Crossing the bottleneck of rain formation

TL;DR

This paper presents a low-dimensional model to predict the time required for droplets to cross the growth bottleneck in rain formation, validated by experiments on binary fluid mixtures with good quantitative agreement.

Contribution

The study introduces a simple yet accurate model capturing droplet growth dynamics across the bottleneck in rain formation, incorporating key physical parameters and validated experimentally.

Findings

Model accurately predicts crossing time t for various conditions.

Good agreement between model and experiments on binary mixtures.

Droplet growth involves diffusive accretion and sedimentation collisions.

Abstract

The demixing of a binary fluid mixture, under gravity, is a two stage process. Initially droplets, or in general aggregates, grow diffusively by collecting supersaturation from the bulk phase. Subsequently, when the droplets have grown to a size, where their Peclet number is of order unity, buoyancy substantially enhances droplet growth. The dynamics approaches a finite-time singularity where the droplets are removed from the system by precipitation. The two growth regimes are separated by a bottleneck of minimal droplet growth. Here, we present a low-dimensional model addressing the time span required to cross the bottleneck, and we hence determine the time, \Delta t, from initial droplet growth to rainfall. Our prediction faithfully captures the dependence of \Delta t on the ramp rate of the droplet volume fraction, \xi, the droplet number density, the interfacial tension, the mass diffusion coefficient, the mass density contrast of the coexisting phases, and the viscosity of the bulk phase. The agreement of observations and the prediction is demonstrated for methanol/hexane and isobutoxyethanol/water mixtures where we determined \Delta t for a vast range of ramp rates, \xi, and temperatures. The very good quantitative agreement demonstrates that it is sufficient for binary mixtures to consider (i) droplet growth by diffusive accretion that relaxes supersaturation, and (ii) growth by collisions of sedimenting droplets. An analytical solution of the resulting model provides a quantitative description of the dependence of \Delta t on the ramp rate and the material constants. Extensions of the model that will admit a quantitative prediction of \Delta t in other settings are addressed.