Re: [Libmesh-users] Navier-Stoke equation in ex13

[Roy S. <roy...@ic...>]

On Thu, 24 Jan 2008, li pan wrote:

> I've worked with Newton type flow equation. To make it
> sure, I would like to know the exact expression of
> equation in ex13. Can you tell me?

The system of equations with variables (u,p) is :
(partial u)/(partial t) = - (u * grad)u - div(sigma)
div(u) = 0

Where sigma is the stress tensor (normalized to have unit viscosity)
sigma = ((grad(u) + transpose(grad(u)))/2 - pI)

Then the weak form we use in ex13 and ex18, with test functions (v,q)
is:
((partial u)/(partial t), v)_Omega = - ((u * grad)u, v)_Omega
   + (sigma, grad v)_Omega + (sigma * n, v)_dOmega
(div(u), q) = 0

In ex13 we use Dirichlet boundaries everywhere, so v = 0 on the
boundary and we drop the dOmega term.  Otherwise, you'd substitute
into that term the natural boundary condition:
sigma * n = 0

which is actually what David wanted in the first place.  ;-)

You know, we probably ought to have something like this in the
comments heading examples 13 and 18.  "The Navier-Stokes equations" is
definitive enough, but the fact that we integrate all of sigma
(including the pressure term) by parts isn't set in stone.
---
Roy