In the usual models of thermocapillary flows, a vorticity singularity occurs at the contact free surface-solid boundaries. The steady axisymmetric hydrodynamics of the side-heated liquid bridge of molten metal is addressed here for its sensitivity to the size delta of a length scale explicitly introduced to regularize the problem. By linear stability analysis of the flows, various stable steady states are predicted: The already known steady states which are reflection-symmetric about the mid-plane, but also others which do not possess this property. The thresholds in Ma of the associated bifurcations are strongly dependent on delta, and converge with delta-->0 towards low values. Published data give these results some physical relevance. (C) 2002 American Institute of Physics.