Measurements of rod climbing in Al give rise to an apparent linear relation between the height rise h and the angular velocity OMEGA of the rod. We use a White-Metzner model to fit the data and we find that the height rise on the rod deviates from the quadratic dependence on the angular velocity when the viscosity or relaxation time vary with shear rate. When both the relaxation time and the viscosity change simultaneously with the shear rate, the climbing profile on the rod deviates more from the standard theory for rod climbing. A simplified argument based on the upper-convected Maxwell model and using power laws for the viscosity and the relaxation time gives rise to h(OMEGA) infinity OMEGA(n+m), where n and m are power-law indices which can be chosen to fit the data. The height rise data for Al, which is linear rather than quadratic in the shear rate at low shears, can be fitted to a White-Metzner model using measured values for the viscosity function and a long time of relaxation which decreases strongly with the rate of shear. This result suggests that a shear-decreasing relaxation time function may be useful for describing the rheology of fluids with small quadratic range.