New analytical models are introduced to describe the motion of a Herschel–Bulkley fluid slumping under gravity in a narrow fracture and in a porous medium. A useful self-similar solution can be derived for a fluid injection rate that scales as time $t$ ; an expansion technique is adopted for a generic injection rate that is power law in time. Experiments in a Hele-Shaw cell and in a narrow channel filled with glass ballotini confirm the theoretical model within the experimental uncertainty.

Most models of sediment transport are based on the hypothesis of a weak interaction between the fluid and the sediment phase, where the main flow is subjected to the mass and momentum conservation of the fluid phase, with small corrections due to the presence of the sediments. These models usually give a correct answer for conditions where the concentration of sediment is really low. In the case of high sediment transport, some models analyze the two subdomains separately, the high and the low concentration, and employ different constitutive equations. In the present model there is a two-phase description of the whole domain. The closure of the turbulence and the interaction between the sediments and the fluid introduce approximations, but results are consistent with experiments and previous models.