Boundary conditions

The computational domain $\Omega$ is embedded in a 3D cuboid domain $\Omega_R$. In the current version the following boundary conditions are implemented:

• periodic boundary conditions
  These conditions hold for ${\bf u},~\tilde{\bf u}$, $p^{n+1}$, $T$ and $C$ on opposite faces of $\Omega_{R}$.

• Dirichlet boundary conditions
  On subsets $\Gamma \subset \partial \Omega$ Dirichlet conditions ${\bf u}={\bf u}_0$ may be defined. The same b.c. hold for $\tilde{\bf u}$, i.e. $\tilde{\bf u}={\bf u}_0$. These b.c. lead to homogeneous Neumann conditions for the pressure $p^{n+1}$ on $\Gamma$. If $T$ or $C$ is calculated, Dirichlet b.c. for these quantities have to be specified on $\Gamma$.

• Slip conditions
  Here, homogeneous Dirichlet conditions hold for the normal velocity component and homogeneous Neumann conditions for the tangential velocity components. The same b.c. hold for $\tilde{\bf u}$. These b.c. lead to homogeneous Neumann conditions for the pressure $p^{n+1}$.

• Outflow boundary condition I
  On faces $\Gamma$ of $\Omega_R$ homogeneous Neumann b.c. for the velocity may be defined as
  $$\frac{\partial {\bf u}}{\partial n}=0~.$$

• Outflow boundary condition II
  Alternatively a boundary condition of convective type can be applied at outflow boundaries, i.e. it consists of a discretization of
  $$\frac{\partial {\bf u}}{\partial t} + {\bf u} \cdot \nabla {\bf u} = 0$$

• If no Dirichlet or periodic boundary conditions are specified, then homogeneous Neumann conditions are assumed for $T$ and $C$.