Rivulet cascade from falling liquid films with side contact lines

We investigate the dynamics of films of partially wetting liquids falling down a vertical plane in the presence of air. We focus on the little-studied case where the film is not in contact with lateral walls and thus has two side contact lines. By way of 3D direct numerical simulations, we study the fingering instability and the distribution of rivulets. Our results show that the instability develops as a cascade: two twin rivulets first develop at the free sides, followed by a series of twin rivulets growing closer to the center. This self-sustained fall is caused by the formation of stagnation zones at the front contact line as the liquid behind the ridge escapes toward a newly created rivulet. We also show that the established rivulet distance follows the prediction of linear instability, despite the cascade. The dynamics of the cascade is modified by inertial and film thickness effects, as well as by initial shape perturbations of the contact line. On the contrary, varying the contact angle does not affect the development of the cascade, apart from the average wavelength.

Figure 1

Sketch of the considered problem and used notations: a film of partially wetting fluid with side contact lines down a vertical plane in contact with air.

Figure 2

Snapshots of 3D view of a liquid film of fluid A down a vertical plane at ${\mathrm{Re}}_l=2.92$, $\mathrm{Ca}=0.04$, ${\theta }_s=82.5^{\circ }$. (a)–(d): ${\ell }_z/h_0=55.6$, i.e., ${\ell }_z/W=0.45$; (e)–(h): ${\ell }_z/h_0=105.1$, i.e., ${\ell }_z/W=0.85$; (i)-(l): ${\ell }_z/h_0=121.2$, i.e., ${\ell }_z/W=0.98$. Time steps: $t^{★}=9.78$ (leftmost panels); $t^{★}=16.3$ (second panels from the left); $t^{★}=22.8$ (third panels from the left); $t^{★}=29.3$ (rightmost panels).

Figure 3

Time evolution of finger dynamics for the case ${\ell }_z/W=0.85$ [Figs. 2]. (a) Length of main rivulets ${\ell }_I$ (empty square), secondary rivulets ${\ell }_{II}$ (empty triangle), and tertiary rivulets ${\ell }_{III}$ (empty circle). Wavelength of main rivulets ${\mathrm{\Lambda }}_I$ (filled square) and secondary rivulets ${\mathrm{\Lambda }}_{II}$ (filled triangle). (b) Rim profile at the spanwise center at different streamwise positions, from left to right: $x/L=0.001,0.02,0.06,0.1,0.15$. Filled red circles mark the position of maximum and minimum film thicknesses.

Figure 4

3D view of a liquid film down a vertical plate. (a) Fluid A: ${\mathrm{Re}}_l=1.2$, $h_0=0.60\mathrm{mm}$, ${\ell }_z/h_0=116.7$, $t^{★}=29.3$. (b) Fluid A: ${\mathrm{Re}}_l=30$, $h_0=1.76\mathrm{mm}$, ${\ell }_z/h_0=39.8$, $t^{★}=29.3$. (c) Fluid B: ${\mathrm{Re}}_l=28.3$, $h_0=0.60\mathrm{mm}$, ${\ell }_z/h_0=116.7$, $t^{★}=75.8$.

Figure 5

Sketch of a falling film with side contact lines and of the lateral rim (zoomed area).

Figure 6

(a) Capillary number based on the contact-line velocity as a function of $\mathrm{Re}\mathrm{Bo}$: DNS results (circle); fit curve: $0.04\sqrt{\mathrm{Re}\mathrm{Bo}}-0.025$ (solid thick line). Dashed line: capillary number based on $U_0$. (b) Angle $\psi$ made by the decrease in film width in a falling film of partially wetting fluid (${\theta }_s=82.5^{\circ }$) as a function of $\mathrm{Re}\mathrm{Bo}$, obtained via Eq. (22) (black solid line: fluid A; red solid line: fluid B). Filled circles mark the DNS results (black: fluid A; red: fluid B). The dashed line indicates the viscous-capillary limit [Eq. (22) without inertial term and with $\theta ={\theta }_s$] whereas the dotted line the purely inertial limit [Eq. (22) with ${\mathrm{Ca}}_{cl}=\mathrm{Ca}$]. Inset shows the log-log plot of $sin{\psi }^2$ as a function of $\mathrm{Re}\mathrm{Bo}$ for fluid A.

