In this paper, the method, which is based on the results of numerical solution of Hill's variational principles, is presented for the prediction of the terminal falling velocity of spherical particles moving slowly through a Carreau viscosity model fluid. The suitability of this method is documented by the comparison of calculated data of terminal falling velocity with experimental data measured for the fall of glass, steel? and lead balls through polymer solutions characterised by different measure of shear-thinning and elasticity.