An efficient split-step scheme for fluid–structure interaction involving incompressible viscous flows

The paper proves Theorem 2.1: the ALE velocity–pressure system (22)–(27) is equivalent to the momentum-plus-pressure–Poisson system (30)–(34). The forward direction follows by rewriting the stress term and deriving the PPE and pressure boundary conditions via identities (35)–(37) and curl–curl/variable-viscosity calculus, matching equations (30)–(34) exactly . For the reverse direction, the paper does not assume incompressibility; instead it subtracts the PPE from the divergence of the momentum balance to obtain a parabolic ALE equation for Φ:=∇·u with mixed boundary conditions (38)–(40). Using the geometric conservation law and the initial condition ∇·u0=0, it concludes Φ≡0, and thus recovers (22)–(27) . By contrast, the model’s B⇒A step incorrectly claims that subtracting the PPE from the divergence of momentum yields ∇·(ρ ∂t u|A)=0; the missing μ-dependent diffusion term means essential terms were dropped. The correct result is the heat-type equation (38), not a pure conservation law, so the model’s reverse implication is flawed .