Abstract

In the present analysis, the effects of an asymmetric peristaltic movement on the bifurcations of stagnation points have been investigated. An exact analytic solution for a flow of an incompressible micropolar fluid has been established under long wavelength and vanishing Reynolds number assumptions in a moving frame of reference. The stagnation points are located through a system of autonomous differential equations. The behavior and bifurcations of these stagnation points and corresponding streamline patterns have been epitomized through dynamical system methods. Different flow situations manifesting in the flow are characterized as follows: backward flow and trapping and augmented flow. Two possible bifurcations encountered in the flow because of the transitions between these flow regions, where nonhyperbolic degenerate points appear and heteroclinic connections between saddles are conceived. The micropolar parameter, coupling number, amplitude ratios, and phase difference have significant impacts on the bifurcations of the stagnation points and the ranges of the flow rate, which are explored graphically by local bifurcation diagrams. The backward flow region is observed to shrink by increasing the micropolar parameter up to an optimal value, and later an opposite trend is found. Furthermore, the increment in the coupling number causes the trapping region to expand. A reduction in the trapping phenomenon is encountered by enlarging the phase difference, while the augmented flow region becomes smaller for large amplitudes of peristaltic waves propagating along the walls of the channel. At the end, global bifurcation diagrams are used to summarize the obtained results.

Highlights

  • The study of the peristaltic flow problems has attracted a lot of attention from researchers due to its wide range of physiological, industrial, and engineering applications

  • Two critical conditions are manifested themselves at the centerline ((h1 + h2)/2) of the channel: first one is appeared between the waves crests of the lower and upper channel walls because of the conversion of the backward flow region into the trapping region, while the second bifurcation condition happens in the middle of the wave troughs as a result of the transition of the trapping region into the augmented flow region

  • Panel (ii) of Fig. 3 shows that the trapping phenomenon occurs at low volumetric flow rates for larger values of the coupling number n

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Summary

Introduction

The study of the peristaltic flow problems has attracted a lot of attention from researchers due to its wide range of physiological, industrial, and engineering applications. In this phenomenon, the propagation of transverse waves along the flexible walls of a duct is responsible for the movement of fluid. After the remarkable works of Shapiro et al. and Fung and Yih, several attempts have been made to analyze the peristaltic flow phenomenon under various situations.

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