Williamson Fluid Model for the Peristaltic Flow of Chyme in Small Intestine

Abstract

Mathematical model for the peristaltic flow of chyme in small intestine along with inserted endoscope is considered. Here, chyme is treated as Williamson fluid, and the flow is considered between the annular region formed by two concentric tubes (i.e., outer tube as small intestine and inner tube as endoscope). Flow is induced by two sinusoidal peristaltic waves of different wave lengths, traveling down the intestinal wall with the same speed. The governing equations of Williamson fluid in cylindrical coordinates have been modeled. The resulting nonlinear momentum equations are simplified using long wavelength and low Reynolds number approximations. The resulting problem is solved using regular perturbation method in terms of a variant of Weissenberg number 𝑊_𝑒. The numerical solution of the problem is also computed by using shooting method, and comparison of results of both solutions for velocity field is presented. The expressions for axial velocity, frictional force, pressure rise, stream function, and axial pressure gradient are obtained, and the effects of various emerging parameters on the flow characteristics are illustrated graphically. Furthermore, the streamlines pattern is plotted, and it is observed that trapping occurs, and the size of the trapped bolus varies with varying embedded flow parameters.

1. Introduction

The object of this study is to investigate the flow induced by peristaltic action of chyme (treated as Williamson fluid) in small intestine with an inserted endoscope. Williamson fluid is characterized as a non-Newtonian fluid with shear thinning property (i.e., viscosity decreases with increasing rate of shear stress). Since many physiological fluids behave like a non-Newtonian fluid [1], so chyme in small intestine is assumed to behave like Williamson fluid. Peristaltic motion is one of the most characteristics fluid transport mechanism in many biological systems. It pumps the fluids against pressure rise. It involves involuntary movements of the longitudinal and circular muscles, primarily in the digestive tract but occasionally in other hollow tubes of the body, that occur in progressive wavelike contractions. The waves can be short, local reflexes or long, continuous contractions that travel the whole length of the organ, depending upon their location and what initiates their action.

In human gastrointestinal tract, the peristaltic phenomenon plays a vital role throughout the digestion and absorption of food. The small intestine is the largest part of the gastrointestinal tract and is composed of the duodenum which is about one foot long, the jejunum (5–8 feet long), and the ileum (16–20 feet long). The rhythmic muscular action of the stomach wall (peristalsis) moves the chyme (partially digested mass of food) into the duodenum, the first section of the small intestine, where it stimulates the release of secretin, a hormone that increases the flow of pancreatic juice as well as bile and intestinal juices. Nutrients are absorbed throughout the small intestine. There are blood vessels and vessels contained a fluid called lymph inside the villi. Fat-soluble vitamins and fatty acids are absorbed into the lymph system. Glucose, amino acids, water-soluble vitamins, and minerals are absorbed into the blood vessels. The blood and lymph then carry the completely digested food throughout the body [2].

The endoscope effect on peristaltic motion occurs in many medical applications. Direct visualization of interior of the hollow gastrointestinal organs is one of the most powerful diagnostic and therapeutic modalities in modern medicine. A flexible tube called an endoscope is used to view different parts of the digestive tract. The tube contains several channels along its length. The different channels are used to transmit light to the area being examined, to view the area through a camera lens (with a camera at the tip of the tube), to pump fluids or air in or out, and to pass biopsy or surgical instruments through [3]. When passed through the mouth, an endoscope can be used to examine the esophagus, the stomach, and first part of the small intestine. When passed through the anus, an endoscope can be used to examine the rectum and the entire large intestine.

After the pioneering work of Latham [4], a number of analytical, numerical and experimental studies [5–13] of peristaltic flows of different fluids have been reported under different conditions with reference to physiological and mechanical situations. Several mathematical and experimental models have been developed to understand the chyme movement aspects of peristaltic motion. But less attention has been given to its relevance with endoscope effect. Lew et al. [14] discussed the physiological significance of carrying, mixing, and compression, accompanied by peristalsis. Srivastava [15] devoted his study to observe the effects of an inserted endoscope on chyme movement in small intestine. The important studies of recent years include the investigations of Saxena and Srivastava [16, 17], L.M. Srivastava and V.P. Srivastava [18], Srivastava et al. [19], Cotton and Williams [20], and Abd El-Naby and El-Misery [21].

