N₂-instability in suspensions undergoing Couette flow

In this study, we consider the possibility of N₂-instability in suspensions, as proposed by Carpen and Brady [J. NonNewt. Fluid Mech. 102(2), 2002]. This transverse instability arises due to the jump in the second normal stress difference between two fluids. We perform a linear stability analysis in a two-layer Couette flow using a classical Morris-Boulay constitutive law for the suspension. Interestingly, this leads to an exact closed-form theoretical solution. Our results are slightly different from Carpen and Brady, suggesting that this instability depends much on the exact constitutive law for the suspension. We have also performed 3D numerical simulations of this instability in the linear regime and were able to validate our theoretical expressions. Although this two-layer Couette flow is appealing due to its simplicity, it is likely to show shear-induced resuspension and migration with time scales similar to the N₂-instability. This could possibly explain why it has not been definitely attested in experiments so far. We discuss configurations in which this instability could prevail over migration.