂𝑟₂−𝐼₂𝑟₁−𝐼₃−12𝑟²₂−𝑟²₁,𝐼₃=^𝑟2_𝑟1𝑟³𝜇(𝑟)𝑑𝑟. (2.22)(2.23) Solving (2.22) for 𝑑𝑝/𝑑𝑧, we obtain𝑑𝑝=𝑟𝑑𝑧4𝐹+2²₂−𝑟²₁−(1/4)𝑀²𝐼₁𝑟₂−𝐼₁𝑟₁²/𝐼₂𝑟₂−𝐼₂𝑟₁−𝐼₃𝐼₁𝑟₂−𝐼₁𝑟₁²/𝐼₂𝑟₂−𝐼₂𝑟₁−𝐼₃. (2.24) The pressure rise Δ𝑃_𝜆 and friction force on inner and outer tubes 𝐹_𝜆^(𝑖) and 𝐹_𝜆^(𝑜), in their nondimensional forms, are given byΔ𝑃_𝜆=¹₀𝑑𝑝𝐹𝑑𝑧𝑑𝑧,_𝜆(𝑖)=¹₀𝑟²₁−𝑑𝑝𝐹𝑑𝑧𝑑𝑧,_𝜆(𝑜)=¹₀𝑟²₂−𝑑𝑝𝑑𝑧𝑑𝑧. (2.25) The effect of viscosity variation on peristaltic transport can be investigated through (2.25) for any given viscosity function 𝜇(𝑟).

For the present instigation, we assume viscosity variation in the dimensionless form following Srivastava et al. [1] as follows:𝜇(𝑟)=𝑒^−𝑎𝑟, (2.26) or 𝜇(𝑟)=1−𝛼𝑟for𝛼≪1, (2.27) where 𝛼 is viscosity parameter. The assumption is reasonable for the following physiological reason. Since a normal person of animal or similar size takes 1 to 2 L of fluid every day, another 6 to 7 L of fluid are received by the small intestine daily as secretion from salivary glands, stomach, pancreas, liver, and the small intestine itself. This implies that concentration of fluid is dependent on the radial distance. Therefore, the above choice of 𝜇(𝑟)=𝑒^−𝑎𝑟 is justified.

Substituting (2.27) into (2.21), and (2.23), and using (2.24), we obtain𝑑𝑝=𝑑𝑧16Θ−81−𝜀²+𝜙²/2+8𝑟²₂−𝑟²₁/𝑟²₂−𝑟²₁²log𝑟₂/𝑟₁−𝑟⁴₂−𝑟⁴₁×1−4𝛼+𝑀²𝑟²₂−𝑟²₁𝑟³₂−𝑟³₁3log𝑟₂/𝑟₁−𝑟²₂−𝑟²₁²𝑟₂−𝑟₁4ln𝑟₂/𝑟₁²−𝑟⁵₂−𝑟⁵₁5/𝑟²₂−𝑟²₁²log𝑟₂/𝑟₁−𝑟⁴₂−𝑟⁴₁. (2.28) Substituting (2.28) in (2.25) yieldΔ𝑃_𝜆=¹₀16Θ−81−𝜀²+𝜙²/2+8𝑟²₂−𝑟²₁/𝑟²₂−𝑟²₁²log𝑟₂/𝑟₁−𝑟⁴₂−𝑟⁴₁×1−4𝛼+𝑀²𝑟²₂−𝑟²₁𝑟³₂−𝑟³₁3log𝑟₂/𝑟₁−𝑟²₂−𝑟²₁²𝑟₂−𝑟₁4log𝑟₂/𝑟₁²−𝑟⁵₂−𝑟⁵₁5/𝑟²₂−𝑟²₁²log𝑟₂/𝑟₁−𝑟⁴₂−𝑟⁴₁𝐹𝑑𝑧,_𝜆(𝑖)=¹₀−𝑟²₁16Θ−81−𝜀²+𝜙²/2+8𝑟²₂−𝑟²₁/𝑟²₂−𝑟²₁²log𝑟₂/𝑟₁−𝑟⁴₂−𝑟⁴₁×1−4𝛼+𝑀²𝑟²₂−𝑟²₁𝑟³₂−𝑟³₁3log𝑟₂/𝑟₁−𝑟²₂−𝑟²₁²𝑟₂−𝑟₁4log𝑟₂/𝑟₁²−𝑟⁵₂−𝑟⁵₁5/𝑟²₂−𝑟²₁²log𝑟₂/𝑟₁−𝑟⁴₂−𝑟⁴₁𝐹𝑑𝑧,_𝜆(𝑜)=¹₀−𝑟²₂16Θ−81−𝜀²+𝜙²/2+8𝑟²₂−𝑟²₁/𝑟²₂−𝑟²₁²log𝑟₂/𝑟₁−𝑟⁴₂−𝑟⁴₁×1−4𝛼+𝑀²𝑟²₂−𝑟²₁𝑟³₂−𝑟³₁3log𝑟₂/𝑟₁−𝑟²₂−𝑟²₁²𝑟₂−𝑟₁4log𝑟₂/𝑟₁²−𝑟⁵₂−𝑟⁵₁5/𝑟²₂−𝑟²₁²log𝑟₂/𝑟₁−𝑟⁴₂−𝑟⁴₁𝑑𝑧. (2.29)(2.30)(2.31)

