This is my current understanding:
The no-slip condition at a boundary means that there is no velocity relative to the boundary, this means that the individual components is zero. So we have (in two dimensions) $\vec{u}|_{{\partial\Omega}_{no-slip}} = [u_1 = 0, u_2 = 0]|_{{\partial\Omega}_{no-slip}}$.
However, in some papers I've read, they state that the no-slip condition can be imposed by $\vec{u}\cdot \hat{t} = 0$, where $\hat{t}$ is the unit tangent on the boundary $\partial\Omega_{no-slip}$. But this does not mean that the velocity at $\partial\Omega_{no-slip}$ is zero, it just means that there is no tangential component. Examples of this terminology can be seen in Layton 1999 "Weak imposition of the no-slip condition finite element methods", I can provide more examples if need be. Is this definition incorrect? Then what would you call the condition $\vec{u}\cdot \hat{t} = 0$ on $\partial\Omega$ if it is not called no-slip?
Now for "no-penetration", I know that this can be described as $\vec{u}\cdot\hat{n} = 0$ on $\partial\Omega_{no-pene}$, where $\hat{n}$ is the unit normal to the boundary $\partial\Omega_{no-pene}$, but I've also seen this condition called the "slip condition". I've also seen the slip condition described as $\vec{u}\cdot\hat{n} = g$, and also in two parts as per the attachment (equations 1.1c and 1.1d).
.
My question is, what is just $\vec{u}\cdot\hat{n} = 0$? – is this no-penetration or slip condition?
Last question, I've read that the slip condition is related to the wall velocity, what is this? (I have a vague idea but would like to solidify my understanding).
Thank you all.
Best Answer
I would say it simply depends on the context, like often in physics. For "usual" fluid mechanics (usual = what I am usually in contact with :-) ), borders are watertight and thus non-penetrative bu default, so no-slip condition is u=0 and slip condition is u.n=0. But it you have porosity, or maybe a grid, then you have situations where a flux can go through, but not tangentially. Then it makes sense to distinguish no-slip alone (no tangential component) independantly of non-penetrating.