  The paper addresses the question if there exists a finite stopping time for
an unforced motion of a yield stress fluid with free surface. A variation
inequality formulation is deduced for the problem of yield stress fluid
dynamics with a free surface. Free surface is assumed to evolve with a normal
velocity the flow. We also consider capillary forces acting along the free
surface. Based on the variational inequality formulation an energy equality is
obtained, where kinetic and free energy rate of change is in a balance with the
internal energy viscoplastic dissipation and the work of external forces.
Further, the paper considers free small-amplitude oscillations of a droplet of
Herschel-Bulkley fluid under the action of surface tension forces. Under
certain assumptions it is shown that the finite stopping time $T_f$ of
oscillations exists once the yield stress parameter is positive and the flow
index $\alpha$ satisfies ($\alpha\ge1$). Results of several numerical
experiments illustrate the analysis, reveal the dependence of $T_f$ on problem
parameters and suggest an instantaneous transition of the whole drop from
yielding state to the rigid one.
