In a fluid handling machine, such as a pump or motor, or a hydrostactic transmission, a presser body abuts at one end a rotor and forms at the other end outer and inner annular pressure chambers with a housing part. The outer periphery of the outer pressure chamber and the inner periphery of the inner pressure chamber are coaxial with the rotor, but slidingly engaged cylindrical face portions along the inner periphery of the outer pressure chamber and outer periphery of the inner pressure chamber have a common axis eccentric to, and parallel with the rotor axis so that the widths of the effective annular pressure faces of the outer and inner pressure chambers varies. Fluid at different pressures is located in the inner and outer pressure chambers, and pressure forces are transmitted from the pressure chambers to the rotor to press the rotor against a staionary control face of the housing.