An arbitrary lagrangian eulerian formulation with exact mass conservation for the numerical simulation of a rising bubble in a viscoelastic fluid

The arbitrary Lagrangian Eulerian (ALE) framework presented in [Sahin and Guventurk, An Arbitrary Lagrangian-Eulerian framework with exact mass conservation for the numerical simulation of 2D rising bubble problem. International Journal for Numerical Methods in Engineering, 112:2110-2134, (2017)] has been initially extended to three-dimensional multiphase ows. In the present formulation, the governing equations are discretized over the unstructured moving meshes using the divergence-free side-centered nite volume formulation with the exact jump conditions across the interface. Therefore, the pressure eld is treated to be discontinuous with the discontinuous treatment of density and viscosity. The surface tension term at the interface is handled as a force tangent to the interface. A special attention is given to the application of the kinematic boundary condition to be compatible with the local and global discrete geometric conservation laws (DGCL) as well as the discrete form of the continuity equation in order to conserve the total mass of both species at machine precision. The mesh deformation is achieved by solving the linear elasticity equations with the modi ed material properties based on the minimum distance to the interface. Then, the numerical method has been further extended to viscoelastic multiphase ows using the approach in [M. Sahin, A stable unstructured nite volume method for parallel large-scale viscoelastic uid ow calculations. Journal of non-Newtonian Fluid Mechanics, 166:779-791, (2011)]. The resulting algebraic equations are solved in a fully coupled (monolithic) manner and a one-level restricted additive Schwarz preconditioner with a block-incomplete factorization is utilized within each partitioned sub-domain. The proposed method is initially validated by simulating the classical three-dimensional benchmark problems of a single rising bubble in a Newtonian uid and then it will be applied to a rising bubble in an Oldroyd-B uid. The mass of the bubble is conserved and discontinuous pressure eld is obtained in order to avoid errors due to the incompressibility condition in the vicinity of the interface.