CAREER: Boundary Layers and Hydrodynamic Stability

About This Grant

The Navier-Stokes (NS) equations are fundamental equations that describe the motion of viscous fluid flows. Boundary layers form for low viscosity flows (such as air and water) in thin regions near physical boundaries where the flow experiences a sharp transition from nearly inviscid flow in the bulk to the no-slip boundary condition at the wall. These boundary layers exist in a variety of flows in nature and society, such as flows over an airplane wing or fluid flow near the nose of a submarine. The mathematical questions of characterization and stability of boundary layers, as well as large time limits (hydrodynamic stability), are classical and notoriously challenging. They pose many novel analytical challenges within the Navier-Stokes equations, such as multiscale phenomena, mixed-type phenomena, and singular perturbations. The project will develop of a systematic theory of hydrodynamic stability and boundary layers which develops the mathematical tools to address these challenges. These mathematical results have large downstream scientific impacts on a variety of applications (such as airfoil modeling, modeling the nozzle of a submarine). More precisely, a better understanding of reduced models and their stability enhances computational algorithms and gives rise to new algorithms to model these important phenomena. The boundary layer theory for stationary NS flows naturally trifurcates into the three regimes: local-in-x, favorable, and adverse. For the local-in-x and favorable cases, the investigator studies the stability of the boundary layer ansatz in the inviscid limit, focusing on physical scenarios of wedge flows as a starting point which feature family of well-known self-similar profiles, the Falkner-Skan profiles. For the adverse cases, the investigator first develops fully the boundary layer equations through separation and reversal. The investigator then studies how these phenomena manifest in the full NS equations. Within hydrodynamic stability, there is a celebrated literature recently in which the subtle stability mechanisms of mixing, through inviscid damping and enhanced dissipation, are propagated at the nonlinear level. These mechanisms have been developed through delicate frequency side techniques on domains without boundaries. There has been a major gap to develop these mechanisms on domains with boundaries, the prototype being the channel, where the physical inhomogeniety of the vertical boundary obstructs the known techniques. The principal investigator develops new techniques that help fill this gap, in the canonical scenario of the Couette flow with Dirichlet boundary conditions. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.