A viscoplastic slump on a horizontally oscillating plate

A thin, two-dimensional deposit of viscoplastic fluid is studied as it slumps atop a horizontal plate exhibiting in-plane oscillations. Numerical results are reported for a range of Bingham and oscillatory Reynolds numbers, and comparison is made with asymptotic results obtained for low oscillatory Reynolds numbers. When inertia is small, these flows become arrested as time $t\rightarrow \infty$ , and approach a symmetric state where the maximum basal stress in a period of oscillation balances with the yield stress. An exact expression for this final shape is obtained. The deposit slumps further across the plate at small Bingham numbers and results in a thinner profile. The arrested shape is equivalent to the down-slope arrested state attained by viscoplastic fluid on an inclined plane, for an inclination angle that is related to the plate oscillation parameters and gravity. For a Herschel–Bulkley fluid, it is shown that small perturbations to the final shape decay as $t^{-2N/(N+2)}$ at late times, where $N$ is the fluid’s strain-rate power-law index. This decay rate is slower than from slumps driven solely by gravity, which is due to the differing physical mechanisms driving the approach to the arrested state. With forced oscillations, yielding occurs only within progressively shorter intervals of time about the maximum basal stress, whereas the inclined plane deposit is always yielded in some diminishing region at the base.