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Quasiincompressible Cahn–Hilliard fluids and topological transitions
 Proc. R. Soc. Lond. A
, 1998
"... One 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. F ..."
Abstract

Cited by 38 (3 self)
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One 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 nonlocality (dispersion) associated with an internal length scale and localized dissipation due
A phase field model for the mixture of two incompressible fluids and its approximation by a Fourierspectral method
 PHYSICA D
, 2003
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Conservative multigrid methods for Cahn–Hilliard fluids
 J. Comput. Phys
"... We develop a conservative, second order accurate fully implicit discretization in two dimensions of the NavierStokes NS and CahnHilliard CH system that has an associated discrete energy functional. This system provides a diffuseinterface description of binary fluid flows with compressible or inco ..."
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Cited by 15 (3 self)
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We develop a conservative, second order accurate fully implicit discretization in two dimensions of the NavierStokes NS and CahnHilliard CH system that has an associated discrete energy functional. This system provides a diffuseinterface description of binary fluid flows with compressible or incompressible flow components [44,4]. In this work, we focus on the case of flows containing two immiscible, incompressible and densitymatched components. The scheme, however, has a straightforward extension to multicomponent systems. To efficiently solve the discrete system at the implicit timelevel, we develop a nonlinear multigrid method to solve the CH equation which is then coupled to a projection method that is used to solve the NS equation. We analyze and prove convergence of the scheme in the absence of flow. We demonstrate convergence of our scheme numerically in both the presence and absence of flow and perform simulations of phase separation via spinodal decomposition. We examine the separate effects of surface tension and external flow on the decomposition. We find surface tension driven flow alone increases coalescence rates through the retraction of interfaces. When there is an external shear flow, the evolution
Modelling Pinchoff and Reconnection in a HeleShaw Cell I: The Models and their Calibration
, 2000
"... This is the first paper in a twopart series in which we analyze two model systems to study pinchoff and reconnection in binary fluid flow in a HeleShaw cell. The systems stem from a simplification of a general system of equations governing the motion of a binary fluid (NSCH model [69]) to flow ..."
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Cited by 10 (1 self)
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This is the first paper in a twopart series in which we analyze two model systems to study pinchoff and reconnection in binary fluid flow in a HeleShaw cell. The systems stem from a simplification of a general system of equations governing the motion of a binary fluid (NSCH model [69]) to flow in a HeleShaw cell. The system takes into account the chemical diffusivity between different components of a fluid mixture and the reactive stresses induced by inhomogeneity. In one of the systems we consider (HSCH), the binary fluid may be compressible due to diffusion. In the other system (BHSCH), a Boussinesq approximation is used and the fluid is incompressible. In this paper, we motivate, present and calibrate the HSCH/BHSCH equations so as to yield the classical sharp interface model as a limiting case. We then analyze their equilibria, one dimensional evolution and linear stability. In the second paper (Part II [66]), we analyze the behavior of the models in the fully nonline...
Retrieving topological information for phase field models
 SIAM J. Appl. Math
, 2005
"... Abstract. The phase field approach has become a popular tool in modeling interface motion, microstructure evolution, and more recently the shape transformation of vesicle membranes under elastic bending energy. While it is advantageous to employ phase field models in numerical simulations to automat ..."
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Cited by 7 (4 self)
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Abstract. The phase field approach has become a popular tool in modeling interface motion, microstructure evolution, and more recently the shape transformation of vesicle membranes under elastic bending energy. While it is advantageous to employ phase field models in numerical simulations to automatically handle topological changes to the microstructures or the configurations of vesicle membranes, detecting topological events may also become important for many applications such as those in the simulation of blood cells. Motivated by such considerations, a new quantity is formulated to retrieve some topological information based on the phase field formulation and to capture the occurrence of topological events. It can also be used as a control method to avoid unphysical changes of topology due to the numerical methods, should it become necessary for particular practical applications. Through numerical experiments, we demonstrate the effectiveness and the robustness of the new quantity in detecting the topology of fluid bubbles and vesicle membranes.