Derivation of one form of Navier-Stokes equation using another form of it

Question

There are two forms of Navier-Stokes equations, which are $$\frac{\partial (\rho \boldsymbol U)}{\partial t} + \boldsymbol U\cdot \nabla(\rho \boldsymbol U) = \boldsymbol f$$ $$\frac{\partial (\rho \boldsymbol U)}{\partial t} + \nabla \cdot (\rho \boldsymbol U \boldsymbol U) = \boldsymbol f$$ I tried to go through the derivation to prove that both vector terms are equal but this calculation is all I have reached. This attempt is based on considering the scalar component of the velocity and trying and to observe how both equations will behave in addition to running from dealing with tensors, which is quite tedious.I know that both equations are two forms of Navier-Stokes, but my issue is more related to vector analysis. In other words, how did we change the second term of the equation from the first form in the first equation to the second form in the second equation. I hope my inquiry is clear enough.

Answer 1

The two are equal due to the incompressibility constraint, which requires $\nabla\cdot U = 0$. Under this, you have, by the Leibniz rule

$$ \nabla\cdot (\rho \color{red}{U} \color{green}{U}) = \color{red}{U}\cdot \nabla(\rho \color{green}{U}) + (\rho \color{green}{U}) \underbrace{\nabla\cdot \color{red}{U}}_{=0}$$