22/02/2019

Dissertation creates new technique in the field of fluid dynamics.

Karine Rodrigues

Researchers have been trying for centuries to understand the movement of fluids, described mathematically by the Euler and Navier-Stokes equations. But essential points for understanding phenomena such as blood circulation, the dynamics of stars, and the dispersion of atmospheric pollution remain open.

A new step towards this understanding was taken by engineer Ciro Campolina and is described in the master's thesis that he will defend next Tuesday (26), at 2 pm, in room 224 at IMPA. He was advised by Alexei Mailybaev, a researcher in the area of fluid dynamics.

Titled “Fluid Dynamics on Logarithmic Lattices and Singularities of Euler Flow”, the work sheds light on the three-dimensional Euler equations for incompressible fluids, such as water, and, under certain conditions, air, which preserve volume in their motion.

Campolina, 25 years old, tried to discover whether, given an initial condition of the fluid, there exists a definite solution for these equations at all instants of time.

"It's an issue that exists for both Euler's equations and Navier-Stokes' equations," he says, referring, lastly, to the equations considered one of the seven Millennium Problems, the subject of the $1 million prize from the Clay Mathematics Institute.

A native of Rio de Janeiro with a typical accent from the Zona da Mata region of Minas Gerais, where he grew up, he explains that the difference between the equations mentioned lies in the type of fluid being analyzed. Euler's equations are used for fluids in which internal friction (viscosity) is so small that it is disregarded. They describe turbulent flows, such as those seen in storms. When friction exists, Navier-Stokes equations are used. Both describe how changes resulting from, for example, pressure variations affect the fluid particles.

To verify whether or not a type of singularity, called a blow-up, exists—a singularity that prevents the existence of a solution to these equations at any given time—Campolina needed to develop a new analytical technique.

As he explains, the results obtained with the current method, state-of-the-art direct simulations (DNS), are ambiguous and do not account for the complexity and variety of the observed structures.

“The blow-up would be the formation of a point in the fluid where the local circulation of particles reaches very large values and goes to infinity. The solution, therefore, only exists up to a certain instant and, from there, cannot be extended because this local circulation described by the solution of the equation has grown and, at that instant, 'exploded,' that is, it went to infinity,” he explains, having entered the master's program in 2017, shortly after completing his undergraduate degree in mechanical engineering at the Federal University of Juiz de Fora (UFJF).

It was during his studies that he became interested in research, while dealing with problems of fundamental practical importance, such as measuring air resistance on an aircraft, identifying transition points to turbulence in an aerodynamic system, and quantifying combustion efficiency in an automotive engine. All of these require the same physical theory: the description of fluid motion.

“Many phenomena involving fluids lack a complete and satisfactory scientific theory. Part of this difficulty lies in the mathematical complexity of the equations that describe them,” he observes.

Nevertheless, Campolina, the third of four children of Claudia, a homemaker, and Antonio Henrique, a professor of Philosophy and Theology – “two highly educated people, both of whom served as examples for me to pursue an academic career” – took a step further. He found clear evidence of blow-up, explained as a chaotic attractor in a renormalized system, after developing a new mathematical model that simplifies Euler's equations while preserving their properties. Formally identical to Euler's equations, the model was created by restricting the fluid dynamics equations to a 3D logarithmic lattice.

In addition to observing the occurrence of the singularity, the study, according to the master's student, reveals something he considers even more important: the singularities of Euler's equations cannot be observed in computer simulations using currently employed techniques. Not even by supercomputers in large research centers.

“The problem was investigated from a mathematical point of view, through proofs and rigorous analysis, but they were unable to arrive at an answer. A second possibility was to try to simulate it on a computer, but even using super powerful machines with months of computing time, they did not arrive at clear results, precisely because of the complexity of the equations. With this work, we obtained clear results, and we characterized the dynamics of this problem,” he states.

According to Mailybaev, the work introduces a new mathematical modeling formalism, through which fundamental questions about turbulence have a greater chance of being answered. "Such a formalism may indicate the next steps on the path to Turbulence Theory, a theory of great interest to physicists and engineers, but still far from being fully developed."

The discovery was the subject of an article – “Chaotic Blow up in the 3D Incompressible Euler Equations on a Logarithmic Lattice ” – published in partnership with Mailybaev, in the renowned “Physical Review Letters”.

After completing her master's degree, Campolina will remain at IMPA for her doctorate.

“When I decided to pursue an academic career, I started looking for research groups that worked on problems involving fundamental fluid theory. I met the Fluid Dynamics Laboratory at IMPA, and Professor Alexei (Mailybaev), who was working on turbulence problems, a theory I had a great interest in,” he says, highlighting that he was warmly welcomed. Judging by the studies done so far, the encounter was providential for Mathematics.