The aim of present investigation is to investigate the peristaltic motion of chyme, by treating it as Williamson fluid, in the small intestine with an inserted endoscope. For mathematical modeling, we consider the flow in the annular space between two concentric tubes (i.e., outer tube as small intestine and inner tube as endoscope). Moreover, the flow is induced by two sinusoidal peristaltic waves of different wave lengths, traveling along the length of the intestinal wall. The solution of the problem is calculated by two techniques: (i) analytical technique (i.e., regular perturbation method in terms of a variant of Weissenberg number 𝑊_𝑒), (ii) numerical technique (i.e., shooting method). The expressions for axial velocity, frictional force, pressure rise, axial pressure gradient and stream function are obtained and the effects of various emerging parameters on the flow characteristics are illustrated graphically. Streamlines are plotted, and trapping is also discussed. Trapping is an important fluid dynamics phenomenon inherent in peristalsis. At high flow rates and occlusions, there is a region of closed stream lines in the wave frame, and thus, some fluid is found trapped within a wave of propagation. The trapped fluid mass (called bolus) is found to move with the mean speed equal to that of the wave [9].

2. Mathematical Development

We consider the flow of an incompressible, non-Newtonian fluid, bounded between small intestine (outer boundary) and inserted cylindrical endoscope (inner boundary). A physical sketch of the problem is shown in the Figure 1(a). We assume that the peristaltic wave is formed in nonperiodic rush mode composing of two sinusoidal waves of different wave lengths, traveling down the intestinal wall with the same speed 𝑐. We consider the cylindrical coordinate system (𝑅,𝑍) in the fixed frame, where 𝑍- axis lies along the centerline of the tube, and 𝑅 is transverse to it. Also a symmetry condition is used at the centre.

(a)

(b)

Figure 1 

(a) Physical sketch of the problem. (b) Comparison of numerical and perturbation solutions of axial velocity 𝑤(𝑟,𝑧) for 𝑊𝑒=0.02, 𝑑𝑝/𝑑𝑧=0.7, and 𝛿=0.1.

The geometry of the outer wall surface is described asℎ𝑍,𝑡=𝑟₀+𝐴₁sin2𝜋𝜆₁𝑍−𝑐𝑡+𝐴₂sin2𝜋𝜆₂𝑍−𝑐𝑡, (2.1) where 𝑟₀ is the radius of the outer tube (small intestine), 𝐴₁ and 𝜆₁ are the amplitude and the wave length of first wave, 𝐴₂ and 𝜆₂ are the amplitude and the wave length of the second wave, 𝑐 is the propagation velocity, 𝑡 is the time, and 𝑍 is the axial coordinate.

The governing equations in the fixed frame for an incompressible Williamson fluid model [6] are given as follows:𝜕𝑈𝜕𝑅+𝑈𝑅+𝜕𝑊𝜕𝑍𝜌𝜕=0,𝜕𝑡+𝑈𝜕𝜕𝑅+𝑊𝜕𝜕𝑍𝜕𝑈=−𝑃𝜕𝑅+1𝑅𝜕𝜕𝑅𝑅𝝉_𝑅𝑅+𝜕𝜕𝑍𝝉_𝑅𝑍,𝜌𝜕𝜕𝑡+𝑈𝜕𝜕𝑅+𝑊𝜕𝜕𝑍𝜕𝑊=−𝑃𝜕𝑍+1𝑅𝜕𝜕𝑅𝑅𝝉_𝑅𝑍+𝜕𝜕𝑍𝝉_𝑍𝑍, (2.2) where 𝑃 is the pressure, and 𝑈,𝑊 are the respective velocity components in the radial and axial directions in the fixed frame, respectively. Further, the constitutive equation of extra shear stress tensor 𝝉 for Williamson fluid [22] is expressed as𝝉=𝜇_∞+𝜇₀+𝜇_∞1−Γ𝛾̇⁻¹𝛾̇, (2.3) where 𝜇_∞ is the infinite shear rate viscosity, 𝜇₀ is the zero shear rate viscosity, Γ is characteristic time and the generalized shear rate 𝛾̇ is expressed in terms of second invariant strain tensor Π as,𝛾̇=12_𝑖𝑗𝛾̇_𝑖𝑗𝛾̇_𝑗𝑖=12Π, (2.4) in which Π=tr(grad𝐕+(grad𝐕)^𝑇)², and 𝐕 denotes the velocity vector.

