Interplay of viscosity and surface tension for ripple formation by laser melting

A model for ripple formation on liquid surfaces exposed to an external laser or particle beam and a variable ground is developed. The external incident beam is hereby mechanically coupled to the liquid surface due to surface roughness. Starting from the Navier-Stokes equation, the coupled equations for the velocity potential and the surface height are derived in a shallow-water approximation with special attention to viscosity. The resulting equations obey conservation laws for volume and momentum where characteristic potentials for gravitation and surface tension are identified analogously to conservative forces. The approximate solutions are discussed in the context of ripple formation in laser-materials processing involving melting of a surface by a laser beam. Linear stability analysis provides the formation of a damped wave modified by an interplay between the external beam, the viscosity, and the surface tension. The limit of small viscosity leads to damped gravitational and the limit of high viscosity to capillary waves. The resulting wavelengths are in the order of the ripples occurring in laser welding experiments, hinting at the involvement of hydrodynamic processes in their origin. By discussing the response of the system to external periodic excitations with the help of Floquet multipliers, we show that the ripple formation could be triggered by a a periodically modulated external beam, e.g., appropriate repetition rates of an incident laser beam. The weak nonlinear stability analysis provides ranges where hexagonal or stripe structures can appear. The orientation of stripe structures and ripples are shown to be dependent on the incident angle of the laser or particle beam where a minimal angle is reported. Numerical simulations confirm the findings and allow us to describe the influence of variable grounds.