The Muskat Problem
Part of my research focuses on the flow of two incompressible fluids in porous media. These fluids are assumed to be immiscible and, as such, they are separated by a sharp interface. This is known in the literature as the Muskat problem (named after M. Muskat in the 1930's). Here, due to the effect of the solid matrix of the porous medium, the usual fluid equations for the conservation of momentum (Euler or Navier-Stokes equations) are replaced with the empirical Darcy's Law.
Remarkably, the Muskat problem is mathematically analogous to the Hele-Shaw cell problem (named after H. Hele-Shaw's classical 1898 paper) that studies the movement of a fluid trapped between two parallel vertical plates, which are separated by a very narrow distance. Besides its mathematical interest, the Muskat problem is also relevant in geosciences (aquifers, oil wells or geothermal reservoirs). Moreover, it has also been considered as a model for the velocity of cells in tumour growth.
In my thesis (you can download it here) I studied the effect of impervious walls on the qualitative behaviour of the interface separating both fluids.
Here you can see some simulations of the interface. I also considered the case where the porous medium has a permeability given by two different values (i.e. the porous medium is inhomogeneous). For that case you can also see some videos.
One of my (favourite) results for the Muskat problem is related to wave breaking. In the case of water waves in the ocean, breaking of waves may occur anywhere where the amplitude is sufficient. In particular, water waves may break in mid-ocean. However, it is particularly common on beaches. This occurrence is due to the effect of the ocean floor: as a wave approaches the coastline and comes into contact with the ocean floor, the waves' velocity decreases. Under stationary conditions, a decrease in speed must be compensated by an increase in height in order to maintain a constant energy flux. This process is called shoaling and results in wave breaking.
Our result studies the effect of the porous medium's floor in the breaking of the interface. In particular, we, using a computer assisted proof, we were able to prove that there exist initial interfaces such that if the depth is finite, they lead to solutions that turn, while if the depth is infinite, they lead to solutions that do not turn. Or, in other words, the finiteness of the depth enhances the turning behaviour of the system. This could be though as a mathematical proof of the fact that
Smooth runs the water where the brook is deep
(Shakespeare's Henry VI. Part II, 1592.)
Due in part to this contributions, I was awarded with the 2015 'Vicent Caselles' prize from the Royal Spanish Mathematical Society (RSME).