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?
Best Answer
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!