A numerical investigation is performed for non Newtonian fluids flow between two concentric cylinders. The D2Q9 lattice Boltzmann model developed from the Bhatangar-Gross-Krook (LBGK) approximation is used to obtain the flow field for fluids obeying to the power-law model. The inner and outer cylinders rotate in the same and the opposite direction while the end walls are maintained at rest. The combined effects of the Reynolds number (Re) of the inner and outer cylinders, the radius ratio (η) as well as the power-law index (n) on the flow characteristics are analyzed for an annular space of a finite aspect ratio (Γ). Two flow modes are obtained: a primary mode (laminar stable regime) and a secondary mode (laminar unstable regime). The so obtained flow structures are different from one mode to another. The transition critical Reynolds number Rec from the primary to the secondary mode is analyzed for the co-courant and counter-courant flows. This critical value increases as n increases. The prediction of the swirling flow of non Newtonians fluids in axisymmetric geometries is shown in the present work., {"references":["G. I. Taylor, \"Stability of viscous liquid contained between two rotating\ncylinders\", Phil. Trans. R. Soc. London, A223:289–343, (1923).","R. C. Diprima and P.M. Eagles, \"The effect of radius ratio on the\nstability of Couette flow and Taylor vortex flow\", Phys. Fluids, 27,\n2403-2411, (1984).","H. -S. Dou, B. C. Khoo, K. S. Yeo, \"Energy loss distribution in the plane\nCouette flow and the Taylor–Couetteflow between concentric rotating\ncylinders\", Int. J. of Thermal Sciences 46, 262–275,(2007).","T. B. Benjamin, \"Bifurcation phenomena in steady flow of a viscous\nfluid Part I and II\", Proc. R. Soc. London, Ser. A 359, 1 (1978).","V. Sinevic, R. Kuboi, and A. W. Nienow, \"Power numbers, Taylor\nnumbers and Taylor vortices in viscous Newtonian and non-Newtonian\nfluids\", Chemical. Engineering. Science, 41:2915–2923, (1986).","S. T. Wereley, and R.M. Lueptow, \"Spatio-temporal character of nonwavy\nand wavy Taylor-Couette flow\", J. Fluid Mech. 364, 59–80,\n(1998).","T. J. Lockett, S. M. Richardson, and W. J. Worraker, \"The stability of\ninelastic non-Newtonian fluids in Couette flow between concentric\ncylinders: a finite element study\", J. Non-Newtonian Fluid Mech.\n43:165–177, (1992).","M. P. Escudier, I. W. Gouldson, and D. M. Jones, \"Taylor vortices in\nNewtonian and shear-thinning liquids\", Proceedings - Royal Society of\nLondon, A 449 (1935):155–176, (1995).","S. Khali, R. Nebbali, D. E. Ameziani, and K. Bouhadef, \"Numerical\ninvestigation of non-Newtonian fluids in annular ducts with finite aspect\nratio using lattice Boltzmann method\", Physical Review E 87, 053002\n(2013).\n[10] H. Huang, T. S. Lee, and C. Shu, \"Hybrid lattice Boltzmann finitedifference\nsimulation of axisymmetric swirling and rotating flows\", Int.\nJ. Num. Meth. Heat & Fluid Flow, Vol. 17, p. 587-607, (2007).\n[11] I. Halliday, and L. A. Hammond, \"Care, Enhanced closure scheme for\nlattice Boltzmann equation hydrodynamics\", J. Phys. A: Math. Gen, Vol.\n35, p. 157–166, (2002).\n[12] X. D. Niu, C. Y. Shu, and T. Chew, \"An axisymmetric lattice Boltzmann\nmodel for simulation of Taylor-Couette flows between two concentric\ncylinders\", Int. J. of Modern Physics, Vol. 14, No 6, p. 785–796,\n(2003).\n[13] A. Brahim, C. Lemaitre, C. Nouar, and N. Ait-Moussa, \"Revisiting the\nstability of circular Couette flow of shear-thinning fluids\", J. Non-\nNewtonian Fluid Mech. 183-184, 37 (2012)."]}