We consider the case 𝜇_∞=0 and Γ|𝛾̇|<1 for constitutive equation (2.3). Hence, the extra stress tensor can be written as𝝉=𝜇₀1−Γ𝛾̇⁻¹𝛾̇,=𝜇₀1+Γ𝛾̇𝛾̇, (2.5) such that𝝉_𝑅𝑅=2𝜇₀1+Γ𝜕𝛾̇𝑈𝜕𝑅,𝝉_𝑅𝑍=𝜇₀1+Γ𝜕𝛾̇𝑈𝜕𝑍+𝜕𝑊𝜕𝑅,𝝉_𝑍𝑍=2𝜇₀1+Γ𝜕𝛾̇𝑊𝜕𝑍,=2𝜕𝛾̇𝑈𝜕𝑅²+𝜕𝑈𝜕𝑍+𝜕𝑊𝜕𝑅²𝜕+2𝑊𝜕𝑍^21/2. (2.6)

In the fixed coordinates (𝑅,𝑍), the flow is unsteady. It becomes steady in a wave frame (𝑟,𝑧) moving with the same speed as the wave moves in the 𝑍-direction. The transformations between the two frames are𝑟=𝑅,𝑧=𝑍−𝑐𝑡,𝑢=𝑈,𝑤=𝑊−𝑐, (2.7) in which 𝑢 and 𝑤 are the velocities in the wave frame. The corresponding boundary conditions in the wave frame are𝑤=−𝑐,at𝑟=𝑟₁,𝑤=−𝑐,at𝑟=ℎ=𝑟₀+𝐴₁sin2𝜋𝜆₁(𝑧)+𝐴₂sin2𝜋𝜆₂(𝑧), (2.8) where 𝑟₁ is the radius of the inner tube (endoscope). In order to reduce the number of variables, we introduce the following nondimensional variables: 𝑅=𝑅𝑟₀,𝑟=𝑟𝑟₀,𝑍=𝑍𝜆₁,𝑧=𝑧𝜆₁,𝑊=𝑊𝑐,𝑤=𝑤𝑐𝑟,𝝉=⁰𝝉𝑐𝜇₀,𝜆𝑈=¹𝑈𝑟₀𝑐𝜆,𝑢=¹𝑢𝑟₀𝑐𝑟,𝑝=⁰𝑃𝑐𝜆₁𝜇₀𝑐,𝑡=𝑡𝜆₁,𝑅_𝑒=𝜌𝑐𝑟₀𝜇₀,𝑊_𝑒=Γ𝑐𝑟₀,ℎ=ℎ𝑟₀=1+𝜙₁sin2𝜋𝑧+𝜙₂sin2𝜋𝛾𝑧,𝑟₁=𝑟₁𝑟₀𝑟,𝛿=¹𝑟₀,𝜙₁=𝐴₁𝑟₀,𝜙₂=𝐴₂𝑟₀𝜆,𝛾=¹𝜆₂𝑟,𝜖=⁰𝜆₁. (2.9) Here 𝑅_𝑒, 𝑊_𝑒, and 𝜖 represent the Reynolds number, Weissenberg number, and wave number, respectively. Moreover, 𝜙₁ and 𝜙₂ are nondimensional amplitudes of the waves, 𝛿 is the annulus aspect ratio, and 𝛾 represents the wave length ratio between two waves. By using (2.7) and (2.9), we get𝜕𝑢+𝑢𝜕𝑟𝑟+𝜕𝑤𝑅𝜕𝑧=0,_𝑒𝜖³𝑢𝜕𝜕𝜕𝑟+𝑤𝜕𝑧𝑢=−𝜕𝑝+𝜖𝜕𝑟𝑟𝜕𝜕𝑟𝑟𝝉_𝑟𝑟+𝜖²𝜕𝝉𝜕𝑧^𝑟𝑧,𝑅_𝑒𝜖𝑢𝜕𝜕𝜕𝑟+𝑤𝜕𝑧𝑤=−𝜕𝑝+1𝜕𝑧𝑟𝜕𝜕𝑟𝑟𝝉_𝑟𝑧𝜕+𝜖𝜕𝑧𝑟𝝉_𝑧𝑧, (2.10) and components of the extra stress tensor take the following form: 𝝉_𝑟𝑟=2𝜖1+𝑊_𝑒𝛾̇𝜕𝑢,𝝉𝜕𝑟_𝑟𝑧=1+𝑊_𝑒𝛾̇𝜕𝑢𝜖𝜕𝑧²+𝜕𝑤,𝝉𝜕𝑟_𝑧𝑧=2𝜖1+𝑊_𝑒𝛾̇𝜕𝑤,=𝜕𝑧𝛾̇2𝜖²𝜕𝑢𝜕𝑟²+𝜕𝑤+𝜕𝑟𝜕𝑢𝜖𝜕𝑧²²+2𝜖²𝜕𝑤𝜕𝑟^21/2. (2.11)

