This paper describes experiments related to the settling of a small solid particle in quiescent non-Newtonian fluids which are confined in a circular duct. Measurements of the pressure drop created by the settling particle were conducted, in order to verify the validity of a conjecture by Brenner. This conjecture, already confirmed for Newtonian fluids, predicts that for a very small particle flowing in the Stokes and Oseen regimes, the pressure drop force would not be equal to the drag force on the sphere. as would be suggested by momentum considerations in the limiting situation of an "unbounded" fluid. The present experiments indicate that the conjecture is valid for the case of non-viscoelastic power law fluids. For fluids exhibiting normal stresses effects and a power-law viscosity function, the validity of Brenner’s results depends on the balance of normal and viscous stresses. The predictions seem to hold for situations where the effects of normal stresses are small compared to those of viscous stresses.