||A theoretical analysis of the fast transients of turbomachineries, based on the study of unsteady and incompressible fluids mechanics equations applied to an impeller, is proposed. It leads to internal torque, internal power, and impeller head of an impeller during transient periods. The equations show that the behavior of a pump impeller is not only depending on the acceleration rate and flow rate, as it is usually admitted, but also on velocity profiles and their evolution during the transient. Some hypotheses on the flow in a radial flow pump are proposed. They are validated by comparison with the experimental results of a single stage, single volute radial flow pump during some fast acceleration periods. The model is also used to analyze the behavior of the pump during a fast startup.