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Rayleigh–Taylor instability in a finite cylinder: linear stability analysis and long-time fingering solutions

Published online by Cambridge University Press:  09 October 2013

H. Sweeney ,
R. R. Kerswell  and
T. Mullin
H. Sweeney
Affiliation:
School of Mathematics, University of Bristol, Bristol BS8 1TW, UK
R. R. Kerswell*
Affiliation:
School of Mathematics, University of Bristol, Bristol BS8 1TW, UK
T. Mullin
Affiliation:
School of Physics and Astronomy, University of Manchester. Manchester M13 9PL, UK
*
†Email address for correspondence: [email protected]
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Abstract

We consider the Rayleigh–Taylor instability problem of two initially stationary immiscible viscous fluids positioned with the denser above the less dense in a finite circular cylinder, such that their starting fluid–fluid interface is the horizontal midplane of the cylinder. The ensuing linear instability problem has a five-dimensional parameter space – defined by the density ratio, the viscosity ratio, the cylinder aspect ratio, the surface tension between the fluids and the ratio of viscous to gravitational time scales – of which we explore only part, motivated by recent experiments where viscous fluids exchange in vertical tubes (Beckett et al., J. Fluid Mech., 2011, vol. 682, pp. 652–670). We find that for these experiments, the instability is invariably ‘side-by-side’ (of azimuthal wavenumber 1 type) but we also uncover parameter regions where the preferred instability is axisymmetric. The fact that both ‘core-annular’ (axisymmetric) and ‘side-by-side’ (asymmetric) long-time flows are seen experimentally highlights the fact that the initial Rayleigh–Taylor instability of the interface does not determine the long-time flow configuration in these situations. Finally, long-time flow solutions are presented on the basis that they will be slowly varying fingering solutions.

JFM classification

Convection: Buoyancy-driven instability Interfacial Flows (free surface): Fingering instability Low-Reynolds-number flows: Low-Reynolds-number flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 734 , 10 November 2013 , pp. 338 - 362
Copyright
©2013 Cambridge University Press 

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Sweeney et al. supplemenatry material

This video shows the motion induced by overturning a 15 cm long, 6.3 cm diameter glass cylinder which is filled 50:50 with golden syrup (ρ1 = 1.1 g/cm3, ν1 = 1200cm2/s) and silicone fluid (ρ2 = 0.98g/cm3, ν2 = 600cm2/s). The video starts just as the motion begins which is ≈ 2 secs. after overturning. Note the creation of a pair of cusps in the middle of the cylinder.

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