Instability and dripping of electrified liquid films flowing down inverted substrates

We consider the gravity-driven flow of a perfect dielectric, viscous, thin liquid film, wetting a flat substrate inclined at a nonzero angle to the horizontal. The dynamics of the thin film is influenced by an electric field which is set up parallel to the substrate surface—this nonlocal physical mechanism has a linearly stabilizing effect on the interfacial dynamics. Our particular interest is in fluid films that are hanging from the underside of the substrate; these films may drip depending on physical parameters, and we investigate whether a sufficiently strong electric field can suppress such nonlinear phenomena. For a non-electrified flow, it was observed by Brun et al. [Phys. Fluids 27, 084107 (2015)] that the thresholds of linear absolute instability and dripping are reasonably close. In the present study, we incorporate an electric field and analyze the absolute and convective instabilities of a hierarchy of reduced-order models to predict the dripping limit in parameter space. The spatial stability results for the reduced-order models are verified by performing an impulse-response analysis with direct numerical simulations (DNS) of the Navier–Stokes equations coupled to the appropriate electrical equations. Guided by the results of the linear theory, we perform DNS on extended domains with inflow and outflow conditions (mimicking an experimental setup) to investigate the dripping limit for both non-electrified and electrified liquid films. For the latter, we find that the absolute instability threshold provides an order-of-magnitude estimate for the electric-field strength required to suppress dripping; the linear theory may thus be used to determine the feasibility of dripping suppression given a set of geometrical, fluid, and electrical parameters.

Figure 1

Schematic for the problem of a liquid film influenced by a parallel electric field. As the inclination angle $\theta$ (measured from the horizontal) varies, the substrate and axes rotate, with overlying films found for $\theta <\pi /2$, and hanging films (as in the schematic) for $\theta >\pi /2$.

Figure 2

Linear growth rates as a function of wave number for ${\mathcal{E}}^{★}=0,10,20$. The parameters taken are $\theta =\pi /2$ ($cot\theta =0$) and $\mathit{\text{C}}=0.01$. Panels (a) and (b) take the cases $\mathrm{Re}=1,4$, respectively. The dotted, dashed, and solid lines correspond to the Benney, WIBL1, and WIBL2 models, respectively.

Figure 3

Critical wave number dependence on $\mathrm{Re}$ for $\theta =\pi /2, \mathit{\text{C}}=0.01$, and ${\mathcal{E}}^{★}=0,40$. The dotted, dashed, and solid lines correspond to the results for the Benney/WIBL1, WIBL2, and Orr–Sommerfeld models, respectively. The electrical permittivities used in the Orr–Sommerfeld model are ${\epsilon }^{\text{S}}=1.5, {\epsilon }^{\text{I}}=2, {\epsilon }^{\text{II}}=1$.

Figure 4

Curves of ${\omega }_{\text{r}}=0$ in the right half plane and A/C regions of the Benney and WIBL models. Panels (a) and (b) plot ${\omega }_{\text{r}}=0$ for $\mathit{\text{C}}=0.01, \mathrm{Re}=4, cot\theta =-8, {\mathcal{E}}^{★}=0,20,40$, for the Benney and WIBL1 models, respectively. The black dots correspond to the nontrivial real roots. Panel (c) displays the transition curves for the Benney (dotted line), WIBL1 (dashed line), and WIBL2 (solid line) models with $\mathit{\text{C}}=0.01, {\mathcal{E}}^{★}=0,20,40$, in terms of $\mathrm{Re}$ and $cot\theta$; convective instabilities occur for parameters on the right of the corresponding curve, with absolute on the left. The transition points at zero Reynolds number are marked with diamonds for the Benney and WIBL1 models, and a square for the WIBL2 model. The fold point for the WIBL1 (WIBL2) model is marked with a down-pointing (up-pointing) triangle. The star in panel (c) corresponds to the parameters used in panels (a) and (b). Panel (d) shows continuation of the $\mathrm{Re}=0$ transition points and the fold points for the WIBL1 and WIBL2 models.

Figure 5

Effect of electric-field strength on A/C transition surface for WIBL2. Panels (a) and (b) show the A/C transition surface for the choices of ${\mathcal{E}}^{★}=0,5$. Panel (c) shows the minima of these surfaces, and those corresponding to other electric-field strengths, over $\mathit{\text{C}}$ plotted against $\mathrm{Re}$. The dotted lines signify that the minimum is attained in the $\mathit{\text{C}}\to \infty$ limit, and the diamonds denote minima of curves where the minimum is found at $\mathrm{Re}>0$. Panel (d) plots the minima in the Reynolds number of the curves in panel (c) against ${\mathcal{E}}^{★}sin\theta$.

