Shaken not stirred –On internal flow patterns in oscillating sessile drops

We use numerical (volume of fluid) simulations to study the flow in an oscillating sessile drop immersed in an ambient immiscible fluid. The drop is excited by a sinusoidal variation of the contact angle at variable frequency. We identify the eigenfrequencies and eigenmodes of the drops and analyze the internal flow fields by following the trajectories of tracer particles. The flow fields display an oscillatory component as well as a time-averaged mean component. The latter is oriented upward along the surface of the drop from the contact line towards the apex and downward along the symmetry axis. It vanishes at high and low frequencies and displays a broad maximum around f = 200–300Hz. We show that the frequency dependence of the mean flow can be described in terms of Stokes drift driven by capillary waves that originate from the contact line, in agreement with recent experiments (Mugele F. et al., Lab Chip, 11 (2011) 2011). Introduction. – The dynamics of oscillating sessile drops are of great interest to many fundamental and practical problems including the impact of drops on solid surfaces, the generation of drops (e.g. in inkjet printers), and the manipulation of drops in microfluidic and optofluidic applications. Compared to the classical fluid dynamics problem of oscillating free drops, for which Rayleigh [1] developed the first systematic solution, the problem of sessile drops is more complex because the presence of the solid surface reduces the symmetry and gives rise to additional dissipation due to the adhesion of the liquid to the solid surface [2,3]. Moreover, it introduces a contact line that can be either pinned or mobile and a contact angle with a velocity-dependent dynamic value. As a consequence, the absolute values of the eigenfrequencies of the drops shift while their scaling with surface tension, density, and drop size remains unaffected [4]. Novel applications inspired by microfluidics trigger questions beyond the previous focus on eigenfrequencies and eigenmodes including the coupling between drop resonances, contact angle hysteresis and drop motion [4–6] as well as timeaveraged streaming flows that promote mixing within the (a)E-mail: [email protected] (b)E-mail: [email protected] drops [7–10] and affect the deposition of residues from evaporating drops [11]. In those experiments drop oscillations are typically excited either by mechanical shaking or by a periodic modulation of the equilibrium contact angle using electrowetting (EW). The goal of the present numerical study is to characterize the flow patterns in oscillating drops and to elucidate their physical origin. Inspired by EW experiments [7–11], we adopt a numerical scheme based on the volume of fluid (VOF) method with a diffuse interface that is able to deal with both moving contact lines and viscous ambient fluids [12]. This approach extends the scope of earlier numerical studies [3,13] that focused on shape oscillations, eigenfrequencies, and instantaneous flow fields in the numerically simpler situation of drops with pinned contact lines in ambient air. We analyze the drop oscillations in terms of eigenmodes and characterize instantaneous and mean flows by following the trajectories of passive tracer particles. Despite the presence of pronounced resonances the simulations are found to reproduce all important characteristics including the frequency dependence of the mean flow reported in the experiments. The origin of the mean flow can be described consistently with a Stokes drift model that is based on propagating capillary waves rather than discrete eigenmodes [9].