The influence of axial flow on stability of the Taylor-Couette flow is carried for shear thinning fluids. The fluid is assumed to follow the Carreau-Bird model and mixed boundary conditions are imposed. The four-dimensional low-order dynamical system, resulted from Galerkin projection of the conservation of mass and momentum equations, includes additional nonlinear terms in the velocity components originated from the shear-dependent viscosity. In absence of axial flow the base flow loses its radial flow stability to the vortex structure at a lower critical Taylor number, as the shear thinning effects increases. The emergence of the vortices corresponds to the onset of a supercritical bifurcation which is also seen in the flow of a linear fluid. However, unlike the Newtonian case, shear-thinning Taylor vortices lose their stability as the Taylor number reaches a second critical number corresponding to the onset of a Hopf bifurcation. Existence of an axial flow, manifested by a pressure gradient appears to further advance each critical point on the bifurcation diagram. Complete flow field together with viscosity maps are given for different scenarios in the bifurcation diagram.

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