Anomalous mixing times in turbulent binary mixtures at high Prandtl number

A theory of turbulent binary fluid mixtures is applied to situations where the diffusive Prandtl number, $P=\frac{\nu }{D}$, 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 $\tau =\frac{t_{0}P}{R}$, where $t_0$ is a characteristic stirring time. The spectrum of concentration fluctuations rapidly acquires a peak at wave vector $k^{*}\sim k_{0}R$, where $k_0^{-1\mathrm{}}$ is the scale of the initial inhomogeneity. Mixing times of ten minutes or more are possible.