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Asymptotic theory of resonant flow in a spheroidal cavity driven by latitudinal libration

Published online by Cambridge University Press:  06 January 2012

Keke Zhang ,
Kit H. Chan  and
Xinhao Liao
Keke Zhang*
Affiliation:
Department of Mathematical Sciences, University of Exeter, Exeter EX4 4QF, UK Institute of Mathematical Sciences, Chinese University of Hong Kong, Shatin, Hong Kong
Kit H. Chan
Affiliation:
Department of Mathematics, University of Hong Kong, Pokfulam, Hong Kong
Xinhao Liao
Affiliation:
Shanghai Astronomical Observatory, Chinese Academy of Sciences, Shanghai 200030, China
*
†Email address for correspondence: kzhang@ex.ac.uk
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Abstract

We consider a homogeneous fluid of viscosity confined within an oblate spheroidal cavity, , with eccentricity . The spheroidal container rotates rapidly with an angular velocity , which is fixed in an inertial frame and defines a small Ekman number , and undergoes weak latitudinal libration with frequency and amplitude , where is the Poincaré number quantifying the strength of Poincaré force resulting from latitudinal libration. We investigate, via both asymptotic and numerical analysis, fluid motion in the spheroidal cavity driven by latitudinal libration. When , an asymptotic solution for and in oblate spheroidal coordinates satisfying the no-slip boundary condition is derived for a spheroidal cavity of arbitrary eccentricity without making any prior assumptions about the spatial–temporal structure of the librating flow. In this case, the librationally driven flow is non-axisymmetric with amplitude , and the role of the viscous boundary layer is primarily passive such that the flow satisfies the no-slip boundary condition. When , the librationally driven flow is also non-axisymmetric but latitudinal libration resonates with a spheroidal inertial mode that is in the form of an azimuthally travelling wave in the retrograde direction. The amplitude of the flow becomes at and the role of the viscous boundary layer becomes active in determining the key property of the flow. An asymptotic solution for describing the librationally resonant flow is also derived for an oblate spheroidal cavity of arbitrary eccentricity. Three-dimensional direct numerical simulation in an oblate spheroidal cavity is performed to demonstrate that, in both the non-resonant and resonant cases, a satisfactory agreement is achieved between the asymptotic solution and numerical simulation at .

JFM classification

Geophysical and Geological Flows: Geophysical and Geological Flows Geophysical and Geological Flows: Rotating flows

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Type
Papers
Information
Journal of Fluid Mechanics , Volume 692 , 10 February 2012 , pp. 420 - 445
Copyright
Copyright © Cambridge University Press 2012

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