Stochastic particle level set simulations of buoyancy-driven droplets in non-Newtonian fluids

Abstract

The aim of this paper is to present stochastic, micro–macro simulations of droplets rising in non-Newtonian fluids by means of a particle level set method implemented using the Finite Element Method (FEM) and semi-Lagrangian schemes. A set of dumbbells scattered throughout the domain conveys the molecular information from the microscopic scale to the macroscopic scale, first by integrating the stochastic-equivalent of the Fokker–Planck equations, then by taking second moments from the internal configurations to produce the polymer stress tensor to be incorporated into the macroscopic momentum equation. The kinetic models considered in this work are the simpler Hooke (Oldroyd-B constitutive) model, and the more realistic ‘Finitely Extensible Nonlinear Elastic’ (FENE) model. First, we carry out microscopic and macroscopic convergence analyses to validate the numerical calculations of 2D rising droplets, obtaining satisfactory results for grids of size 1/h={40,80,160},and a number of particles ranging from Nd=105up to 1.75 · 108, making use of both kinetic models along with mildly viscoelastic effects in solutions with relatively low polymeric concentrations and relaxation times (De=1,c=1). We then perform a series of increasingly elastic simulations that ends up with the arrival of two purely non-Newtonian features: the onset of a cusp-like tail and the appearance of downward velocities at a certain distance of the droplet (negative wake).