Under the assumption of long wavelength 𝜖≪1 and low Reynolds number approximations, the above equations are further reduced to𝜕𝑝𝜕𝑟=0, (2.12)−𝜕𝑝+1𝜕𝑧𝑟𝜕𝜕𝑟𝑟𝝉_𝑟𝑧=0, (2.13)𝝉_𝑟𝑧=𝜕𝑤𝜕𝑟+𝑊_𝑒𝜕𝑤𝜕𝑟². (2.14) Equation (2.12) shows that 𝑝=𝑝(𝑧). Now putting expression of 𝝉_𝑟𝑧 in (2.13) we get−𝜕𝑝+1𝜕𝑧𝑟𝑟𝜕𝑤𝜕𝑟+𝑊_𝑒𝜕𝑤𝜕𝑟²=0. (2.15) The corresponding nondimensional boundary conditions thus obtained are(𝑧)𝑤=−1at𝑟=𝛿,𝑤=−1at𝑟=ℎ=ℎ=1+𝜙₁sin2𝜋𝑧+𝜙₂sin2𝜋𝛾𝑧. (2.16)

3. Solution of the Problem

3.1. Perturbation Solution

Since (2.15) is a nonlinear equation and the exact solution may not be possible, therefore, in order to find the solution, we employ the regular perturbation method in terms of a variant of Weissenberg number 𝑊_𝑒. For perturbation solution, we expand 𝑤 and 𝑝 as𝑤=𝑤₀+𝑊_𝑒𝑤₁+𝑂𝑊²_𝑒,𝑝=𝑝₀+𝑊_𝑒𝑝₁+𝑂𝑊²_𝑒. (3.1) To first order, the expressions for axial velocity and axial pressure gradient satisfying boundary conditions (2.16) directly yield𝑤(𝑟,𝑧)=−1+0.25𝑑𝑝𝑎𝑑𝑧⁴⁵+𝑊_𝑒𝑑𝑝𝑎𝑑𝑧⁴⁶−𝑎₄₇, (3.2)𝑑𝑝𝑑𝑧=𝑎₄₈𝑎₄₉+𝑎₅₀−𝑎₅₁+𝑎₅₂, (3.3) where the involved quantities are defined in Appendix.

The expressions of pressure rise Δ𝑝 and the frictional forces 𝐹₀ and 𝐹₁ at the outer and inner boundaries, respectively, in their nondimensional forms, are given asΔ𝑝=¹₀𝑑𝑝𝑑𝑧𝑑𝑧, (3.4)𝐹₀=¹₀𝛿²−𝑑𝑝𝑑𝑧𝑑𝑧, (3.5)𝐹₁=¹₀ℎ²−𝑑𝑝𝑑𝑧𝑑𝑧. (3.6) where 𝑑𝑝/𝑑𝑧 is defined through (3.3), and flow rate 𝑄 in dimensionless form is defined as𝑄=𝑞−𝜋ℎ²−𝛿², (3.7) where 𝑞 is flow rate in the fixed frame of reference, and ⟨ℎ²⟩ is square of displacement of the walls over the length of an annulus, defined asℎ²=¹₀ℎ²𝑑𝑧. (3.8) Also in order to establish stream lines, we obtain stream function by using the following relation:1𝑢=−𝑟𝜕𝜓1𝜕𝑧,𝑤=𝑟𝜕𝜓𝜕𝑟, (3.9) which yields𝑟𝜓=𝑎480³⁰+𝑎₃₁𝑎₃₂−𝑎₃₃−𝑎₃₄+5𝑎₂₆𝑎²₁₂−𝑎₃₅×𝑎₃₆−𝑎₃₇−𝑎₃₈+𝑎₃₉+𝑎₄₀+𝑎₄₁−𝑎₄₂+𝑎₄₃+𝑎₄₄, (3.10) where the involved quantities are defined in Appendix.

