This paper gives a synthesis of the expected individual and coupled effects, with the governing parameters, of two mechanical non-linearities (shear-thinning and yield stress) on stress reduction and flow rate enhancement due to longitudinal and transverse vibrations applied at the wall of two basic flows: Couette plane and Poiseuille pipe flows in conditions when the vibrational Reynolds number remains sufficiently small. Longitudinal vibrations may nevertherless show noticeable effects for both types of flows whereas transverse vibrations are mostly efficient for velocity-driven flows. Shear-thinning and yield stress effects are enhanced in all cases by their coupling like in Herschel-Bulkley models. In Couette plane flows, such coupling significantly increases stress reductions and decreases power-dissipated relative variations almost to limit values corresponding to Od = infinite or n = 0. In Poiseuille flows, decreasing n values from I to 0. 1 increases by up to one or two decades both flow rate enhancements and relative dissipated power variations due to yield stress effects. These theoretical results compare well with our experimental data on pipe extrusion Of uncured rubber compounds. A useful guide is then given for choosing suitable ultrasonic wall vibrations for industrial extruders in connection with the die design and behaviour of the extruded material. (c) 2007 Elsevier B.V. All rights reserved.