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