My research focuses on the mathematical modelling of fluid flows. The
two problems I am currently studying are: a) thin film
spreadings on solid surfaces and b)
singularities in fluids.
Below is a short list of some recent research problems we
tackled with my collaborators:
Spreading of power law fluids:
some paints are not newtonian,
and their mechanical properties can be approximately described with
power law rheology. We developed models for the spreading of these
fluids, and found a number of interesting phenomena, such as the non
uniqueness of self-similar solutions, and the absence of the paradox of
the contact line. Below is a graph of the drop profile given by the
many self-similar solutions that describe the spreading of a circular
Spreading of charged fluids:
droplets can carry an electrical charge in many natural and man made
settings. The electric charge is a powerful driving force of these
flows: the electric field, pulls the drop, spreading it over the solid
(see figure below). We found that if a conducting fluid spreads on a
dielectric, the flow can be described with selfsimilar solutions that
have the same exponents than capillary spreading.
of charged water droplets: they are common in thunderclouds, and
their dynamics are not well understood. The charge deforms the drop
(see figure below) and finally splits it. We developed numerical
methods to study this problem and we are currently exploring the
effects of finite conductivity and external electric fields.
Self-similar solutions and singularities on fluid flows. In this
setting, singularities represent the divergence of the pressure or the
velocity in a given point and time. We recently found numerical
evidence indicating that the fluid flow becomes singular at tips
generated in finite time in droplets.
Allen cahn dynamics. Recently,
I started a new collaboration with Professors Nicholas
Alikakos (UNT) and Xinfu Chen (University of Pittsburgh). We are
studying the problem of the curvature flow of multiple curves joined by
triple junctions. The problem is indirectly related to the fluid flows
mentioned above, the connection lies in that under certain boundary
conditions, the shape of the free surfaces have the same steady states.
In this research, I was able to use my experience in computational
differential equations in order to solve the vector Allen Cahn
equations. We found the different connections between local
minima produce two different kinds of interfaces. This animation shows the formation of
interfaces with different connections: on the left we show the
interfaces and on the right the three different phases.
One can also solve three dimensional problems. In the image below, we
show a surface evolving by curvature flow inside
a sphere, obtained with the three dimensional vector Allen Cahn
simulation of cusp formation:
Click to see animation
the problem is whether a cusp can
form at the free surface of a liquid at the molecular scale. Koplik and
Banavar found a negative result (Phys. of Fluids 1994 Volume
6, Issue 2, pp. 480-488). However, this was due because they studied
interface between two fluids of similar viscosity. If one repeats the
simulation with a
fluid-vacuum interface, cusps do
The simulation is tricky
because the temperature must be carefully controlled. If it is too
high, the interface becomes diffuse. In the right range of temperature,
cusps as shown in the link below (be patient, the animation has
Convergent shallow water flows.
Click link to see animation
Multiphase flow with
diffuse interfases: creation of triple junctions
Click on image to see animation (slow)
Convergent flows in the Porous Medium
of a two dimensional fluid with Navier Stokes equations.
Click link to see animation
Done mostly in Minnesota, Kansas and Tandil. I am still engaged in some
of these projects.