Interface-dominated sliding compound drops

Abstract

We investigate compound drops composed of two immiscible nonvolatile partially wetting liquids that slide down an inclined homogeneous smooth solid substrate based on a mesoscopic hydrodynamic two-layer model in full-curvature formulation. First, drops of one liquid stationarily sliding on a layer of the other liquid are briefly investigated with a focus on the dependence of drop velocity and interface profiles on inclination and mean thickness of the adaptive substrate. Then, stationary sliding compound drops are studied with a focus on the dependence of their configuration, velocity, dynamic Young and Neumann angles on three control parameters, namely, the inclination, the volume ratio and the viscosity ratio. The reasons for the encountered dependence of the velocity on configuration are clarified based on a discussion of the lateral dissipation profile. Finally, we briefly consider the time-periodic fusion-overtaking-splitting behavior found outside the existence range of the stationary sliding compound drops as determined by saddle-node bifurcations.

Figures (8)

• Figure 1: Sketches of the considered geometry showing as an example an asymmetric compound drop in 2-1 configuration (drop of liquid 1 in front of drop of liquid 2) on an inclined plane in the (a) macroscopic and (b) mesoscopic picture. The substrate is tilted by an angle $\alpha$ w.r.t. the horizontal $x^\prime$-direction, yielding a rotation between the laboratory system (dashed coordinates, blue) and the substrate coordinates (coordinates without dashes, red). Gravity is acting in the negative $z^\prime$-direction. The macroscopic description allows for direct Young and Neumann constructions at the three-phase contacts. The mesoscopic description features various adsorption layers and contact angles are determined at the inflection point of the respective profile (see Fig. 1 in DiTh2025prf).
• Figure 2: Behavior of a drop of liquid 2 sliding stationary down on a layer of liquid 1. Panel (a) shows the dependence of the drop velocity $U$ on the mean thickness of the liquid substrate (liquid 1) $\bar{h}_{12}$ for a number of different inclination parameters $\beta$ as given in the legend. Panel (b) gives selected corresponding drop profiles at the $\bar{h}_{12}$-values marked in (a) by thin vertical lines (with the same left-to-right order), including streamlines and the color-encoded absolute value of the velocity field (both in the comoving frame). The vertical order of the profiles in (b) mirrors the vertical order of the curves in (a), i.e., the top [bottom] row are for the largest [smallest] $\beta$. In each panel the value of an asymmetry measure is given. It is defined as $A_{h_{12}}=\frac{1}{2V_1}\int xh_{12}\mathrm{d}x$ (cf. Ref. HeST2021sm), i.e., here, based on the profile of the liquid substrate. The domain size is $L=500$, the mean thickness of the upper layer is $\bar{h}_{23}-\bar{h}_{12}=10$, and its viscosity is $\eta_2=1$. All other parameters are as described in section \ref{['subsec:nondim-parameters-setup']}.
• Figure 3: Characterization of stationary sliding compound drops in dependence of the inclination parameter $\beta$. Panel (a) shows the drop velocity $U$ (horizontal axis) as a function of $\beta$ (vertical axis) while (b) and (c) show the absolute values of the three Neumann angles and the three Young angles, respectively (see legends), all as a function of $U$. In (a), solid [dashed] black lines indicate stable [unstable] states obtained by continuation, while crosses mark data from time simulations. Compound drops in 2-1 and 1-2 configuration are indicated by orange and purple coloring. Panels (d)-(g) give example profiles for both configurations at selected $\beta$. The corresponding values of $\beta$ and $U$ are indicated by thin solid vertical and horizontal lines in (a)-(c). The dotted, dashed and dot-dashed lines in (b) and (c) mark the various equilibrium angles (see legends). The volume ratio and the viscosity ratio are fixed to one, $\nu=\eta=1$. The remaining parameters are given in section \ref{['subsec:nondim-parameters-setup']}.
• Figure 4: (a)-(c) Vertically integrated dissipation and (d)-(e) velocity fields of the stationary sliding compound drops in (left and right) 2-1 configuration and (middle) 1-2 configuration. Panels (a)-(c) give the laterally spatially resolved dissipation for the two liquids individually and the total value along with the integrated dissipation $D$. Panels (d)-(e) give the height profiles together with the streamlines and the color-coded magnitude of the velocity fields (both in the comoving frame). Note that the velocity $U$ of the compound drop (and thus the comoving frame) differs for each column. Datasets in the two leftmost and rightmost columns are obtained at the same inclination parameter $\beta=10^{-5}$ and integrated dissipation $D\approx3.1\times10^{-4}$, respectively (see labels). Remaining parameters are as in section \ref{['subsec:nondim-parameters-setup']}.
• Figure 5: Dependence of the properties of stationary sliding compound drops on the viscosity ratio $\eta$ at fixed $\nu = 1$ and $\beta = 2.5 \times 10^{-5}$. Panel (a) shows the drop velocity $U$ (horizontal axis) as a function of $\eta$ (vertical axis) while panels (b) and (c) give the absolute values of the three Neumann angles and the three Young angles, respectively (see legends). Compound drops in 2-1 and 1-2 configuration are indicated by orange and purple coloring. Panels (d)-(e), and (f)-(g) present stable, and unstable states, respectively, at selected $\eta$ as indicated by thin solid horizontal lines in (a). The corresponding $U$-values are marked by vertical lines in (a)-(c). Remaining parameters, lines styles etc. are as in Fig. \ref{['fig:plot_beta_variation']}.