Bounds on Surface Stress-Driven Shear Flow

The background method is adapted to derive rigorous limits on surface speeds and bulk energy dissipation for shear stress driven flow in two and three dimensional channels. By-products of the analysis are nonlinear energy stability results for plane Couette flow with a shear stress boundary condition: when the applied stress is gauged by a dimensionless Grashoff number Gr, the critical Gr for energy stability is 140 in two dimensions, and 51.7 in three dimensions. We derive upper bounds on the friction (a.k.a. dissipation) coefficient Cf = τ/u , where τ is the applied shear stress and u is the mean velocity of the fluid at the surface, for flows at higher Gr including developed turbulence: Cf ≤ 1/32 in two dimensions and Cf ≤ 1/8 in three dimensions. This analysis rigorously justifies previously computed numerical estimates.