An exact solution for the thermocapillary motion of a Newtonian droplet in a viscoelastic fluid

The steady-state thermocapillary motion of a Newtonian droplet translating in an otherwise quiescent Oldroyd-B fluid has been investigated analytically in the framework of a perturbation technique and in the limit of small Deborah numbers. The analysis has been carried out assuming the absence of convective transport effects and decoupling the solution of the energy equation from the velocity field. Specific non-Newtonian correction formulae for the droplet migration velocity have been obtained in the limit of small Capillary numbers, i.e., assuming a spherical drop, as well as in the presence of small boundary deformations (small but finite Capillary numbers). Equations describing the droplet shape have also been obtained. The results show that, in the absence of deformation, the migration speed decreases monotonically with the Deborah number irrespective of the other parameters. In particular, when the viscosity and thermal conductivity of the drop are much smaller than the corresponding values for the continuous phase, the effect of elasticity becomes increasingly more important and the migration velocity is significantly decreased. When shape deformations are allowed, the velocity, evaluated as a function of the Deborah number, either initially increases with respect to the Newtonian value, or takes a behavior qualitatively similar to that observed for the spherical particle depending on the specific value of the viscosity ratio.