For temporal normal-mode stability of a viscous parallel shear flow, any unstable three-dimensional mode maps by the Squire transformation to an unstable two-dimensional mode at a no larger Reynolds number. Hence the first temporal modal instability occurs among two-dimensional disturbances. The theorem does not exclude three-dimensional transient growth or forced responses.
For positive streamwise wavenumber , replace a three-dimensional Orr-Sommerfeld equation by a two-dimensional one with the displayed parameters. Both the transverse operator and the coefficient remain unchanged, preserving the eigenfunction and phase-speed eigenvalue. This proves the modal comparison in Squire's theorem.
Articles by others on the same topic
Squire's theorem is a fundamental result in the field of fluid dynamics, particularly in the study of boundary layer theory. Named after the mathematician and engineer, it establishes a relationship between the stability of flow in a two-dimensional incompressible boundary layer and the stability of flow in the corresponding inviscid (non-viscous) flow.