Navier slip boundary condition ($u_b={\ell }_s{u}_z$)

A Navier slip boundary condition relates fluid velocity relative to a stationary flat wall to the wall-normal tangential velocity gradient, $u_b={\ell }_s{u}_z$, with positive slip length ${\ell }_s$. For a Newtonian fluid, it is equivalent to linear resisting wall traction of magnitude $\mu u_b/{\ell }_s$. The normal points from the wall into the fluid; this fixes the sign convention.

The slip length is the distance obtained by extrapolating a locally linear tangential velocity profile back to zero velocity beneath a wall. A Navier slip boundary condition has $u_b={\ell }_s{\partial }_{n}u$. Zero slip length gives a no-slip boundary condition; large slip permits approximately plug-like motion when the surrounding geometry and forces allow it.