The dissolution of a miscible drop rising or falling in another liquid at low Reynolds number

TL;DR

This paper develops a theoretical framework to predict the behavior of miscible drops in another liquid at low Reynolds numbers, including velocity, volume, and composition, validated by experimental data.

Contribution

It introduces the first comprehensive theory for the evolution of miscible drops in another liquid at low Reynolds numbers, accounting for diffusion and initial conditions.

Findings

Universal scaling law for negligible diffusion cases

Closed-form solution for general diffusion cases

Validation against experimental data with water drops in syrup

Abstract

'A basic and basically unsolved problem in fluid dynamics is to determine the evolution of rising bubbles and falling drops of one miscible liquid in another' [1]. Here, we address this important literature gap and present the first theory predicting the velocity, volume and composition of such drops at low Reynolds numbers. For the case where the diffusion out of the drop is negligible, we obtain a universal scaling law. For the more general case where diffusion occurs into and out of the drop, the full dynamics is governed by a parameter-free first-order ordinary differential equation, whose closed form solution exists, and only depends on the initial condition. Our analysis depends primarily on 'drop-scale' effective parameters for the diffusivity through the interfacial boundary layer. We validate our results against experimental data for water drops suspended in syrup, corresponding to certain regimes of the mass exchange ratio between water and syrup, and by this explicitly identify the drop-scale parameters of the theory. [1] Joseph, D.D. and Renardy, Y.Y., 2013. Fundamentals of two-fluid dynamics: part II: lubricated transport, drops and miscible liquids (Vol. 4). Springer Science & Business Media.