A motion of freely oscillating droplet of a yield stress fluid: analysis and numerical studies

This dissertation studies the problem of free small-amplitude oscillations of a droplet of a yield stress fluid under the action of surface tension forces. The problem is treated both analytically and numerically. In particular, we address the question if there exists a finite stopping time for an unforced motion of a yield stress fluid with free surface.

In this thesis, 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 balance with the internal energy viscoplastic dissipation and the work of external forces. Further, we consider 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 $\alpha$ satisfies $\alpha \ge 1$. 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.

In Charpter 1, we review the Navier-Stokes equations for motion of incompressible viscous fluid and consider different boundary conditions. We also discuss several approaches to recover the evolution of free interface.

In Charpter 2, we derive a variational inequality formulation for the problem of yield stress fluid dynamics with a free surface. An energy balance follows from the variational inequality. In this chapter, we also describe a numerical method to simulate a non-Newtonian fluid flow with free surface.

In Charpter 3, we apply the method of viscous velocity potentials to study the problem of small-amplitude oscillations of a fluid droplet driven by surface tension forces. First the Newtonian fluid is treated and some well-known results are derived. Numerical experiments are provided to illustrate our results.

In Charpter 4, we apply the method of visco-plastic velocity potentials to study the problem of small-amplitude oscillations of a non-Newtonian droplet driven by free surface tension forces. For a yield stress fluid we prove that oscillations have a finite stopping time. We describe the motion of a single harmonic ($2nd$ order harmonic) of oscillating droplet by an ODE. Numerical experiments illustrated our results.

In Charpter 5, we give the conclusion and outlook.