Electrohydrodynamic stresses

The EHD force density, \mathbf{f}_e, can be computed as the divergence of the Maxwell stress tensor \mathbf{M}, \displaystyle M_{ij} = \varepsilon (E_i E_j - \frac{E^2}{2}\delta_{ij}) where E_i is the i-component of the electric field, \mathbf{E}=-\nabla \phi and \delta_{ij} is the Kronecker delta.

We need to add the corresponding acceleration to the Navier–Stokes solver.

If the acceleration vector a (defined by the Navier–Stokes solver) is constant, we make it variable.

event defaults (i = 0) {
if (is_constant (a.x))
a = new face vector;
}

We overload the acceleration event of the Navier–Stokes solver to add the electrohydrodynamics acceleration term.

event acceleration (i++) {
assert (dimension <= 2); // not 3D yet
vector f[];
foreach_dimension() {
face vector Mx[];
foreach_face(x)
Mx.x[] = epsilon.x[]/2.*(sq(phi[] - phi[-1,0]) -
sq(phi[0,1] - phi[0,-1] +
phi[-1,1] - phi[-1,-1])/16.)/sq(Delta);
foreach_face(y)
Mx.y[] = epsilon.y[]*(phi[] - phi[0,-1])*
(phi[1,0] - phi[-1,0] +
phi[1,-1] - phi[-1,-1])/sq(2.*Delta);

The electric force is the divergence of the Maxwell stress tensor \mathbf{M}.

    foreach()
f.x[] = (Mx.x[1,0] - Mx.x[] + Mx.y[0,1] - Mx.y[])/(Delta*cm[]);
}

If axisymmetric cylindrical coordinates are used an additional term must be added.

#if AXI
foreach()
f.y[] += (sq(phi[1,0] - phi[-1,0]) +
sq(phi[0,1] - phi[0,-1]))/(8.*cm[]*sq(Delta))
*(epsilon.x[]/fm.x[] + epsilon.y[]/fm.y[] +
epsilon.x[1,0]/fm.x[1,0] + epsilon.y[0,1]/fm.y[0,1])/4.;
#endif

To get the acceleration from the force we need to multiply by the specific volume \alpha.

  face vector av = a;
foreach_face()
av.x[] += alpha.x[]/fm.x[]*(f.x[] + f.x[-1])/2.;
}