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

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” [D. D. Joseph and Y. Y. Renardy, Fundamentals of Two-Fluid Dynamics: Part II: Lubricated Transport, Drops and Miscible Liquids (Springer Science & Business Media, 2013), Vol. 4.]. 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 a 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.