# Correct form of the convection term in the Navier-Stokes equation

Usually the NS momentum equation for an compressible fluid is written in its convecting form, in the absence of external forces, as

$$\rho\frac{\partial \vec u}{\partial t} +\rho(\vec u\cdot\nabla)\vec u+\nabla P = \mu\nabla^2\vec u +\frac{\mu}{3}\nabla(\nabla\cdot \vec u)$$

However, from the Wikipedia, this can be written in "conservation form", as

$$\rho\frac{\partial \vec u}{\partial t} +\rho\nabla\cdot (\vec u\otimes\vec u)+\nabla P = \mu\nabla^2\vec u +\frac{\mu}{3}\nabla(\nabla\cdot \vec u)$$

I can't see how these two are consistent, the convection term in the lower one is

$$\nabla_\alpha(u_\alpha u_\beta) = \vec u\nabla\cdot \vec u + (\vec u\cdot \nabla)\vec u$$

So there's an extra term $$\vec u\nabla\cdot \vec u$$ with respect to the first equation.

What am I not understanding here? I see that for incompressible fluids, there's a pressure equations that constraints the system to obey $$\nabla\cdot \vec u=0$$, but I'm still missing something about the general case.

## 1 Answer

You have changed the equation from what is in Wikipedia. You have $$\rho\frac{\partial{\bf v }}{\partial t}+\rho \nabla\cdot ({\bf v}\otimes {\bf v})+\ldots$$ Wiki's conservation form has an expression with the $$\rho$$'s in the correct places $$\frac{\partial{\rho \bf v }}{\partial t}+\nabla\cdot (\rho({\bf v}\otimes {\bf v}))+\ldots$$ The extra bit they add to the plain Euler equation is
$${\bf v}\left( \frac{\partial \rho }{\partial t} + \nabla \cdot (\rho {\bf v})\right)$$ which is zero by the continuity equation.

• I guess, for further clarification, that I assumed that the pressure is constant along the fluid which is pretty much the definition of incompressible. Thanks! – MyUserIsThis Jun 11 '20 at 20:02
• Mmmm... No. In an incompressible fluid the pressure can be very big. It's the density that does not change! I guess that's what you means though, – mike stone Jun 12 '20 at 13:15
• Yes I was thinking density and wrote pressure... sorry, and thanks again. – MyUserIsThis Jun 13 '20 at 17:51