Abstract
Stability is an important consideration for fluid flow just as it is for any mechanical system. It is often said that in order to be observable, a flow must be stable since an unstable flow is merely a state of transition to some other flow, or possibly to turbulence. But if this were the whole story, hydrodynamic stability would be a dull pursuit in which we restrict our attention to only stable results. Instead, we turn our attention to the instabilities themselves, hoping that their form will help us understand the flows that support them.
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Notes
- 1.
An operator, or matrix, is non-normal when it does not commute with its adjoint.
- 2.
The Blasius functions, as the solutions are called, and their generalization to flow over wedges, known as Falkner–Skan solutions, are built-in functions in MATLAB, for example.
- 3.
Found in the work On floating bodies written by Archimedes of Syracuse circa 250 B.C.
- 4.
Allowing to decompose any sufficiently smooth vector field to the sum of a curl (∇×) free component plus a divergence (∇⋅ ) free component
- 5.
The reader is referred to the stability texts by Drazin, Drazin and Reid, or Charru listed in the Bibliographical Notes to examine in detail G.I. Taylor’s graphical comparison of theoretical and experimental results.
- 6.
Note that the condition that the vertical velocity is zero at the boundaries is equivalent to \(dU/dr = 0\) at \(r = r_{{1}},r_{{2}}\) since U = 0 at those points too. This is a consequence of the continuity equation.
- 7.
One of the simplest effects to consider is rotation. In Chandrasekhar’s book (reference 1) the effect of uniform rotation on Jeans instability is given.
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Regev, O., Umurhan, O.M., Yecko, P.A. (2016). Hydrodynamic Stability. In: Modern Fluid Dynamics for Physics and Astrophysics. Graduate Texts in Physics. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-3164-4_7
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