A viscoplastic slump on an oscillating plate

Motivated by recent experimental work exploring the oscillation of drops of molten chocolate and carbopol, we study a thin, two-dimensional deposit of viscoplastic fluid as it slumps atop a horizontal plate exhibiting in-plane oscillations. Numerical results are obtained for a range of Bingham and Stokes numbers, and are compared to asymptotic results for relatively low oscillation frequencies. When inertia is small, these flows become arrested asymptotically and approach a static, symmetric shape for which an exact expression is reported. Interestingly, this shape is equivalent to the down-slope arrested state obtained for a viscoplastic fluid on an inclined plane, for an inclination angle that depends on the plate oscillation parameters and gravity. For a Herschel-Bulkley fluid, 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 result differs from the gravitationally driven case, which exhibits a t^-N decay, so forced oscillations result in a slower decay to the arrested state.