Dissolving together

I am continuing to play around with systems that start to separate into two phases (the green and the red phase in the movie above) but don’t get very far before one phase (the green one) dissolves. I have tweaked some parameters above, so that green droplets form, and the system is a bit bigger (I also changed the orientation, sorry if that is confusing). if you follow the movie carefully, it seems clear that the dissolution of the green droplets is via fronts that are pretty straight (horizontal), one front that moves upwards, dissolving droplets as it goes, while another moves downwards. When they meet in the middle, the green droplets are all gone.

I think that the droplets in the middle of the strip of droplets are protected from the encroaching red phase by the outer layers of droplets. In other words, the red phase has to dissolve the green droplets layer-by-layer. As the initial formation of the droplets is much faster than their later dissolution, when I made the movie, the simulation time between successive frames increases with exponentially with time. This allows me to fit all the fun physics into a 19 s movie, but this does hide the fact that in real times the advancing fronts are actually continually slowing.

In fact the speed of the dissolution fronts varies as one over the square root of time, which means that you have to wait four times as long to see the front move twice the distance. This slowing-as-time-passes behaviour is common when the molecules move via diffusion, which is what is happening here. It is not obvious from the movie, but as the green droplets dissolve, the red phase becomes a bit less red and bit more green, and this decreases the concentration gradients. Smaller concentration gradients mean slower diffusion down these gradients, and I think it is this that causes the dissolution fronts to slow down. Anyway, I am not sure that these results are very useful at the moment, but at least I am having some fun making pretty movies.