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

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?

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