EVAPORATION OF LIQUID DROPLETS ON FLAT AND CURVED SOLID SUBSTRATES
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We study the evaporation of liquid droplets on flat and curved heated substrates. We derive an evolution equation for the drop thickness to evaluate its shape as it evaporates on a curved substrate. We adopt a one-sided model with thermal control which, together with the lubrication approximation, results in an evolution equation for the local height of the droplet. Without requiring any pre-assumption for the shape of the drop, we formulate the problem for the two modes of evaporation: (i) pinned contact line and (ii) moving contact line with fixed contact angle. For each evaporation mode, we solve the problem numerically for flat and curved solid substrates, separately. We first analyze the simplified evolution equation for the case of flat solid substrates. For the pinned contact line case, we observe that after a time interval the contact angle, i.e. the angle between the solid and the interface at the contact line, dynamics become nonlinear and, interestingly, the local contact angle goes to zero in advance of total evaporation of the drop. For the case of moving contact line, in which the singularity at the contact line is treated by a numerical slip model, we find that the droplet nearly keeps its initial circular shape and that the contact line recedes with constant speed. We then analyze the problem for various periodic and quasi-periodic substrate profiles. For the case of pinned contact line, we find the results to be very similar to the flat case. For the case of moving contact line, we study the dynamics of the contact line and the apparent contact angle, i.e. the angle that the liquid appears to make with the solid surface when viewed at such a coarse resolution that the substrate appears flat (the angle between the horizon and the interface at the contact line). In contrast with our results for a flat substrate, for which the contact line recedes at a nearly constant speed, we observe that the contact line speed and position show significant time variation and that the contact line moves in an approximately step-like fashion on relatively steep substrates. For the simplest case of a periodic substrate, we find that the apparent contact angle is periodic in time. For doubly periodic substrates, we find that the apparent contact angle is again periodic and that the problem exhibits a phase-locking behavior. For multi-mode quasi-periodic substrates, we find the contact line behavior to be temporally complex and not only limited to an approximately step-like motion. In all cases, we find that the overall evaporation is increased relative to the flat substrate.
- Mechanical engineering