How does dye move in water?

Question

My understanding is that dye moves through water primarily through diffusion. The introduction to these lecture notes seems to confirm:

If you we put a drop of red dye in water, it will slowly diffuse throughout the water. Why does this happen? How fast does it happen? What is going on microscopically? The microscopic mechanism of diffusion is very simple: the dye molecules start densely concentrated near one point. Then they get bumped by neighboring molecules until they are spread out all over.

But later, the note do a short computation and conclude the following:

For example, taking a dye molecule in water with D = $10^{−9} \frac{m^2}{s}$, to move $1$ $m$ would take $31$ years. So clearly diffusion is not the main mechanism by which dyes move around in water.

My intuition suggests that if I drop some dye in a shallow vat of water with $1$ $m$ radius, then the dye will saturate the water and hit the sides in far less than 31 years. If diffusion does not cause this, what does?

Answer 1

If diffusion does not cause this, what does?

Diffusion IS an important phenomenon but in the situation you describe it's somewhat 'obscured' by other transport phenomena.

Fick's Second Law (see link) really only works when there's no convection currents at all, or no currents caused by air drafts.

At room temperature in a low viscosity fluid that is a big ask: smallish temperature differences cause small density variations and thus convection currents, which provide a much faster transport mode than pure diffusion.

Diffusion would be better observed in a carefully thermostated, draft-free medium, possibly thickened somewhat with glycerine (e.g.)... assuming you've got a lot of time on your hands!

Answer 2

The notes also say

These numbers are characteristic of diffusion processes: diffusion in air over macroscopic distances is generally very slow, and almost always dominated by convection (see Section 4.1).

And if you watch dye droplets mixing with water, you can plainly see the convection taking place. Sometimes “blobs” of dye are carried from place relatively intact. Other blobs of dye are interrupted by smaller convection eddies, and become separated except for delicate little tendrils.