Why is there a density instead of mass in the Navier-Stokes Equation, if it's analogue to Newton's Second Law?

Question

I read in Ian Stewart's 17 Equations that Changed the World book that Navier-Stokes equation (I know it's not exactly a scientific book, but still, I'd like clarification on what is wrong if it's the case):

$\rho \left( \dfrac{\partial v}{\partial t} + v \cdot \nabla v\right) = -\nabla p + \nabla \cdot T + f$

is an analogue of Newton's Second Law, for fluids.

Newton's second Law states force is equal to mass times acceleration, and I can see that inside the parenthesis in NS equation there are derivatives of velocity, so it is acceleration. The RHS are the forces on the fluid. But so why is $\rho$ (density) multiplying acceleration instead of $m$? What is the property of fluids that allows for this?

Answer 1

The Navier-Stokes equation describes the motion of some infinitesimal volume of the fluid. That is we divide the fluid up into tiny volumes $dV$ and the equation tells us how these tiny volumes move. The overall motion of the fluid comes from combining the motions of all these tiny volumes.

So the equation is really:

$$ \rho \left( \dfrac{\partial v}{\partial t} + v \cdot \nabla v\right) dV = -\nabla p dV + \nabla \cdot T dV + f dV $$

and of course $\rho dV$ is the mass of the volume element $dV$, so the left hand side is mass times acceleration just like Newton's second law.

However we don't want the equation to depend on the exact value of $dV$, and we will be taking the limit $dV \to 0$ anyway, so we divide through by $dV$ to get the form of the equation that Stewart gives.