Viscous Withdrawal of Miscible Liquid Layers

TL;DR

This paper develops a scaling law for viscous withdrawal of miscible liquid layers, accounting for global flow geometry to predict liquid entrainment more accurately.

Contribution

It introduces a new scaling law that incorporates global flow geometry, resolving degeneracy in local models of viscous withdrawal.

Findings

Scaling law accounts for global flow geometry effects.

Degeneracy in local models is resolved by including global information.

Logarithmic dependence on global parameters is demonstrated.

Abstract

In viscous withdrawal, a converging flow imposed in an upper layer of viscous liquid entrains liquid from a lower, stably stratified layer. Using the idea that a thin tendril is entrained by a local straining flow, we propose a scaling law for the volume flux of liquid entrained from miscible liquid layers. A long-wavelength model including only local information about the withdrawal flow is degenerate, with multiple tendril solutions for one withdrawal condition. Including information about the global geometry of the withdrawal flow removes the degeneracy while introducing only a logarithmic dependence on the global flow parameters into the scaling law.