3.2. Numerical Solution

Equation (2.15) is solved numerically by using shooting method [23]. The numerical result for axial velocity 𝑤(𝑟,𝑧) is compared with the perturbation result, and both results reveal a very good agreement with each other, as demonstrated in Table 1 and Figure 1(b).

Table 1 

Numerical and perturbation solutions for axial velocity 𝑤(𝑟,𝑧).

4. Graphical Results and Discussion

In this section, the graphical representations of the obtained solutions are demonstrated along with their respective explanation. The expressions for pressure rise and frictional forces are not found analytically; therefore, MATHEMATICA software is used to perform the integration in order to analyze their graphical behavior. It is also pertinent to mention that the values of all embedded flow parameters are considered to be less than 1.

Figures 2(a) to 2(d) are graphs of pressure rise Δ𝑝 versus flow rate 𝑄 to show the effects of different parameters on pumping rate. For peristaltic pumping, we divide the whole region into three parts. The region corresponding to Δ𝑝>0 and 𝑄>0 is known as the peristaltic pumping region. At Δ𝑝=0 is the free pumping region. And the region at Δ𝑝<0 and 𝑄>0 is called augmented pumping. Figure 2(a) shows that the pumping rate decreases by increasing the values of Weissenberg number 𝑊_𝑒, and this behavior remains the same in all three pumping regions. Figure 2(b) indicates the effect of the wavelength ratio 𝛾 on Δ𝑝. Here, pressure rise decreases with an increase in 𝛾 peristaltic pumping region, and after a critical value of 𝑄=1.4, it increases in the free and augmented pumping regions. Figure 2(c) explains the effect of the amplitude ratio 𝜙₂ on Δ𝑝. Here, pressure rise decreases with an increase in value of 𝜙₂ in the free and peristaltic pumping region, and after a critical value of 𝑄=1.5, it increases in the augmented pumping region. Figure 2(d) shows that pressure rise decreases when annulus aspect ratio 𝛿 increases.

(a)

(b)

(c)

(d)

Figure 2 

Variation of pressure rise per wavelength (Δ𝑝) for different values of (a)  𝑊𝑒 with 𝜙1=0.03,  𝜙2=0.05,  𝑄=0.2,  𝛿=0.7 and 𝛾=0.9,  (b)  𝛾 with 𝜙1=0.03,  𝜙2=0.05,  𝑄=2,  𝛿=0.7, and 𝑊𝑒=0.01,  (c)  𝜙2 with 𝜙1=0.01,  𝑄=0.2,  𝛾=0.9,  𝛿=0.7, and 𝑊𝑒=0.01, and (d)  𝛿 with 𝜙1=0.03,  𝜙2=0.05,  𝑄=0.2,  𝛾=0.9, and 𝑊𝑒=0.01.

Similarly the effects of 𝑊_𝑒, 𝛾, 𝜙₂, and 𝛿 on frictional forces are plotted in Figures 3-4. Figures 3(a) to 3(d) represent the variation of the frictional force at the outer wall(𝐹₀) and Figures 4(a) to 4(d) indicate the variation of the frictional force at the inner wall (𝐹₁) with flow rate 𝑄. It can be noted that the phenomena presented in these figures possess opposite character to the pressure rise for any given set of parameters. Also the graphs of both 𝐹₀ and 𝐹₁ show similar behavior when compared to their respective parameters. It is significant to mention that inner friction force 𝐹₁ attains higher magnitude than outer friction force 𝐹₀ with increasing values of any given set of parameters.

