An exact mass-conserving arbitrary Lagrangian-Eulerian framework for viscoelastic multiphase fluid flows

An arbitrary Lagrangian Eulerian (ALE) framework presented in A mass conserving arbitrary Lagrangian-Eulerian formulation for three-dimensional multiphase fluid flows , International Journal for Numerical Methods in Fluids 94 (4), 346–376 has been extended to solve incompressible multiphase viscoelastic flow problems in two- and three-dimensions. The incompressible, isothermal linear momentum balance equations, coupled with the viscoelastic constitutive models Oldroyd-B and FENE-CR, are discretized using a div-stable, side-centered finite volume approach in which the velocity components are defined at the mid-points of element faces, the displacement vector is defined at the vertices, and the pressure and modified conformation tensor are defined at the element centroids. At the interface, the surface tension force is treated as a force tangential to the interface, and its normal vector is evaluated by using the mean weighted by sine and edge length reciprocals (MWSELR) approach. In order to ensure mass conservation of both species at machine precision, special attention is given to enforcing the kinematic boundary condition at the interface in the normal direction, while obeying the discrete geometric conservation law (DGCL). The numerical approach allows discontinuities in material properties, including density and viscosity, as well as in the pressure and modified conformation tensor across the interface. The discrete algebraic equations arising from the incompressible linear momentum balance equations are solved monolithically using a block preconditioner based on the BoomerAMG parallel algebraic multigrid solver from the HYPRE library, interfaced through PETSc. To validate the numerical algorithm, the benchmark problem of a single Newtonian or viscoelastic bubble (modeled using Oldroyd-B and FENE-CR) rising through a quiescent Newtonian or viscoelastic fluid is examined in both two- and three-dimensions. The numerical simulations exhibit excellent agreement with previous results in the literature and show strong consistency with mesh refinement. Positive and negative transient wakes are observed behind the bubble, demonstrating that the formation of a transient negative wake does not require a viscoelastic fluid model with shear-thinning behavior. The numerical approach successfully preserves the volume of the bubble to nearly machine precision and accurately captures discontinuities in the pressure and modified conformation tensor across the interface, where there are jumps in density and viscosity.