So I've recently started looking into where do Navier-Stokes equations come from and I've learned lots of interesting stuff so far but I stopped at Reynold's transport equation, which states that: $$\frac{d}{dt}\iiint L dV = \iiint \frac{\delta L}{\delta t} dV + \iint L(\vec{u} \cdot \vec{n})dA$$, where L is some property transferred with the fluid, $\vec{u}$ is velocity field, $\vec{n}$ is the normal vector pointing outside of given volume bounded by infinitesimal $dA$.
So I tried deriving this by myself starting from just the ordinary derivative of mass over dt. Note that I'd like to derive transport of density of fluid assuming incompressibility condition.
First let's define our mass scalar field as $$m(x(t), y(t), z(t), t)$$ Then: $$\frac{dm}{dt} = \frac{\delta m}{\delta t} + \frac{\delta m}{\delta x} \frac{\delta x}{\delta t} + \frac{\delta m}{\delta y} \frac{\delta y}{\delta t} + \frac{\delta m}{\delta z} \frac{\delta z}{\delta t}$$ For fluids $\vec{u} = (\frac{\delta x}{\delta t}, \frac{\delta y}{\delta t}, \frac{\delta z}{\delta t})$ Therefore: $$\frac{dm}{dt} = \frac{\delta m}{\delta t} + (\nabla m)\vec{u}$$ which is material derivative... Now we want to enforce the conservation of mass so that the following is true: $$\frac{dm}{dt} = 0$$ $$\frac{\delta m}{\delta t} = - (\nabla m)\vec{u} = -\nabla (\rho V)\vec{u}$$ And now comes the surprising thing which I'm not sure is true (maybe I did some mistake sin my understanding?)... My question is whether the following is actually true: $$ -\nabla (\rho V)\vec{u} = - \iiint \nabla \cdot (\rho \vec{u} ) dV$$ If this is true indeed then we can use divergence theorem to get: $$\iiint \frac{\delta \rho}{\delta t} dV = \frac{\delta m}{\delta t} = -\nabla (\rho V)\vec{u} = - \iiint \nabla \cdot (\rho \vec{u} ) dV = - \iint \rho \vec{u} \cdot \vec{n} dA$$ $$\iiint \frac{\delta \rho}{\delta t} dV = - \iint \rho \vec{u} \cdot \vec{n} dA$$ which is Reynold's transport equation for fluid density without $\frac{d}{dt}\iiint L dV $ term since it zeroes out due to conservation of mass principle. Is my reasoning correct with all of the above? If so, how would you intuitively (or mathematically) justify the equation below? $$ -\nabla (\rho V)\vec{u} = - \iiint \nabla \cdot (\rho \vec{u} ) dV$$ Thanks for any time you spend reading my post ;)