(a)

(b)

(c)

(d)

Figure 3 

Variation of frictional force at the inner wall (𝐹1) for different values of (a) 𝑊𝑒 with 𝜙1=0.03, 𝜙2=0.05, 𝑄=2, 𝛿=0.7, and 𝛾=0.9, (b)  𝛾  with 𝜙1=0.03, 𝜙2=0.05, 𝑄=0.2, 𝛿=0.7, and 𝑊𝑒=0.01, (c) 𝜙2 with 𝜙1=0.01, 𝑄=0.2,𝛾=0.9, 𝛿=0.7 and 𝑊𝑒=0.01, and (d)  𝛿  with 𝜙1=0.03, 𝜙2=0.05, 𝑄=0.2, 𝛾=0.9 and 𝑊𝑒=0.01.

(a)

(b)

(c)

(d)

Figure 4 

Variation of frictional force at the outer wall (𝐹0) for different values of (a)  𝑊𝑒  with 𝜙1=0.03, 𝜙2=0.05, 𝑄=0.2, 𝛿=0.7, and 𝛾=0.9, (b)  𝛾 with  𝜙1=0.03, 𝜙2=0.05, 𝑄=2, 𝛿=0.7, and 𝑊𝑒=0.01, (c)  𝜙2 with 𝜙1=0.01,𝑄=0.2, 𝛾=0.9, 𝛿=0.7, and 𝑊𝑒=0.01, and (d)   𝛿  with 𝜙1=0.03, 𝜙2=0.05, 𝑄=0.2, 𝛾=0.9 and 𝑊𝑒=0.01.

In order to discuss the effects of variation of various parameters on the axial pressure gradient 𝑑𝑝/𝑑𝑧, MATHEMATICA has been used for the numerical evaluation of the analytical results, and the results are graphically presented in Figures 5(a) to (5(e). In these figures, the pressure gradient distribution for various values of 𝜙₁, 𝜙₂, 𝑊_𝑒, 𝛾, and 𝛿 is depicted. It is observed that pressure gradient increases with increasing the values of previously mentioned parameters.

(a)

(b)

(c)

(d)

(e)

Figure 5 

Pressure gradient 𝑑𝑃/𝑑𝑧 verses 𝑧 for different values of (a)  𝜙1  with 𝜙2=0.06, 𝑄=0.2, 𝛾=0.9, 𝛿=0.5, and 𝑊𝑒=0.05, (b)  𝜙2  with 𝜙1=0.06, 𝑄=0.2, 𝛾=0.9, 𝛿=0.5, and 𝑊𝑒=0.05, (c)   𝑊𝑒. with 𝜙1=0.04, 𝜙2=0.06, 𝑄=0.2, 𝛿=0.5, and 𝛾=0.9, (d)  𝛿  with 𝜙1=0.04, 𝜙2=0.06, 𝑄=0.2, 𝛾=0.9 and 𝑊𝑒=0.05, and (e)  𝛾 with  𝜙1=0.04, 𝜙2=0.06, 𝑄=0.2, 𝛿=0.5 and 𝑊𝑒=0.05.

The influence of various parameters on streamlines pattern is depicted in Figures 6, 7, 8, 9, 10. It is noted that trapping is observed in all these cases. Figures 6, 7, 8, 9 show that the size of trapped bolus increases for higher values of Weissenberg number (𝑊_𝑒), wave length ratio (𝛾), and amplitude (𝜙₂). Figure 9 exhibits the effects of radius ratio (𝛿) on streamlines pattern. It can be seen that the size and number of the trapped bolus increase with an increase in 𝛿. However, the size and number of trapped bolus decrease with increasing values of flow rate as shown in Figure 10.

(a)

(b)

Figure 6 

Streamlines pattern for (a) 𝜙2=0.08 (b)  𝜙2=0.09  with 𝜙1=0.04, 𝑄=0.2, 𝛿=0.4, 𝛾=0.9 and 𝑊𝑒=0.03.

(a)

(b)

Figure 7 

Streamlines pattern for (a) 𝑊𝑒=0.01 (b) 𝑊𝑒=0.02 with 𝜙1=0.04, 𝜙2=0.06, 𝑄=0.2, 𝛾=0.9 and 𝛿=0.4.

(a)

(b)

Figure 8 

Streamlines pattern for (a) 𝛾=0.8 (b) 𝛾=0.9 with 𝜙1=0.04, 𝜙2=0.06, 𝑄=0.2, 𝛿=0.4 and 𝑊𝑒=0.03.

