Viscous flow in bistable elastic channels - analogy to Stefan's solidification problem

The presented work examines the effect of cross-section bi-stability on the dynamics of viscous flow propagating into slender elastic channels. In the examined configuration, the channel boundaries include a compressed and curved elastic sheet which is stable in two different deformation patterns for a given pressure. The state of the elastic arc is determined by the history of the pressure changes and the initial conditions of the system. The viscous flow is analyzed via applying the lubrication approximation and examining self-similarity. For the case of constant inlet pressure, the propagation rate of the location in which snap between the two states occurs, is presented for various physical limits. An analogy between the examined bi-stable configuration and Stefan's solidification problem is presented.