  In this work we consider theoretically the problem of a Newtonian droplet
moving in an otherwise quiescent infinite viscoelastic fluid under the
influence of an externally applied temperature gradient. The outer fluid is
modelled by the Oldroyd-B equation, and the problem is solved for small
Weissenberg and Capillary numbers in terms of a double perturbation expansion.
  We assume microgravity conditions and neglect the convective transport of
energy and momentum. We derive expressions for the droplet migration speed and
its shape in terms of the properties of both fluids. In the absence of shape
deformation, the droplet speed decreases monotonically for sufficiently viscous
inner fluids, while for fluids with a smaller inner-to-outer viscosity ratio,
the droplet speed first increases and then decreases as a function of the
Weissenberg number. For small but finite values of the Capillary number, the
droplet speed behaves monotonically as a function of the applied temperature
gradient for a fixed ratio of the Capillary and Weissenberg numbers. We
demonstrate that this behaviour is related to the polymeric stresses deforming
the droplet in the direction of its migration, while the associated changes in
its speed are Newtonian in nature, being related to a change in the droplet's
hydrodynamic resistance and its internal temperature distribution. When
compared to the results of numerical simulations, our theory exhibits a good
predictive power for sufficiently small values of the Capillary and Weissenberg
numbers.
