Transport Models

1. Monophasic Scalar Transport

Fixed geometry (Ω, Γ fixed):

\[\partial_t \phi + \nabla\cdot(\mathbf{u}_\omega\,\phi) = s,\]

with steady limit

\[\nabla\cdot(\mathbf{u}_\omega\,\phi) = s.\]

Moving geometry (Ω(t), Γ(t)):

\[\partial_t \phi + \nabla\cdot(\mathbf{u}_\omega\,\phi) = s \quad \text{in } \Omega(t),\]

with interface velocity wγ; moving embedded-interface transport is assembled with relative interface velocity (uγ - wγ), producing the discrete coefficient κrel used by closure logic.

uω is bulk velocity, uγ is interface-sampled advective velocity, and φγ are interface unknowns used for local closure rows.

2. Two-Phase Scalar Transport

Fixed geometry:

\[\partial_t \phi_1 + \nabla\cdot(\mathbf{u}_{\omega,1}\,\phi_1) = s_1,\]

\[\partial_t \phi_2 + \nabla\cdot(\mathbf{u}_{\omega,2}\,\phi_2) = s_2.\]

Moving geometry uses Ω1(t), Ω2(t) and per-phase relative interface velocities (uγ,1-wγ) and (uγ,2-wγ) to assemble discrete coefficients κ1rel, κ2rel for closure switching.

Unknown ordering is always:

(ω1, γ1, ω2, γ2)

This is a hyperbolic inflow/outflow closure problem, not a diffusion-style double-jump constraint.

3. Outer Boundary Conditions

Supported advection BCs:

• Inflow(value)
• Outflow()
• Periodic()

Only inflow boundaries require imposed scalar values.

4. Embedded-Interface Convention

The implementation does not branch from a pointwise u·n probe. Inflow/outflow switching is done from the same discrete embedded-interface coefficient used by the ω rows:

\[\kappa_i = \sum_d \mathrm{diag}(K_d)_i.\]

Fixed geometry sign gate

\[\text{fixed gate}:\quad \kappa_i < 0.\]

Moving geometry sign gate

\[\text{moving gate}:\quad \kappa^{rel}_i < 0, \quad \kappa^{rel} \text{ assembled from } (u_\gamma-w_\gamma).\]

Mono closure

• fixed: if κ < 0 and interface data exist, impose inflow Dirichlet Tγ = g
• moving: if κrel < 0 and interface data exist, impose inflow Dirichlet Tγ = g
• otherwise use continuity closure Tγ = Tω

|κ| / |κrel| near machine zero is treated as non-inflow (continuity/outflow behavior).

Two-phase closure

Fixed uses (κ1, κ2), moving uses (κ1rel, κ2rel) with the same local logic:

• one inflow / one outflow: conservative transport coupling row on inflow side, continuity closure on outflow side
• both outflow: continuity closure on both phases
• both inflow at the same interface location: rejected as ill-posed (ArgumentError)

5. Moving-Slab Conservative Interpretation

Moving models are assembled from reduced space-time slabs with physical volumes Vn and Vn1.

• time terms use M0 = diag(Vn) and M1 = diag(Vn1)
• geometric sweep is represented by Vn1 - Vn
• no standalone extra geometric flux term is added

This is why moving interface logic uses the relative discrete coefficient in closure and interface advection treatment.

6. Callback Conventions

Accepted scalar/velocity inputs:

• constants
• space callbacks (x...)
• time callbacks (x..., t)

Velocity input layout:

• mono: uω, uγ, and for moving also wγ
• two-phase: u1ω, u1γ, u2ω, u2γ, and for moving wγ
• each component can be a vector of length ntotal, constant, or callback

7. Scheme Conventions

All unsteady APIs (fixed and moving) accept:

• :BE (θ = 1)
• :CN (θ = 1/2)
• numeric θ with 0 <= θ <= 1

Anything else throws ArgumentError.