J. Fluid Mech. (2013), vol. 714, pp. 95–126. c  Cambridge University Press 2013 95 doi:10.1017/jfm.2012.461 The sharp-interface limit of the Cahn–Hilliard/Navier–Stokes model for binary fluids F. Magaletti 1 , F. Picano 1,2 , M. Chinappi 3 , L. Marino 1 and C. M. Casciola 1 † 1 Dipartimento di Ingegneria Meccanica e Aerospaziale, Universit` a di Roma La Sapienza, Via Eudossiana 18, 00184 Roma, Italy 2 Linn´ e Flow Center, KTH Mechanics, Osquars Backe 18, SE-100 44 Stockholm, Sweden 3 Dipartimento di Fisica, Universit` a di Roma La Sapienza, P. le Aldo Moro 5, 00185 Roma, Italy (Received 2 April 2012; revised 3 August 2012; accepted 14 September 2012) The Cahn–Hilliard model is increasingly often being used in combination with the incompressible Navier–Stokes equation to describe unsteady binary fluids in a variety of applications ranging from turbulent two-phase flows to microfluidics. The thickness of the interface between the two bulk fluids and the mobility are the main parameters of the model. For real fluids they are usually too small to be directly used in numerical simulations. Several authors proposed criteria for the proper choice of interface thickness and mobility in order to reach the so-called ‘sharp-interface limit’. In this paper the problem is approached by a formal asymptotic expansion of the governing equations. It is shown that the mobility is an effective parameter to be chosen proportional to the square of the interface thickness. The theoretical results are confirmed by numerical simulations for two prototypal flows, namely capillary waves riding the interface and droplets coalescence. The numerical analysis of two different physical problems confirms the theoretical findings and establishes an optimal relationship between the effective parameters of the model. Key words: capillary flows, drops and bubbles, diffuse interface methods, interfacial flows (free surface) 1. Introduction Binary flows appear in several natural phenomena and in a large number of engineering applications ranging from spray and aerosol formation in turbulent flows (Marmottant & Villermaux 2004) to droplet and bubble transport in microfluidic devices (Thorsen et al. 2001). In all of these problems surface tension acts on a very thin layer between the two bulk fluids (typically three or four times the characteristic size of the fluid molecules) and plays a crucial role on macroscopic dynamics. Two classes of approaches are used to describe binary flows: sharp- and diffuse- interface models. In the former, the narrow zone separating the two immiscible fluids is described as a discontinuity in the fluids properties. The classical Navier–Stokes equations govern the hydrodynamic behaviour on both sides of the interface and jump conditions on stresses and velocities are prescribed across the interface. † Email address for correspondence: carlomassimo.casciola@gmail.com