Analysis of Asymptotic Solutions for Cusp Problems in Capillarity

Abstract

The capillary surface $u(x,y)$ near a cusp region satisfies the boundary value problem: \begin{eqnarray} \nabla \cdot \frac{\nabla u}{\sqrt{1+\left|\nabla u \right|^2}}&=&\kappa u \qquad \textrm{in }\left\{(x,y): 0<x,f_2(x)<y<f_1(x)\right\}\,, \label{0.1}\\ \nu \cdot \frac{\nabla u}{\sqrt{1+\left|\nabla u \right|^2}}&=& \cos \gamma_1 \qquad \textrm{on } y=f_1(x)\,,\\ \nu \cdot \frac{\nabla u}{\sqrt{1+\left|\nabla u \right|^2}}&=& \cos \gamma_2 \qquad \textrm{on } y=f_2(x)\,, \label{0.3} \end{eqnarray} where $\lim_{x\rightarrow 0}f_1(x),f_2(x)=0$, $\lim_{x\rightarrow 0}f'_1(x),f'_2(x)=0$.

It is shown that the capillary surface is unbounded at the cusp and satisfies $u(x,y)=O\left(\frac{1}{f_1(x)-f_2(x)}\right)$, even for types of cusp not investigated previously (e.g. exponential cusps).

By using a tangent cylinder coordinate system, we show that the exact solution $v(x,y)$ of the boundary value problem: \begin{eqnarray} \nabla \cdot \frac{\nabla v}{\left|\nabla v \right|}&=&\kappa v \qquad \textrm{in }\left\{(x,y): 0<x,f_2(x)<y<f_1(x)\right\}\,,\\ \nu \cdot \frac{\nabla v}{\left|\nabla v \right|}&=& \cos \gamma_1 \qquad \textrm{on } y=f_1(x)\,,\\ \nu \cdot \frac{\nabla v}{\left|\nabla v \right|}&=& \cos \gamma_2 \qquad \textrm{on } y=f_2(x)\,, \end{eqnarray} exhibits sixth order asymptotic accuracy to the capillary equations~\eqref{0.1}$-$\eqref{0.3} near a circular cusp.

Finally, we show that the solution is bounded and can be defined to be continuous at a symmetric cusp ($f_1(x)=-f_2(x)$) with the supplementary contact angles ($\gamma_2=\pi-\gamma_1$). Also it is shown that the solution surface is of the order $O\left(f_1(x)\right)$, and moreover, the formal asymptotic series for a symmetric circular cusp region is derived.