# Derive conservation law using divergence theorem

Material scientists have discovered a new fluid property called "radost" that is carried along with a fluid as it moves from one place to the next (just like a fluid's mass or momentum). Let $r(x,y,z,t)$ be the amount of radost/unit mass in a fluid. Let $\rho(x,y,z,t)$ be the mass density of the fluid. Let $\vec{v}(x,y,z,t)$ be the velocity vector of the fluid. Use the divergence theorem to derive a conservation law for radost.

We did an example like this in class, but for conserving mass, so it was a little different. What we ended up with was the following expression $$\frac{\partial \rho}{\partial t}+\nabla \cdot (\rho \vec{v})=0$$

We started by writing, $$dM=\rho\, dV$$ Thus, $$M=\int_V \rho\, dV$$ Then we applied the divergence theorem and that was basically it.

I'm just kind of confused how to start this one.

You begin it the same way as for mass. $r$ is radost per unit mass, so you can say that:

$$dR = \rho r dV$$

where $R$ is the total radost. You then integrate to get:

$$R = \int_V \rho r dV$$

and apply the divergence theorem like you did for mass.

Note -- there are several steps between determining $R$ and applying the divergence theorem. But you stated you were unsure where to start and had an example from that point onwards so I left them out.

• Thank you, that makes sense. So does that make the flux $\int_S \rho r \vec{v}dV$? – Lefty Sep 20 '13 at 5:04
• For my final expression, I get: $\frac{\partial \rho r}{\partial t} + \nabla \cdot (\rho r \vec{v})$ – Lefty Sep 20 '13 at 12:41
• @Lefty You got it! Glad I could help. – tpg2114 Sep 20 '13 at 14:52