Turning Lord Kelvin's 1875 argument into a general approach to determine stability of fluid flow
Over a century ago, Lord Kelvin proposed an argument for stability of flows with negligible dissipation. The practical application of Kelvin’s ideas has been the subject of extensive debate among researchers from Caltech, Princeton, and Cambridge University, and elsewhere. As a result, it is presently believed that Kelvin’s argument cannot be used to determine stability, for most flows of practical interest. In our Letter, we overturn what has been, so far, the accepted wisdom on the use of Kelvin’s argument.
Flows with negligible dissipation are important in a wide variety of fields; examples include vortices and waves in the atmosphere and oceans, plasmas in nuclear fusion reactors, liquid helium at temperatures near absolute zero, and the dynamics of quantum condensates. Determining whether such a fluid flow is stable usually requires laborious calculations, if standard techniques are used. On the other hand, the approach that we introduce can be used with little effort, since no additional stability calculations are needed, beyond those required for finding the steady flows. Instead, we rely on the construction of a special bifurcation diagram involving the velocity and impulse of the equilibrium flows. In addition, we discover new families of solutions, originating at otherwise hidden bifurcations, by effectively promoting symmetry-breaking behavior in the fluid.
We have applied the resulting approach to a wide range of fluid flows. For all cases considered, we find precise agreement with more involved stability approaches. In addition, for each flow examined, we discover new, symmetry-breaking, fluid flows.
