Abstract: Methods and apparatus for processing materials employ two cylindrical members, one mounted within the other, defining an annular processing chamber. Preferably, the outer member is stationery (stator), while the inner rotates (rotor). The radial spacing between the stator inner surface and the rotor outer surface is equal to or less than the back-to-back radial thicknesses of the two laminar boundary layers formed on the two surfaces by the material being processed. The surfaces are made smooth, as by buffing to a finish of not more than 10 microinches. This structure inhibits formation of Taylor vortices in the processing passage, which cause unstable flow and consequent incomplete mixing. Preferably, the relative velocity between rotor and stator surfaces is at least 1.2 meters per second. The surfaces may be coated with catalysts. Transducers may be provided to apply processing energy, such as microwave, light or ultrasonic waves, through the stator wall.