Boundary Conditions

We will consider the so-called inflow-outflow problem. The problem of a fluid moving through a channel or the problem of a flow past an air profile belong to this class.

We distinguish the inlet part of the boundary $\partial\Omega$

$$\Gamma_I=\left\{ {\bf x}\in\partial\Omega,\quad {\bf u(x)\cdot n(x)}<0 \right \},$$

through which the fluid enters to the domain $\Omega$, the outlet part

$$\Gamma_O=\left\{ {\bf x}\in\partial\Omega,\quad {\bf u(x)\cdot n(x)}\gt \right \},$$

through which the fluid leaves the domain $\Omega$ and the rest of the boundary $\partial\Omega$, called the solid impermeable wall, denoted by $\Gamma_W$. (${\bf n}$ is the unit outer normal to $\partial\Omega$).

(i) Viscous case, $\lambda <0,\mu \gt,k\gt$

In this case, the system (15) is of the hyperbolic-parabolic type. The first equation is hyperbolic and the others are parabolic. Due to the viscosity of the fluid the particles adhere to the solid wall and one has to prescribe the no-slip boundary condition

$${\bf u} = 0\qquad \mbox{on} \ \Gamma_W\times (0,T).$$ (17)

Some theoretical results and also our numerical experiments lead us to the use of the following boundary conditions:

(ii) Inviscid case, $\lambda =0,\mu =0,k=0$

The system of Euler equations (17) is hyperbolic. In this case the number of boundary conditions on $\Gamma_I$ and $\Gamma_O$ is different if the flow is subsonic ($\vert{\bf v}\vert < a$) or supersonic ($\vert{\bf v}\vert \gt a$), where a is the local sound speed given by

$$a= \sqrt{\frac{\kappa p}{\rho}}.$$

More precisely we will speak about this in the following section.

On $\Gamma_W$ we prescribe a zero normal component of the velocity because the wall is impermeable, i.e.

$${\bf u\cdot n =0}\qquad \mbox{on}\ \Gamma_W\times(0,T).$$