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Models and Equations

This page documents the PDE models represented by the current implementation. The source/tests/examples are treated as the ground truth for supported scope.

1. Monophasic Steady Stokes

For velocity u and pressure p in fluid region Ω:

\[-\nabla\cdot\left(2\mu D(u)\right) + \nabla p = f, \qquad \nabla\cdot u = 0,\]

with D(u) = (\nabla u + \nabla u^T)/2.

Discrete representation in StokesModelMono:

  • MAC staggered unknowns on per-component velocity grids (uomega) and pressure grid (pomega),
  • cut/interface trace unknowns ugamma,
  • momentum diffusion-like blocks from cut-cell operators,
  • split pressure-gradient/divergence coupling,
  • row overwrite for boundary/interface constraints,
  • pressure gauge replacement to remove nullspace.

2. Monophasic Unsteady Stokes

\[\rho \partial_t u - \nabla\cdot\left(2\mu D(u)\right) + \nabla p = f, \qquad \nabla\cdot u = 0.\]

Implemented via assemble_unsteady! / solve_unsteady! with theta scheme:

  • :BE (\theta = 1),
  • :CN (\theta = 1/2),
  • numeric theta in [0,1].

Mass terms are added on momentum omega rows; continuity and gauge remain algebraic constraints.

3. Moving-Boundary Monophasic Stokes

MovingStokesModelMono supports prescribed moving embedded boundaries:

  • geometry callback body(x..., t) (or static body(x...)),
  • per-component cut velocity BC bc_cut_u,
  • slab-based assembly over [t_n, t_{n+1}] using SpaceTimeCartesianGrid reduction,
  • strong end-time trace enforcement on ugamma rows,
  • activity masking based on end-time support.

This is a prescribed-motion moving-boundary formulation, not deformable-body FSI.

4. Fixed-Interface Two-Phase Stokes

For phases k in {1,2} in fixed subdomains Ω_k:

\[-\nabla\cdot\left(2\mu_k D(u_k)\right) + \nabla p_k = f_k, \qquad \nabla\cdot u_k = 0.\]

Current interface formulation in StokesModelTwoPhase:

  • shared interface velocity trace unknowns ugamma (velocity continuity path),
  • phase-wise pressure blocks pomega1, pomega2,
  • traction-balance rows on ugamma with optional interface forcing interface_force,
  • per-phase mu1, mu2, rho1, rho2 supported.

Current scope is fixed-interface (no geometric interface evolution in two-phase constructor path).

5. Pressure Gauges

A gauge is required to remove pressure nullspace.

  • PinPressureGauge(index=...): pin one pressure DOF.
  • MeanPressureGauge(): enforce volume-weighted zero mean over active pressure cells.

Applied for mono/two-phase/moving models by row replacement in assembled systems.

6. Boundary and Interface Scope (Current)

Implemented:

  • outer velocity BCs: Dirichlet, Neumann, Periodic, plus Stokes traction-family side BCs (Traction, PressureOutlet, DoNothing, Symmetry),
  • optional outer pressure BC bc_p with Dirichlet/Neumann/Periodic handling,
  • cut/interface velocity Dirichlet enforcement on ugamma.

Missing/unsupported in current code path:

  • cut/interface velocity Neumann and Periodic (ArgumentError),
  • moving two-phase interface model,
  • deformable-body coupling,
  • multi-body/contact models in FSI.

7. FSI Scope and Limits

Rigid-body FSI wrappers are implemented beyond translation-only:

  • 2D/3D rigid states and parameters,
  • rotational state updates (2D scalar spin, 3D angular velocity + orientation),
  • split one-pass coupling (step_fsi!),
  • strong fixed-point coupling with optional Aitken relaxation (step_fsi_strong!).

Current limits:

  • single rigid body,
  • no contact/collision handling,
  • no deformable body model.