Quasi–incompressible Cahn–Hilliard fluids and topological transitions

Abstract

Restricted accessMoreSectionsView PDF ToolsAdd to favoritesDownload CitationsTrack Citations ShareShare onFacebookTwitterLinked InRedditEmail Cite this article Lowengrub J. and Truskinovsky L. 1998Quasi–incompressible Cahn–Hilliard fluids and topological transitionsProc. R. Soc. Lond. A.4542617–2654http://doi.org/10.1098/rspa.1998.0273SectionRestricted accessQuasi–incompressible Cahn–Hilliard fluids and topological transitions J. Lowengrub J. Lowengrub Department of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA Google Scholar Find this author on PubMed Search for more papers by this author and L. Truskinovsky L. Truskinovsky Department of Aerospace Engineering and Mechanics, Minneapolis, MN 55455, USA Google Scholar Find this author on PubMed Search for more papers by this author J. Lowengrub J. Lowengrub Department of Mathematics, University of Minnesota, Minneapolis, MN 55455, USA Google Scholar Find this author on PubMed Search for more papers by this author and L. Truskinovsky L. Truskinovsky Department of Aerospace Engineering and Mechanics, Minneapolis, MN 55455, USA Google Scholar Find this author on PubMed Search for more papers by this author Published:08 October 1998https://doi.org/10.1098/rspa.1998.0273AbstractOne of the fundamental problems in simulating the motion of sharp interfaces between immiscible fluids is a description of the transition that occurs when the interfaces merge and reconnect. It is well known that classical methods involving sharp interfaces fail to describe this type of phenomena. Following some previous work in this area, we suggest a physically motivated regularization of the Euler equations which allows topological transitions to occur smoothly. In this model, the sharp interface is replaced by a narrow transition layer across which the fluids may mix. The model describes a flow of a binary mixture, and the internal structure of the interface is determined by both diffusion and motion. An advantage of our regularization is that it automatically yields a continuous description of surface tension, which can play an important role in topological transitions. An additional scalar field is introduced to describe the concentration of one of the fluid components and the resulting system of equations couples the Euler (or Navier–Stokes) and the Cahn–Hilliard equations. The model takes into account weak non–locality (dispersion) associated with an internal length scale and localized dissipation due to mixing. The non–locality introduces a dimensional surface energy; dissipation is added to handle the loss of regularity of solutions to the sharp interface equations and to provide a mechanism for topological changes. In particular, we study a non–trivial limit when both components are incompressible, the pressure is kinematic but the velocity field is non–solenoidal (quasi–incompressibility). To demonstrate the effects of quasi–incompressibility, we analyse the linear stage of spinodal decomposition in one dimension. We show that when the densities of the fluids are not perfectly matched, the evolution of the concentration field causes fluid motion even if the fluids are inviscid. In the limit of infinitely thin and well–separated interfacial layers, an appropriately scaled quasi–incompressible Euler–Cahn–Hilliard system converges to the classical sharp interface model. In order to investigate the behaviour of the model outside the range of parameters where the sharp interface approximation is sufficient, we consider a simple example of a change of topology and show that the model permits the transition to occur without an associated singularity. 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