Topics of Interest & Research

As I read Thomas Kuhn in The Structure of Scientific Revolutions, a science — any discipline of science — is shaped by the people who practice it, and by the perspective on life they bring to it. So it matters that I say where I come from. If someday I make a mistake in my research because of a bias I carry without seeing it, readers should be able to find its source here and hold it against me.

I was born in 1999, the eldest son of a middle-class family in India. My father was one of the first engineers from his village. He comes from Bihar, a state the rest of India looks down on; I will not go into why Bihar is where it is. What my father understood, and ingrained in me, was that for people like us knowledge would be the only savior.

That conviction has shaped my life heavily, and now that I am grown I am trying to change it. I do not want to chase knowledge for financial security, or for security in life. I want to chase it because I find it extraordinary that we have developed what Carl Jung, in Psychology of the Unconscious, calls directed thinking. As I see it, we spend our mental energy either in directed thinking or in fantasy thinking, and directed thinking is best done in the study of science.

So over the years I was drawn to three aspects of science. The first is its language, mathematics. We must be able to communicate the concepts we think with to other humans, because it is very easy to fool yourself; that is why we came up with mathematics, to share concepts of advanced directed thinking among us. The second is physics: the observation and understanding of nature. Not controlling it, not predicting it — some may lean toward those more than I do — but marveling at why it works the way it does. The third, in recent times, is computing, which I expect to change a great deal within my own lifetime.

I always wanted to study aerospace engineering, and I got into it at IIT Kharagpur. Right away I was drawn to linear algebra and robotics. Everything was clean and neat; every result could be derived without approximation; code for robotic simulation was simple to write; everything was deterministic, at least in the introductory courses. I am not claiming that this is the frontier of research. Fluid mechanics was the opposite of discrete mechanics and robotics: partial differential equations and continuous fields. I revered the subject, but I was not drawn to it then. I felt I would have to study a great deal before I could do anything in it.

When it was time to choose the topic of my bachelor’s thesis, I applied to all three openings in control theory, and was rejected from all three, on the grounds that an undergraduate was not fit for that kind of research. I had filled in no other choice, because I was cocky enough to assume I would get one of them. Few options were left. The one I got was working on CFD code for flapping flight with Dr. Sunil Manohar Dash — aerodynamics, which is to say fluid mechanics. Maintaining the code and running simulations in ANSYS, I found an entirely different computational world, very unlike robotics. Yet underneath, there was still a great deal of arithmetic, and it was still linear algebra. That left me with a question I kept coming back to: could linear algebra be spoken more fluently in the world of fluid mechanics?

In the summer of 2022, during an internship at TU Munich, working with Dr. Sophie Armanini (now at IC London), I was building a thrust model for a rigid tandem flapping-wing dragonfly. I reached out to Prof. Haithem Taha to ask whether he had an idea of how to build a state-space model for flapping flight. I was already interested in his research, and it was around the time his variational theory of lift was making the rounds in the scientific community and in the news. When we talked, we hit it off. I think he liked that I was more interested in mathematics than the average engineering student, and that I had a background in fluid mechanics, dynamics and robotics. So I started my PhD with him.

It was in one of our lab meetings at the ADC Lab that Prof. Taha introduced Udwadia and Kalaba’s equation of constrained motion, and it blew my mind. Gauss’s principle exists. Its closed-form analytical solution exists, and it is easy to understand — the paper is four pages long. It could make so many of the problems we solve in mechanics and dynamics with Newtonian, vectorial methods simpler and more intuitive, and yet it is not taught in our textbooks. That filled me with rage. I cannot change the education system in my lifetime, so I decided to direct the rage somewhere I could.

What if we applied it to the Principle of Minimum Pressure Gradient? That principle, which Prof. Taha has published, is the continuous analog of Gauss’s principle. If Gauss’s principle has a closed-form analytical solution, could we use it to get a closed-form analytical discrete solution of the incompressible Navier–Stokes equation, which is the first-order optimality condition of the Principle of Minimum Pressure Gradient? It turns out that the idea works. But there are many bridges still to build and gaps to fill, and that is what I foresee working on for the next two to four years of my life.

Last Updated on Sept 26, 2026

In mathematics, I am currently focused heavily on studying analysis from the basics. I am also reading more advanced numerical linear algebra, along with the details of its implementation on a computer, especially parallel implementation. This is to develop the finite element solver we call the Variational Projection of Navier–Stokes, which applies the idea of Udwadia and Kalaba’s closed-form solution of constrained motion to the Principle of Minimum Pressure Gradient, or PMPG (IEEE Control Systems Letters, 2025; ASME IDETC-CIE 2025).

We have already demonstrated the idea as a proof of concept (Physical Review Fluids, 2026; AIAA SciTech 2026), and all our efforts are now directed toward scaling it: making a meaningful, industrial-grade solver that handles real-life geometries. The solver is built entirely in physical space, and it is capable of direct numerical simulation (DNS).

An interesting open question: how does the problem size scale in the framework of the Variational Projection of Navier–Stokes, if we pay close attention to how the PMPG cost functional density is distributed over the flow domain? My intuition is that this density is far from uniform — concentrated in some regions and negligible in others — so the degrees of freedom that actually govern the solution may be far fewer than the mesh count.

Last Updated on Sept 26, 2026

I plan on sharing articles related to Gauss’s principle during the months of Oct – Dec 2026.