The usual modelling of thermocapillarity introduces a vorticity singularity along the contact of free surfaces with solid boundaries. The liquid bridge hydrodynamics is adressed here, for its sensitivity to the size of a filtering length, δ, introduced by an explicit regularization of the singularity. Systematically following the convergence of the numerical results with δ shows that the stability properties of the axisymmetric flows, and their bifurcation maps, are correctly identified provided that this length scale is small enough. Although the singularity treatment is localized near the boundaries, the flow stability is controlled by an accumulation of mechanical power in the vicinity of the mid plane of the liquid bridge, that is far from the boundaries.
