AUTHOR=Berres Stefan , Castañeda Pablo 

TITLE=Bifurcation of solutions through a contact manifold in bidisperse models

JOURNAL=Frontiers in Applied Mathematics and Statistics

VOLUME=Volume 9 - 2023

YEAR=2023

URL=https://www.frontiersin.org/journals/applied-mathematics-and-statistics/articles/10.3389/fams.2023.1199011

DOI=10.3389/fams.2023.1199011

ISSN=2297-4687

ABSTRACT=This work deals with a hyperbolic system of two nonlinear conservation laws that models bidisperse suspensions comprising two types of small particles that disperse in a viscous fluid and differ in size and viscosity. We examine the solution dependence for the relative position to the two contact manifolds in the phase space. When the initial condition crosses the first contact manifold, a bifurcation occurs. Another structure appears to bifurcate as the initial condition approaches the second manifold; however, the triple shock rule shows that this phenomenon does not represent a bifurcation.

The first analysis is based on the classification of elementary waves that begin at the origin of phase space. Analytical solutions to prototypical Riemann problems connecting the origin to any point in the state space are provided. The second analysis is based on semi-analytical solutions to Riemann problems connecting any state in the phase space with the maximum packing concentration line, which are the two Riemann problems that occur in standard batch sedimentation tests.

A small appendix discusses the emerging quasi-umbilic points that appear in this and earlier works. These points can develop new types of bifurcations as relevant members of the elliptic/hyperbolic boundary of the system of partial differential equations.