Local projection stabilized finite element modeling of viscoelastic two-phase flows

A three-field local projection stabilized finite element method is developed for computations of a 3D-axisymmetric buoyancy driven bubble rising in a liquid column in which either the bubble or the liquid column can be viscoelastic. The two-phase flow is described by the time-dependent incompressible Navier--Stokes equations, whereas the viscoelasticity is modeled by the Giesekus constitutive equation in a time-dependent domain. The arbitrary Lagrangian Eulerian~(ALE) formulation with finite elements is used to solve the governing equations in the time-dependent domain. The interface-resolved moving meshes in ALE allows to incorporate the interfacial tension force and jumps in the material parameters accurately. An one-level Local Projection Stabilization~(LPS), which is based on an enriched approximation space and a discontinuous projection space, where both spaces are defined on a same mesh is used to stabilize the model equations. The stabilized numerical scheme allows us to use equal order interpolation spaces for the velocity and the viscoelastic stress, whereas inf-sup stable finite elements are used for the velocity and the pressure. A comprehensive numerical investigation is performed for a Newtonian bubble rising in a viscoelastic fluid and a viscoelastic bubble rising in a Newtonian fluid. The influence of the viscosity ratio, Newtonian solvent ratio, Giesekus mobility factor and the E\"{o}tv\"{o}s number on the bubble dynamics are analyzed. The numerical study shows that a Newtonian bubble rising in a viscoelastic fluid experiences an extended trailing edge with a cusp-like shape and also exhibits the negative wake phenomena. However, a viscoelastic bubble rising in a Newtonian fluid develops an indentation around the rear stagnation point with a dimpled shape.