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.

在这项工作中，我们从理论上考虑了牛顿液滴在外部施加的温度梯度的影响下在静止的无限粘弹性流体中移动的问题。外部流体通过 Oldroyd-B 方程进行建模，并且针对较小的 Weissenberg 数和毛细管数，该问题通过双摄动展开式求解。我们假设微重力条件并忽略能量和动量的对流传输。我们根据两种流体的特性推导了液滴迁移速度及其形状的表达式。在没有形状变形的情况下，对于足够粘稠的内部流体，液滴速度单调减小，而对于具有较小内外部粘度比的流体，液滴速度作为魏森伯格数的函数先增大然后减小。对于毛细管数较小但有限的值，对于毛细管数和魏森伯格数的固定比率，液滴速度单调地表现为所施加的温度梯度的函数。我们证明，这种行为与聚合物应力使液滴在其迁移方向上变形有关，而其速度的相关变化本质上是牛顿的，与液滴的流体动力学阻力及其内部温度分布的变化有关。与数值模拟的结果相比，我们的理论对毛细管数和魏森伯格数足够小的值表现出良好的预测能力。