Stabilities in plane Poiseuille flow of Herschel-Bulkley fluid

Linear stability in plane Poiseuille flow of a yield-stress shear-thinning fluid is considered. The rheological behavior of the fluid is described by the Herschel Bulkley model. The effect of shear-thinning on the stability is investigated using the energy method and the nonmodal stability theory. The result of the energy method shows that with the increase of shear thinning, the critical energy Reynolds number decreases for both the streamwise and spanwise disturbances. For the nonmodal stability, we focus on the response to initial conditions by examining the energy growth function G(t). For a Herschel Bulkley fluid, it is found that there can be a rather large transient growth even though the linear operator of the plan Poiseuille flow has no unstable eigenvalue. The results show that the shear thinning plays an important role in determining the energy growth rate and the structure of the disturbance with optimal transient growth.