Hydrodynamic stability and bifurcation

It is shown that the difficulties involved in arriving at a correct mathematical interpretation of the mechanisms involved in the observed instability, bifurcation and transition of shearing flows are enormous. In addition to the difficulty of the linear stability problem, the proof of even so important a point as the linear stability of laminar flow in pipes at all finite values of the Reynolds number has yet to be established in a mathematically secure way. The nonlinear problem is more difficult than the linear, and nearly all analytical results are restricted to small amplitudes. This restriction is especially serious in the case of problems where the condition may be one of instability for disturbances of a certain magnitude, and stable for smaller disturbances. Numerical methods have until recently been only slightly more successful than ordinary analytical ones.