Anomalous mixing times in turbulent binary mixtures at high Prandtl number

Abstract

A theory of turbulent binary fluid mixtures is applied to situations where the diffusive Prandtl number, $P=\frac{\ensuremath{\nu}}{D}$, far exceeds the Reynolds number $R$. This regime is accessible just above the consolute point in binary mixtures, where Prandtl numbers in excess of ${10}^{6}$ have been observed. We find that sizable large-scale inhomogeneities are mixed to uniformity quite slowly, in a time of order $\ensuremath{\tau}=\frac{{t}_{0}P}{R}$, where ${t}_{0}$ is a characteristic stirring time. The spectrum of concentration fluctuations rapidly acquires a peak at wave vector ${k}^{*}\ensuremath{\sim}{k}_{0}R$, where ${k}_{0}^{\ensuremath{-}1}$ is the scale of the initial inhomogeneity. Mixing times of ten minutes or more are possible.