An investigation into the stability of a shear thinning fluid

Experiments [T. Mullin, R.R. Kerswell (Eds.), Laminar Turbulent Transition and Finite Amplitude Solutions, Proceedings of the UITAM Symposium, Springer, Bristol, UK, Dodrecht, 2005.] show that transition to turbulence can start at Reynolds numbers lower than it is predicted by the linear stability analysis - the subcritical transition to turbulence. To give a possible qualitative explanation of these observations we suggest that the onset of subcritical instability can be related to failure of the fluid viscosity: friction between fluid layers drops with the increase of the velocity gradient. To describe the drop of friction theoretically we relax the assumption of the stability of the fluid material and introduce a constant of the fluid strength. Particularly, we enhance the Navier-Stokes model with a failure description by introducing the fluid strength in the constitutive equation for the viscous stress. The classical model is obtained from the enhanced one when the strength goes to infinity. We use the modified Navier-Stokes model to analyze the Couette flow between two parallel plates and find that the lateral perturbations can destabilize the flow and the critical Reynolds number is proportional to the fluid strength. The latter means that the classical Navier-Stokes model of a stable material with the infinite strength does not capture the subcritical transition to turbulence while the modified model does.