Understanding Navier Stokes

schettel

Consider a stationary body within the flow of some fluid. I want to calculate pressure on the surface of the body. From the Navier Stokes (incompressible, stationary, no volume forces) equations, you would get something like:
dp/dx=-rho(u du/dx+v du/dy+w du/dz)+eta(d²u/dx²+d²u/dy²+d²u/dz²)
...likewise for the other coordinates.
This means that, when calculating pressure on the surface of my body, impulse (rho(..) on the right hand side of the equation) and wall shear stress (eta(..)) come into play. That seams quite logical. But my body is not moving, so, assuming a no-slip condition for its surface, u=v=w=0, which leaves me only with shear stress.
If the body is moving and the fluid is not, you should still get the same pressure on the surface. But in this case, the first term on the right hand side of the equation (impulse) is not zero.
There must be a mistake in my reasoning. Can anybody tell me what it is?
Maybe it has something to do Euler and Lagrange?

arildno

You're asking if N-S is Galilean invariant, right? It is.
The solution lies in that you cannot neglect the local temporal derivative if you shift to a description in which the body has a non-zero velocity.

This makes sense, because roughly, the dynamics in the fluid at a field point should primarily depend upon the distance from the field point to the body.

Clausius2

Welcome to Mech&Aero forum. Glad to see you here, Arildno!!!

I do agree with your answer. The local variation of the velocity [tex]\partial u/\partial t[/tex] is non zero if the body is in motion. The problem is transformed into an unsteady one.

arildno

I do poke my nose into here on occasion..
Thanks for the welcome, though..