= Pressure compatibility at a no-slip wall
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 $\partial_np=\nu\mathbf n\cdot\Delta\mathbf u-\mathrm{Ri}_B\theta\,\mathbf n\cdot\hat{\mathbf y}$. 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>.
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