Linear stability analysis of pressure-driven channel flow of a Newtonian and a Herschel-Bulkley fluid

The linear stability of pressure-driven channel flow of a Newtonian layer past a non-Newtonian fluid is studied; the latter is assumed to possess a finite yield stress and to exhibit a power-law behaviour. Coupled Orr-Sommerfeld-type eigenvalue equations are derived and solved using a spectral collocation method in the absence of unyielded regions. The numerical solutions of these equations are in agreement with analytical predictions valid in the long-wave limit. Our results indicate that increasing the yield stress (prior to the formation of unyielded regions) and shear thickening tendency of the non-Newtonian fluid promote instability. An analysis of the disturbance `energy' illustrates the presence of an unstable, `interfacial' mode at all Reynolds numbers studied, and an additional, less unstable `shear' mode at relatively high Reynolds numbers. The influence of non-Newtonian rheology on the stability characteristics of these modes is elucidated.