Trying to explain why laminar (smooth) fluid flows suddenly become turbulent (chaotic, disordered) when a certain flow speed is reached is still an outstanding unsolved problem in fluid mechanics. There is an enormous literature on turbulence, yet a precise physical mechanism for this transition to turbulence has not yet emerged.
Mathematical accounts of turbulence, in textbooks and papers, almost all start from the premise that the Navier-Stokes equations are sufficient to describe the phenomenon. With these equations as the starting point, a range of explanations has then been proposed, from fairly simple averaging arguments right through to sophisticated accounts based on dynamical systems theory.
In this talk, I want to do a little "thought experiment", and ask the simple question: Do the Navier-Stokes equations really describe the transition to turbulence, and if not, what should replace them? After all, Navier-Stokes theory makes a crucial linearizing approximation, similar to Hooke's Law in solid mechanics. If this assumption is relaxed, it is possible to see behaviour very similar to turbulence. This only requires classical linear stability theory, which is simple in concept even if rather challenging in application. Does this provide a rational explanation for the transition to turbulence?