2-D prototype, round 4: CONJUGATE transport with the probe-flux cut-cell FV.
Geometry: solid core r < a (diffusivity Ds, capacity gs), fluid annulus a < r < R (Df = 1, capacity
1), u = 0 at r = R (immersed Dirichlet, probe-flux). Interface at r = a:
flux continuity q = -Ds du_s/dr = -Df du_f/dr (q: radial, solid -> fluid)
q = hc (u_s/K - u_f) (hc = inf -> u_s = K u_f: partition coefficient K)
Exact: slowest mode of g du/dt = div(D grad u) via a Bessel determinant.
Discretization (two fields on one grid; every cell with fluid fraction > 0 has a fluid unknown,
every cell with solid-core fraction > 0 has a solid unknown):
faces : aperture-weighted two-point fluxes, separately per phase (fluid apertures / core apertures)
facet : per cut cell, wall area A_w (aperture area vector), centroid x_w (radial projection),
normal n (into the fluid). Fluid probe x_w + c h n (bilinear over fluid unknowns), solid
probe x_w - c h n (bilinear over solid unknowns). Two interface equations
Df (u_pf - u_Gf)/s = Ds (u_Gs - u_ps)/s (flux continuity, q = -Df (u_pf-u_Gf)/s)
q = hc (u_Gs/K - u_Gf) or u_Gs = K u_Gf
are solved for (u_Gf, u_Gs) -> q is a linear combination of probe-stencil unknowns;
+q A_w enters the fluid cell, -q A_w the solid cell (exactly conservative).
The fluid facet and the solid facet of a cell are the same facet; cells cut by the core interface
carry both unknowns. Outer boundary: Dirichlet probe flux as in jc.py.
1.9.8