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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: francois.gallaire@epfl.ch
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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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References

Baggett, J. S., Driscoll, T. A. & Trefethen, L. N. 1995 A mostly linear model of transition to turbulence. Phys. Fluids 7 (4), 833–838.Google Scholar
Balasubramanian, K. & Sujith, R. I. 2008 Thermoacoustic instability in a Rijke tube: non-normality and nonlinearity. Phys. Fluids 20 (4), 044103.Google Scholar
Batchelor, G. K. 2000 An Introduction to Fluid Dynamics. Cambridge University Press.CrossRefGoogle Scholar
Bertozzi, A. L. & Brenner, M. P. 1997 Linear stability and transient growth in driven contact lines. Phys. Fluids 9 (3), 530–539.Google Scholar
de Bruyn, J. R. 1992 Growth of fingers at a driven three-phase contact line. Phys. Rev. A 46 (8), R4500.Google Scholar
Davis, J. M. & Troian, S. M. 2003 On a generalized approach to the linear stability of spatially nonuniform thin film flows. Phys. Fluids 15 (5), 1344–1347.Google Scholar
Hadamard, J. 1911 Mouvement permanent lent d’une sphère liquide et visqueuse dans un liquide visqueux. C. R. Acad. Sci. Paris 152, 1735–1738.Google Scholar
Huppert, H. E. 1982 Flow and instability of a viscous current down a slope. Nature 300 (5891), 427–429.Google Scholar
Johnson, R. A. & Borhan, A. 2000 Stability of the shape of a surfactant-laden drop translating at low Reynolds number. Phys. Fluids 12 (4), 773–784.CrossRefGoogle Scholar
Jovanović, M. R. & Kumar, S. 2010 Transient growth without inertia. Phys. Fluids 22 (2), 023101.Google Scholar
Juniper, M. P. 2011 Triggering in the horizontal Rijke tube: non-normality, transient growth and bypass transition. J. Fluid Mech. 667, 272–308.Google Scholar
Koh, C. J. & Leal, L. G. 1989 The stability of drop shapes for translation at zero Reynolds number through a quiescent fluid. Phys. Fluids 1 (8), 1309–1313.CrossRefGoogle Scholar
Kojima, M., Hinch, E. J. & Acrivos, A. 1984 The formation and expansion of a toroidal drop moving in a viscous fluid. Phys. Fluids 27 (1), 19–32.CrossRefGoogle Scholar
Leal, L. G. 2007 Advanced Transport Phenomena: Fluid Mechanics and Convective Transport Processes. Cambridge University Press.CrossRefGoogle Scholar
Reddy, S. C. & Henningson, D. S. 1993 Energy growth in viscous channel flows. J. Fluid Mech. 252, 209–238.Google Scholar
Rybzynski, W. 1911 Über die fortschreitende Bewegung einer flüssingen Kugel in einem zähen Medium. Bull. Int. Acad. Sci. Cracovie 1A, 40–46.Google Scholar
Schmid, P. J. 2001 Tools for nonmodal stability analysis. In Notes from a Ladhyx Tutorial, p. 24.Google Scholar
Schmid, P. J. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129–162.Google Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and transition in shear flows. In Applied Mathematical Sciences, vol. 142. Springer.Google Scholar
Trefethen, L. N. & Embree, M. 2005 Spectra and Pseudospectra: the Behavior of Nonnormal Matrices and Operators. Princeton University Press.Google Scholar
Trefethen, L. N., Trefethen, A. E., Reddy, S. C. & Driscoll, T. A. 1993 Hydrodynamic stability without eigenvalues. Science 261 (5121), 578–584.Google Scholar
Wu, H., Haj-Hariri, H. & Borhan, A. 2012 Stability of the shape of a translating viscoelastic drop at low Reynolds number. Phys. Fluids 24 (11), 113101.Google Scholar