Linear stability of Taylor–Couette flow of shear-thinning fluids: modal and non-modal approaches

Abstract

In this paper, the response of circular Couette flow of shear-thinning fluids between two infinitely long coaxial cylinders to weak disturbances is addressed. It is highlighted by transient growth analysis. Both power-law and Carreau models are used to describe the rheological behaviour of the fluid. The first part of the paper deals with the asymptotic long-time behaviour of three-dimensional infinitesimal perturbations. Using the normal-mode approach, an eigenvalue problem is derived and solved by means of the spectral collocation method. An extensive description and the classification of eigenspectra are presented. The influence of shear-thinning effects on the critical Reynolds numbers as well as on the critical azimuthal and axial wavenumbers is analysed. It is shown that with a reference viscosity defined with the characteristic scales $\hat{{\it\mu}}_{ref}=\hat{K}(\hat{R}_{1}\hat{{\it\Omega}}_{1}/\hat{d})^{(n-1)}$ for a power-law fluid and $\hat{{\it\mu}}_{ref}=\hat{{\it\mu}}_{0}$ for a Carreau fluid, the shear-thinning character is destabilizing for counter-rotating cylinders. Moreover, the axial wavenumber increases with $\mathit{Re}_{2}$ and with shear-thinning effects. The second part investigates the short-time behaviour of the disturbance using the non-modal approach. For the same inner and outer Reynolds numbers, the amplification of the kinetic energy perturbation becomes much more important with increasing shear-thinning effects. Two different mechanisms are used to explain the transient growth, depending on whether or not there is a stratification of the angular momentum. On the Rayleigh line and for Newtonian fluids, the optimal perturbation is in the form of azimuthal streaks, which transform into Taylor vortices through the anti-lift-up mechanism. In the other cases, the optimal perturbation is initially oriented against the base flow, then it tilts to align with the base flow at optimal time. The scaling laws for the optimal energy amplification proposed in the literature for Newtonian fluids are extended to shear-thinning fluids.