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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

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 [

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 [

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 [

After the pioneering work of Latham [

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

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

(a) Physical sketch of the problem. (b) Comparison of numerical and perturbation solutions of axial velocity

The geometry of the outer wall surface is described as

The governing equations in the fixed frame for an incompressible Williamson fluid model [

We consider the case

In the fixed coordinates

Under the assumption of long wavelength

Since (

The expressions of pressure rise

Equation (

Numerical and perturbation solutions for axial velocity

Numerical Sol. | Perturbation Sol. | Error | |
---|---|---|---|

0.10 | −1.0000 | −1.0000 | 0.0000 |

0.15 | −1.0284 | −1.0286 | 0.0002 |

0.20 | −1.0470 | −1.0473 | 0.0003 |

0.25 | −1.0599 | −1.0601 | 0.0002 |

0.30 | −1.0687 | −1.0689 | 0.0002 |

0.35 | −1.0745 | −1.0748 | 0.0003 |

0.40 | −1.078 | −1.0781 | 0.0001 |

0.45 | −1.0794 | −1.0795 | 0.0001 |

0.50 | −1.0790 | −1.0790 | 0.0000 |

0.55 | −1.0770 | −1.0770 | 0.0000 |

0.60 | −1.0734 | −1.0734 | 0.0000 |

0.65 | −1.0685 | −1.0684 | 0.0001 |

0.70 | −1.0622 | −1.0622 | 0.0000 |

0.75 | −1.0546 | −1.0547 | 0.0001 |

0.80 | −1.0461 | −1.0460 | 0.0001 |

0.85 | −1.0362 | −1.0361 | 0.0001 |

0.90 | −1.0251 | −1.0252 | 0.0001 |

0.95 | −1.0128 | −1.0131 | 0.0003 |

1.0 | −1.0000 | −1.0000 | 0.0000 |

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

Variation of pressure rise per wavelength

Similarly the effects of

Variation of frictional force at the inner wall

Variation of frictional force at the outer wall

In order to discuss the effects of variation of various parameters on the axial pressure gradient

Pressure gradient

The influence of various parameters on streamlines pattern is depicted in Figures

Streamlines pattern for (a)

Streamlines pattern for (a)

Streamlines pattern for (a)

Streamlines pattern for (a)

Streamlines pattern for (a)

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:

the peristaltic pumping rate decreases with increasing the values of

frictional forces show an opposite behavior to that of pressure rise in peristaltic transport;

the inner friction force

Pressure gradient increases with increasing the values of all embedded parameters that is,

an increase in radius ratio

moreover, it is observed that the size and number of trapping bolus decrease with increasing values of flow rate

The values of quantities appearing in the expressions (

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.