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

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 smal...

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Bibliographic Details
Published in:Journal of fluid mechanics 2020-01, Vol.883, Article A35
Main Authors: Martínez-Calvo, A., Rivero-Rodríguez, J., Scheid, B., Sevilla, A.
Format: Article
Language:English
Subjects:
Approximation
Aquatic reptiles
Droplets
Elasticity
Experiments
Initial conditions
Interfaces
Newtonian liquids
Numerical analysis
Parameters
Reynolds number
Satellites
Scaling
Scaling laws
Simulation
Strutt, John William (Lord Rayleigh) (1842-1919)
Surface tension
Surfactants
Viscosity
Wavelength
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Summary: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
ISSN:0022-1120
1469-7645
DOI:10.1017/jfm.2019.874