  We explore the convergence rate of the Ka\v{c}anov iteration scheme for
different models of shear-thinning fluids, including Carreau and power-law type
explicit quasi-Newtonian constitutive laws. It is shown that the energy
difference contracts along the sequence generated by the iteration. In
addition, an a posteriori computable contraction factor is proposed, which
improves, on finite-dimensional Galerkin spaces, previously derived bounds on
the contraction factor in the context of the power-law model. Significantly,
this factor is shown to be independent of the choice of the cut-off parameters
whose use was proposed in the literature for the Ka\v{c}anov iteration applied
to the power-law model. Our analytical findings are confirmed by a series of
numerical experiments.
