In
fluid dynamics
In physics and engineering, fluid dynamics is a subdiscipline of fluid mechanics that describes the flow of fluids— liquids and gases. It has several subdisciplines, including ''aerodynamics'' (the study of air and other gases in motion) an ...
, Squire's theorem states that of all the
perturbations that may be applied to a
shear flow The term shear flow is used in solid mechanics as well as in fluid dynamics. The expression ''shear flow'' is used to indicate:
* a shear stress over a distance in a thin-walled structure (in solid mechanics);Higdon, Ohlsen, Stiles and Weese (1960) ...
(i.e. a
velocity field
In continuum mechanics the flow velocity in fluid dynamics, also macroscopic velocity in statistical mechanics, or drift velocity in electromagnetism, is a vector field used to mathematically describe the motion of a continuum. The length of the f ...
of the form
), the perturbations which are least stable are two-dimensional, i.e. of the form
, rather than the three-dimensional disturbances. This applies to
incompressible flows which are governed by the
Navier–Stokes equations
In physics, the Navier–Stokes equations ( ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician Geo ...
. The theorem is named after
Herbert Squire, who proved the theorem in 1933.
[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.]
Squire's theorem allows many simplifications to be made in
stability theory
In mathematics, stability theory addresses the stability of solutions of differential equations and of trajectories of dynamical systems under small perturbations of initial conditions. The heat equation, for example, is a stable partial diffe ...
. 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 The Orr–Sommerfeld equation, in fluid dynamics, is an eigenvalue equation describing the linear two-dimensional modes of disturbance to a viscous parallel flow. The solution to the Navier–Stokes equations for a parallel, laminar flow can become ...
for
viscous
The viscosity of a fluid is a measure of its resistance to deformation at a given rate. For liquids, it corresponds to the informal concept of "thickness": for example, syrup has a higher viscosity than water.
Viscosity quantifies the inter ...
flow, and by
Rayleigh's equation for inviscid flow.
References
Fluid dynamics
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