Mass-flux-controlled diffuse stationary jumps in dense granular flows upstream of a finite-width obstacle

Granular flows impacting obstacles may develop jump-like structures whose height, upstream extent, and deposit morphology depend sensitively on the incoming flow state and obstacle geometry. For finite-width obstacles, however, part of the incoming material can bypass the obstacle laterally, so that the stationary jump is governed not only by a local front condition but also by system-scale mass redistribution. Here, we develop a depth-averaged control-volume model for diffuse stationary jumps (DSJ) formed upstream of a finite-width obstacle in dense granular flows down an incline. The model is derived from mass and momentum balances across the reverse-propagating jump front and retains the effects of gravity, effective friction, finite jump length, and weak compressibility. We show that, once a statistically stationary DSJ is reached, the normalized mass flux at the jump front is in one-to-one correspondence with the local Froude number under the dense-flow approximation. This relation allows the conventional Froude-based jump relation to be reformulated in terms of a mass-flux variable that is directly linked to the imposed inflow and lateral redistribution in the finite-width-obstacle configuration. Comparisons with discrete element method simulations and available experimental data show that the model captures the jump-height ratio in the dense DSJ regime. By coupling the jump relation with geometric closures for the post-jump deposit, the model further predicts the deposit length and height within the range of configurations considered here. The results provide a compact framework for connecting inflow supply, jump-front adjustment, lateral bypass, and stationary deposit morphology in dense granular flows interacting with finite-width obstacles.