pnp.h
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Test cases (1): debye
Ohmic conduction flux of charged species
This function computes the fluxes due to ohmic conduction appearing in the Nernst--Planck equation. The species charge concentrations are then updated using the explicit scheme
$$
c^{n+1}_i = c^n_i +\Delta t \, \nabla \cdot( K_i c^n_i \nabla \phi^n)
$$
where $c_i$ is the volume density of the $i$-specie, $K_i$ its volume electric conductivity and $\phi$ the electric potential.
extern scalar phi;
void ohmic_flux (scalar * c, // A list of the species concentration...
int * z, // ... and their corresponding valences
double dt,
vector * K = NULL) // electric mobility (default the valence)
{If the volume conductivity is not provided it is set to the value of the valence.
if (!K) { // fixme: this does not work yet
int i = 0;
for (scalar s in c) {
const face vector kc[] = {z[i], z[i]}; i++;
K = vectors_append (K, kc); // fixme: K should be freed eventually
}
}
scalar s;
(const) face vector k;
for (s, k in c, K) {The fluxes of each specie through each face due to ohmic transport are
face vector f[];
foreach_face()
f.x[] = k.x[]*(s[] + s[-1])*(phi[] - phi[-1])/(2.*Delta);The specie concentration is updated using the net amount of that specie leaving/entering each cell through the face in the interval $dt$
foreach()
foreach_dimension()
s[] += dt*(f.x[1] - f.x[])/Delta;
}
}