This Spring 2015 semester, I teach a graduate topics course on “boundary layers in fluid dynamics“ at Penn State. The purpose of this topics course is to introduce a spectral analysis approach to analyze boundary layers and investigate the inviscid limit problem of Navier-Stokes equations. The problem of small viscosity limit or high Reynolds number (mathematically equivalent; see next lectures) has a very long story. Indeed, it is one of the most classical subjects in fluid dynamics. It interests most prominent physicists such as Lord Rayleigh, W. Orr, A. Sommerfeld, Heisenberg, W. Tollmien, H. Schlichting, among many others. It was already noted by Reynolds himself in his seminal experiment (1883) that the Reynolds number governs the transition from laminar to turbulent flows. The studies became active around 1930, motivated by the study of the boundary layer around wings. In airplanes design, it is crucial to study the boundary layer around the wing, and more precisely the transition between the laminar and turbulent regimes, and even more crucial to predict the point where boundary layer splits from the boundary. A large number of papers has been devoted to the estimation of the critical Rayleigh number of classical shear flows (Blasius profile, exponential suction/blowing profile, etc…). It was Heisenberg in 1924 who first estimated the critical Reynolds number of parallel shear flows. C. C Lin and then Tollmien around 1940s completed the picture with lower and upper stability branches, respectively for parallel flows and boundary layers. Most of the physical literature, together with many mathematical insights, on the subject is well documented by Drazin and Reid in their famous book on hydrodynamics instability.

Many substantial mathematical works follow to justify the formal asymptotic expansions used by the physicists, notably the work of Wasow in the 50s; see also his book on linear turning point theory. Despite many efforts, the linear stability theory has been mathematically incomplete. Recently, together with E. Grenier and Y. Guo, we provide a complete proof of the linear stability theory discovered by Heisenberg, C. C. Lin, and Tollmien; see our preprint on arxiv: arXiv:1402.1395. My ultimate plan is to present the spectral approach that we have developed to study boundary layers. Tentatively, I plan to cover

1. Derivation of fluid dynamics equations: Euler and Navier-Stokes equations.

2. The inviscid limit problem and an introduction to Prandtl boundary layers.

3. Singular perturbations: basic ODE theory.

4. Classical stability theory of shear flows: Orr-Sommerfeld equations.

5. Semigroups of linear operators, with applications to the linearized Navier-Stokes equations near a boundary layer.

6. Nonlinear stability theory: Arnold’s stability theorem, Guo-Strauss’ linear to nonlinear instability, Grenier’s nonlinear iterative scheme.

7. Time permitting, possible applications to stratified fluids, compressible flows, and thermal boundary layers.

As for references, I will mostly use the following sources:

• A. Majda and A. Bertozzi, Vorticity and Incompressible Flow, Cambridge University Press, 2002
• O.A. Oleinik and V.N. Samokhin, Mathematical Models in Boundary Layer Theory, Applied Mathematics and Mathematical Computation. Chapman & Hall/CRC, Boca Raton, FL, 1999.
• {Drazin, P. G.; Reid, W. H.} Hydrodynamic stability. Cambridge Monographs on Mechanics and Applied Mathematics. Cambridge University, Cambridge–New York, 1981.
• {E. Grenier,}
  On the nonlinear instability of {E}uler and {P}randtl equations.
  Comm. Pure Appl. Math. 53, 9 (2000), 1067–1091.
• {E. Grenier, Y. Guo and T. Nguyen}, {Spectral instability of characteristic boundary layer flows}, preprint 2014.
• G. Métivier, Small viscosity and boundary layer methods, Birkhäuser Boston, Inc., Boston, MA, 2004.