The heat transfer principle of power maximization in power plants with heat transfer irreversibilities is extended to fluid flow. It is shown that when a stream flows between two pressure reservoirs (P-1 > P-2) across linear flow resistances, a piston delivers maximum power when the pressure difference across its faces is (P-1-P-2)/2. The energy conversion efficiency at maximum power is eta(max) = (1/2) (1-P-2/P-1), as an analog to the efficiency for maximum power in power plants, eta(max) = 1-(T-2/T-1)(1/2). These results are generalized to fluid flow with nonlinear relations of pressure drop vs flow rate. Depending on overall size constraints, the power delivery can be further maximized by balancing the flow resistances upstream and downstream of the piston. The paper concludes with applications to steady-flow shaft-power components. It is shown that turbines can be optimized for maximum power output by selecting the inlet or outlet pressure drop, or the flowrate. Compressors and pumps do not have a power input minimum with respect to pressure drop or flowrate.