Projection method for co-incompressible two-phase mixtures

Develop a projection method for the co-incompressible two-phase mixture model that projects the network and solvent velocity fields in a manner consistent with both phase momentum equations while enforcing the co-incompressibility constraint.

Background

In single-phase incompressible flow, projection-based methods are widely used to enforce the incompressibility constraint efficiently by solving an auxiliary Poisson problem. In the two-phase mixture setting considered here, each phase has its own velocity, and the model enforces a volume-averaged co-incompressibility constraint coupling the phase velocities.

Because both phase momentum equations must remain consistent after projection, it is not straightforward to construct a projection that simultaneously enforces the constraint and preserves consistency with both momentum equations. The paper explicitly notes that how to perform such a projection is currently unclear.

References

For the co-incompressible condition used herein, it remains unclear how to project the velocity fields of individual phases that gives fields consistent with both momentum equations.

— Adaptive Mesh Refinement for Two-Phase Viscoelastic Fluid Mixture Models  (2409.19974 - Nagda et al., 2024) in Section 1, Introduction