3. Results and Discussions

The dimensionless pressure rise (𝑃_𝜆) and the friction forces on the inner and outer tube for different given values of the dimensionless flow rate Θ, amplitude ratio 𝜙, radius ratio 𝜀, Hartmann number 𝑀, and viscosity parameter 𝛼 are computed using the (2.29) to (2.31). As the integrals in (2.29) to (2.31) are not integrable in the closed form so they are evaluated using 𝑎₂₀𝑎=1.25cm,𝜆=0.156. (3.1)

The values of viscosity parameter 𝛼 as reported by Srivastava et al. [1] are 𝛼 = 0.0 and 𝛼 = 0.1. Furthermore, since most routine upper gastrointestinal endoscopes are between 8–11 mm in diameter as reported by Cotton and Williams [20] and radius ratio 1.25 cm reported by L. M. Srivastava and V. P. Srivastava [12].

Figure 1 Shows the pressure rise against the flow rate; here it is observed that the pressure increases with the increase of flow rate for different values of radius ratio 𝜀=0.32,𝜀=0.38,  and 𝜀=0.44 and pressure decreases for the viscosity 𝛼=0.0 and 𝛼=0.1. Figure 2 shows that as the viscosity 𝛼 increases the pressure decreases. And for the different values of amplitude ratio 𝜙=0.0 and 𝜙=0.4, the pressure decreases.

Figure 1 

The pressure rise versus flow rate for 𝜙=0.4 and 𝛼=0.0 and 𝛼=0.1.

Figure 2 

The pressure rise versus flow rate for 𝜀=0.32, 𝛼=0.0, and 𝛼=0.1.

Figures 3 and 4 show the friction force on the outer tube for different values of radius ratio and amplitude ratio; here it is observed that as radius ratio increases the friction force also decreases and they are independent of radius ratio at certain values of the flow rate (for the values 𝜑=0.4 and 𝛼=0.0 and 𝛼=0.1). In Figures 5 and 6, it is noticed that the friction force on the inner tube (endoscope) and on outer tube is plotted against the flow rate for different values of amplitude ratio 𝜑 and for different values radius ratio 𝜀=0.32, 𝜀=0.38,  and 𝜀=0.44 and for the values of viscosity 𝛼=0.0 and 𝛼=0.1. It is noticed that as the amplitude ratio 𝜑 increases the friction force on the outer tube and inner tube decreases and as the viscosity increases the friction force on the outer tube and inner tube decreases.

Figure 3 

The friction force on the outer tube versus flow rate for 𝜙=0.2 and 𝛼=0.0, 𝛼=0.1.

Figure 4 

The friction force on the outer tube versus flow rate for 𝜀=0.32, 𝛼=0.0, 𝛼=0.1.

Figure 5 

The friction on the inner tube (endoscope) versus flow rate for 𝜙=0.2, and 𝛼=0.0, 𝛼=0.1.

Figure 6 

The friction on the inner tube (endoscope) versus flow rate for 𝜀=0.32, 𝛼=0.0, and 𝛼=0.1.

From Figure 7, it is noticed that the pressure increases for different values of magnetic field 𝑀 = 1, 3, and 5. From Figures 8 and 9, it is noticed that the friction force decreases on endoscope and on the outer tube as magnetic field increases.

Figure 7 

The pressure versus flow rate for 𝜀=0.32, 𝛼=0.0, 𝛼=0.1, and 𝜙=0.2.

Figure 8 

The friction on the inner tube (endoscope) versus flow rate for different magnetic field for 𝜙=0.2, 𝜀=0.32, 𝛼=0.0, and 𝛼=0.1.

Figure 9 

The friction on the outer tube versus flow rate for different magnetic field for 𝜀=0.32, 𝜙=0.2, 𝛼=0.0, and 𝛼=0.1.