Template:Equations of motion time domain without body condition: Difference between revisions

Latest revision as of 23:14, 22 April 2010

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]

@@ Line 1: / Line 1: @@
 The equations
-of motion in the time domain, in non-dimensional
+of motion in the time domain are
-form (so that the fluid density and gravity are both unity are
 Laplace's equation through out the fluid
 <center><math>
-\Delta\Phi\left(  \mathbf{x,}t\right)  =0,\ \ \mathbf{x}\in\Omega,
+\Delta\Phi\left(  \mathbf{x,}t\right)  =0,\ \ \mathbf{x}\in\Omega.
 </math></center>
 At the bottom surface we have no flow
@@ Line 13: / Line 12: @@
 surface we have the kinematic condition
 <center><math>
-\partial_{t}\zeta=\partial_{n}\Phi,\ \ z=0,\ x\in F,
+\partial_{t}\zeta=\partial_{n}\Phi,\ \ z=0,\ x\in \partial\Omega_{F},
 </math></center>
 and the dynamic condition (the linearized Bernoulli equation)
 <center><math>
-\zeta = -\partial_{t}\Phi,\ \ z=0,\ x\in F,
+\partial_{t}\Phi = -g\zeta ,\ \ z=0,\ x\in \partial\Omega_{F}.
 </math></center>