Dynamic free-surface deformations in thermocapillary liquid bridges

Flow-induced dynamic free-surface deformations in thermocapillary liquid bridges are investigated in the limit of small capillary number. The asymptotic solution in lowest order is calculated numerically. Dynamic deformations caused by steady axisymmetric flows and by time-dependent three-dimensional critical modes are considered for representative Prandtl numbers (Pr  =  0.02 and 4.38). The magnitude and phase of the dynamic surface-deformation wave relative to the magnitude of the temperature field of a hydrothermal wave are predicted and discussed. It is shown that the total dynamic deformation can be decomposed to elucidate the relative importance of the hydrodynamic pressure, the viscous stress, the change of the volume due to thermal expansion, the temperature-dependence of the surface tension, and the temperature-dependence of the hydrostatic pressure. Since the dynamic deformations are caused by flow fields which arise at lower orders of the capillary number, the leading-order dynamic deformations do not feed back to the leading-order thermocapillary flow. This conclusion holds for steady axisymmetric flows as well as for non-axisymmetric three-dimensional flows close to the critical onset of hydrothermal waves.