In collaboration with Prof. Coti Zelati (Imperial College) and Prof. El Gindi (UC San Diego), we connected the concept of mixing with enhanced dissipation. To illustrate consider the problem of mixing milk into your morning coffee, depending on the way you stir the two fluids we can find the associated time scale on which you will attain an homogenized mixture.
In collaboration with Prof. Maggi (UT Austin), we showed that the unique critical point of perimeter at fixed volume is a union of equal sized balls. This validates perimeter as a model for capillarity induced motion. As shown in our theorem, we observe in everyday life that tiny droplets or soap bubbles are always round.
In collaboration with Prof. Mellet (U. of Maryland), we showed that the thin-film equation (a fourth order model for the motion of thin droplets) approximates a super-solution of a free boundary problem. This is known in the fluid mechanics literature as Tanner's law, namely the motion of a droplet is almost fully determined by the shape of its wetted region. |
