The transition from two-dimensional thermoconvective steady flow to a time-dependent flow is considered for a liquid with a high Prandtl number(Pr=105)in a liquid bridge with a curved free surface. Both thermocapillary and buoyancy mechanisms of convection are taken into account. The computer program developed for this simulation transforms the original nonrectangular physical domain into a rectangular computational domain. To solve the problem in body-fitted curvilinear coordinates, the time-dependent Navier–Stokes equations were approximated by central differences on a stretched mesh. For liquid bridges with a flat interface, the instability corresponding to an azimuthal wave number ofm=0is not found for the investigated range of Marangoni numbers. The instability corresponding to anm=0is found for relatively low Marangoni numbers only in liquid bridges with a nonflat, free surface, and nonzero Rayleigh number. The steady state becomes unstable to axially running waves. It is shown that the onset of instability depends strongly upon the volume of the liquid. The stability boundary is reported for the aspect ratio&Ggr;=height/radius=4/3and for a wide range of liquid bridge volumes. The physical mechanism of the oscillations is based on the temporal interaction of the temperature sensitive free surface with the small local disturbances, created by temperature distribution inside the liquid bridge. ©1998 American Institute of Physics.