Let $\displaystyle \rho = \rho_0 + \rho_1 \sin(\omega t)$ be the density of oxygen in a spherical shell with inner radius a and outer radius b.

Find the total amount of oxygen, Q(t) in this shell for any time t.

Find the most general flux vector of the form

$\displaystyle \vec{J}(x,y,x,t) = f(r,t)(\vec{r}) \ \mbox{where} \ r = \sqrt{x^2 +y^2 + z^2} $ and $\displaystyle \vec{r} = x\vec{i} + y\vec{j} + z\vec{k}$ such that

$\displaystyle \frac{\delta \rho}{\delta t} + \ \mbox{div} \ \vec{J} = 0 $

If at any time and anywhere on the the outer surface of the shell $\displaystyle (r=b)$ the flux vector satisfies $\displaystyle \vec{J}$.$\displaystyle \vec{r} = \frac{\omega}{3b} \sin (\omega t)$ then

$\displaystyle f(r,t) = \frac{\omega}{3r^3} (\rho_1 (b^3 - r^3) \cos (\omega t) + \sin (\omega t))$

Show that the total flux into the spherical shell per unit time is equal to

$\displaystyle \frac{d Q(t)}{dt}$