The effect of contact lines on the Rayleigh instability with anisotropic surface energy

We determine the linear stability of a rod or wire on a substrate subject to capillary forces arising from an anisotropic surface energy for a range of contact angles between -π/2 and π/2. The unperturbed rod is assumed to have infinite length with a uniform cross-section given by a portion of the two-dimensional equilibrium shape. We examine the effect of surface perturbations on the total energy. The stability of the equilibrium interface is reduced to determining the eigenvalues of a coupled system of ordinary differential equations. This system is solved both asymptotically and numerically for several types of anisotropic surface energies. We find that, in general, the presence of the substrate tends to stabilize the rod.