Hydrodynamic limit to penetration of a material by a high-power beam

In laser (or electron-beam) welding a high-intensity beam is directed on to a metal surface causing melting and evaporation. If the rate of evaporation is sufficiently high, then the laser will drill a 'keyhole' into the molten metal, thereby depositing power deep into the material. This drilling process will be opposed by the flow of molten metal into the keyhole, and in the steady state the two effects balance each other over the entire surface of the hole. Steady-state hole profiles are obtained for a vertical beam including the effects of gravity and surface tension. It is shown that surface tension reduces the depth of penetration typically by a factor of about three.