Template:Equations for fixed bodies in the time domain

We consider a two-dimensional fluid domain of constant depth, which contains a finite number of fixed bodies of arbitrary geometry. We denote the fluid domain by [math]\displaystyle{ \Omega }[/math], the boundary of the fluid domain which touches the fixed bodies by [math]\displaystyle{ \partial\Omega }[/math], and the free surface by [math]\displaystyle{ F. }[/math] The [math]\displaystyle{ x }[/math] and [math]\displaystyle{ z }[/math] coordinates are such that [math]\displaystyle{ x }[/math] is pointing in the horizontal direction and [math]\displaystyle{ z }[/math] is pointing in the vertical upwards direction (we denote [math]\displaystyle{ \mathbf{x}=\left( x,z\right) ). }[/math] The free surface is at [math]\displaystyle{ z=0 }[/math] and the sea floor is at [math]\displaystyle{ z=-h }[/math]. The fluid motion is described by a velocity potential [math]\displaystyle{ \Phi }[/math] and free surface by [math]\displaystyle{ \zeta }[/math].

The equations of motion in the time domain 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 \partial\Omega_{F}, }[/math]

and the dynamic condition (the linearized Bernoulli equation)

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

The body boundary condition for a fixed body is

[math]\displaystyle{ \partial_{n}\Phi=0,\ \ \mathbf{x}\in\partial\Omega, }[/math]

The initial conditions are

[math]\displaystyle{ \left.\zeta\right|_{t=0} = \zeta_0(x)\,\,\,\mathrm{and}\,\,\, \left.\partial_t\zeta\right|_{t=0} = \partial_t\zeta_0(x). }[/math]