Squire theorem

It is worth noticing that Tlapa and Bernstein [71] have proven that the Squire theorem holds true for the PoiseuiUe flow of m upper-convected Maxwell fluid. It means that any instability, which may be present for three dimensional disturbances, is also present for two dimensional ones at a lower value of the Reynolds number. This property is not true, in general, for non-Newtonian fluids [72]. 

Fig. 5. Illustration of the convolution theorem applied to a crystal structure and its diffraction pattern. (A) is a lattice and (B) is the motif or repeating unit on the lattice. The full crystal (C) is a convolution of (A) and (B). The diffraction pattern (F) of the crystal (C) is the product of the diffraction patterns (Fourier transforms) (D) and (E) from (A) and (A), respectively. For details, see text. (Based on Squire, 1981.)...

In essence, Eqns. (2.3.20) and (2.3.23) are identical - expressed for different parameters, where the parameters are related via Eqn. (2.3.22). The mean flow U is real and unchanged and if a and (3 are real, then a three-dimensional stability problem at a Reynolds number Re has been reduced to a two-dimensional problem at the lower Reynolds number Re. This is known as the Squire s theorem, and formally stated as ... 

We must, however, note that the utility of the Squire s theorem is lost if the mean flow is three dimensional or if a and j3 are complex, as in spatial stability problems. 

The last section in this chapter is a brief introduction to stability of parallel shear flows. We consider three basic issues (i) Rayleigh s equation for inviscid flows, (ii) Rayleigh s necessary condition on an inflection point for inviscid instability, and (iii) a derivation of the Orr-Sommerfeld equation and Squire s theorem. 

The Equations (12-321) are an obvious generalization of the inviscid, linearized disturbance equations (12-307), and it is therefore not surprising that Squire s theorem turns out to be applicable. To see this, we apply Squire s transformation (12-310), plus the one additional condition... 

These equations are the same form as (12-321) with ay = v = 0 and thus define a mathematically equivalent 2D problem. According to Squire s theorem, to determine the minimum critcal Reynolds number for stability, it is sufficient to consider only 2D disturbances. This follows immediately from the fact a > ax so that Re < Re. 

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