Is the slip condition equivalent to the zero velocity condition for the Euler equations?

Question

When doing an inviscid fluid simulation, one typically uses the Euler equations. With these equations it is important to specify that there may be no normal velocity on the surface of an object (e.g. the cylinder below). Typically, this condition is specified as

$$u \cos(\theta) + v \sin(\theta)=0$$

with $\theta$ the angle of the normal vector of the surface. However, I am not very clear what the difference is between the slip condition and the zero velocity condition, which is

$$u=0,\ \ \ v=0$$

I understand that this also sets the tangential velocity on the surface to zero. But because the Euler equations are inviscid, the zero velocity condition would just create a "slipping" velocity field just outside of the surface right? So perhaps the only effect is that the object would appear slightly larger (the width of a cell)?

Answer 1

The issue is that if you impose zero velocity at the boundary, you system is over determined. Intuitively, this is because the Euler equations are first order in space. However, due to the non-linearity, it is not that obvious.

You can build intuition in specific case. Take for example potential flow: $$ \vec v = \nabla \phi $$ In this cas, incompressibility gives the Laplace equation: $$ \Delta \phi = 0 $$ and the boundary conditions amount to Neumann boundary conditions, which makes the problem well posed. If you were to add more conditions, such as imposing a zero tangential velocity, this would overdetermine the system.

Note the zero velocity contain is used however for the Navier-Stokes equation. This is because this time, dissipation introduces the second order spatial derivative term $\nu\Delta \vec v$. In this case, you can build intuition in the high viscosity limit: Stoke flow. The problem becomes once again a Laplace equation (up to the pressure term) in velocity, only this time, the boundary condition is a Dirichlet condition. The problem is again well-posed.

Hope this helps.