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.

References

1. Drazin, P. G., & Reid, W. H. (2004). Hydrodynamic stability. Cambridge university press.
2. Squire, H. B. (1933). On the stability for three-dimensional disturbances of viscous fluid flow between parallel walls. Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, 142(847), 621-628.