No-slip boundary condition

No slip requires fluid velocity to equal solid-boundary velocity.

For a smooth incompressible Navier-Stokes equation trajectory with zero velocity at a fixed flat wall, the normal momentum equation determines the normal pressure gradient. In a perturbation model with buoyancy term ${\mathrm{Ri}}_B\theta \hat{\mathbf{y}}$, it gives ${\partial }_{n}p=\nu \mathbf{n}\cdot \Delta \mathbf{u}-{\mathrm{Ri}}_B\theta \mathbf{n}\cdot \hat{\mathbf{y}}$. An extra homogeneous pressure Neumann boundary condition is valid only when the right side vanishes; it does not follow from the no-slip boundary condition.

For a displaced wall $y=ϵh_1(x,t)$, write fluid velocity as $\mathbf{u}=ϵ{\mathbf{u}}_1+{ϵ}^2{\mathbf{u}}_2+\cdots$ and prescribed wall velocity as $ϵ{\mathbf{v}}_1+{ϵ}^2{\mathbf{v}}_2+\cdots$. A Taylor expansion about $y=0$ gives ${\mathbf{u}}_1(0)={\mathbf{v}}_1$ and ${\mathbf{u}}_2(0)={\mathbf{v}}_2-h_1{\partial }_y{\mathbf{u}}_1(0)$. The second-order correction converts first-order velocity gradients into a mean boundary flow.