Single singular shock formation via Hugoniot asymptotics during transition to the ridged regime

Determine whether, in the hyperbolic system of three conservation laws for film height and the two depth-averaged particle volume fractions governing bidisperse particle-laden thin-film flow down an incline, the transition from the settled regime to the ridged regime corresponds to Hugoniot loci that approach one another asymptotically so as to produce a single singular shock, rather than the multiple-shock structure observed in the bidensity case; and characterize the parameter conditions under which this behavior occurs.

Background

The paper derives a three-by-three hyperbolic conservation law model for bidisperse suspensions on an incline and discusses solution structures via Riemann problems and Hugoniot loci.

In earlier monodisperse and bidensity studies, singular shocks were associated with Hugoniot curves approaching asymptotically in the far field. Here, the authors note that in the ridged regime they observe a single singular shock and explicitly pose the open problem of establishing that, during the transition from settled to ridged regimes, the Hugoniot curves approach asymptotically in a manner that yields a single singular shock, in contrast to the multiple-shock structure seen in the bidensity case.