Present numerical investigation describes the thermo-capillary flow in a high Prandtl number liquid of 5 cSt silicone oil that evolves due to appearance of surface temperature gradients and interaction of the surface flow with the viscous stresses exerted by surrounding air motion. Steady axisymmetric and unsteady three-dimensional computations are carried out in the liquid and air domains by employing the finite volume method (FVM). The influence of ambient air temperature, surrounding air velocity, and the volume ratio on the evolution of stationary flow state inside the liquid bridge is investigated in detail. The present study reveals that interfacial heat transfer is found to have paramount importance in dictating the flow and thermal states inside the liquid zone, which is enhanced by the parallel air motion. The influence of various free surface heat transfer conditions on the bifurcation from the basic steady axisymmetric state to the 3D oscillatory state is discussed in terms of critical azimuthal wavenumber, m. The existence of both symmetric and asymmetric modes of the thermal convection for different values of Biot number, Bi is elaborately discussed. The dominant fluid dynamic information of 3D oscillatory state is extracted from a specific set of disturbance temperature data by employing a data-driven computational technique, ‘dynamic mode decomposition’ (DMD). The identification of the most energetic modes that govern the appearance of the oscillatory structures in the liquid zone is performed.