# Navier-Stokes Equation

1. Sep 30, 2009

### iasc

I was wondering if someone could help me this Navier-Stokes Equation.

f[(δv/δt) + v.Dv] = -DP + Dt + f

Could someone maybe explain the symbols and what it means.
I'm not sure but I think Navier-Stokes equations describe fluid motion.

(The P could be ρ. I'm not too sure)

Thanks

2. Sep 30, 2009

### Djinn

You are correct that the equations describe fluid motion. I have a beautiful proof that solutions always exist in three dimensions, but unfortunately it is too big for this marginal comment.

3. Oct 1, 2009

### Andy Resnick

I can't completely parse what you wrote, but some of it I can decipher:

The term in [] (not sure what that 'f' is doing there), is the "total" or "convective" derivative. It simply means that a spatial quantity (the velocity field vector 'v') is allowed to vary both in time and in space. I am assuming 'D' is a nabla (del) operator.

The term DP could be the gradient of pressure term, but the Dt term is normally the divergence of the stress tensor, so those two terms are a little ambiguous. The stress tensor consists of both an isotropic part (the pressure) and the off-diagonal antisymmetric components (the shear stress). The final 'f' is used if there is an external body force: gravity, centripetal forces, electromagnetic, etc. etc.

The Navier-Stokes equation you wrote is nothing more than ma=F for a continuous material. If ma(or []) = 0, then you have conservation of momentum.