Numerical Simulation of a Rising Bubble in Viscoelastic Fluids

In this paper we discuss simulation techniques for a rising bubble in viscoelastic fluids via numerical methods based on high order FEM. A level set approach based on the work in (Sethian, Level set methods and fast marching methods: evolving interfaces in computational geometry, fluid mechanics, computer vision, and material science, 2nd edn. Cambridge University Press, 1999) is used for interface tracking between the bubble and the surrounding fluid. The two matters obey the Newtonian and the Oldroyd-B constitutive law in the case of a viscoelastic fluid while the flow model is given by the Navier-Stokes equations. The total system of equations is discretized in space by the LBB-stable finite element Q 2 P 1, and in time by the family of θ-scheme integrators. The solver is based on Newton-multigrid techniques (Damanik et al., J Comput Phys 228:3869–3881, 2009; J Non-Newton Fluid Mech 165:1105–1113, 2010) for nonlinear fluids. First, we validate the multiphase flow results with respect to the benchmark results in (Hysing et al., Int J Numer Methods Fluids 60(11):1259–1288, 2009), then we perform numerical simulations of a bubble rising in a viscoelastic fluid and show cusp formation at the trailing edge.