Three-dimensional instability of a two-layer Dean flow

Stability of a two-layer Dean flow in a cylindrical annulus with respect to three-dimensional perturbations is studied by a global Galerkin method. It is shown that for large inner radius of the annulus (i) the instability becomes three-dimensional if one of the fluid layers is thin, (ii) its onset is not affected by possible small deformations of the interface, and (iii) multiple three-dimensional flow states are expected in a slightly supercritical flow regime. Stability diagrams and patterns of the three-dimensional perturbations are reported. It is concluded that even when the axisymmetric perturbation is the most dangerous, the resulting supercritical flow is expected to be three-dimensional. Possible multiplicity of supercritical three-dimensional states is predicted. The basis functions of the global Galerkin method are constructed so as to satisfy analytically the boundary conditions on no-slip walls and at the liquid-liquid interface. A modification of the numerical approach, accounting for small deformations of the interface which is subject to the action of the capillary force, is proposed. The results are of potential importance for development of novel bioseparators employing Dean vortices for enhancement of mass transfer of a passive scalar (say, a protein) through the interface. The developed numerical approach can be used for stability analysis in other two-fluid systems.