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The stability of a rising droplet: an inertialess non-modal growth mechanism

Published online by Cambridge University Press:  24 November 2015

Giacomo Gallino ,
Lailai Zhu  and
François Gallaire
Giacomo Gallino
Affiliation:
Laboratory of Fluid Mechanics and Instabilities, École Polytechnique Fédérale de Lausanne, Lausanne, CH-1015, Switzerland
Lailai Zhu
Affiliation:
Laboratory of Fluid Mechanics and Instabilities, École Polytechnique Fédérale de Lausanne, Lausanne, CH-1015, Switzerland
François Gallaire*
Affiliation:
Laboratory of Fluid Mechanics and Instabilities, École Polytechnique Fédérale de Lausanne, Lausanne, CH-1015, Switzerland
*
†Email address for correspondence: [email protected]
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Abstract

Prior modal stability analysis (Kojima et al., Phys. Fluids, vol. 27, 1984, pp. 19–32) predicted that a rising or sedimenting droplet in a viscous fluid is stable in the presence of surface tension no matter how small, in contrast to experimental and numerical results. By performing a non-modal stability analysis, we demonstrate the potential for transient growth of the interfacial energy of a rising droplet in the limit of inertialess Stokes equations. The predicted critical capillary numbers for transient growth agree well with those for unstable shape evolution of droplets found in the direct numerical simulations of Koh & Leal (Phys. Fluids, vol. 1, 1989, pp. 1309–1313). Boundary integral simulations are used to delineate the critical amplitude of the most destabilizing perturbations. The critical amplitude is negatively correlated with the linear optimal energy growth, implying that the transient growth is responsible for reducing the necessary perturbation amplitude required to escape the basin of attraction of the spherical solution.

JFM classification

Low-Reynolds-number flows: Boundary integral methods Drops and Bubbles: Drops Instability: Nonlinear instability
Type
Rapids
Information
Journal of Fluid Mechanics , Volume 786 , 10 January 2016 , R2
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Copyright
© 2015 Cambridge University Press 

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