Viscous Taylor droplets in axisymmetric and planar tubes: from Bretherton's theory to empirical models

TL;DR

This paper develops and validates models for the dynamics of viscous droplets in capillaries, extending Bretherton's theory to account for viscosity ratios and higher capillary numbers, with implications for pressure drop predictions.

Contribution

It introduces new empirical models for droplet film thickness, curvature, and velocity that incorporate viscosity effects and capillary number ranges beyond classical theory.

Findings

Film thickness agrees with Bretherton's scaling for low viscosity ratios and small Ca.

At high viscosity ratios, film thickness saturates at a larger value.

Droplet velocity depends strongly on Ca and viscosity ratio.

Abstract

The aim of this study is to derive accurate models for quantities characterizing the dynamics of droplets of non-vanishing viscosity in capillaries. In particular, we propose models for the uniform-film thickness separating the droplet from the tube walls, for the droplet front and rear curvatures and pressure jumps, and for the droplet velocity in a range of capillary numbers, $Ca$, from $10^{-4}$ to $1$ and inner-to-outer viscosity ratios, $\lambda$, from $0$, i.e. a bubble, to high viscosity droplets. Theoretical asymptotic results obtained in the limit of small capillary number are combined with accurate numerical simulations at larger $Ca$. With these models at hand, we can compute the pressure drop induced by the droplet. The film thickness at low capillary numbers ($Ca<10^{-3}$) agrees well with Bretherton's scaling for bubbles as long as $\lambda<1$. For larger viscosity ratios, the film thickness increases monotonically, before saturating for $\lambda>10^3$ to a value $2^{2/3}$ times larger than the film thickness of a bubble. At larger capillary numbers, the film thickness follows the rational function proposed by Aussillous \& Qu\'er\'e (2000) for bubbles, with a fitting coefficient which is viscosity-ratio dependent. This coefficient modifies the value to which the film thickness saturates at large capillary numbers. The velocity of the droplet is found to be strongly dependent on the capillary number and viscosity ratio. We also show that the normal viscous stresses at the front and rear caps of the droplets cannot be neglected when calculating the pressure drop for $Ca>10^{-3}$.