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

Jan Martin Nordbotten, Endre Joachim Lerheim Mossige

Published 2022-11-02Version 1

The dissolution of a buoyant drop rising or falling through a viscous environment is a fundamental mixing problem; yet, it has enjoyed relatively little attention. In this work, we analyze the evolution of a freely suspended, miscible drop. 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 characterizes the drop size, velocity and composition, and 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.