Junior Analysis Seminar – Daniel Abraham (Bath)

In this talk we study steady viscous flow governed by the incompressible Navier–Stokes equations, on a two-dimensional domain with a lower inclined bed and a free upper surface. Waves driven by gravity down an incline are a fascinating physical phenomenon, and can be observed for instance when rain falls on window panes and street gutters. We find a local branch of travelling periodic ‘roll wave’ solutions close to shear flow, under natural assumptions on the related Orr—Sommerfeld equation. Using techniques from analytic global bifurcation theory, we extend this branch to a global curve of solutions. A key step of the proof is reformulating the problem an elliptic system (in the sense of Agmon–Douglis–Nirenberg). We verify the hypotheses on the Orr–Sommerfeld equation for two regimes: small wavenumber and low Reynolds number. This work is joint with Miles Wheeler.