Figure 6

Effect of electric-field strength on A/C transition for a flow with negligible surface tension. Panel (a) shows the A/C transition surface, and panel (b) gives the minimum of the surface over ${\mathcal{E}}^{★}$ as a function of $\mathrm{Re}$.

Figure 7

Comparison of A/C regions for Stokes flow (solid) with WIBL2 at zero Reynolds number (dotted). Panel (a) plots the transition curves for ${\mathcal{E}}^{★}=0,5$. Panel (b) plots the minimum critical angle against ${\mathcal{E}}^{★}sin\theta$; the values to the right of the stars are attained in the weak-surface-tension limit, as for the ${\mathcal{E}}^{★}=5$ case in panel (a). The permittivities used for the Stokes flow computations are ${\epsilon }^{\text{S}}=1.5, {\epsilon }^{\text{I}}=2$, and ${\epsilon }^{\text{II}}=1$.

Figure 8

Panels (a) and (b) plot interfacial profiles starting from Gaussian pulse initial data for $E_0=0$ and $6\times {10}^5$ V/m, respectively, for a film with inclination $\theta =5\pi /6$. The system is represented with the wall-normal $z$ coordinate upwards, and the flow from left to right. We include an off-vertical arrow indicating the direction of gravity at an angle of $\pi /6$ (clockwise) from the $z$ axis. Note that the long streamwise scale produces an arrow which looks close to vertical. For the height axes on the left-hand sides, the substrate is at $-2.5$, since the dimensional film thickness is $\ell =2.5$ mm. Panel (c) shows the evolution of the upstream edge of the wave packet; from left to right, the lines correspond to $E_0=0, 1.5\times {10}^5, 3\times {10}^5, 4.5\times {10}^5, 6\times {10}^5$, and $7.5\times {10}^5$ V/m. In all panels, the profiles are shifted so that the initial Gaussian pulse is located at the origin.

Figure 9

Snapshots of the fluid interface for $\theta =7\pi /8$ and a range of inflow film thicknesses. Panel (a) shows a bounded wave-train for $\ell =1.35$ mm, panel (b) shows a wave train with dripping due to coalescence (transient) for $\ell =1.40$ mm, and panel (c) displays a solution which is dripping due to instability of solitary waves (not coalescence) for $\ell =1.45$ mm. The wall-normal $z$ coordinate is directed downwards, with the flow from right to left. The direction of gravity is indicated in the panels with an arrow at $\pi /8$ (clockwise) to the downwards $z$ direction. Note that the long streamwise scale produces an arrow which looks close to vertical. The dashed line denotes the inflow film thicknesses $\ell$, and the dotted line is given by $h_{\text{max}}=1.649\ell$ (see text). The waves with peaks below the dotted lines in panels (b) and (c) eventually yield dripping events.

Figure 10

Time series of $h_{\text{max}}-\ell$ for a range of $E_0$. Panels (a)–(g) plot the time series for $E_0=0.0, 1.5\times {10}^5, 3.0\times {10}^5, 4.5\times {10}^5, 6.0\times {10}^5, 7.5\times {10}^5, 9.0\times {10}^5$ V/m, respectively. The angle of inclination is $\theta =5\pi /6$, the undisturbed film thickness $\ell =2.5$ mm, and the domain length is $L_0=300\ell$.

Figure 11

Snapshots of waves and dripping for electrified films. Panels (a) and (b) show cases corresponding to $E_0=3\times {10}^5$ V/m and $E_0=6\times {10}^5$ V/m, respectively. The angle of inclination is $\theta =5\pi /6$, the undisturbed film thickness $\ell =2.5$ mm, and the domain length is $L_0=300\ell$. The flow is from right to left, and the direction of gravity is indicated in the figure with an arrow at an angle of $\pi /6$ (clockwise) to the downwards pointing $z$ direction. The interfacial profile at $t\approx 67$ s is represented with a black solid curve. The lower plots are enlargements of the outlined rectangular regions in the upper plots of each panel. The color bar indicates pressure values and the faint solid curves in the lower plots are voltage equipotentials.