Infinite-wavelength analysis for two-phase flow: A three-parameter computer-assisted study of global bifurcations

A computer-assisted analysis of the long-wave solutions of a two-phase flow model in packed beds is presented, based on the numerical study of global bifurcations. A time-dependent, pulsing flow arises spontaneously for relatively high flow rates of gas and liquid through a packed bed. A recently developed nonlinear model for this phenomenon invokes a travelling wave assumption, and associates the observed wavelike pulses with the existence of long-period periodic solutions in the travelling wave frame. In parameter space, such periodic solutions are located in the neighborhood of certain infinite-period (global) bifurcations, such as homoclinic and double heteroclinic loops. In order to completely map out the travelling wave flow behavior for a given set of fluids and packing, it is necessary to perform a three parameter analysis of these global bifurcations. Using techniques for the continuation of homoclinic and heteroclinic connections, we have examined the relation between homoclinic, heteroclinic (both codimension-1) and double heteroclinic (codimension-2) connections in the model. In parameter space, the families of these bifurcations interact and undergo further qualitative changes. These changes include a crossing of homoclinic connection branches, and the birth of a pair of double heteroclinic connections. We discuss briefly how double heteroclinic loops apparently arise from a local codimension-three singularity where a double-zero eigenvalue occurs simultaneously with the vanishing of a higher (second) order term. We also discuss the relation between the nature of the predicted solutions and experimental observations. This computational study of global bifurcations provides a concise description of the operating region in which pulsing flow can be observed in packed beds.