(a)

(b)

Figure 9 

Streamlines pattern for (a) 𝛿=0.4 (b) 𝛿=0.5 with 𝜙1=0.04, 𝜙2=0.06, 𝑄=0.2, 𝛾=0.9 and 𝑊𝑒=0.03.

(a)

(b)

Figure 10 

Streamlines pattern for (a) 𝑄=0.3 (b) 𝑄=0.5 with 𝜙1=0.04, 𝜙2=0.06, 𝛾=0.9, 𝛿=0.4 and 𝑊𝑒=0.03.

5. Conclusion

The peristaltic flow of chyme (treated as Williamson fluid) in small intestine with an inserted endoscope is investigated. The flow is considered between annular space of small intestine and inserted endoscope and is induced by two sinusoidal peristaltic waves of different wave lengths, traveling along the length of the intestinal wall. Long wavelength and low Reynolds number approximations are used to simplify the resulting equations. The solution of the problem is calculated using analytical technique (i.e., regular perturbation method) and numerical technique (i.e., shooting method). Also results of axial velocity for both solutions are compared and found a very good agreement between them.

The performed analysis can be concluded as follows:(1)the peristaltic pumping rate decreases with increasing the values of 𝜙₂, 𝑊_𝑒, 𝛾, and 𝛿. This shows that the effects of these parameters on the pressure rise are qualitatively similar;(2)frictional forces show an opposite behavior to that of pressure rise in peristaltic transport;(3)the inner friction force 𝐹₁ attains higher magnitude than outer friction force 𝐹₀ with increasing values of any given set of parameters.(4)Pressure gradient increases with increasing the values of all embedded parameters that is, 𝜙₁, 𝜙₂, 𝑊_𝑒, 𝛾, and 𝛿;(5)an increase in radius ratio (𝛿) results in the increase of the size and number of trapped bolus. Also the size of trapped bolus increases for higher values of amplitude rate (𝜙₂), Weissenberg number (𝑊_𝑒), and wave length ratio (𝛾);(6)moreover, it is observed that the size and number of trapping bolus decrease with increasing values of flow rate (𝑄).

