Stability and bifurcation analysis of stagnation/equilibrium points for peristaltic transport in a curved channel

Abstract

The stability of equilibrium points and their bifurcations for a peristaltic transport of an incompressible viscous fluid through a curved channel have been studied when the channel width is assumed to be very small as compared to the wavelength of peristaltic wave and inertial effects are negligible. An analytic solution for the stream function has been obtained in a moving coordinate system which is translating with the wave velocity. Equilibrium points in the flow field are located and categorized by developing a system of nonlinear autonomous differential equations, and the dynamical system methods are used to investigate the local bifurcations and corresponding topological changes. Different flow situations, encountered in the flow field, are classified as backward flow, trapping, and augmented flow. The transition of backward flow into a trapping phenomenon corresponds to the first bifurcation, where a nonsimple degenerate point bifurcates under the wave crest and forms a saddle-center pair with the homoclinic orbit. The second bifurcation appears when the saddle point further bifurcates to produce the heteroclinic connection between the saddle nodes that enclose the recirculating eddies. The third bifurcation point manifests in the flow field due to the transition of trapping into augmented flow, in which a degenerate saddle bifurcates into saddle nodes under the wave trough. The existence of second critical condition is exclusive for peristaltic flow in a curved channel. This bifurcation tends to coincide with the first one with a gradual reduction in the channel curvature. Global bifurcation diagrams are utilized to summarize these bifurcations.