Squire's theorem

In fluid dynamics, Squire's theorem states that of all the perturbations that may be applied to a shear flow (i.e. a velocity field of the form ${\displaystyle \mathbf {U} =(U(z),0,0)}$), the perturbations which are least stable are two-dimensional, i.e. of the form ${\displaystyle \mathbf {u} '=(u'(x,z,t),0,w'(x,z,t))}$, rather than the three-dimensional disturbances.^[1] This applies to incompressible flows which are governed by the Navier–Stokes equations. The theorem is named after Herbert Squire, who proved the theorem in 1933.^[2]

Squire's theorem allows many simplifications to be made in stability theory. If we want to decide whether a flow is unstable or not, it suffices to look at two-dimensional perturbations. These are governed by the Orr–Sommerfeld equation for viscous flow, and by Rayleigh's equation for inviscid flow.

An extension to the classical Squire's theorem was proposed by Turkac.^[3] Unlike the classical theorem, which is limited to the temporal stability framework, this extension addresses also the spatial development of modes. Turkac showed that the classical Squire's theorem does not apply to spatial instability modes.

In the framework of the extension, Turkac demonstrated that the critical Reynolds number for 3D instability modes can be lower than that of 2D modes. This finding contrasts with the classical Squire's theorem, which predicts that two-dimensional perturbations dominate. Instead, in the spatial instability framework, 3D perturbations can occur at subcritical Reynolds numbers, implying they might precede 2D disturbances in certain flows.

This extended framework introduces a complex spanwise wave number (${\displaystyle \beta \in \mathbb {C} }$), allowing disturbances to propagate and grow not only in the streamwise direction but also in the spanwise direction. The extension sheds new light on hydrodynamic stability and broadens the applicability of Squire's theorem to previously unexplored scenarios in fluid dynamics.

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