The Dean instability for shear-thinning fluids

• We consider the Dean instability for a generalised non-Newtonian fluid using an approximate power-law viscosity model.
• The model incorporates a low-shear Newtonian plateau region.
• The effect of channel aspect ratio is considered.
• The results differ significantly from those for a Newtonian fluid.
• The sensitivity of the results on the plateau region are considered.

We investigate the Dean instability for a generalised Newtonian fluid which satisfies an approximately power-law viscosity model, albeit modified to incorporate a low-shear Newtonian plateau. Infinite aspect ratio linear stability results are presented for both a narrow-gap width and a finite radius of curvature. These results reveal a surprising sensitivity to the details of the low-shear Newtonian region. Finite element solutions of the axisymmetric Navier–Stokes equations for flow through a finite aspect ratio duct confirm this sensitivity and, in addition, demonstrate the potential for hysteresis on the primary branch of vortices. A detailed bifurcation analysis over a range of the aspect ratio reveals that the nonlinear structure of the problem is qualitatively similar to that for a Newtonian fluid despite the apparently quite distinctive behaviour when a comparison is made at a fixed aspect ratio.