The empirical equation for viscosity as a function of shear rate contains a hyperbolic cosine function and predicts infinite viscosity at either zero or infinite shear rate, with a minimum viscosity at an intermediate shear rate. It contains three parameters; each is a constant multiplied by a positive or negative power of the free volume fraction. This is defined as the difference between the critical particle volume fraction for random close-packing and the actual volume fraction augmented by the volume of a stabilizing layer around each particle. The equations are applied to data for acrylic dispersions in hydrocarbon. The critical volume fraction increases as the particle radius decreases because the particles are slightly deformable; the stabilizing layer thickness agrees with a literature value (6 nm) from a different method. Of the remaining three constants, the viscosity constant is independent of radius, showing it is determined by hydrodynamics; the time constant is proportional to the fifth power of the radius, probably reflecting Brownian motion; and the dimensionless constant, which is inversely proportional to the radius squared, is possibly associated with steric interaction forces. The reduction in viscosity caused by blending different particle sizes is described, by adding an excess term to the free volume fraction and then applying the empirical equations with an average particle radius intermediate between the number and weight averages. Discontinuous or abrupt shear thickening occurs, in very concentrated dispersions, when the shear rate is raised to a value roughly proportional to the sixth power of the free volume fraction; it may be associated with particle deformability and the rate of acceleration of the motion.