= Navier slip boundary condition
{c}
{title2=$u_{\mathrm{rel}}=b\partial_nu$}
= Navier slip
{c}
{synonym}
The <Navier slip boundary condition> allows a <fluid> to slide tangentially past a solid wall while resisting that motion through <wall friction in the Navier slip boundary condition>. For parallel flow past a flat, impermeable wall translating uniformly with tangential <velocity> $U_w$, write the fluid's tangential <velocity> as $u$ and its relative tangential <velocity> as $u_{\mathrm{rel}}=u-U_w$. Then
$$
u_{\mathrm{rel}}=b\partial_nu,\qquad b\ge0,
$$
where $b$ is the <slip length>, also written $\ell_s$. The <unit normal> $\mathbf n$ points from the wall into the <fluid>, and $\partial_nu=\mathbf n\cdot\nabla u$ is the <normal derivative>. The separate <no-penetration boundary condition> requires $(\mathbf u-\mathbf U_w)\cdot\mathbf n=0$: tangential sliding does not allow flow through the wall.
The sign convention matters. In a channel $0<z<H$, the lower wall has inward <unit normal> $+\mathbf e_z$ and the upper wall has inward <unit normal> $-\mathbf e_z$. If both walls obey the <Navier slip boundary condition>, their tangential <velocities> satisfy
$$
u(0)-U_0=b_0u'(0),\qquad
u(H)-U_H=-b_Hu'(H).
$$
Here the prime means the <derivative> with respect to $z$. Using $+b_Hu'(H)$ at the upper wall would reverse the friction law. The examples <Couette flow with a slipping lower wall> and <planar Poiseuille flow with equal wall slip> use different upper-wall <boundary conditions>.
This flat-wall formula is a local linear model. On a curved wall the <Newtonian fluid stress tensor> must supply the tangential <traction>; the <normal derivative> alone generally misses curvature terms. The author-written review by Lauga, Brenner and Stone states the full strain-based condition with the <unit normal> directed into the liquid. https://arxiv.org/pdf/cond-mat/0501557[Lauga, Brenner and Stone, Microfluidics: The No-Slip Boundary Condition (2005), §2.1, equation (2)]{ref}
The same review credits Navier's 1823 treatise with the linear slip law. Its history is a reminder that the <no-slip boundary condition> is an experimentally successful model, whose validity depends on the interface and scale. The history here follows that review rather than a new reading of Navier's original memoir. https://arxiv.org/pdf/cond-mat/0501557[Lauga, Brenner and Stone (2005), §2.1: the nineteenth-century boundary-condition debate]{ref}
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