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 ${\displaystyle \mathbf {u} '=(u'(x,z,t),0,w'(x,z,t))}$ 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.