Physics: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 [math]\displaystyle{ \mathbf{U} = (U(z), 0, 0) }[/math]), the perturbations which are least stable are two-dimensional, i.e. of the form [math]\displaystyle{ \mathbf{u}' = (u'(x,z,t),0,w'(x,z,t)) }[/math], 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
- ↑ Drazin, P. G., & Reid, W. H. (2004). Hydrodynamic stability. Cambridge university press.
- ↑ 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.
 |  |
