Absolute and Convective Instabilities of a Viscous Film Flowing Down a Vertical Fiber

The stability of a viscous film flowing down a vertical fiber under the action of gravity is analyzed both experimentally and theoretically. At large or small film thicknesses, the instability is convective, whereas an absolute instability mode is observed in an intermediate range of film thicknesses for fibers of small enough radius. The onset of the experimental irregular wavy regime corresponds precisely to the theoretical prediction of the threshold of the convective instability.

Figure 1

(a) Experimental setup. (b) and (c) Snapshots of the liquid film at the top of the fiber and at 10 cm from the entrance valve, respectively, showing a nonlinear regular wave train ($R=0.23\mathrm{mm}$, $q=0.0348\mathrm{g}/\mathrm{s}$ and $h_N=0.48\mathrm{mm}$).

Figure 2

Left: spatiotemporal diagrams of the film thickness along the fiber. $x$ is the distance from the top of the fiber. Right: power spectra. The locations at which the time series have been recorded are indicated by arrows. $R=0.32\mathrm{mm}$; (a) $q=0.024\mathrm{g}/\mathrm{s}$ ($h_N=0.42\mathrm{mm}$); (b) $q=0.077\mathrm{g}/\mathrm{s}$ ($h_N=0.58\mathrm{mm}$).

Figure 3

Healing length (left ordinate) and frequency (right ordinate) versus flow rate at $R=0.32\mathrm{mm}$. Solid and dashed lines correspond to the absolute frequency and the frequency of the maximum spatial growth rate, respectively. Both are obtained from the OS analysis.

Figure 4

AI and CI regions in the parameter plane ${\alpha }_N=h_N/R$ versus $R/l_c$. Thick solid lines refer to the OS solutions; label 1 refers to Rhodorsil silicon oil v50 and label 2 to water. Dashed and thin solid lines correspond to Eqs. (7, 3), respectively. Regular and irregular primary wave trains reported in the experiment are indicated by crosses and dots, respectively. The dependence of $h_N$ (and thus ${\alpha }_N$) with the flow rate $q_N$ is given in (5). The inset is a blowup of the diagram for small values of ${\alpha }_N$.