Capturing nonlinear dynamics of two-fluid Couette flows with asymptotic models

The nonlinear stability of two-fluid Couette flows is studied using a novel evolution equation whose dynamics are validated by direct numerical simulations (DNS). The evolution equation incorporates inertial effects at arbitrary Reynolds numbers through a non-local term arising from the coupling between the two fluid regions, and is valid when one of the layers is thin. The equation predicts asymmetric solutions and exhibits bistability, features that are essential observations in the experiments of Barthelet et al. (1995). Related low-inertia models have been used in qualitative predictions rather than the direct comparisons carried out here, and ad hoc modifications appear to be necessary in order to predict asymmetry and bistability. Comparisons between model solutions and DNS show excellent agreement at Reynolds numbers of O(10³) found in the experiments. Direct comparisons are also made with the available experimental results of Barthelet et al. (1995) when the thin layer occupies 1/5 of the channel height. Pointwise comparisons of the travelling wave shapes are carried out and once again the agreement is very good.