A diffuse-interface method for simulating two-phase flows of complex fluids
TL;DR: In this paper, the authors proposed a diffuse-interface approach to simulating the flow of two-phase systems of microstructured complex fluids, where the energy law of the system guarantees the existence of a solution.
Abstract: Two-phase systems of microstructured complex fluids are an important class of engineering materials. Their flow behaviour is interesting because of the coupling among three disparate length scales: molecular or supra-molecular conformation inside each component, mesoscopic interfacial morphology and macroscopic hydrodynamics. In this paper, we propose a diffuse-interface approach to simulating the flow of such materials. The diffuse-interface model circumvents certain numerical difficulties in tracking the interface in the classical sharp-interface description. More importantly, our energy-based variational formalism makes it possible to incorporate complex rheology easily, as long as it is due to the evolution of a microstructure describable by a free energy. Thus, complex rheology and interfacial dynamics are treated in a unified framework. An additional advantage of our model is that the energy law of the system guarantees the existence of a solution. We will outline the general approach for any two-phase complex fluids, and then present, as an example, a detailed formulation for an emulsion of nematic drops in a Newtonian matrix. Using spectral discretizations, we compute shear-induced deformation, head-on collision and coalescence of drops where the matrix and drop phases are Newtonian or viscoelastic Oldroyd-B fluids. Numerical results are compared with previous studies as a validation of the theoretical model and numerical code. Finally, we simulate the retraction of an extended nematic drop in a Newtonian matrix as a method for measuring interfacial tension.
Citations
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TL;DR: In this paper, stability analyses and error estimates for a number of commonly used numerical schemes for the Allen-Cahn and Cahn-Hilliard equations were carried out and it was shown that all the schemes were either unconditionally energy stable, or reasonably stable with reasonable stability conditions in the semi-discretized versions.
Abstract: Stability analyses and error estimates are carried out for a number of commonly used
numerical schemes for the Allen-Cahn and Cahn-Hilliard equations. It is shown that
all the schemes we considered are either unconditionally energy stable, or
conditionally energy stable with reasonable stability conditions in the
semi-discretized versions. Error estimates for selected schemes with a
spectral-Galerkin approximation are also derived. The stability analyses and error
estimates are based on a weak formulation thus the results can be easily extended to
other spatial discretizations, such as Galerkin finite element methods, which are
based on a weak formulation.
734 citations
TL;DR: A comprehensive review of the lattice Boltzmann (LB) method for thermofluids and energy applications, focusing on multiphase flows, thermal flows and thermal multi-phase flows with phase change, is provided in this paper.
618 citations
Cites methods from "A diffuse-interface method for simu..."
...[65, 300, 306], the interface thickness is calculated from 0....
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TL;DR: Numerical results are presented to show that the accuracy and effectiveness of the SAV approach over the existing methods are superior.
596 citations
Cites background from "A diffuse-interface method for simu..."
...[26]), hence, this scheme is extremely efficient and easy to implement....
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...Moreover, it can be decoupled into two sequential second-order equations with constant coefficients [26]....
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...The total cost is essentially solving 2k four-order equations with constant coefficients, each of which can be further reduced to two decoupled Poisson type equations (cf [26])....
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TL;DR: The convective Cahn-Hilliard equation and the condition that the velocity field is divergence-free are derived from the conservation law of mass of binary mixtures in a straightforward way, for fluids with large density and viscosity ratios.
572 citations
Cites background from "A diffuse-interface method for simu..."
...Differences in the equations of motion with a previously derived quasi-incompressible model are shown to result from the respective assumptions made regarding the relationship between the diffuse fluxes of two species....
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TL;DR: In this article, the authors review the recent development of phase-field models and their numerical methods for multi-component fluid flows with interfacial phenomena, and provide practical applications to illustrate the usefulness of using a phasefield method.
Abstract: In this paper, we review the recent development of phase-field models and their numerical methods for multi-component fluid flows with interfacial phenomena. The models consist of a Navier-Stokes system coupled with a multi-component Cahn-Hilliard system through a phase-field dependent surface tension force, variable density and viscosity, and the advection term. The classical infinitely thin boundary of separation between two immiscible fluids is replaced by a transition region of a small but finite width, across which the composition of the mixture changes continuously. A constant level set of the phase-field is used to capture the interface between two immiscible fluids. Phase-field methods are capable of computing topological changes such as splitting and merging, and thus have been applied successfully to multi-component fluid flows involving large interface deformations. Practical applications are provided to illustrate the usefulness of using a phase-field method. Computational results of various experiments show the accuracy and effectiveness of phase-field models.
433 citations
Cites background from "A diffuse-interface method for simu..."
