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Weakly nonlinear instability of a Newtonian liquid jet

Published online by Cambridge University Press:  03 October 2018

Marie-Charlotte Renoult ,
Günter Brenn Open the ORCID record for Günter Brenn [Opens in a new window] ,
Gregor Plohl  and
Innocent Mutabazi Open the ORCID record for Innocent Mutabazi [Opens in a new window]
Marie-Charlotte Renoult
Affiliation:
Normandie Université, Université du Havre, CNRS – LOMC, 76058 Le Havre, France
Günter Brenn
Affiliation:
Institute of Fluid Mechanics and Heat Transfer (ISW), Graz University of Technology, Inffeldgasse 25/F, 8010 Graz, Austria
Gregor Plohl
Affiliation:
Institute of Fluid Mechanics and Heat Transfer (ISW), Graz University of Technology, Inffeldgasse 25/F, 8010 Graz, Austria
Innocent Mutabazi
Affiliation:
Normandie Université, Université du Havre, CNRS – LOMC, 76058 Le Havre, France
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Abstract

A weakly nonlinear stability analysis of an axisymmetric Newtonian liquid jet is presented. The calculation is based on a small-amplitude perturbation method and performed to second order in the perturbation parameter. The obtained solution includes terms derived from a polynomial approximation of a viscous contribution containing products of Bessel functions with different arguments. The use of such an approximation is not needed in the inviscid case and the planar case, since the equations of those problems can be solved in an exact form. The developed model depends on three dimensionless parameters: the initial perturbation amplitude, the perturbation wavenumber and the liquid Ohnesorge number, the latter being the dimensionless liquid viscosity. The influence of the approximate terms was shown to be relatively small for a large range of Ohnesorge numbers so that they can be ignored. This simplification provides a jet model as simple to use as the previous ones, but taking into account the liquid viscosity and the cylindrical geometry. The jet model is used to reveal the effect of both the wavenumber and the Ohnesorge number on the formation of satellite drops, which is known as a nonlinear effect. Results are found in good agreement with direct numerical simulations and forced liquid jet experiments for wavenumbers lower than a threshold value. Satellite drop formation is retarded with increasing Ohnesorge number and wavenumber, as expected by the damping and size effects of viscosity. The threshold number corresponds to the maximum wavenumber for which satellite drop formation is predicted before jet breakup, and for which volume conservation is satisfied within a certain amount. The volume conservation criterion is imposed to ensure that the conclusions inferred by our model are safe.

JFM classification

Drops and Bubbles: Drops Wakes/Jets: Jets Instability: Nonlinear instability
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 856 , 10 December 2018 , pp. 169 - 201
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Copyright
© 2018 Cambridge University Press 

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Footnotes

†

Present address: Normandie Université, Université et INSA de Rouen, CNRS – CORIA, 76801 Saint-Etienne du Rouvray, France. Email address for correspondence: renoultm@coria.fr

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