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

• Published in: Journal of non-Newtonian fluid mechanics Vol. 239; pp. 73 - 84
• Main Authors: Cheng, Wanli, Olshanskii, Maxim A.
• Format: Journal Article
• Language: English
• Published: Amsterdam Elsevier B.V 01-01-2017; Elsevier BV
• Subjects: Capillary flow; Computational fluid dynamics; Dissipation; Experiments; Finite stopping time; Flow velocity; Free energy; Free surface flows; Internal energy; Oscillating drop; Oscillations; Surface tension; The Herschel-Bulkley model; Viscoplastic fluid; Yield strength; Yield stress; The Herschel-Bulkley model; Oscillating drop; Viscoplastic fluid; Finite stopping time; Free surface flows
• ISSN: 0377-0257; 1873-2631
• DOI: 10.1016/j.jnnfm.2016.12.001

•We devise variational inequality and energy balance for a free-surface flow of a yield-stress fluid.•The method of viscous potentials is used to study small oscillations of a viscoplastic droplet.•Oscillations of a viscoplastic droplet driven by surface tension forces have finite stopping times.
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 variational inequality formulation is deduced for the problem of yield stress fluid dynamics with a free surface. The free surface is assumed to evolve with a normal velocity of 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 Tf of oscillations exists once the yield stress parameter is positive and the flow index α satisfies α ≥ 1. Results of several numerical experiments illustrate the analysis, reveal the dependence of Tf on problem parameters and suggest an instantaneous transition of the whole drop from yielding state to the rigid one.