The Two-Dimensional Stability of a Viscous Fluid between Rotating Cylinders

Abstract

The problem of a steady motion of a viscous incompressible fluid between two Ž . rotating coaxial cylinders Couette flow is considered. It is shown by operator theory that it is linearly stable with respect to two dimensional disturbances under all circumstances. The proof is based on a lemma which can be generalized to apply to other problems with a similar structure.

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“…To complete the proof we need the following two lemmas. The lemmas generalize and unify the approaches in ( [7], [8], [2]). Lemma 1.…”

“…Problem (iii) [2], is similar to problems (i) and (ii), the basic flow being circumferential rather than axial. Nevertheless, an equation like (2.1) also occurs.…”

“…In order to prove linear stability, use will be made of the von Neumann extension. We have developed an equivalent representation in earlier studies for problem (i) in [7], for problem (ii) in ( [7], [8]), and for problem (iii) in [2]. To complete the proof we need the following two lemmas.…”

“…A combination of functional analysis and asymptotic methods were employed, but application of that method to pipe flow has not been successful. There is another problem, (iii) the stability of Couette flow to plane disturbances [2], which can also be solved by the method to be presented here. Synge [26] discussed this problem but did not directly compare it to (i) and (ii).…”

“…It was possible to further the analysis of the underlying operators. However, the proofs of stability advanced earlier ( [7], [8], [2]) were incomplete. The introduction of the square root operator unifies and cinches the proofs.…”