I have the (typical) Navier--Stokes system for incompressible fluid:
$$div(u)=0$$ $$\rho(u_t+u\cdot\nabla u)=-\nabla p+div(\nu\nabla u)+\rho g$$
In a paper that I'm reading says that the term
is an acceleration. I can understand that $u_t$ (the derivative of the velocity $u$ with respect to time) is in fact an acceleration, but, why "$u_t+u\cdot\nabla u$" is also an acceleration?
My second question is: why the right side terms $$-\nabla p+div(\nu\nabla u)+\rho g$$ represent the sum forze?