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

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 We. 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.


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  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 .

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.
The geometry of the outer wall surface is described as where r 0 is the radius of the outer tube small intestine , A 1 and λ 1 are the amplitude and the wave length of first wave, A 2 and λ 2 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: 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 as where μ ∞ is the infinite shear rate viscosity, μ 0 is the zero shear rate viscosity, Γ is characteristic time and the generalized shear rateγ is expressed in terms of second invariant strain tensor Π as, in which Π tr gradV gradV T 2 , and V denotes the velocity vector.

Mathematical Problems in Engineering 5
We consider the case μ ∞ 0 and Γ|γ| < 1 for constitutive equation 2.3 . Hence, the extra stress tensor can be written as

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 are in which u and w are the velocities in the wave frame. The corresponding boundary conditions in the wave frame are where r 1 is the radius of the inner tube endoscope . In order to reduce the number of variables, we introduce the following nondimensional variables: Mathematical Problems in Engineering Here R e , W e , and represent the Reynolds number, Weissenberg number, and wave number, respectively. Moreover, φ 1 and φ 2 are nondimensional amplitudes of the waves, δ is the annulus aspect ratio, and γ represents the wave length ratio between two waves. where the involved quantities are defined in Appendix. The expressions of pressure rise Δp and the frictional forces F 0 and F 1 at the outer and inner boundaries, respectively, in their nondimensional forms, are given as where dp/dz is defined through 3.3 , and flow rate Q in dimensionless form is defined as where q is flow rate in the fixed frame of reference, and h 2 is square of displacement of the walls over the length of an annulus, defined as  where the involved quantities are defined in Appendix. 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 .

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. 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. 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 F 0 and F 1 show similar behavior when compared to their respective parameters. It is significant to mention that inner friction force F 1 attains higher magnitude than outer friction force F 0 with increasing values of any given set of parameters.
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 φ 1 , φ 2 , W e , γ, and δ is depicted. It is observed that pressure gradient increases with increasing the values of previously mentioned parameters. The influence of various parameters on streamlines pattern is depicted in Figures 6,7,8,9,and 10. It is noted that trapping is observed in all these cases. Figures 6, 7, 8, and 9 show that the size of trapped bolus increases for higher values of Weissenberg number W e , wave length ratio γ , and amplitude φ 2 . 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.

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 φ 2 , 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;