The commonly used incompressible phase field models for non-reactive,binary fluids,in which the Cahn-Hilliard equation is used for the transport of phase variables(volume fractions),conserve the total volume of each p...The commonly used incompressible phase field models for non-reactive,binary fluids,in which the Cahn-Hilliard equation is used for the transport of phase variables(volume fractions),conserve the total volume of each phase as well as the material volume,but do not conserve the mass of the fluid mixture when densities of two components are different.In this paper,we formulate the phase field theory for mixtures of two incompressible fluids,consistent with the quasi-compressible theory[28],to ensure conservation of mass and momentum for the fluid mixture in addition to conservation of volume for each fluid phase.In this formulation,the mass-average velocity is no longer divergence-free(solenoidal)when densities of two components in the mixture are not equal,making it a compressible model subject to an internal constraint.In one formulation of the compressible models with internal constraints(model 2),energy dissipation can be clearly established.An efficient numerical method is then devised to enforce this compressible internal constraint.Numerical simulations in confined geometries for both compressible and the incompressible models are carried out using spatially high order spectral methods to contrast the model predictions.Numerical comparisons show that(a)predictions by the two models agree qualitatively in the situation where the interfacial mixing layer is thin;and(b)predictions differ significantly in binary fluid mixtures undergoing mixing with a large mixing zone.The numerical study delineates the limitation of the commonly used incompressible phase field model using volume fractions and thereby cautions its predictive value in simulating well-mixed binary fluids.展开更多
We merge classical kinetic theories [M. Doi and S. F. Edwards, The Theoryof Polymer Dynamics, 1986] for viscous dispersions of rigid rods, extended to semi-flexibility [A. R. Khokhlov and A. N. Semenov, Macromolecules...We merge classical kinetic theories [M. Doi and S. F. Edwards, The Theoryof Polymer Dynamics, 1986] for viscous dispersions of rigid rods, extended to semi-flexibility [A. R. Khokhlov and A. N. Semenov, Macromolecules, 17 (1984), pp. 2678-2685], and for Rouse flexible chains to model the hydrodynamics of polymer nano-rodcomposites (PNCs). A mean-field potential for the polymer-rod interface provides thekey coupling between the two phases. We restrict this first study to two-dimensionalconformational space. We solve the coupled set of Smoluchowski equations for threebenchmark experiments. First we explore how rod semi-flexibility and the polymer-rod interface alter the Onsager equilibrium phase diagram. Then we determine mon-odomain phase behavior of PNCs for imposed simple elongation and shear, respec-tively. These results inform the effects that each phase has on the other as parametricstrengths of the interactions are varied in the context of the most basic rheological ex-periments.展开更多
基金partially supported by NSF grants DMS-0915066 and AFOSR FA9550-11-1-0328partially supported by NSF grants DMS-0819051,DMS-0908330,SC EPSCOR award+1 种基金the USC startup fundpartially supported by the ARO grant W911NF-09-1-0389 and the USC startup fund.
文摘The commonly used incompressible phase field models for non-reactive,binary fluids,in which the Cahn-Hilliard equation is used for the transport of phase variables(volume fractions),conserve the total volume of each phase as well as the material volume,but do not conserve the mass of the fluid mixture when densities of two components are different.In this paper,we formulate the phase field theory for mixtures of two incompressible fluids,consistent with the quasi-compressible theory[28],to ensure conservation of mass and momentum for the fluid mixture in addition to conservation of volume for each fluid phase.In this formulation,the mass-average velocity is no longer divergence-free(solenoidal)when densities of two components in the mixture are not equal,making it a compressible model subject to an internal constraint.In one formulation of the compressible models with internal constraints(model 2),energy dissipation can be clearly established.An efficient numerical method is then devised to enforce this compressible internal constraint.Numerical simulations in confined geometries for both compressible and the incompressible models are carried out using spatially high order spectral methods to contrast the model predictions.Numerical comparisons show that(a)predictions by the two models agree qualitatively in the situation where the interfacial mixing layer is thin;and(b)predictions differ significantly in binary fluid mixtures undergoing mixing with a large mixing zone.The numerical study delineates the limitation of the commonly used incompressible phase field model using volume fractions and thereby cautions its predictive value in simulating well-mixed binary fluids.
基金This research has been supported in part by the Air Force Office of Scientific Research,Air Force Materials Command,USAF,under grant number FA9550-06-1-0063,FA9550-08-1-0107 and the National Science Foundation through grants DMS-0605029,0626180,0548511 and 0604891,0724273 the Army Research Office contract 47089-MS-SR,and NASA URETI BIMat award No.NCC-1-0203.
文摘We merge classical kinetic theories [M. Doi and S. F. Edwards, The Theoryof Polymer Dynamics, 1986] for viscous dispersions of rigid rods, extended to semi-flexibility [A. R. Khokhlov and A. N. Semenov, Macromolecules, 17 (1984), pp. 2678-2685], and for Rouse flexible chains to model the hydrodynamics of polymer nano-rodcomposites (PNCs). A mean-field potential for the polymer-rod interface provides thekey coupling between the two phases. We restrict this first study to two-dimensionalconformational space. We solve the coupled set of Smoluchowski equations for threebenchmark experiments. First we explore how rod semi-flexibility and the polymer-rod interface alter the Onsager equilibrium phase diagram. Then we determine mon-odomain phase behavior of PNCs for imposed simple elongation and shear, respec-tively. These results inform the effects that each phase has on the other as parametricstrengths of the interactions are varied in the context of the most basic rheological ex-periments.
