Solitary waves on superconfined falling liquid films

Solitary traveling waves are prominent features covering the surface of a falling liquid film and are known to promote heat and mass transfer. We focus on the little studied case where they are subject to an extremely confined countercurrent gas flow, and we identify two secondary instabilities. At high gas velocities, a catastrophic instability develops, leading to flooding through wave reversal and liquid arrest. At lower gas velocities, an oscillatory instability occurs, producing a high-frequency periodic modulation of the wave height. Conjunction of this self-sustained oscillatory state and vortices forming in the liquid is shown to enhance mixing. We also show that the gas flow can cause extreme local film thinning, leading to almost dry patches where the liquid thickness is very small.

Figure 1

Solitary traveling waves subject to an increasingly strong extremely confined gas flow: $H^{★}=1\mathrm{mm}$; $\mathrm{Ka}=509.5$; ${\mathrm{Re}}_{\mathrm{l}}=15$; $f=f_0=0.027$. (a) Effect of the gas Reynolds number ${\mathrm{Re}}_{\mathrm{g}}$ on the maximal height $h_{\max }$ (solid) and minimal height $h_{\min }$ (dashed). Black (right of cross, ${\mathrm{Re}}_{\mathrm{g}}>-13.5$), stable solutions; blue (between cross and asterisk, $-78<{\mathrm{Re}}_{\mathrm{g}}<-13.5$), oscillatory instability; red (between asterisk and circle, $-168<{\mathrm{Re}}_{\mathrm{g}}<-78$), catastrophic instability. Limit point S coincides with linear stability cutoff. [(b)–(d)] Streamlines in the reference frame of the wave for the three solutions marked by arrows in graph (a). Thick lines represent film surface, blue streamlines represent the liquid (bottom), and red ones represent the gas (top): (b) ${\mathrm{Re}}_{\mathrm{g}}=-1$; (c) ${\mathrm{Re}}_{\mathrm{g}}=-16$; and (d) ${\mathrm{Re}}_{\mathrm{g}}=-80$.

Figure 2

Transient periodic computations started from traveling-wave solutions marked by arrows 1, 2, and 3 in Fig. 1. The total flow rate $q_{\mathrm{tot}}$ and the liquid volume are fixed. (a) Time evolution of maximal wave height $h_{\max }$ over several nominal wave periods 1/$f_0$. Black dashed: stable regime (arrow 1); blue solid: oscillatory instability regime (arrow 2); (b) space-time plot of film height contours $h(x,t)$/$H$ for the regime of catastrophic instability (arrow 3), displaying wave reversal. Profiles superimposed on left and right correspond to start and end times; (c) time traces of the wave-averaged liquid flow rate ${\overline{q}}_{\mathrm{l}}={\mathrm{\Lambda }}^{-1}{\int }_0^{\mathrm{\Lambda }}{q}_{\mathrm{l}}dx$ for computations from graphs (a) (black and blue lines) and (b) (thick red line), showing liquid arrest for the catastrophic instability.

Figure 3

Regime of oscillatory instability. [(a), (b)] Transient computations on an open domain with inlet-outlet conditions: $H^{★}=1$ mm, ${\mathrm{Re}}_{\mathrm{l}}=15$, $f_0=0.027$. Displayed length is 0.8 m. Black lines, snapshots of the film profile $y=h(x,t=\text{const})$ (Eulerian view); red, path followed by the wave maxima, as the oscillating wave humps propagate through the domain (Lagrangian view); green, snapshot from DNS. (a) ${\mathrm{Re}}_{\mathrm{g}}=-19$, single-mode oscillation; (b) ${\mathrm{Re}}_{\mathrm{g}}=-25$, multimode oscillation; [(c), (d)] heat and mass transport data obtained from DNS in graph (a): $\mathrm{Pe}=4590$. See also the Supplemental Movie [31]. Instantaneous color plots of the field of the transported scalar $\mathrm{\Theta }$ over the course of one nominal wave period 1/$f_0$. (c) $t{f}_0=0$; (d) $t{f}_0=0.65$; (e) $t{f}_0=1.0$.

Figure 4

Extreme film thinning under an increasing countercurrent gas flow. Traveling-wave solutions at fixed $\mathrm{\Lambda }=46.8H$ and liquid volume (per unit width) $V_{\mathrm{l}}=0.31\mathrm{\Lambda }H$. (a) Grayscale plot of film surface profiles $h\left(x\right)$/$H$ as a function of ${\mathrm{Re}}_{\mathrm{g}}$ (vertical axis); (b) corresponding evolution of the minimal film thickness $h_{\min }$ (vertical axis is logarithmic). The lowest thickness reached is $h_{\min }^{★}=62$ nm. In both graphs, two representative cross-sectional views are also included.