Finite stopping times for freely oscillating drop of a yield stress fluid

TL;DR

This paper investigates whether a yield stress fluid droplet with a free surface can come to rest in finite time, deriving conditions for finite stopping times and illustrating the results through numerical experiments.

Contribution

It introduces a variational inequality formulation for yield stress fluid dynamics with free surfaces and establishes the existence of finite stopping times for droplet oscillations under certain conditions.

Findings

Finite stopping time exists for positive yield stress and flow index ≥ 1.

Numerical experiments show dependence of stopping time on parameters.

Droplet transitions instantaneously from yielding to rigid state.

Abstract

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.