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Homework Help: Ink diffusing in water - partial diff equations

  1. Dec 5, 2008 #1
    1. The problem statement, all variables and given/known data

    The particles of an ink blob dropped into a large container of water diffuse outward and obey the radial diffusion equation:

    dn/dt = (D/r2) (d/dr) (r2* (dn/dr) )

    where n(r,t) is the density of ink particles at point r at time t and D is the diffusion constant.

    Verify, by direct differentiation that:

    ns = N*(1 / (4*pi*D*t) )3/2 * er2/4Dt

    is a solution of this equation and satisfies the condition that the total number of ink particles is N for any value of t.

    2. Relevant equations



    3. The attempt at a solution

    I have no idea?
     
  2. jcsd
  3. Dec 5, 2008 #2
    No idea...? Not so good!!
    For
    [tex]
    n_s(r,t)=N\frac{e^{r^2/4Dt}}{(4\pi Dt)^{3/2}}
    [/tex]
    what is
    [tex]
    \frac{d}{dt}n_s(r,t)
    [/tex],
    [tex]
    \frac{d}{dr}n_s(r,t)
    [/tex]
    and
    [tex]
    \frac{d^2}{dr^2}n_s(r,t)
    [/tex]
    ?
    Do these expression satisfy
    [tex]
    \frac{d}{dt}n_s(r,t)=\frac{D}{r^2}\frac{d}{dr}\left[r^2\frac{d}{dr}n_s(r,t)\right]=\frac{D}{r^2}\left[2r\frac{d}{dr}n_s(r,t)+r^2\frac{d^2}{dr^2}n_s(r,t)\right\
    [/tex]?
     
  4. Dec 6, 2008 #3
    I did it :D
    I won't type it all out because it would take forever, but I got both sides of the equation to equal :)
     
  5. Dec 6, 2008 #4
    Congratulations, no need to type it all out. :smile:
     
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