What boundary condition is necessary for the water surface ?

We can impose a kinematic boundary condition on the water surface. We assume that the water surface is at $y = \eta(x,z,t)$ ($t$ is the time parameterization), and the kinematic boundary condition in this case will be

$$\nabla\phi.\vec{n} = \frac{\partial \eta}{\partial t} \mbox{\ on \ } y =\eta(x,z,t) \notag$$

where $\vec{n} = \frac{\nabla(y-\eta)}{\vert\vert\nabla(y-\eta)\vert\vert}$. With this expression for $\vec{n}$, the condition above reduces to

$$\frac{\partial \phi}{\partial y} = \frac{\partial \eta}{\part... ...\partial\eta}{\partial z} \mbox{\ on \ } y =\eta(x,z,t) \notag$$