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- Minimum dissipation (1)
- Transition (1)
- Turbulence (1)

The “1. Stokes’ problem”, the “suddenly accelerated flat wall”, is the oldest application of the Navier-Stokes equations. Stokes’ solution of the “problem” does not comply with the mathematical theorem of Cauchy and Kowalewskaya on the “Uniqueness and Existence” of solutions of partial differential equations and violates the physical theorem of minimum entropy production/dissipation of the Thermodynamics of Irreversible Processes. The result includes very high local shear stresses and dissipation rates. That is of special interest for the theory of turbulent and mixed turbulent/laminar flow. A textbook solution of the “1. Stokes Problem” is the Couette flow, which has a constant sheer stress along a linear profile. A consequence is that the Navier-Stokes equations do not describe any S-shaped part of a turbulent profile found in any turbulent Couette experiment. The paper surveys arguments referring to that statement, concerning the history of >150 years. Contrary to this there is always a Navier-Stokes solution near the wall, observed by a linear part of the Couette profile. There a turbulent description (e.g. by the logarithmic law-of-the-wall) fails completely. That is explained by the minimum dissipation requirement together with the Couette feature τ = const. The local co-existence of a turbulent zone and a laminar zone near the wall is stable and observed also at high Reynolds-Numbers.

Analyzing thermodynamic non-equilibrium processes, like the laminar and turbulent fluid flow, the dissipation is a key parameter with a characteristic minimum condition. That is applied to characterize laminar and turbulent behaviour of the Couette flow, including its transition in both directions. The Couette flow is chosen as the only flow form with constant shear stress over the flow profile, being laminar, turbulent or both. The local dissipation defines quantitative and stable criteria for the transition and the existence of turbulence. There are basic results: The Navier Stokes equations cannot describe the experimental flow profiles of the turbulent Couette flow. But they are used to quantify the dissipation of turbulent fluctuation. The dissipation minimum requires turbulent structures reaching maximum macroscopic dimensions, describing turbulence as a “non-local” phenomenon. At the transition the Couette flow profiles and the shear stress change by a factor ≅ 5 due to a change of the “apparent” turbulent viscosity by a calculated factor ≅ 27. The resulting difference of the laminar and the turbulent profiles results in two different Reynolds numbers and different loci of transition, which are identified by calculation.

The minimum dissipation requirement of the thermodynamics of irreversible processes is applied to characterize the existence of laminar and non-laminar, and the co-existence of laminar and turbulent flow zones. Local limitations of the different zones and three different forms of transition are defined. For the Couette flow a non-local “corpuscular” flow mechanism explains the logarithmic law-of-the-wall, maximum turbulent dimensions and a value x=0,415 for the v. Kármán constant. Limitations of the logarithmic law near the wall and in the centre of the experiment are interpreted.