A new variable-density plastic flow model is developed, in which the Drucker Prager yield condition holds identically, but the corresponding flow condition contains the time derivative of density (or the divergence of mass flux), in order to satisfy mass conservation. This "softening" model is applied to the steady radial flow of a cohesionless granular material from steep-walled wedge and conical hopper. Density is assumed to vary with pressure. The variation of density within the hopper is shown to decrease the mass discharge rate, relative to the incompressible model, by a similar amount to the fractional reduction in voidage about the orifice. The predicted mass discharge decreased with increasing internal friction angle. This paper assumed that the inclination of the stagnant region in hopper flow is described by regression curves fitted to data from Brown and Richards. Approximate agreement between the theory of this paper and voidage measurements by Fickie et al. was obtained. Approximate agreement was also obtained with the published mass discharge rates of Nedderman and Beverloo for wedge and conical hoppers, respectively, and our results were insensitive to variations in internal angles of friction between about 25degrees and 35degrees. The steady equations considered here can only be satisfied approximately, supporting observations that granular flows are intrinsically transient. (C) 2003 Elsevier Ltd. All rights reserved.