Creeping thermocapillary motion of a Newtonian droplet suspended in a viscoelastic fluid

TL;DR

This paper presents a theoretical analysis of the creeping thermocapillary motion of a Newtonian droplet in a viscoelastic fluid, deriving expressions for droplet speed and shape under microgravity and small deformation conditions.

Contribution

The study develops a perturbation-based theoretical model for droplet motion in viscoelastic fluids, incorporating Oldroyd-B fluid dynamics and thermocapillary effects, with validation against numerical simulations.

Findings

Droplet speed decreases monotonically with viscosity for large inner fluid viscosity.

For smaller viscosity ratios, droplet speed first increases then decreases with Weissenberg number.

Droplet speed varies monotonically with temperature gradient at small Capillary numbers.

Abstract

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.