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Fluid MechanicsA Geometrical Point of View$
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S. G. Rajeev

Print publication date: 2018

Print ISBN-13: 9780198805021

Published to Oxford Scholarship Online: October 2018

DOI: 10.1093/oso/9780198805021.001.0001

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(p.91) 8 Instabilities
Fluid Mechanics

S. G. Rajeev

Oxford University Press

The most well-known of the many instabilities of a fluid is the Rayleigh–Taylor instability. A denser fluid sitting on top of a lighter fluid is in unstable equilibrium, much like a pendulum standing on its head. Kapitza showed that rapidly oscillating the point of support of a pendulum can counteract this instability. The Rayleigh–Taylor instability can also be inhibited by shaking the two fluid layers rapidly. The Orr–Sommerfeld equations are a linear model of instabilities of a steady solution of Navier-Stokes. The Orr–Sommerfeld operator is not normal (does not commute with its adjoint). This means that there are transients (solutions that grow large before dying out) even if the linear equations predict stability. A simple nonlinear model with transients due to Trefethen et al. is studied to gain intuition into fluid instabilities.

Keywords:   Instability, Rayleigh–Taylor instability, Kapitza pendulum, Orr–Sommerfeld equations, transients, normal operator

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