Droplet Footprint Control

Abstract

Controlling droplet shape via surface tension has numerous technological applications, such as droplet lenses and lab-on-a-chip. This motivates a partial differential equationconstrained shape optimization approach for controlling the shape of droplets on flat substrates by controlling the surface tension of the substrate. We use shape differential calculus to derive an L 2 gradient flow approach to compute equilibrium shapes for sessile droplets on substrates. We then develop a gradient-based optimization method to find the substrate surface tension coefficient yielding an equilibrium droplet shape with a desired footprint (i.e., the liquid-solid interface has a desired shape). Moreover, we prove a sensitivity result with respect to the substrate surface tensions for the free boundary problem associated with the footprint. Numerical results are also presented to showcase the method.

Optimal control in ink-jet printing via instantaneous control

“…In some situations, a reformulation on a fixed, reference domain can help reducing the computational effort (see, e.g., [4,5,6]), but if big deformations are involved, this strategy loses its suitability. The same arguments hold also for free surface problems, like in the present case, where the motion of the contact line [7,8,9,10,11] is a further source of complexity.…”

An algorithm for prescribed mean curvature using isogeometric methods

“…One would then need to prescribe the contact angle of the first surface (with respect to the substrate) on its boundary curve. This would be desireable for simulating equilibrium shapes of droplets interacting with rigid walls (see [14,15]).…”

Optimal control of volume-preserving mean curvature flow

“…There exists a host of applications that involve large deformations or geometric flows: the motion of droplets [21], fluid structure interaction [22,6], mean curvature flow [3,14,18], deformable elastic bodies [31,34,37], liquid thin films on curved surfaces [32] to name a few. Such problems can often be formulated as the evolution of sets under a gradient flow.…”

“…The type of problem considered in this paper shares many similarities with the optimal control of free boundaries for stationnary problems. Recent contributions on this topic have been obtained using tools from shape calculus including [16,33] for the shape controllability of the free boundary of an obstacle problem, [17] where the control of the Bernoulli free boundary problem has been investigated, and [21] for the control of the footprint of a sessile droplet via substrate surface tension. In these works, the free boundary depends implicitely on the control and a perturbation analysis allows to compute the sensitivity of the free boundary with respect to the control.…”

“…The problem considered in the present paper is similar in the sense that the space-time tube also depends implicitely on the control. Thus, on one hand several ideas of [17,21] may be used, but on the other hand the control of the geometric evolution problem presents many specificities that require a different analysis and novel ideas. To the best of our knowledge this is the first investigation of optimal control of geometric evolution problems and MCF.…”