#include "poisson.h"

Include dependency graph for implicit.h:

Functions
void	event_defaults (void)
	Event: defaults (i = 0)

void	event_tracer_diffusion (void)
	Event: tracer_diffusion (i++)

Variables
double	dt

scalar	phi []
	The electric potential and the volume charge density are scalars while the permittivity and conductivity are face vectors.

scalar	rhoe []

vector	epsilon []

vector	K []

mgstats	mgphi

Function Documentation

◆ event_defaults()

void event_defaults ( void )

Event: defaults (i = 0)

Boundary conditions for VOF-advected tracers usually depend on boundary conditions for the VOF field.

Event: defaults (i = 0)

Initialisation

We set the default values.

The restriction/refine attributes of the charge density are those of a tracer otherwise the conservation is not guaranteed.

By default the permittivity is unity and other quantities are zero.

Definition at line 39 of file implicit.h.

References _i, epsilon, fm, K, refine_linear(), restriction_volume_average(), rhoe, vector::x, and x.

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◆ event_tracer_diffusion()

void event_tracer_diffusion ( void )

Event: tracer_diffusion (i++)

The r.h.s of the poisson equation is set to $-\rho_e^n$,

We compute the coefficients of the Laplacian: $(K dt +\epsilon)$.

The poisson equation is solved to obtain $\phi^{n+1}$.

Finally $\rho_e^{n+1}$ is computed as $\rho_e^{n+1} = -\nabla \cdot (\epsilon \nabla \phi^{n+1})$.

In the absence of conductivity, the electric potential only varies if the electrical permittivity varies,

Definition at line 59 of file implicit.h.

References _i, cm, dimension, dt, epsilon, f, scalar::i, K, mgphi, phi, poisson, rhoe, sq(), vector::x, and x.

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Variable Documentation

◆ dt

double dt

extern

Ohmic conduction

This function approximates implicitly the EHD equation set given by the electric Poisson equation and the ohmic conduction term of the charge density conservation (the advection term is computed elsewhere using a tracer advection scheme),

\[ \nabla \cdot( \epsilon \nabla \phi^{n+1}) = -\rho_e^{n+1} \quad \mathrm{and} \quad (\rho_e^{n+1}-\rho_e^n) = \Delta t\nabla \cdot (K\nabla \phi^{n+1}) \]

where $\rho_e$ is the charge density, and $\phi$ is the electric potential. $K$ and $\epsilon$ are the conductivity and permittivity respectively.

Substituting the Poisson equation into the conservation equation gives the following time-implicit scheme,

\[ \nabla \cdot [(K \, \Delta t + \epsilon) \nabla \phi^{n+1}] = -\rho_e^{n} \]

We need the Poisson solver and the timestep dt.