Hydrothermal waves in sessile droplets evaporating at a constant contact angle mode

Abstract

Although various Marangoni instability patterns have been observed in evaporating sessile droplets, most are limited to a small contact angle. As a result, the droplets resemble a flat liquid layer, where the vertical temperature gradient dominates, and the hydrothermal waves (HTWs) predominantly driven by the tangential temperature gradient do not occur. In this paper, we focus on the possibility of HTWs in droplets with moderate contact angles using numerical simulation. For this purpose, a three-dimensional numerical model is proposed, in which the deformation of the droplet surface induced by evaporation can be directly calculated from the local evaporation flux. We predict fan-like HTWs with three blades in the droplet. The waves are located at the center region of the droplet and are uniformly distributed throughout the circumference; they simultaneously propagate along the anticlockwise and radial directions. With evaporation, the blades become shorter, whereas the fan-like shape and wave number remain without significant variation. The behavior of the fluid flow inside the droplet and its oscillatory characteristics were analyzed carefully. The numerical results were verified by experiments, and confirm that HTWs would only occur under moderate contact angles. Additionally, a higher substrate temperature or a larger initial radius would promote the appearance of the irregular fan-like HTWs with two blades, whereas a sufficient small initial radius will inhibit the appearance of HTWs. The effects of the substrate temperature, the initial radius and the contact angle on the critical radius and the corresponding critical normal Marangoni number (Mav,c) for the disappearance of the HTWs were also studied.