Ink diffusing in water - partial diff equations

[mikehibbert]

Homework Statement

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:

n_s = N*(1 / (4*pi*D*t) )^3/2 * e^r2/4Dt

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

Homework Equations

The Attempt at a Solution

I have no idea?

 

[Pere Callahan]

No idea...? Not so good!
For
<br /> n_s(r,t)=N\frac{e^{r^2/4Dt}}{(4\pi Dt)^{3/2}}<br />
what is
<br /> \frac{d}{dt}n_s(r,t)<br />,
<br /> \frac{d}{dr}n_s(r,t)<br />
and
<br /> \frac{d^2}{dr^2}n_s(r,t)<br />
?
Do these expression satisfy
<br /> \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\<br />?

 

[mikehibbert]

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 :)

 

[Pere Callahan]

Congratulations, no need to type it all out.