Modeling the dynamics of an oil drop driven by a surface acoustic wave in the underlying substrate

TL;DR

The paper develops a two-dimensional thin-film model for millimetric oil drops driven by a leaky surface acoustic wave (SAW) propagating in a solid substrate. By coupling a damped first-order acoustic field to a time-averaged second-order streaming force and applying a long-wave (lubrication) approximation, the authors derive a closed evolution equation for the film height that includes capillarity, gravity, and SAW-induced forcing. The results reveal two dynamical regimes: at low SAW amplitudes, capillary and gravitational stresses dominate, while at higher amplitudes the SAW-driven Eckart streaming yields a translated drop with a trailing wetting film and a quasi-steady speed; a traveling-wave reduction further shows v_f scales as $v_f \sim A^2$ and $h_{\max} \sim A$, aligning with experimental trends. The work provides a tractable framework for predicting SAW-driven drop dynamics in thick films and points to future 3D extensions and boundary-layer effects for improved quantitative agreement.

Abstract

We present a theoretical study, supported by simulations and experiments, on the spreading of a silicone oil drop under MHz-frequency surface acoustic wave (SAW) excitation in the underlying solid substrate. Our time-dependent theoretical model uses the long wave approach and considers interactions between fluid dynamics and acoustic driving. While similar methods have analyzed micron-scale oil and water film dynamics under SAW excitation, acoustic forcing was linked to boundary layer flow, specifically Schlichting and Rayleigh streaming, and acoustic radiation pressure. For the macroscopic drops in this study, acoustic forcing arises from Reynolds stress variations in the liquid due to changes in the intensity of the acoustic field leaking from the SAW beneath the drop and the viscous dissipation of the leaked wave. Contributions from Schlichting and Rayleigh streaming are negligible in this case. Both experiments and simulations show that after an initial phase where the oil drop deforms to accommodate acoustic stress, it accelerates, achieving nearly constant speed over time, leaving a thin wetting layer. Our model indicates that the steady speed of the drop results from the quasi-steady shape of its body. The drop speed depends on drop size and SAW intensity. Its steady shape and speed are further clarified by a simplified traveling wave-type model that highlights various physical effects. Although the agreement between experiment and theory on drop speed is qualitative, the results' trend regarding SAW amplitude variations suggests that the model realistically incorporates the primary physical effects driving drop dynamics.

Figures (10)

• Figure 1: (A) Upper schematic view of the experimental setup: Surface acoustic wave (SAW) propagates from the interdigitated transducer (IDT) until it reaches the acoustic absorber (comprised of glycerol-soaked paper placed on the actuator under and to the right of the needle). The needle is of known diameter (510 $\mu$m), placed for identifying spatial resolution in the images. (a-e) Successive snapshots are taken from an experiment monitoring the flow of a silicone oil film. During the experiment, a drop of silicone oil is placed on the horizontal surface (a), it deforms due to the application of SAW (b), and moves in the direction of SAW propagation (c-e); time is shown in seconds, and the vertical lines serve as a reference.
• Figure 2: Two sets of images from experiments monitoring silicone oil drops powered by SAWs of three different measured amplitudes $A_{\rm n}$ (1.2 nm--1.4 nm), where (A-C) silicone oil starts from the same location, moves in the direction of the SAW, and reaches different distances (a-c) after 4.09 seconds. The corresponding side-view videos at different acoustic power levels are used to analyze the speed and profile of silicone oil during the movement. (D, d) show top view snapshots (corresponding to (C, c)) of the oil. A thin film of oil, hardly visible in the side view, can be seen more clearly behind the main body of silicone oil in the top views.
• Figure 3: Time evolution of (a) the front position and (b) the maximum drop height for several values of $A_{\rm n}$, for silicone oil drops of volume $\mathcal{V}_{\rm d}=8\, \mu$l and kinematic viscosity $\nu=50$ cSt. The symbols correspond to the measurements, and the lines are simple fits to guide the eye. We expect errors of $\pm 50~\mu$m in the values of the film height due to the limited resolution of the side view image and tracker software. Note that $t=0$ corresponds to the time instant at which SAW is applied, which is 2 s after deposition of the oil. A video showing the evolution at various SAW amplitudes is available as a supplementary material (movie 1).
• Figure 4: Asymptotic front speed of drop profiles for several values of the SAW amplitude $A_{\rm n}$ obtained using (a) silicone oil drops of volume $\mathcal{V}_{\rm d}=8\,\mu$l and different kinematic viscosities $\nu$; and (b) different volumes $\mathcal{V}_{\rm d}$ of silicone oil of the same viscosity $\nu=50$ cSt. We present two example videos comparing typical evolution at various kinematic viscosities and volumes in the supplementary materials (movie 2, movie 3).
• Figure 5: Schematic of a liquid drop (blue curve) driven by a leaky SAW of amplitude $A_{\rm n}$. The SAW travels from left to right in the solid substrate. The drop extends from the rear to the front contact lines at $x_{\rm r}(t)$ and $x_{\rm f}(t)$, respectively. The thin red line represents the amplitude $A_{\rm n}$ of the SAW at the substrate and shows how it is attenuated due to the presence of the liquid (the black dotted line is the envelope of the decaying amplitude in the liquid). The Rayleigh angle is denoted by $\theta_R$. The thickness $h^\ast$ and the positions $x^{\ast}_{1,2}$ are defined later in the text, see Eq. (\ref{['eq:phis_hcut']}).