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 $\operatorname{Gr}$ , the critical $\operatorname{Gr}$ for energy stability is 139.5 in two dimensions, and 51.73 in three dimensions. We derive upper bounds on the friction (a.k.a. dissipation) coefficient $C_{f} = \tau/\overline{u}^{2}$ , where τ is the applied shear stress and $\overline{u}$ is the mean velocity of the fluid at the surface, for flows at higher $\operatorname{Gr}$ including developed turbulence: C _f ≤1/32 in two dimensions and C _f ≤1/8 in three dimensions. This analysis rigorously justifies previously computed numerical estimates.