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Natural break-up and satellite formation regimes of surfactant-laden liquid threads

Published online by Cambridge University Press:  26 November 2019

A. Martínez-Calvo Open the ORCID record for A. Martínez-Calvo [Opens in a new window] ,
J. Rivero-Rodríguez Open the ORCID record for J. Rivero-Rodríguez [Opens in a new window] ,
B. Scheid Open the ORCID record for B. Scheid [Opens in a new window]  and
A. Sevilla Open the ORCID record for A. Sevilla [Opens in a new window]
A. Martínez-Calvo*
Affiliation:
Grupo de Mecánica de Fluidos, Departamento de Ingeniería Térmica y de Fluidos,Universidad Carlos III de Madrid, Av. Universidad 30, 28911 Leganés (Madrid), Spain
J. Rivero-Rodríguez
Affiliation:
TIPs, Université Libre de Bruxelles, CP 165/67, Avenue F. D. Roosevelt 50, 1050Bruxelles, Belgium
B. Scheid
Affiliation:
TIPs, Université Libre de Bruxelles, CP 165/67, Avenue F. D. Roosevelt 50, 1050Bruxelles, Belgium
A. Sevilla
Affiliation:
Grupo de Mecánica de Fluidos, Departamento de Ingeniería Térmica y de Fluidos,Universidad Carlos III de Madrid, Av. Universidad 30, 28911 Leganés (Madrid), Spain
*
†Email address for correspondence: amcalvo@ing.uc3m.es
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Abstract

We report a numerical analysis of the unforced break-up of free cylindrical threads of viscous Newtonian liquid whose interface is coated with insoluble surfactants, focusing on the formation of satellite droplets. The initial conditions are harmonic disturbances of the cylindrical shape with a small amplitude $\unicode[STIX]{x1D716}$, and whose wavelength is the most unstable one deduced from linear stability theory. We demonstrate that, in the limit $\unicode[STIX]{x1D716}\rightarrow 0$, the problem depends on two dimensionless parameters, namely the Laplace number, $La=\unicode[STIX]{x1D70C}\unicode[STIX]{x1D70E}_{0}\bar{R}/\unicode[STIX]{x1D707}^{2}$, and the elasticity parameter, $\unicode[STIX]{x1D6FD}=E/\unicode[STIX]{x1D70E}_{0}$, where $\unicode[STIX]{x1D70C}$, $\unicode[STIX]{x1D707}$ and $\unicode[STIX]{x1D70E}_{0}$ are the liquid density, viscosity and initial surface tension, respectively, $E$ is the Gibbs elasticity and $\bar{R}$ is the unperturbed thread radius. A parametric study is presented to quantify the influence of $La$ and $\unicode[STIX]{x1D6FD}$ on two key quantities: the satellite droplet volume and the mass of surfactant trapped at the satellite’s surface just prior to pinch-off, $V_{sat}$ and $\unicode[STIX]{x1D6F4}_{sat}$, respectively. We identify a weak-elasticity regime, $\unicode[STIX]{x1D6FD}\lesssim 0.05$, in which the satellite volume and the associated mass of surfactant obey the scaling law $V_{sat}=\unicode[STIX]{x1D6F4}_{sat}=0.0042La^{1.64}$ for $La\lesssim 2$. For $La\gtrsim 10$, $V_{sat}$ and $\unicode[STIX]{x1D6F4}_{sat}$ reach a plateau of about $3\,\%$ and $2.9\,\%$, respectively, $V_{sat}$ being in close agreement with previous experiments of low-viscosity threads with clean interfaces. For $La<7.5$, we reveal the existence of a discontinuous transition in $V_{sat}$ and $\unicode[STIX]{x1D6F4}_{sat}$ at a critical elasticity, $\unicode[STIX]{x1D6FD}_{c}(La)$, with $\unicode[STIX]{x1D6FD}_{c}\rightarrow 0.98$ for $La\lesssim 0.2$, such that $V_{sat}$ and $\unicode[STIX]{x1D6F4}_{sat}$ abruptly increase at $\unicode[STIX]{x1D6FD}=\unicode[STIX]{x1D6FD}_{c}$ for increasing $\unicode[STIX]{x1D6FD}$. The jumps experienced by both quantities reach a plateau when $La\lesssim 0.2$, while they decrease monotonically as $La$ increases up to $La=7.5$, where both become zero.

JFM classification

Interfacial Flows (free surface): Capillary flows Drops and Bubbles: Breakup/coalescence
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 883 , 25 January 2020 , A35
Copyright
© 2019 Cambridge University Press

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