Corner solutions of the Laplace–Young equation

Journal article

The upper free surface $z=u(x,y)$ of a static fluid with gravity acting in the $z$ direction, occupying a volume $V$, satisfies the Laplace–Young equation. The fluid wets the vertical boundaries of $V$ so that the usual capillary contact conditions hold. This paper considers wedge shaped volumes $V$ with corner angle $2\alpha$, that belong to the intermediate corner angle case of $\pi/2-\gamma<\alpha<\pi/2$, where $\gamma$ is the contact angle and determines explicitly a regular power series expansion for the height $u(r,\theta)$ of the fluid near the corner, $r=0$, to all orders in $r$. Miersemann (1988, Pacific J. Math., 134, 99–311), shows that it is possible to have logarithmic terms for a general corner expansion of the Laplace–Young equation, with appropriate boundary conditions. However, we suggest that the usual practical cases do not possess any singular terms near the corner, and we analytically and explicitly produce a non-singular series to any order in $r$, and propose that near the corner the far-field effects are lost through any ‘interior or inner flat’ region in exponentially small terms. We give computational solutions for these regular (energy minimizing) cases based on a numerical finite volume method on an unstructured mesh, which fully support our assertions and our analytical series results, including the (minor) influence of the far field on local corner behaviour.

Authors: Norbury, J; Sander, G; Scott, C

• Publication date: 2005-01-01