Transport and mixing of miscible fluids inside a capillary tube

The dynamics of mixing of two miscible fluids has been widely modeled by Fick’s Law, while interfacial and surface tension effects were neglected. However, these effects are not negligible for the dynamics of mixing of miscible fluids when the interface persists for a long period before the two fluids are fully mixed. In addition, Fick’s Law is not suitable for the case of a large concentration gradient, which has been confirmed to fail to reproduce the experimental results of the dynamics of mixing of water and glycerol in a capillary tube. We propose a Phase-Field model, where the diffusion flux is modeled by the gradient of a chemical potential that consists of a single-well potential, to model the mixing, and we incorporate the gradient of the Phase-Field parameter to capture interfacial effects. When coupling this Phase-Field model to the Navier-Stokes equations, we reproduce similar behavior as in experiments, where the interface of two miscible fluids is more inclined when the capillary tube is wider, or the diffusivity is smaller. The long-term behavior of the displacement of the interface scales as t^2/3 at intermediate times and then as t^1/3 at later times, which is consistent with the experiments.