A new indirect method of determining the rheological behavior of a non-Newtonian fluid, modeled by a two parameter power law, flowing in a tube with a geometrical singularity is proposed which makes it possible to indirectly estimate the rheology. Due to this singularity, the shape of the velocity profiles can be strongly dependent on the fluid rheology, which makes it possible to obtain indirectly the required viscosity estimation. Non-Newtonian fluid flows were simulated numerically with various flow conditions. Based on the results of these calculations, plots were drawn emphasizing the dependence of the dimensionless velocity on the dimensionless rheological parameters. On the other hand, dimensionless velocities were also obtained by measuring velocity profiles with an ultrasonic Doppler velocimeter. These experimental values were put on the plots and the rheological parameters of the actual fluid were thus determined. Comparisons were made with viscometer measurements in order to assess the accuracy of the method and its range of validity. This method is of great potential interest for application to industrial processes when it is necessary to know the viscosity of a fluid undergoing a transformation without interrupting the process by taking fluid samples.