Adopted Navier friction law for a stationary wall

Let $\mathrm{\Omega }\subset {\mathbb{R}}^3$ be a bounded domain with $C^2$ boundary, fixed in an inertial Euclidean frame, and let $n$ be its outward unit normal. On $\mathrm{\partial }\mathrm{\Omega }$, let $u$ be the trace of a $C^1$ fluid-velocity field in metres per second and let $\sigma$ be the Cauchy-stress tensor in pascals, with traction on the fluid defined by $\sigma n$. For a vector $a$, define its tangential projection by $a_{\tau }:=a-(a\cdot n)n$. Adopt impermeability $u\cdot n=0$ and the linear friction wall law

$(\sigma n{)}_{\tau }=-\beta u_{\tau },$

where $\beta \ge 0$ is a constant with units pascal-seconds per metre. In the incompressible Newtonian branch, write the constant shear-viscosity coefficient as $\eta =\mu >0$ in pascal-seconds. For $\beta >0$, define the corresponding finite slip-length parameter by ${\mathrm{\ell }}_s:=\eta \mathrm{/}\beta$ metres; equivalently, $\beta =\eta \mathrm{/}{\mathrm{\ell }}_s$. The endpoint $\beta =0$ is zero tangential traction and has no finite ${\mathrm{\ell }}_s$. This is a selected wall model, not a consequence of interior balance equations, a measured value of $\beta$, or a claim that any material or wall universally obeys it.

Chen and Qian, A Study of the Navier–Stokes Equations with the Kinematic and Navier Boundary Conditions, arXiv:0901.0147v1, §§1–2