Global optimality of mode three for shear-free Lagrangian streaming

Determine whether azimuthal mode m=3 globally maximizes the peak Lagrangian streamfunction for a shear-free circular interface driven at fixed radial wall-velocity amplitude over all positive integer modes.

Background

For a shear-free interface, the reference-boundary streaming solution vanishes at m=1, while the Lagrangian correction multiplies the reference solution by (5m+4)/(m+2). The resulting computed peak Lagrangian streamfunction is larger at m=3 than at m=2 by 4.75%.

The authors report that a numerical sweep through m≤100 found no larger value, but they explicitly note that this finite sweep does not establish optimality over all positive integers. Thus, the unresolved problem is to prove or disprove that m=3 is the global maximizing mode under the stated amplitude constraint.