Using the lubrication approximation we investigate the self-similar axisymmetric flow of a power-law liquid towards a central circular cavity. It is shown that this problem has a self-similar solution of the second kind. The self-similarity exponent is found by solving a non-linear eigenvalue problem arising from the requirement that the integral curve that represents the solution must join the appropriate singular points in the phase plane of the governing equation. The eigenvalues for different values of the theological index are computed. Numerical integration of the equations allows us to determine the shape of the solution in terms of the physical variables. We make a detailed analysis of the influence of the rheology on the properties of the solutions. (C) 2009 Elsevier B.V. All rights reserved.