# Ink diffusing in water - partial diff equations

1. Dec 5, 2008

### mikehibbert

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. Dec 5, 2008

### Pere Callahan

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

3. Dec 6, 2008

### 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 :)

4. Dec 6, 2008

### Pere Callahan

Congratulations, no need to type it all out.