Linear stability of circular Couette flow of inelastic viscoplastic fluids

Results for the fully non-axisymetric, wide gap, linear stability of the circular Couette flow for two models of inelastic viscoplastic fluids are presented.1 For a given rheological behaviour, an analytical technique is used to obtain the matrix problem associated with the linear stability. This problem is then solved using a standard collocation technique. For power law fluids, as long as the index nn is larger than 0.2, the first unstable regime is axisymetric, and the non newtonian behaviour of the fluid has a stabilizing influence on the flow. For indexes smaller than 0.2, however, a new non-axisymmetric regime is obtained, consisting of a large number of columnar vortices. This flow is similar to those observed in some granular and fiber suspension flows. The same stability analysis is performed for a regularized Bingham model, with similar findings. Indeed, when the Bingham number is not too large, the first unstable regime is again axisymetric. When the Bingham becomes large enough, the columnar vortices regime is obtained as in the power law case. More generally, the stability of the circular Couette flow is little dependent on the details of the rheological laws. The stability results using a regularized Bingham case not only compare well to the results using a non-regularized Bingham model, but they also compare qualitatively well to the flow stability of a power law fluid with small indexes. Finally, a scaling analysis of the linear stability in the Bingham case indicates that for these fluids, the instability occurs when the supplied mechanical energy is larger than the energy necessary to “fluidize” the material.