On the shape of a hydrostatic meniscus attached to a corrugated plate or wavy cylinder

The shape of a hydrostatic meniscus attached at a fixed contact angle to a vertical plate or circular cylinder with periodic corrugations is studied by analytical and numerical methods, and the effect of wall irregularities on the shape of the contact line and vertical component of the capillary force is discussed. An asymptotic analysis for a plate with small-amplitude sinusoidal corrugations is carried out to first order with respect to the corrugation amplitude, and a boundary-value problem is formulated and solved by a shooting method to determine the meniscus shape and elevation of the contact line. The meniscus attached to a corrugated plate with rounded corners produced by a Schwarz-Christoffel mapping function for a triangular wave is considered by numerical methods. The Laplace-Young equation determining the meniscus shape is solved in orthogonal curvilinear coordinates generated by conformal mapping using a finite-difference method. The numerical results are successfully compared with the predictions of the perturbation expansion for small amplitudes and discussed with reference to the rise of a meniscus inside a dihedral angle for large amplitudes. A companion asymptotic analysis is presented for a meniscus outside a vertical circular cylinder with small-amplitude sinusoidal corrugations. The analytical predictions are successfully compared with numerical solutions of the Laplace-Young equation for a meniscus outside an elliptical cylinder with aspect ratio near unity, regarded as a deformed circle.