Kinematic boundary condition

A material fluid boundary has no relative normal flow; at a fixed impermeable graph $y=\eta (x)$ this is $v=u{\eta }_x$.

For a free surface $z=\eta (x,y,t)$ and velocity $(u,v,w)$, material conservation of $z-\eta$ giveson the surface.

For a small boundary displacement and weak disturbance, evaluate boundary data on the undisturbed surface and discard products of small quantities.