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Math Help - General Flux Vector

  1. #1
    Member Maccaman's Avatar
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    General Flux Vector

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

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


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

    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

    \frac{d Q(t)}{dt}
    Last edited by Maccaman; September 22nd 2009 at 06:17 AM. Reason: Typo in heading
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  2. #2
    Member Maccaman's Avatar
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    Quote Originally Posted by Maccaman View Post
    Let \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.
    [/tex]
    This could be wrong but here is what I have got:

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

    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
    Q(t) = \left( \rho_0 + \rho_1 \sin (\omega t)\right) \frac{4}{3}\pi (b^3-a^3)
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