Template:Equations of motion time domain without body condition

The equations of motion in the time domain, in non-dimensional form (so that the fluid density and gravity are both unity are Laplace's equation through out the fluid

[math]\displaystyle{ \Delta\Phi\left( \mathbf{x,}t\right) =0,\ \ \mathbf{x}\in\Omega, }[/math]

At the bottom surface we have no flow

[math]\displaystyle{ \partial_{n}\Phi=0,\ \ z=-h. }[/math]

At the free surface we have the kinematic condition

[math]\displaystyle{ \partial_{t}\zeta=\partial_{n}\Phi,\ \ z=0,\ x\in F, }[/math]

and the dynamic condition (the linearized Bernoulli equation)

[math]\displaystyle{ \zeta = -\partial_{t}\Phi,\ \ z=0,\ x\in F, }[/math]