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30
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 ..."
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Cited by 101 (4 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
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 47 (7 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
On the theory and computation of surface tension: the elimination of parasitic currents through energy conservation in the secondgradient method
 J. Comput. Phys
, 2002
"... Errors in the computation of fluid flows with surface tension are examined. These errors cause large parasitic flows when the capillary number is large and have often been attributed to truncation error in underresolved interfacial regions. A study using the secondgradient method reveals that when ..."
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Cited by 28 (0 self)
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Errors in the computation of fluid flows with surface tension are examined. These errors cause large parasitic flows when the capillary number is large and have often been attributed to truncation error in underresolved interfacial regions. A study using the secondgradient method reveals that when truncation error is eliminated in the computation of energy exchanges between surface and kinetic energies so that energy is strictly conserved, the parasitic currents are reduced to roundoff. The results are based on general thermodynamic arguments and can be used to guide improvements in other methods, such as the continuumsurfaceforce (CSF) method, which is commonly used with the volumeoffluid (VOF) method. c ○ 2002 Elsevier Science (USA) 1.
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 28 (2 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...
Efficient Numerical Solution of the Density Profile Equation in Hydrodynamics
 J. Sci. Comp
"... Abstract We discuss the numerical treatment of a nonlinear second order boundary value problem in ordinary differential equations posed on an unbounded domain which represents the density profile equation for the description of the formation of microscopical bubbles in a nonhomogeneous fluid. For a ..."
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Cited by 24 (15 self)
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Abstract We discuss the numerical treatment of a nonlinear second order boundary value problem in ordinary differential equations posed on an unbounded domain which represents the density profile equation for the description of the formation of microscopical bubbles in a nonhomogeneous fluid. For an efficient numerical solution the problem is transformed to a finite interval and polynomial collocation is applied to the resulting boundary value problem with essential singularity. We demonstrate that this problem is wellposed and the involved collocation methods show their classical convergence order. Moreover, we investigate what problem statement yields favorable conditioning of the associated collocation equations. Thus, collocation methods provide a sound basis for the implementation of a standard code equipped with an a posteriori error estimate and an adaptive mesh selection procedure. We present a code based on these algorithmic components that we are currently developing especially for the numerical solution of singular boundary value problems of arbitrary, mixed order, which also admits to solve problems in an implicit formulation. Finally, we compare our approach to a solution method proposed in the literature and conclude that collocation is an easy to use, reliable and highly accurate way to solve problems of the present type.
A variational deduction of second gradient poroelasticity Part I: General theory
 J Mech Mater Struct
"... Second gradient theories have to be used to capture how local micro heterogeneities macroscopically affect the behavior of a continuum. In this paper a configurational space for a solid matrix filled by an unknown amount of fluid is introduced. The Euler–Lagrange equations valid for second gradient ..."
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Cited by 15 (6 self)
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Second gradient theories have to be used to capture how local micro heterogeneities macroscopically affect the behavior of a continuum. In this paper a configurational space for a solid matrix filled by an unknown amount of fluid is introduced. The Euler–Lagrange equations valid for second gradient poromechanics, generalizing those due to Biot, are deduced by means of a Lagrangian variational formulation. Starting from a generalized Clausius–Duhem inequality, valid in the framework of second gradient theories, the existence of a macroscopic solid skeleton Lagrangian deformation energy, depending on the solid strain and the Lagrangian fluid mass density as well as on their Lagrangian gradients, is proven. 1.
BOUNDARY CONDITIONS FOR A CAPILLARY FLUID IN CONTACT WITH A WALL
, 802
"... Contact of a fluid with a solid or an elastic wall is investigated. The wall exerts molecular forces on the fluid which is locally strongly nonhomogeneous. The problem is approached with a fluid energy of the second gradient form and a wall surface energy depending on the value of the fluid density ..."
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Cited by 7 (6 self)
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Contact of a fluid with a solid or an elastic wall is investigated. The wall exerts molecular forces on the fluid which is locally strongly nonhomogeneous. The problem is approached with a fluid energy of the second gradient form and a wall surface energy depending on the value of the fluid density at the contact. From the virtual work principle are obtained limit conditions taking into account the fluid density, its normal derivative to the wall and the curvature of the surface. 1.
A new approach for the limit to tree height using a liquid nanolayer model, Continuum Mech. Thermodyn
, 2008
"... Liquids in contact with solids are submitted to intermolecular forces inferring density gradients at the walls. The van der Waals forces make liquid heterogeneous, the stress tensor is not any more spherical as in homogeneous bulks and it is possible to obtain stable thin liquid films wetting vertic ..."
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Cited by 6 (6 self)
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Liquids in contact with solids are submitted to intermolecular forces inferring density gradients at the walls. The van der Waals forces make liquid heterogeneous, the stress tensor is not any more spherical as in homogeneous bulks and it is possible to obtain stable thin liquid films wetting vertical walls up to altitudes that incompressible fluid models are not forecasting. Application to micro tubes of xylem enables to understand why the ascent of sap is possible for very high trees like sequoias or giant eucalyptus. Key words: nanofilms, inhomogeneous liquids, van der Waals forces, ascent of sap, high trees. PACS: 68.65.k, 82.45.Mp, 87.10.+e, 87.15.Kg, 87.15.La 1
Numerical Solution of Singular Two Point BVPs
, 2008
"... An algorithm is described for the efficient numerical solution of singular twopoint boundary value problems. The algorithm is based on collocation at Gauss points, is applied directly to second order equations and uses a transformation of the independent variable to obtain extra smoothness if neede ..."
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Cited by 3 (3 self)
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An algorithm is described for the efficient numerical solution of singular twopoint boundary value problems. The algorithm is based on collocation at Gauss points, is applied directly to second order equations and uses a transformation of the independent variable to obtain extra smoothness if needed. Numerical comparisons between a code based on this approach and codes based on other options which have previously been thought of as possible alternatives such as collocation at Lobatto points or reduction to a first order system, are made and the efficiency of the new approach is clearly demonstrated by numerical results. 1
Liquid Nanofilms. A Mechanical Model for the Disjoining Pressure
, 904
"... Liquids in contact with solids are submitted to intermolecular forces making liquids heterogeneous and, in a mechanical model, the stress tensor is not any more spherical as in homogeneous bulks. The aim of this article is to show that a squaregradient functional taking into account the volume liqu ..."
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Cited by 3 (1 self)
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Liquids in contact with solids are submitted to intermolecular forces making liquids heterogeneous and, in a mechanical model, the stress tensor is not any more spherical as in homogeneous bulks. The aim of this article is to show that a squaregradient functional taking into account the volume liquid free energy corrected with two surface liquid density functionals is a mean field approximation allowing to study structures of very thin liquid nanofilms near plane solid walls. The model determines analytically the concept of disjoining pressure for liquid films of thicknesses of a very few number of nanometers and yields a behavior in good agreement with the shapes of experimental curves carried out by Derjaguin and his successors. Key words: Nanofilms; disjoining pressure; mechanical properties of thin films. PACS: 61.30.Hn; 61.46.w; 68.65.k. 1