...Please refer to [105, 106] for more details about a diffuse-interface formulation of Newtonian and Oldroyd-B fluids....
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...The falling drop continues to stretch the thread, and eventually the Rayleigh instability leads to a pinchoff of the main drop [105]....
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References
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Book•
01 Jan 1986
TL;DR: In this article, the viscoelasticity of polymeric liquids was studied in the context of rigid rod-like polymers and concentrated solutions of rigid rods like polymers.
Abstract: Introduction Static properties of polymers Brownian motion Dynamics of flexible polymers in dilute solution Many chain systems Dynamics of a polymer in a fixed network Molecular theory for the viscoelasticity of polymeric liquids Dilute solutions of rigid rodlike polymers Semidilute solutions of rigid rodlike polymers Concentrated solutions of rigid rodlike polymers Index.
10,225 citations
Book•
01 Feb 1974
TL;DR: In this paper, the authors define an order parameter statistical theories of the nematic order phenomonological description of the nematic-isotopic mixtures and describe the properties of these mixtures.
Abstract: Part 1 Liquid crystals - main types and properties: introduction - what is a liquid crystal? the building blocks nematics and cholesterics smectics columnar phases more on long-, quasi-long and short-range order remarkable features of liquid crystals. Part 2 Long- and short-range order in nematics: definition of an order parameter statistical theories of the nematic order phenomonological description of the nematic-isotopic mixtures. Part 3 Static distortion in a nematic single crystal: principles of the continuum theory magnetic field effects electric field effects in an insulating nematic fluctuations in the alignment hydrostatics of nematics. Part 4 Defects and textures in nematics: observations disclination lines point disclinations walls under magnetic fields umbilics surface disclinations. Part 5 Dynamical properties of nematics: the equations of "nematodynamics" experiments measuring the Leslie co-efficients convective instabilities under electric fields molecular motions. Part 6 Cholesterics: optical properties of an ideal helix agents influencing the pitch dynamical properties textures and defects in cholesterics. Part 7 Smectics: symmetry of the main smectic phases continuum description of smectics A and C remarks on phase and precritical phenomena.
9,683 citations
TL;DR: In this article, it was shown that the thickness of the interface increases with increasing temperature and becomes infinite at the critical temperature Tc, and that at a temperature T just below Tc the interfacial free energy σ is proportional to (T c −T) 3 2.
Abstract: It is shown that the free energy of a volume V of an isotropic system of nonuniform composition or density is given by : NV∫V [f 0(c)+κ(▿c)2]dV, where NV is the number of molecules per unit volume, ▿c the composition or density gradient, f 0 the free energy per molecule of a homogeneous system, and κ a parameter which, in general, may be dependent on c and temperature, but for a regular solution is a constant which can be evaluated. This expression is used to determine the properties of a flat interface between two coexisting phases. In particular, we find that the thickness of the interface increases with increasing temperature and becomes infinite at the critical temperature Tc , and that at a temperature T just below Tc the interfacial free energy σ is proportional to (T c −T) 3 2 . The predicted interfacial free energy and its temperature dependence are found to be in agreement with existing experimental data. The possibility of using optical measurements of the interface thickness to provide an additional check of our treatment is briefly discussed.
8,720 citations
Additional excerpts
...Following Cahn & Hilliard (1958), we express the mixing-energy density as a function of φ and its gradient: fmix(φ, ∇φ) = 12λ|∇φ| 2 + f0(φ)....
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TL;DR: The renormalization group theory has been applied to a variety of dynamic critical phenomena, such as the phase separation of a symmetric binary fluid as mentioned in this paper, and it has been shown that it can explain available experimental data at the critical point of pure fluids, and binary mixtures, and at many magnetic phase transitions.
Abstract: An introductory review of the central ideas in the modern theory of dynamic critical phenomena is followed by a more detailed account of recent developments in the field. The concepts of the conventional theory, mode-coupling, scaling, universality, and the renormalization group are introduced and are illustrated in the context of a simple example---the phase separation of a symmetric binary fluid. The renormalization group is then developed in some detail, and applied to a variety of systems. The main dynamic universality classes are identified and characterized. It is found that the mode-coupling and renormalization group theories successfully explain available experimental data at the critical point of pure fluids, and binary mixtures, and at many magnetic phase transitions, but that a number of discrepancies exist with data at the superfluid transition of $^{4}\mathrm{He}$.
4,980 citations
"A diffuse-interface method for simu..." refers background in this paper
...It has found extensive use in studies of critical phenomena such as phase separation (Hohenberg & Halperin 1977)....
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...This idea can be traced to van der Waals (1892), and is the foundation for the phase-field theory for phase transition and critical phenomena (Hohenberg & Halperin 1977)....
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