Governing equations

In addition to the incompressibility constraint

\[\der{u_i}{x_i} = 0,\]

the momentum balance

\[\der{u_i}{t} + u_j \der{u_i}{x_j} = -\der{p}{x_i} + \frac{\sqrt{Pr}}{\sqrt{Ra}} \der{}{x_j} \der{u_i}{x_j} + T \delta_{x i} + f_i^{st},\]

and the equation of the internal energy (temperature)

\[\der{T}{t} + u_j \der{T}{x_j} = \frac{1}{\sqrt{Pr}\sqrt{Ra}} \der{}{x_j} \der{T}{x_j},\]

I need to solve the advection equation with respect to the indicator function \(H\)

\[\frac{DH}{Dt} = 0\]

or

\[\der{H}{t} + u_j \der{H}{x_j} = 0,\]

where \(H\) is a flag to distinguish the two liquid phases, which takes \(0\) (in phase 0) or \(1\) (in phase 1).

Note that \(f_i^{st}\) is included in the momentum balance, describing the effect of the surface tension force:

\[f_i^{st} \equiv \frac{1}{We} \kappa \delta n_i,\]

where \(We\), \(\kappa\), \(\delta\), and \(n_i\) are the Weber number, local interfacial (mean) curvature, Dirac delta function and surface normal vector, respectively. Here, I assume that the surface is ideally clean and no tangential force (Marangoni stress) is caused.

See also

SimpleNSSolver for details.