Difference between revisions of "Template:Linear elastic plate on water time domain"

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= D\frac{\partial^4 \eta}{\partial x^4} + \rho_i h \frac{\partial^2 \eta}{\partial t^2}
 
= D\frac{\partial^4 \eta}{\partial x^4} + \rho_i h \frac{\partial^2 \eta}{\partial t^2}
 
, \ z=0; </math></center>
 
, \ z=0; </math></center>
We also have Laplace's equation
+
We also have [[Laplace's Equation|Laplace's equation]]
 
<center><math>
 
<center><math>
 
\Delta \Phi = 0,\,\,-h<z<0
 
\Delta \Phi = 0,\,\,-h<z<0

Revision as of 10:16, 7 April 2009

We begin with the linear equations for a fluid. The kinematic condition is the same

[math]\displaystyle{ \frac{\partial\zeta}{\partial t} = \frac{\partial\Phi}{\partial z} , \ z=0; }[/math]

but the dynamic condition needs to be modified to include the effect of the the plate

[math]\displaystyle{ \rho g\zeta + \rho \frac{\partial\Phi}{\partial t} = D\frac{\partial^4 \eta}{\partial x^4} + \rho_i h \frac{\partial^2 \eta}{\partial t^2} , \ z=0; }[/math]

We also have Laplace's equation

[math]\displaystyle{ \Delta \Phi = 0,\,\,-h\lt z\lt 0 }[/math]

and the usual non-flow condition at the bottom surface

[math]\displaystyle{ \partial_z \Phi = 0,\,\,z=-h, }[/math]

where [math]\displaystyle{ \zeta }[/math] is the surface displacement, [math]\displaystyle{ \Phi }[/math] is the velocity potential, and [math]\displaystyle{ \rho }[/math] is the fluid density.