Appendix

The values of quantities appearing in the expressions (3.2), (3.3), and (3.4) are given as 𝑎₁₁=logℎ+log𝑟,𝑎₁₂=logℎ+log𝛿,𝑎₁₃=ℎ²−𝑟²,𝑎₁₄=ℎ²−𝛿²,𝑎₁₅=ℎ−𝑟,𝑎₁₆=ℎ−𝛿,𝑎₁₇𝑎=ℎ+𝛿,₁₈=ℎ²+𝛿²,𝑎₁₉=ℎ²+𝑟²,𝑎₂₀=ℎ⁴−𝛿⁴,𝑎₂₁=ℎ⁶−𝛿⁶,𝑎₂₂=ℎ³+𝛿³,𝑎₂₃=ℎ⁵−𝛿⁵,𝑎₂₄=ℎ²+𝛿²,𝑎₂₅=ℎ³−𝛿³,𝑎₂₆=ℎ⁷−𝛿⁷,𝑎₂₇=ℎ⁴+𝛿⁴,𝑎₂₈=log𝛿³−logℎ³,𝑎₂₉=logℎlog𝛿,𝑎₃₀=240𝑟𝑎₂₈+720𝑟𝑎₂₉𝑎₁₂𝑎³₁₂,𝑎₃₁=𝑟2𝑄+𝑎₁₄300ℎ𝛿𝑎²₁₆𝑎log𝛿³₁₇−𝑎₁₄+𝑎₂₄𝑎₁₂³,𝑎₃₂=−10𝑊_𝑒𝑎³₁₆𝑎³₁₇𝑎₁₂2𝑄+𝑎₁₄,𝑎₃₃=ℎ𝛿𝑎₁₂5𝑎²₁₆𝑎²₁₇−24𝑄𝑊_𝑒+𝑎₁₄𝑎₁₂−12𝑊_𝑒+𝑎₁₇,𝑎₃₄=10𝑎₁₄𝑎₁₂𝑎₂₃−8𝑄𝑊_𝑒𝑎₂₄+ℎ𝛿−4ℎ𝛿𝑊_𝑒𝑎₁₄+𝑎₂₀+𝑎₂₅,𝑎₃₅=32𝑄𝑎²₁₂𝑊_𝑒𝑎₂₇+ℎ𝛿𝑎₁₄+ℎ²𝛿²+16𝑊_𝑒𝑎₂₁−ℎ𝛿𝑎₂₇+5ℎ𝛿𝑎₂₃𝑎³₁₂,𝑎₃₆=30𝑟³−8𝑟⁴𝑊_𝑒−15𝑟𝑊_𝑒𝑎³₁₆𝑎₁₇𝛿𝑎³₁₂𝑎₁₇ℎ,𝑎+logℎ₃₇=1𝑎³₁₂10𝑟ℎ³𝑊_𝑒+𝛿²3−7𝑊_𝑒𝛿+ℎ²−3+6𝑊_𝑒𝛿logℎ²,𝑎₃₈=20𝑟𝛿²3−𝑊_𝑒𝛿logℎ³+5𝑎₁₆𝑟log𝑟3𝑎₁₂𝑎²₁₆𝑎²₁₇ℎ𝛿−12𝑎₁₄𝑎²₁₂,𝑎₃₉=4𝑊²_𝑒𝑎₁₂𝑎₂₄+ℎ𝛿−𝑎₁₇,𝑎₄₀=𝑟𝑎₁₆𝑎³₁₂15𝑊_𝑒ℎ𝑎₁₄log𝛿+80ℎ²𝑊_𝑒+20𝛿−3+4𝑊_𝑒𝛿,𝑎₄₁=20ℎ−3+7𝑊_𝑒𝛿𝑎₂₉+𝑎₂₉logℎ60ℎ−20ℎ³𝑊_𝑒+120𝛿²−40𝛿³𝑊_𝑒,𝑎₄₂=70ℎ²𝑊_𝑒+10𝛿−3+𝑊_𝑒𝛿+10ℎ−3+7𝑊_𝑒𝛿log𝛿²,𝑎₄₃=𝑟𝑎₂₉log𝛿𝑎³₁₂−120ℎ²+40ℎ³𝑊_𝑒−60𝛿²+20𝛿³𝑊_𝑒,𝑎₄₄=20𝑟ℎ²log𝛿³𝑎³₁₂3−ℎ𝑊_𝑒,𝑎₄₅=−𝑎₁₃+𝑎₁₁𝑎₁₆𝑎₁₂𝑎₁₇,𝑎₄₆=𝑎₁₅𝑎12¹⁹+𝑎+ℎ𝑟²¹⁵𝑎₁₆𝑎²₁₇16𝑎²₁₂+𝑎²₁₆𝑎²₁₇𝑎₁₁16𝑎²₁₂,𝑎₄₇=𝑎₁₅𝑎₁₄4𝑎₁₂+𝑎₁₁𝑎₁₆12𝑎₁₂𝑎₁₈+𝑎+ℎ𝛿³¹⁶𝑎²₁₇𝑎₁₁16ℎ𝛿𝑎₁₂³,𝑎₄₈=15ℎ𝛿𝑎²₁₆𝑎³₁₇−𝑎₁₄+𝑎₁₈𝑎₁₂³,𝑎₄₉=82Q+𝑎₁₄−5ℎ𝛿𝑎³₁₆𝑎₁₇𝑎₁₂+10ℎ𝛿𝑎₁₄𝑎₂₃−ℎ𝛿𝑎₂₂𝑎²₁₂+𝑎²₁₂,𝑎₅₀=8𝑊_𝑒−10𝑎³₁₆𝑎³₁₇2𝑄+𝑎₁₄²+60ℎ𝛿𝑎²₁₆𝑎²₁₇𝑎₁₂2𝑄+𝑎₁₄,𝑎₅₁=40ℎ𝛿𝑎₁₄𝑎²₁₂2𝑄𝑎₁₈+𝑎₂₀+𝑎₁₄2𝑄𝑎₁₈+𝑎₂₀+𝑎₁₄,𝑎₅₂=16ℎ𝛿𝑎³₁₂2𝑄𝑎₂₀+ℎ²𝛿𝑎₁₇+ℎ𝛿³+𝑎₂₁+ℎ𝛿𝑎₂₀. (A.1)

Acknowledgments

The first and second author is thankful to higher education of Pakistan for the financial support and the third author extends his appreciation to the deanship of Scientific Research at king Saud University for funding this work through the research group Project no. RGP-VPP-080.