On Well‑Posed Two‑Fluid Interfacial Dynamics Modeling

The fully developed, incompressible Eulerian two‑fluid model with drag as the sole interfacial momentum exchange is mathematically under‑determined: any realizable volume‑fraction profile $\alpha_1(y)\in[0,1]$ yields a steady solution, so uniqueness fails. We propose a Minimum Transverse Force Pair (MTFP)---a balanced collective and dispersive force---as the minimum additional physics required to restore well‑posedness and realizability. This assertion has obvious implications in developing internal and external multi-phase boundary layer modeling. Here, we provide proofs of this assertion, embed the MTFP in contemporary CFD discretizations (coupled dissipation operators with $\nabla\alpha$ consistency, distributed source treatment, staggered mesh correspondence), and demonstrates its necessity and sufficiency on canonical annular, stratified and bubbly‑pipe benchmarks. We show that dispersion \emph{alone} or collective \emph{alone} remain ill‑posed, while a properly discretized MTFP uniquely selects the volume‑fraction field and eliminates spurious grid‑dependent solutions. Recommendations for industrial two‑fluid solvers and open research directions are given.