We review some rigorous bounds on energy dissipation rates and heat transport for incompressible fluid flows. These theoretical limits are valid for both laminar and turbulent flows. In some cases, experiments and theory suggest that turbulence tends to saturate the mathematical estimates. In other cases, most notably turbulent Rayleigh-Benard convection at finite Prandtl number, a considerable gap remaines between experimental and computational observations and the best available bounds.
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