1. ## General Flux Vector

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

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

2. Originally Posted by Maccaman
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.
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This could be wrong but here is what I have got:

$\displaystyle Q(t) = \iiint\limits_V \ \rho \ dv = \rho_0 + \rho_1 \sin (\omega t) \iiint\limits_V \ \rho \ dv = (\rho_0 + \rho_1 \sin (\omega t)) \frac{4}{3}\pi (b-a)^3$

Is that correct? If not, could somebody please show me where I went wrong?
Thanks

3. Originally Posted by Maccaman
This could be wrong but here is what I have got:

$\displaystyle Q(t) = \iiint\limits_V \ \rho \ dv = \rho_0 + \rho_1 \sin (\omega t) \iiint\limits_V \ \rho \ dv = (\rho_0 + \rho_1 \sin (\omega t)) \frac{4}{3}\pi (b-a)^3$

Is that correct? If not, could somebody please show me where I went wrong?
Thanks
I think it should be
$\displaystyle Q(t) = \left( \rho_0 + \rho_1 \sin (\omega t)\right) \frac{4}{3}\pi (b^3-a^3)$