Corner solutions of the Laplace-Young equation

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 2α, that belong to the intermediate corner angle case of π/2 - γ < α < π/2, where γ is the contact angle and determines explicitly a regular power series expansion for the height u(r, θ) 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. © Oxford University Press 2005; all rights reserved