An obstacle approach to rate independent droplet evolution

Abstract

We consider a toy model of rate independent droplet motion on a surface with contact angle hysteresis based on the one-phase Bernoulli free boundary problem. We introduce a notion of solutions based on an obstacle problem. These solutions jump ``as late and as little as possible", a physically natural property that energy solutions do not satisfy. When the initial data is star-shaped, we show that obstacle solutions are uniquely characterized by satisfying the local stability and dynamic slope conditions. This is proved via a novel comparison principle, which is one of the main new technical results of the paper. In this setting we can also show the (almost) optimal $C^{1,1/2-}$-spatial regularity of the contact line. This regularity result explains the asymptotic profile of the contact line as it de-pins via tangential motion similar to de-lamination. Finally we apply our comparison principle to show the convergence of minimizing movements schemes to the same obstacle solution, again in the star-shaped setting.

Figures (6)

• Figure 1: Side view (left) and the top view (right) of the setup for the one-phase free boundary problem.
• Figure 2: Plots of boundaries of obstacle solution (\ref{['d.viscosity_solution']}) simulations. Solid curves represent $\partial \Omega(t) \cap U$ plotted for evenly spaced values of $F(t)$, with the initial shape dashed. $\partial U$ is given by a dotted curve. Top left: Disconnected annuli initial data, jump discontinuity on touching. Top right: Receding situation (decreasing $F(t)$) with initial data given by the last step of the top left image. Note that the jump occurs at a different configuration, as late as possible. Bottom left: Different radius annuli, free boundary peels from the larger annulus after the jump. Bottom right: Stadium type initial data, convexity is not preserved.
• Figure 3: Left: Obstacle from above, slope is larger than $1$ everywhere and saturates where free boundary bends into $O$. Right: Obstacle from below, slope is smaller than $1$ everywhere and saturates where free boundary bends away from $\overline{O}$.
• Figure 4: Asymptotic expansion in non-tangential cone plus monotonicity also gives control in $\{x_n \leq 0\}$.
• Figure 5: Left: velocity $c$ cone touches $\Omega(t)$ from the outside at $(t_0,x_0)$, interpreted as $V_n(t_0,x_0) \geq c$. Right: velocity $c$ cone touches $\Omega(t)$ from the inside at $(t_0,x_0)$, interpreted as $V_n(t_0,x_0) \leq - c$.

Theorems & Definitions (60)

• Definition 1.1
• Theorem 1.2: see Theorem \ref{['t.equivalence']}
• Definition 1.3
• Theorem 1.4: see Theorem \ref{['t.equivalence']} and Theorem \ref{['t.oqe-MM-relation']}
• Definition 3.1
• Definition 3.2
• Remark 3.3
• Theorem 3.4
• Theorem 3.5
• Theorem 3.6