Nonmodal stability in Hagen-Poiseuille flow of a shear thinning fluid

Linear stability in Hagen-Poiseuille flow of a shear-thinning fluid is considered. The non-Newtonian viscosity is described by the Carreau rheological law. The effects of shear thinning on the stability are investigated using the energy method and the nonmodal stability theory. The energy analysis shows that the nonaxisymmetric disturbance with the azimuthal wave number m=1 has the lowest critical energy Reynolds number for both the Newtonian and shear-thinning cases. With the increase of shear thinning, the critical energy Reynolds number decreases for both the axisymmetric and nonaxisymmetric cases. For the nonmodal stability, we focus on two problems: response to external excitations and response to initial conditions. The former is studied by examining the ε pseudospectrum, and the latter by examining the energy growth function G(t). For both Newtonian and shear-thinning fluids, it is found that there can be a rather large transient growth even though the linear operator of the Hagen-Poiseuille flow has no unstable eigenvalue. For the problem of response to external excitations, the optimal response is achieved by disturbance with m=1 for both the Newtonian and non-Newtonian cases. For the problem of response to initial conditions, the optimal disturbance is in the form of streamwise uniform streaks. Being different from the Newtonian case, the azimuthal wave number of the optimal disturbance may be greater than 1 for strongly shear-thinning cases.