Figure 7

Top view of a liquid film of fluid A down a vertical plate at ${\mathrm{Re}}_l=2.92$, ${\theta }_s=82.5^{\circ }$, and ${\ell }_z/W=0.85$ [same case as Figs. 2]. Snapshots of the velocity field along the plane $xz$. (a)–(c): Module of velocity. (d)–(f): Normalized spanwise velocity.

Figure 8

Front view (at the position $x/L=0.008$) of a liquid film of fluid A down a vertical plate at ${\mathrm{Re}}_l=2.92$, ${\theta }_s=82.5^{\circ }$, and ${\ell }_z/W=0.85$ [Figs. 2] at $t^{★}=1.63$. (a) Pressure field. (b) Field of $1/2{\rho }_l{u}^2$. Insets show the film thickness profile and $h_{\max }$ at the spanwise positions identified by dashed lines. The abscissa has been compressed by 3.4 times for visibility purposes.

Figure 9

Effect of width-to-height ratio ${\ell }_z/h_0$ at $t=0.27$ s ($t^{★}=29.3$) on the finger dynamics of a film of fluid A flowing vertically at ${\mathrm{Re}}_l=2.92$, $\mathrm{Ca}=0.04$, with ${\theta }_s=82.5^{\circ }$. (a) Dimensionless primary wavelength ${\mathrm{\Lambda }}_I/h_0$ (filled square), secondary wavelength ${\mathrm{\Lambda }}_{II}/h_0$ (filled triangle), tertiary wavelength ${\mathrm{\Lambda }}_{III}/h_0$ (filled circle), average wavelength $\overline{\mathrm{\Lambda }}$ (filled diamond). The horizontal lines mark the most unstable wavelength (dashed line) and cutoff wavelength (dash-dotted line) from linear theory (Troian et al. [38]). (b) Dimensionless primary rivulet length ${\ell }_I/L$ (empty square), secondary rivulet length ${\ell }_{II}/{\ell }_I$ (empty triangle), tertiary rivulet length ${\ell }_{III}/{\ell }_I$ (empty circle). The number at the top of each graph indicates the number of observed rivulet whereas the vertical dotted lines mark their threshold.

Figure 10

3D view of the flow dynamics at $t^{★}=22$ of a vertical film of fluid A with ${\mathrm{Re}}_l=7.3$, $\mathrm{Ca}=0.07$, ${\theta }_s={82}^{\circ }$, and ${\ell }_z/h_0=63.6$. Effect of the contact-line perturbation $\epsilon cos(2\pi kz/{\ell }_z)$: $k=0$ (a); $k=1$ (b); $k=2$ (c); $k=3$ (d); $k=4$ (e); $k=5$ (f); $k=6$ (g).

Figure 11

Growth rate of the perturbation for a vertical film of fluid A with ${\mathrm{Re}}_l=7.3$, $\mathrm{Ca}=0.07$, ${\theta }_s={82}^{\circ }$, and ${\ell }_z/h_0=63.6$. (a) Dimensionless maximum film thickness $max\left[h(x_0,z_0,t)\right]$ for different streamwise positions ($z_0$ always coincides with the initial perturbation tip closer to the domain center) by varying the spanwise wave number: $k=0$ (plus), $k=1$ (square), $k=2$ (circle), $k=3$ (up-pointing triangle), $k=4$ (down-pointing triangle), $k=5$ (empty diamond), $k=6$ (cross). The inset shows how $h_{\max }$ is computed at two different streamwise positions. (b) Time evolution of the root displacement by varying the initial forcing of the contact line as $\epsilon cos(2\pi kz/{\ell }_z)$. $k=2$ (circle), $k=3$ (up-pointing triangle), $k=4$ (down-pointing triangle), $k=5$ (empty diamond), $k=0$ (plus), $k=1$ (square), $k=6$ (cross). The inset shows the exponential growth as a function of $2\pi k/{\ell }_z$. Here, the solid vertical line marks the most unstable wavelength, whereas the dashed line marks the cutoff wavelength.

