Critical conditions for development of a second pair of Dean vortices in curved microfluidic channels

The centrifugal force in flow through a curved channel initiates a hydrodynamic instability that results in the development of Dean vortices, a pair of counter-rotating roll cells across the channel that deflect the high velocity fluid in the center toward the outer (concave) wall. When this secondary flow toward the concave (outer) wall is too strong to be dissipated by viscous effects, an additional pair of vortices emerges near the outer wall. Combining numerical simulation and dimensional analysis, we find that the critical condition for the onset of the second vortex pair depends on γ^{1/2}Dn (γ: channel aspect ratio; Dn: Dean number). We also investigate the development length for the additional vortex pair in channels with different aspect ratios and curvatures. The larger centrifugal force at higher Dean numbers produces the additional vortices further upstream, with the required development length being inversely proportional to the Reynolds number and increasing linearly with the radius of curvature of the channel.