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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
Computer simulation of glioma growth and morphology. Neuroimage 37
 Suppl
, 2007
"... Despite major advances in the study of glioma, the quantitative links between intratumor molecular/cellular properties, clinically observable properties such as morphology, and critical tumor behaviors such as growth and invasiveness remain unclear, hampering more effective coupling of tumor physic ..."
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Cited by 10 (0 self)
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Despite major advances in the study of glioma, the quantitative links between intratumor molecular/cellular properties, clinically observable properties such as morphology, and critical tumor behaviors such as growth and invasiveness remain unclear, hampering more effective coupling of tumor physical characteristics with implications for prognosis and therapy. Although molecular biology, histopathology, and radiological imaging are employed in this endeavor, studies are severely challenged by the multitude of different physical scales involved in tumor growth, i.e., from molecular nanoscale to cell microscale and finally to tissue centimeter scale. Consequently, it is often difficult to determine the underlying dynamics across dimensions. New techniques are needed to tackle these issues. Here, we address this multiscalar problem by employing a novel predictive threedimensional mathematical and computational model based on firstprinciple equations (conservation laws of physics) that describe mathematically
An Efficient, Energy Stable Scheme for the CahnHilliardBrinkman System
"... We present an unconditionally energy stable and uniquely solvable finite difference scheme for the CahnHilliardBrinkman (CHB) system, which is comprised of a CahnHilliardtype diffusion equation and a generalized Brinkman equation modeling fluid flow. The CHB system is a generalization of the Cah ..."
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We present an unconditionally energy stable and uniquely solvable finite difference scheme for the CahnHilliardBrinkman (CHB) system, which is comprised of a CahnHilliardtype diffusion equation and a generalized Brinkman equation modeling fluid flow. The CHB system is a generalization of the CahnHilliardStokes model and describes two phase very viscous flows in porous media. The scheme is based on a convex splitting of the discrete CH energy and is semiimplicit. The equations at the implicit time level are nonlinear, but we prove that they represent the gradient of a strictly convex functional and are therefore uniquely solvable, regardless of time stepsize. Owing to energy stability, we show that the scheme is stable in the time and space discrete ℓ ∞ ( 0, T; H1) 2 h and ℓ ( 0, T; H2) h norms. We also present an efficient, practical nonlinear multigrid method – comprised of a standard FAS method for the CahnHilliard part, and a method based on the Vanka smoothing strategy for the Brinkman part – for solving these equations. In particular, we provide evidence that the solver has nearly optimal complexity in typical situations. The solver is applied to simulate spinodal decomposition of a viscous fluid in a porous medium, as well as to the more general problems of buoyancy and boundarydriven flows.
unknown title
, 2006
"... Under consideration for publication in J. Fluid Mech. 1 Capillary spreading of a droplet in the partially wetting regime using a diffuseinterface model ..."
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Under consideration for publication in J. Fluid Mech. 1 Capillary spreading of a droplet in the partially wetting regime using a diffuseinterface model
www.elsevier.com/locate/ces On scaling of diffuse–interface models �
, 2005
"... Application of diffuse–interface models (DIM) yields a system of partial differential equations (PDEs) that generally requires a numerical solution. In the analyses of multiphase flows with DIM usually an artificial enlargement of the interface thickness is required for numerical reasons. Replacing ..."
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Application of diffuse–interface models (DIM) yields a system of partial differential equations (PDEs) that generally requires a numerical solution. In the analyses of multiphase flows with DIM usually an artificial enlargement of the interface thickness is required for numerical reasons. Replacing the real interface thickness with a numerically acceptable one, while keeping the surface tension constant, can be justified based on the analysis of the equilibrium planar interface, but demands a change in the local part of the free energy. In a nonequilibrium situation, where the interface position and shape evolve with time, we need to know how to change the mobility in order to still model the same physical problem. Here we approach this question by studying the mixing of two immiscible fluids in a liddriven cavity flow where the interface between the two fluids is stretched roughly linearly with time, before breakup events start. Scaling based on heuristics, where the mobility is taken inversely proportional to the interface thickness, was found to give fairly well results over the period of linear interface stretching for the range of Péclet numbers and viscosity ratios considered when the capillary number is O(10). None of the scalings studied was, however, able to capture the breakup events accurately.
unknown title
, 2004
"... www.elsevier.com/locate/jcp A continuous surface tension force formulation for diffuseinterface models ..."
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www.elsevier.com/locate/jcp A continuous surface tension force formulation for diffuseinterface models