Many of us at Stevens have taken, are taking, or will take a course that involves fluid dynamics. These courses tend to be on the more challenging end, because fluids can behave in immensely complex ways, and it’s difficult even experimentally to understand this behavior, let alone build mathematical models to govern it or computational techniques to simulate it. Even for those of us who don’t have fluids in the curriculum, they nevertheless show up all around us, from the air we breathe and the water we drink, to the blustery winds that make walking to class in the winter no small feat.
To that end, I thought it would be fun to write about two recent advances in the study of fluid dynamics: the first, a body of work on instabilities of ocean waves, and the second, a machine-learning assisted discovery of new types of singularities in several equations that model fluids.
I’ll explain what I mean by instabilities and singularities in a moment, but first, I want to emphasize that we still don’t really know whether the fluid models we use are correct all the time. Indeed, a famous mathematical problem asks whether the Navier-Stokes equations—seen as the most general model of fluid dynamics—have physically realistic solutions in three-dimensional space and for all times. If you are able to rigorously answer this question, you would win $1 million from the Clay Mathematics Institute (and lots of mathematical glory)!
Neither of the works I just mentioned have reached the pinnacle of Navier-Stokes in the setting of the Clay Millennium Problem, but their findings are sure to make waves, not just model them. Regarding the ocean waves, researchers use the Euler equations, a (slight) simplification of Navier-Stokes, where the term modeling viscosity is not included. The inviscid flow of the oceans is nonetheless vastly complex, as the “free” top boundary and interactions with wind, boats, and other disturbances lead to a variety of ocean waves that seem to mysteriously pop into and out of existence.
Now, the mystery is better understood. A few research groups studying this have found that the persistence of ocean waves depends on the frequency of disturbances. More precisely, there are certain bands of frequencies that will “destroy” ocean waves, disturbing in a way that cancels out the propagation of the waves in water. Between these unstable bands are stable ranges of disturbing frequencies that still allow for the ocean waves to propagate.
As for the singularity paper, a team of researchers at Google DeepMind, collaborating with several universities, developed a specialized “physics-informed neural network,” or PINN (pronounced “pin”), to probe certain instabilities in a few different models describing fluids. They discovered a family of new solutions to these models that depend very precisely on the initial conditions, and exhibit “blow-up” behaviors such as the formation of infinite gradients, leading to unphysical dynamics.
The first body of work on the ocean waves shows how a collaboration with theoretical, computational, and experimental fluid dynamicists can yield huge breakthroughs in a challenging area of research: this kind of multidisciplinary work is, I think, crucial to solving the increasingly complex problems we seek to tackle in the world today. And the usage of PINNs here is a great example of machine learning being developed for a specific use case, which, in my opinion, produces much more solid results than a general-purpose AI can (at least for the moment — advances in large AI models also continue to make waves).
In sum, fluids are very challenging to understand, and there’s still a lot we don’t know. This makes the subject a very exciting one in my view, and allows us all to give ourselves some grace when we’re struggling in a fluids class.
