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 small amplitude $\epsilon$, and whose wavelength is the most unstable one deduced from linear stability theory. We demonstrate that, in the limit $\epsilon \to 0$, the problem depends on two dimensionless parameters, namely the Laplace number, $La=\rho \sigma_0 \bar{R}/\mu^2$, and the elasticity parameter, $\beta=E/\sigma_0$, where $\rho$, $\mu$ and $\sigma_0$ 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 $\beta$ 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 $\Sigma_{sat}$, respectively. We identify a weak-elasticity regime, $\beta \lesssim 0.05$, in which the satellite volume and the associated mass of surfactant obey the scaling law $V_{sat} = \Sigma_{sat} = 0.0042 La^{1.64}$ for $La \lesssim 2$. For $La \gtrsim 10$, $V_{sat}$ and $\Sigma_{sat}$ 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 at a critical elasticity $\beta_c (La)$, with $\beta_c \to 0.98$ for $La \lesssim 0.2$, such that $V_{sat}$ and $\Sigma_{sat}$ abruptly increase. 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.

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