Figure 12

Snapshots of the contact-line dynamics in a vertical film of fluid A at ${\mathrm{Re}}_l=7.3$, $\mathrm{Ca}=0.07$, ${\theta }_s={82}^{\circ }$, and ${\ell }_z/h_0=63.6$. Influence of phase shift in the perturbed initial condition (represented in the inset of the leftmost panels, whose amplitude is intentionally not to scale): $\epsilon cos(2\pi kz/{\ell }_z+\chi )$ with $\chi =0$ (top) and $\chi =\pi /2$ (bottom). Time evolution of finger dynamics from left to right: $t^{★}=11$; $t^{★}=13.3$; $t^{★}=17.7$; $t^{★}=22$.

Figure 13

3D dynamics at $t^{★}=27.7$ of a falling film of fluid A down a vertical plane (${\mathrm{Re}}_l=2.92$, $\mathrm{Ca}=0.04$, ${\ell }_z/h_0=121.2$). Effect of varying the contact angle: ${\theta }_s={(40,82.5,120,160)}^{\circ }$ from left to right.

Figure 14

Number of observed rivulets for all simulated cases with fluid A and comparison with linear stability predictions of several models: Johnson et al. [8] (green line), Troian et al. [38] (black line), Silvi and Dussan [5] (red line), Huppert [4] (blue line); the dashed line marks the cutoff wavelength (Kondic and Diez [16]); results of simulated cases are represented by symbols: scenarios with inlet width varied (empty square); scenario ${\mathrm{Re}}_l=30$ (down-pointing triangle); scenario ${\mathrm{Re}}_l=15$ (up-pointing triangle); scenario ${\mathrm{Re}}_l=7.3$ ($k=0,1,2$ filled circle; $k=3$ empty circle; $k=4,6$ filled diamond; $k=5$ empty diamond); scenario ${\mathrm{Re}}_l=1.2$ (filled square); scenario ${\mathrm{Re}}_l=6.47$ (plus).

Figure 15

Grid dependence analysis showing a falling film of fluid A down a vertical wall (${\mathrm{Re}}_l=2.92$, $\mathrm{Ca}=0.04$, ${\theta }_s=82.5^{\circ }$, ${\ell }_z/h_0=121.2$) at $t^{★}=24.4$. Comparison of six different meshes, whose details are given in Table 5. (a) Top view (cut in the plane $xz$ at $y=0.0005$ m). The dashed lines mark the cuts shown in panels (b) and (c). (b) Front view [cut in the plane $yz$ at $x/L=0.09$, horizontal line in panel (a)]. (c) Side view of the lateral rivulet [cut in the plane $xy$ at $z/w=0.88$, vertical line in panel (a)]. (d) Side view of the rivulet shown in panel (c) at the moment where they are at the same position. Mesh 1 gives inconsistent results and therefore is not shown in panels (b), (c), (d).

Figure 16

3D view of a falling film with sides attached to the lateral walls down a vertical plane ($0.1\mathrm{m}$ long and $0.2\mathrm{m}$ wide) at ${\mathrm{Re}}_l=0.13$.

Figure 17

Falling film spreading on a vertical wall: ${\mathrm{Re}}_l=266$, $\mathrm{Ca}=0.015$, ${\theta }_s={30}^{\circ }$, ${\ell }_z=7.62\mathrm{cm}$. (a) Width restriction from our DNS. The dashed line marks the plane located at $10\mathrm{cm}$ from the inlet, where the film thickness profile is computed in (b). The arrow indicates the flow direction. (b) Film thickness profile. Comparison of our DNS (solid line) with Lan et al. [19] (dashed line: numerical; square: experimental).