# Deriving a conservation law using the divergence theorem

Problem:

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.

Attempt at a Solution:

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.

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Office_Shredder
Staff Emeritus
Gold Member
The density of a fluid is the amount of mass of fluid per unit volume. Here we have some radost, R, and the interesting conservation law is going to be saying something about the amount of radost per unit volume. If the function r was radost per unit volume you would literally replace the density with r in the calculations you show in your post and be done - unfortunately it's given in radost per unit mass of fluid, so you need to convert that to radost per unit volume (which can be done with the information they give you)

The density of a fluid is the amount of mass of fluid per unit volume. Here we have some radost, R, and the interesting conservation law is going to be saying something about the amount of radost per unit volume. If the function r was radost per unit volume you would literally replace the density with r in the calculations you show in your post and be done - unfortunately it's given in radost per unit mass of fluid, so you need to convert that to radost per unit volume (which can be done with the information they give you)
Would it be $dr=(r\rho)dV$?

Like OF said, needs to be radost per unit volume.

I'm sorry I'm not really sure how to do that.