The demixing of a two-component fluid can be understood as a gradient flow driven by interfacial energy and limited by viscous dissipation. Bounds on the steepness of the energy landscape translate into bounds on the demixing rate. In order to understand the steepness of the energy landscape one has to understand the distance ``in the large'' on configuration space given by the dissipation metric ``in the small''. It turns out that a transportation distance with logarithmic cost is a good proxy for this distance. This observation builds on a quantitative treatment of the DiPerna-Lions theory by DeLellis-Crippa.