Bifurcations of stagnation points in a micropolar fluent media under the influence of an asymmetric peristaltic movement

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.