J. Fluid Mech. (2013), vol. 737, pp. 146–175. c  Cambridge University Press 2013 146 doi:10.1017/jfm.2013.550 Steady one-dimensional nozzle flow solutions of liquid–gas mixtures S. LeMartelot 1, †, R. Saurel 1,2 and O. Le Métayer 1 1 Aix-Marseille Universit´ e, CNRS, IUSTI UMR 7343, 13013 Marseille, France 2 RS2N, Bastidon de la Caou, 13360 Roquevaire, France (Received 15 February 2013; revised 25 September 2013; accepted 14 October 2013) Exact compressible one-dimensional nozzle flow solutions at steady state are determined in various limit situations of two-phase liquid–gas mixtures. First, the exact solution for a pure liquid nozzle flow is determined in the context of fluids governed by the compressible Euler equations and the ‘stiffened gas’ equation of state. It is an extension of the well-known ideal-gas steady nozzle flow solution. Various two-phase flow models are then addressed, all corresponding to limit situations of partial equilibrium among the phases. The first limit situation corresponds to the two-phase flow model of Kapila et al. (Phys. Fluids, vol. 13, 2001, pp. 3002–3024), where both phases evolve in mechanical equilibrium only. This model contains two entropies, two temperatures and non-conventional shock relations. The second one corresponds to a two-phase model where the phases evolve in both mechanical and thermal equilibrium. The last one corresponds to a model describing a liquid–vapour mixture in thermodynamic equilibrium. They all correspond to two-phase mixtures where the various relaxation effects are either stiff or absent. In all instances, the various flow regimes (subsonic, subsonic–supersonic, and supersonic with shock) are unambiguously determined, as well as various nozzle solution profiles. Key words: compressible flows, gas dynamics, multiphase flow 1. Introduction Multiphase nozzle flows are present in many fundamental and industrial areas, such as cooling systems, propulsion, safety analysis in pressured reactors or oil engineering, to cite a few. The aim of the present paper is to derive exact nozzle flow solutions for various limit models of two-phase flows. These solutions extend the one-dimensional ideal-gas nozzle steady flow solutions, detailed in any compressible fluid mechanics textbook, to various limit two-phase flow models. To be more precise, various reduced two-phase flow models are considered, each one of them corresponding to a limit situation where one or several relaxation effects are infinitely stiff. The first two-phase model considered in the present paper is a reduction of a well-known full non-equilibrium two-phase flow model (Baer and Nunziato) in the limit of stiff mechanical relaxation. The corresponding model was derived by Kapila et al. (2001) and describes multiphase mixtures out of thermal equilibrium but in velocity and † Email address for correspondence: sebastien.lemartelot@polytech.univ-mrs.fr