Fluid mechanics Hydrodynamic stability Theorem

Squire's theorem

Builds on Orr–Sommerfeld equation Reynolds number Navier–Stokes equations Hydrodynamic stability

Statement

Let $U(y)$ , $y_1 \le y \le y_2$ , be a parallel shear flow of a viscous incompressible fluid between rigid walls. Consider three-dimensional normal mode disturbances proportional to $e^{i(\alpha x + \beta z - \alpha c t)}$ with real wavenumbers $\alpha > 0$ , $\beta$ and complex wave speed $c$ , at Reynolds number $\mathrm{Re}$ . Suppose such a disturbance is an unstable mode ( $\operatorname{Im} c > 0$ ) at $(\alpha, \beta, \mathrm{Re})$ . Then there is an unstable two-dimensional mode ( $\beta = 0$ ) with wavenumber $\tilde\alpha = \sqrt{\alpha^2 + \beta^2}$ at the Reynolds number $\widetilde{\mathrm{Re}} = \frac{\alpha}{\sqrt{\alpha^2 + \beta^2}} \, \mathrm{Re} \le \mathrm{Re}.$ Consequently, the critical Reynolds number, the least $\mathrm{Re}$ at which some mode is unstable, is attained by two-dimensional disturbances.

Remarks

The three-dimensional linearised Navier–Stokes equations reduce, for the wall-normal velocity amplitude, to $(U - c)(\varphi'' - k^2 \varphi) - U'' \varphi = \frac{1}{i \alpha \mathrm{Re}} \left( \varphi'''' - 2 k^2 \varphi'' + k^4 \varphi \right), \qquad k^2 = \alpha^2 + \beta^2.$ This is the two-dimensional Orr–Sommerfeld equation with wavenumber $k$ and Reynolds number $\widetilde{\mathrm{Re}}$ defined by $k \widetilde{\mathrm{Re}} = \alpha \mathrm{Re}$ ; this substitution is Squire's transformation. The eigenvalue $c$ is the same, so the two-dimensional mode is unstable exactly when the three-dimensional one is.

For plane Poiseuille flow, the theorem means that the critical value $\mathrm{Re} \approx 5772$ found from two-dimensional waves is the true linear threshold. The theorem concerns only the onset of linear instability. It does not say that two-dimensional modes grow fastest at a given Reynolds number, and it says nothing about transient growth or transition, where three-dimensional disturbances are important.

Going deeper

For a student

Squire's theorem justifies the restriction to two-dimensional disturbances in the classical theory of hydrodynamic stability of parallel flows. The proof is a change of variables, so it extends to other settings where the equations keep the same structure. It fails, for example, when the base flow has a spanwise component or when extra effects such as rotation enter. In practice transition in channels often occurs well below the critical Reynolds number and involves oblique and streamwise structures. This is a nonlinear and non-normal effect outside the scope of the theorem.

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Tag	`ES63E91F6T`
URI	`m2://ES63E91F6T`
Kind	statement, theorem
Writing	W0 stub
MSC	76E05
Status	This copy of the site has no build data. Run `m2 status ES63E91F6T`.
About	Hydrodynamic stability
Uses	Orr–Sommerfeld equation, Reynolds number, Navier–Stokes equations, Hydrodynamic stability
Same as	Wikidata Q25101661 Wikipedia Squire's theorem
Changed	3 November 2026