Template:Equations of motion time domain without body condition - Revision history

Meylan at 23:14, 22 April 2010

2010-04-22T23:14:12Z

← Older revision		Revision as of 23:14, 22 April 2010
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	surface we have the kinematic condition		surface we have the kinematic condition
	<center><math>		<center><math>
	\partial_{t}\zeta=\partial_{n}\Phi,\ \ z=0,\ x\in \partial\~~Omega~~ F,		\partial_{t}\zeta=\partial_{n}\Phi,\ \ z=0,\ x\in \partial\Omega_{F},
	</math></center>		</math></center>
	and the dynamic condition (the linearized Bernoulli equation)		and the dynamic condition (the linearized Bernoulli equation)
	<center><math>		<center><math>
	\partial_{t}\Phi = -g\zeta ,\ \ z=0,\ x\in \partial\~~Omega~~ F.		\partial_{t}\Phi = -g\zeta ,\ \ z=0,\ x\in \partial\Omega_{F}.
	</math></center>		</math></center>

Meylan at 23:13, 22 April 2010

2010-04-22T23:13:53Z

← Older revision		Revision as of 23:13, 22 April 2010
Line 12:		Line 12:
	surface we have the kinematic condition		surface we have the kinematic condition
	<center><math>		<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>		</math></center>
	and the dynamic condition (the linearized Bernoulli equation)		and the dynamic condition (the linearized Bernoulli equation)
	<center><math>		<center><math>
	\partial_{t}\Phi = -g\zeta ,\ \ z=0,\ x\in F.		\partial_{t}\Phi = -g\zeta ,\ \ z=0,\ x\in \partial\Omega F.
	</math></center>		</math></center>

Meylan at 05:37, 22 August 2009

2009-08-22T05:37:21Z

← Older revision		Revision as of 05:37, 22 August 2009
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	The equations		The equations
	of motion in the time domain~~, in non-dimensional~~		of motion in the time domain are
	~~form, so that gravity is unity,~~ are
	Laplace's equation through out the fluid		Laplace's equation through out the fluid
	<center><math>		<center><math>
Line 17:		Line 16:
	and the dynamic condition (the linearized Bernoulli equation)		and the dynamic condition (the linearized Bernoulli equation)
	<center><math>		<center><math>
	~~\zeta = -~~\partial_{t}\Phi,\ \ z=0,\ x\in F.		\partial_{t}\Phi = -g\zeta ,\ \ z=0,\ x\in F.
	</math></center>		</math></center>

Meylan at 11:31, 21 August 2009

2009-08-21T11:31:09Z

← Older revision		Revision as of 11:31, 21 August 2009
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	Laplace's equation through out the fluid		Laplace's equation through out the fluid
	<center><math>		<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>		</math></center>
	At the bottom surface we have no flow		At the bottom surface we have no flow
Line 17:		Line 17:
	and the dynamic condition (the linearized Bernoulli equation)		and the dynamic condition (the linearized Bernoulli equation)
	<center><math>		<center><math>
	\zeta = -\partial_{t}\Phi,\ \ z=0,\ x\in F,		\zeta = -\partial_{t}\Phi,\ \ z=0,\ x\in F.
	</math></center>		</math></center>

Meylan at 11:30, 21 August 2009

2009-08-21T11:30:24Z

← Older revision		Revision as of 11:30, 21 August 2009
Line 1:		Line 1:
	The equations		The equations
	of motion in the time domain, in non-dimensional		of motion in the time domain, in non-dimensional
	form (so that ~~the fluid density and~~ gravity ~~are both~~ unity are		form, so that gravity is unity, are
	Laplace's equation through out the fluid		Laplace's equation through out the fluid
	<center><math>		<center><math>

Meylan at 10:37, 21 August 2009

2009-08-21T10:37:44Z

New page

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
<center><math>
\Delta\Phi\left( \mathbf{x,}t\right) =0,\ \ \mathbf{x}\in\Omega,
</math></center>
At the bottom surface we have no flow
<center><math>
\partial_{n}\Phi=0,\ \ z=-h.
</math></center>
At the free
surface we have the kinematic condition
<center><math>
\partial_{t}\zeta=\partial_{n}\Phi,\ \ z=0,\ x\in F,
</math></center>
and the dynamic condition (the linearized Bernoulli equation)
<center><math>
\zeta = -\partial_{t}\Phi,\ \ z=0,\ x\in F,
</math></center>