The shape and stability of liquid menisci at solid edges

Abstract

The shape and stability of liquid menisci attached to a solid edge with dihedral angle 2α is investigated. It is shown that in addition to the family of cylindrical menisci a family of azimuthally modified unduloids exists. A double Fourier series of the latter with respect to their axis (parallel to the extension of the edge) and with respect to the azimuth is derived. The dispersion relation between the axial wavenumber q, the azimuthal wavenumber s and the waviness parameter d is calculated. When the condition of constant contact angle γ along the contact lines with the solid is applied, a one-dimensional family of modified unduloids fitting to the edge is obtained. Their axial wavenumber q becomes independent of the waviness d at the bifurcation with the family of cylindrical menisci, such that this bifurcation limits the stability. The respective stability criteria are derived and evaluated. For α + γ > ½π the cylindrical menisci are convex. They reveal a maximum stable length, which quadratically tends to infinity when α + γ = ½π is approached. The smallest stable extension arises for the free cylindrical column (the Rayleigh jet), which is covered by the present investigations by assuming α = π, γ = ½π. For α + γ < ½π the cylindrical menisci are concave and stable: no bifurcation with the family of modified unduloids arises.