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The effects of resistive forces on unsteady shallow flows over rigid horizontal boundaries are investigated theoretically. The dynamics of this type of motion are driven by the streamwise gradient of the hydrostatic pressure, which balances the inertia of the fluid and the basal resistance. Drag forces are often negligible provided the fluid is sufficiently deep. However, close to the front of some flows where the depth of the moving layer becomes small, it is possible for drag to substantially influence the motion. Here we consider three aspects of unsteady shallow flows. First we consider a regime in which the drag, inertia and buoyancy (pressure gradient) are formally of the same magnitude throughout the entire current and we construct a new class of similarity solutions for the motion. This reveals the range of solution types possible, which includes those with continuous profiles, those with discontinuous profiles and weak shocks and those which are continuous but have critical points of transition at which the gradients may be discontinuous. Next we analyse one-dimensional dam-break flow and calculate how drag slows the motion. There is always a region close to the front in which drag forces are not negligible. We employ matched asymptotic expansions to combine the flow at the front with the flow in the bulk of the domain and derive theoretical predictions that are compared to laboratory measurements of dam-break flows. Finally we investigate a modified form of dam-break flow in which the vertical profile of the horizontal velocity field is no longer assumed to be uniform. It is found that in the absence of drag it is no longer possible to find a kinematically consistent front of the fluid motion. However the inclusion of drag forces within the region close to the front resolves this difficulty. We calculate velocity and depth profiles within the drag-affected region, and obtain the leading-order expression for the rate at which the fluid propagates when the magnitude of the drag force is modelled using Chézy, Newtonian and power-law fluid closures; this compares well with experimental data and provide new insights into dam-break flows.
Hogg et al. (Mon,) studied this question.