This paper is concerned with the prediction of critical conditions for the behaviour of particles in simulated oil-drilling fluids. In practice, the rheology of many oil-drilling fluids can be represented adequately by a power-law model and the behaviour of spherical particles can give valuable information on the behaviour of less regularly shaped particles. In this paper, the how of a power-law fluid past a single sphere located on the axis of a cylindrical pipe is simulated numerically. Two different regimes are considered. In the settling regime, the spherical particle moves through a quiescent fluid. For this regime, drag coefficients for a wide range of power-law indices and particle Reynolds numbers are computed. Power-law indices range from n = 0.4 to n = 1 and Reynolds numbers (based upon the radius of the sphere and the settling velocity) are in the range 0.2 less than or equal to Re-p less than or equal to 100 for sphere/cylinder diameter ratios of 1/30 and 1/50. An expression is proposed which determines the drag coefficient in power-law fluids as a function of power-law index n and particle Reynolds number Re,. Numerical results are compared with both experimental results and other published numerical results. In the transport regime, the fluid is in motion relative to the pipe and the sphere, so that the fluid in the cylinder is sheared. Drag coefficients in the mid-Reynolds-number range for the transport regime are compared with those obtained in the settling regime. Finally, the numerical results have been presented in the form of two graphs which can be used to predict velocities of particles settling in power-law fluids and also to predict the maximum-sized particle which can be transported upwards by the flow of a power-law fluid.