Template:Coordinate definition in two dimension

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 $\displaystyle{ \Omega }$, the boundary of the fluid domain which touches the fixed bodies by $\displaystyle{ \partial\Omega }$, and the free surface by $\displaystyle{ F. }$ The $\displaystyle{ x }$ and $\displaystyle{ z }$ coordinates are such that $\displaystyle{ x }$ is pointing in the horizontal direction and $\displaystyle{ z }$ is pointing in the vertical upwards direction (we denote $\displaystyle{ \mathbf{x}=\left( x,z\right) ). }$ The free surface is at $\displaystyle{ z=0 }$ and the sea floor is at $\displaystyle{ z=-h }$. The fluid motion is described by a velocity potential $\displaystyle{ \Phi }$ and free surface by $\displaystyle{ \zeta }$.