The dynamics of thermocapillary flows in differentially heated cylindrical liquid bridges is investigated numerically using a mixed finite volume/pseudo-spectral method to solve the Navier–Stokes equations in the Boussinesq approximation. For large Prandtl numbers (Pr = 4 and 7) and sufficiently high Reynolds numbers, the axisymmetric basic flow is unstable to three-dimensional hydrothermal waves. Finite-amplitude azimuthally standing waves are found to decay to travelling waves. Close to the critical Reynolds number, the former may persist for long times. Representative results are explained by computing the coefficients in the Ginzburg–Landau equations for the nonlinear evolution of these waves for a specific set of parameters. We investigate the nonlinear phenomena characteristic of standing and pure travelling waves, including azimuthal mean flow and time-dependent convective heat transport. For Pr [Lt ] 1 the first transition from the two-dimensional basic flow to the three-dimensional stationary flow is inertial in nature. Particular attention is paid to the secondary transition leading to oscillatory three-dimensional flow, and this mechanism is likewise independent of Pr. The spatial and temporal structure of the perturbation flow is analysed in detail and an instability mechanism is proposed based on energy balance calculations and the vorticity distribution.