Streamline topologies of two-dimensional peristaltic flow and their bifurcations

Streamline patterns and their local and global bifurcations in a two-dimensional peristaltic flow for an incompressible Newtonian fluid have been investigated. An analytical solution for the stream-function is found under a long-wavelength and low-Reynolds number approximation. The problem is solved in a moving coordinate system where a system of nonlinear autonomous differential equations can be established for the particle paths. Local bifurcations and their topological changes are inspected using methods of dynamical systems. Three different flow situations manifest themselves: backward flow, trapping or augmented flow. The transition between backward flow to trapping corresponds to a bifurcation of codimension one, in which a non-simple degenerate point changes its stability to form heteroclinic connections between saddle points that enclose recirculating eddies. The transition from trapping to augmented flow is a bifurcation of codimension two, in which heteroclinic saddle connections of adjacent waves coalesce below wavetroughs. The coalescing of saddle nodes on the longitudinal axis produces a degenerate point with six heteroclinic connections (degenerate saddle). As the parameter is increased, the degenerate saddle bifurcates to saddles nodes which lift off the centerline. This new saddle heteroclinic connections enclose recirculating eddies and part of the fluid is able to flow through the center. These bifurcations are summarized in a global bifurcation diagram for the planar and axisymmetric cases. Theoretical results agree with the experimental data available in the literature. Nomenclature a wave amplitude c wave velocity d product of eigenvalues h equation of wall in moving frame (plane) ha equation of wall in moving frame (axisymmetric) H equation of wall in fixed frame (plane) Ha equation of wall in fixed frame (axisymmetric) J Jacobian M,N bifurcation branches (plane) O,P bifurcation branches (axisymmetric) p pressure or trace of J when used with subscripts q flow rate in moving frame (plane) qa flow rate in moving frame (axisymmetric) Q instantaneous flow rate in fixed frame (plane) Qa instantaneous flow rate in fixed frame (axisymmetric) Q0 time-mean flow rate in fixed frame for zero pressure rise