Cahn-Hilliard model for the simulation of unsteady binary flows F. Picano 1 , F. Magaletti 1 , M. Chinappi 2 , L. Marino 1 and C. M. Casciola 1 1 Dipartimento di Ingegneria Meccanica e Aerospaziale, University of Rome “La Sapienza”, Italy E-mail: francesco.picano@uniroma1.it 2 Dipartimento di Scienze Biochimiche “A. Rossi Fanelli”, University of Rome “La Sapienza”, Italy Keywords: binary mixture, multiphase flows, diffuse interface methods, surface tension. SUMMARY. The Cahn-Hilliard model is used in combination with the incompressible Navier- Stokes equations to describe capillary waves in binary fluids. The interface thickness ǫ between the two bulk fluids and the mobility M are the main parameters in Cahn-Hilliard model and for real fluids they are usually so small that they cannot be directly used in numerical simulations. In prac- tice, they are taken small enough that the corresponding results become physically sound, ensuring that a further decrease of ǫ and M do not change the solution, in the spirit of having reached the so called “sharp interface” asymptotic solution. Actually, for unsteady binary flows, an exhaustive criterion for selecting the mobility and the interface thickness is still lacking. This issue is here ad- dressed by discussing a suitable asymptotic expansion of the Cahn-Hilliard/Navier-Stokes equations in the context of capillary waves. It is found that the sharp interface limit can be approached with different scaling laws, M ∝ ǫ α with α ≤ 3. By comparing numerical solutions with the correspond- ing theoretical, sharp interface solutions, it is shown that the convergence to the sharp interface limit can be obtained with all the scaling laws proposed. Nonetheless, from the numerical point of view, the most efficient strategy is found to be M ∝ ǫ 3 . 1 INTRODUCTION Binary flows appear in several engineering applications ranging from spray and aerosol forma- tion in turbulent flows to droplet and bubble transport in micro-fluidic devices [1]. In all these problems surface tension, acting on a very thin layer (3-4 molecular distances) between the two bulk fluids, plays a crucial role on macroscopic dynamics. Two classes of approaches have been used to describe these kind of problems: sharp interface models and diffuse interface models. The former evolve a sharp interface (e.g. front tracking [2]) with the surface tension applied as a concentrated force on the interface and integrated on the Eulerian mesh of the flow. More recently, due to their ability to handle strong topological changes, diffuse interface methods are employed more and more frequently, [3, 4, 5]. These models evolve on the Eulerian mesh a diffuse interface endowed with a distributed volume force, corresponding to the surface tension. Among diffuse interface models, the Cahn-Hilliard equation combined with the incompressible Navier-Stokes equations, see e.g. [3, 6], is especially significant, given its direct derivation from thermodynamic principles. The central object is the scalar phase field Φ(x,t), with x and t position and time, respectively, which describes the binary system, with the two pure fluids corresponding to Φ=1, Φ= −1. The fluids are immiscible, implying that the flow domain is essentially partitioned into disjoined subdomains, each occupied by a single pure fluid, separated by an extremely narrow region, the diffused interface, where −1 < Φ < 1 and the fluids are partially mixed. In fact, in ordinary conditions, the physical thickness of the interface is on the nanometer scale and the model can be numerically exploited for micro and macro scale applications only at the price of an artificial thickening of the interface. 1