# Thin layer equations

Tak Kwong Wong (Croucher Fellowship 2010) works on hydrostatic Euler equations at the University of Pennsylvania.

Setting water to boil, coffee to drip, sparking a gas stove; where the rest of us see basic steps of a very welcome morning caffeine jolt, Dr Tak Kwong Wong sees equations. Thin layer equations, to be exact, which describe the leading order behaviour of fluids moving in very narrow domains or near physical boundaries, such as pipes, ship edges, and plane wings, where rapid changes in fluid velocity are especially important.

Wong’s research focuses primarily on broadening understanding of hydrostatic Euler equations, which describe fluid movement in thin channels such as blood flow and pipelines, and the Prandtl boundary layer equations dealing with viscous fluid moving near and adhering to physical walls, such as air movement along aircraft wings. The equations themselves are obstacles to deeper understanding, with pathological structures of the loss of tangential regularities which resist estimating solutions of underlying equations. Standard practice consists of adjusting variables to make the equations more malleable. However, the information extracted does not necessarily reflect the original physical variables, and is therefore an indirect estimation. As a Croucher fellow at UC Berkeley, this basic conflict piqued Wong’s interest. His research at the time centered on the possibility of deriving estimations without changing the variables while also bypassing the equations’ structural problems. By utilising the nonlinear cancellation between vorticity and velocity, he found that the loss of tangential regularity could be avoided under certain conditions. “Using non-linear cancellation was the key to our breakthrough,” Wong says, “because for the first time, we were able to derive the energy estimate of the Prandtl boundary layer equations with the original variables, which gives us a better understanding of the underlying physics. Of course, how you understand this information is a different story.”

Having opened this door, Wong has his eye on the next chapter as he continues to explore solutions of the Prandtl equations. Boundary layers occur naturally in the real world, and greater understanding of Prandtl equations are closely linked to other mathematical challenges. The motion of a viscous fluid can be described by the Navier-Stokes equations with the Dirichlet boundary condition, which represents real fluid adherence to physical boundaries. The inviscid limit problem asks whether the solutions of the Navier-Stokes equations converge when the viscosity tends to zero. For low viscosity, ideal and inviscid flows can be expected to serve as an approximate solution to the viscous flow. However, this approximation is not uniformly valid throughout the entire domain, as it breaks down near the boundary, to which a real fluid adheres, but ideal and inviscid fluids yield non-zero tangential velocity in general. This discrepancy in boundary conditions causes drastic changes in unknowns in thin regions close to the physical wall. In order to have a better understanding of the inviscid limit problem, a key issue is to study the fluid behaviour in this thin region, formally described by the Prandtl equations. “There are so many different structures and behaviors along boundaries even in ideal circumstances,” Wong explains, “but we must expand our understanding of the Prandtl equations before adding back some factors for real-life applications.”

Asked about applications of his research, Wong expresses mild amusement. “Do you know what would happen if you tried to make an airplane using just pure mathematics?”—surely ‘fly’ is a safe answer—“…it would fall down every time.” Though the impact would be very indirect, he explains, his work on pure mathematics under ideal hypotheses allows deeper understanding of fundamental models. These guidelines could eventually lead to more inventive real-world experimentation, but “product design, weather prediction, gas…” are clearly a cursory footnote to the greater adventure of mathematical possibility.

Even from a mathematical perspective, Wong’s area is difficult, with no set endpoint to the depth of understanding the solutions of thin layer equations. The attractions and purity of this brand of mathematics become increasingly more evident as he explains it as the ultimate challenge, with deep connections to other theoretical problems. “High scores in mathematics at school certainly helped,” Wong says, “but thinking about how to solve problems has been a driving force for me, to extend the physical domains and expand theory.” The field of fluid dynamics is deceptively vast, with many mathematicians working on different facets, but despite the substantial advances, much information still remains to be found. Wong notes that he has always been drawn to areas slightly out of the spotlight, starting from his doctoral work, to see what he can do with just paper, pencil, and brain—his computer is reserved mostly for email and presentations. With this in mind, Wong plans to expand his research from classical Newtonian fluids like water and coffee, which have less complicated structures, to study boundary layer behavior of more viscous non-Newtonian fluids such as petroleum, interested by the different, more complex underlying physics. “Academia really allows me to push this frontier, from the freedom to choose research problems that I’m really interested in to sharing ideas and finding new observations of underlying problems through teaching.”

*Dr Tak Kwong Wong won a Croucher fellowship in 2010 for his postdoctoral research at the University of California, Berkeley. Wong received his Ph.D. from New York University, where he focused on mathematical analysis of partial differential equations in relation to fluid dynamics. He is currently a Hans Rademacher Instructor of Mathematics at the University of Pennsylvania. *

**To view Tak Kwong Wong's personal Croucher profile, please click here. **