Slip-enhanced drop formation in a liquid falling down a vertical fibre. (2nd May 2017) - Halpern, David; Wei, Hsien-Hung
Abstract : For a liquid film falling down along a vertical fibre, classical theory (Kalliadasis & Chang J. Fluid Mech., vol. 261, 1994, pp. 135–168; Yu & Hinch J. Fluid Mech., vol. 737, 2013, pp. 232–248) showed that drop formation can occur due to capillary instability when the Bond number $G=\unicode[STIX]{x1D70C}ga^{3}/\unicode[STIX]{x1D6FE}h_{0}$ is below the critical value $G_{c}\approx 0.60$, where $\unicode[STIX]{x1D70C}$ is the fluid density, $g$ is the gravitational acceleration, $a$ is the fibre radius, $\unicode[STIX]{x1D6FE}$ is the surface tension and $h_{0}$ is the unperturbed film thickness. However, the experiment by Quéré ( Europhys. Lett., vol. 13 (8), 1990, pp. 721–726) found $G_{c}\approx 0.71$, which is slightly greater than the above theoretical value. Here we offer a plausible way to resolve this discrepancy by including additional wall slip whose amount can be measured by the slip parameter $\unicode[STIX]{x1D6EC}=3\unicode[STIX]{x1D706}/h_{0}$, where $\unicode[STIX]{x1D706}$ is the slip length. Using lubrication theory, we find that wall slip promotes capillary instability and, hence, enhances drop formation. In particular, when slip effects are strong ( $\unicode[STIX]{x1D6EC}\gg 1$ ), the transition films and the drop height scale as $(c/\unicode[STIX]{x1D6EC})^{-1/3}$ and $(c/\unicode[STIX]{x1D6EC})^{2/3}$, respectively, distinct from those found by Yu & Hinch for the no-slip case where $c$ is the travelling speed. In addition, for $\unicode[STIX]{x1D6EC}>1$, $G_{c}$ is found to increase with $\unicode[STIX]{x1D6EC}$ according to $G_{c}\approx 0.7\unicode[STIX]{x1D6EC}^{1/3}$, offering a possible explanation why the $G_{c}$ found by Quéré is slightly greater than that predicted by the no-slip model. Using the above expression, the estimated slip length in Quéré's experiment is found to be of the order of several micrometres, consistent with the typical slip length range 1– $10~\unicode[STIX]{x03BC}\text{m}$ for polymeric liquids such as silicone oil used in his experiment. The existence of wall slip in Quéré's experiment is further supported by the observation that the film thinning kinetics exhibits the no-slip result $h\propto t^{-1/2}$ for early times and changes to the strong slip result $h\propto t^{-1}$, where $h$ is the film thickness. We also show that when the film is ultrathin, although capillary instability can become further amplified by strong slip effects, the instability can be arrested by the equally intensified gravity draining in the weakly nonlinear regime whose dynamics is governed by the Kuramoto–Sivashinsky equation.