Binary Fluid Model Research Articles

Consistency-enhanced SAV BDF2 time-marching method with relaxation for the incompressible Cahn–Hilliard–Navier–Stokes binary fluid model

In recent years, the scalar auxiliary variable (SAV) approach is popular to design decoupled, linear, and energy dissipation-preserving scheme for phase-field models. By introducing an appropriate time-dependent auxiliary variable, the original equations are transformed into equivalent forms and the nonlinear and coupling terms can be easily handled in an explicit manner. The estimation of energy stability is easy to prove and the numerical implementation is efficient. By utilizing the “zero-energy-contribution” property and defining another time-dependent auxiliary variable, the SAV approach can be extended to develop linear and energy-stable schemes for various phase-field fluid models. However, the discrete version of energy stability resulting from SAV approach corresponds to a modified energy instead of the original one. Generally, the discrete modified energy and discrete original energy are not consistent. Moreover, the discrete auxiliary variable related to “zero-energy-contribution” property might be not consistent with its continuous value 1. To enhance the consistency, we herein develop a simple and practical relaxation technique for the SAV BDF2 scheme of Cahn–Hilliard–Navier–Stokes model. The relaxation technique does not obviously increase the computational costs and still leads to provable energy dissipation law. In each time step, the calculation is efficient because we only need to solve several elliptic equations with constant coefficients. Numerical experiments indicate that the proposed method has desired accuracy, energy stability, and capability for incompressible two-phase fluid flows.

Thermodynamically Consistent Models for Coupled Bulk and Surface Dynamics.

We present a constructive paradigm to derive thermodynamically consistent models coupling the bulk and surface dynamics hierarchically following the generalized Onsager principle. In the model, the bulk and surface thermodynamical variables are allowed to be different and the free energy of the model comprises the bulk, surface, and coupling energy, which can be weakly or strongly non-local. We illustrate the paradigm using a phase field model for binary materials and show that the model includes the existing thermodynamically consistent ones for the binary material system in the literature as special cases. In addition, we present a set of such phase field models for a few selected mobility operators and free energies to show how boundary dynamics impart changes to bulk dynamics and vice verse. As an example, we show numerically how reactive transport on the boundary impacts the dynamics in the bulk using a reactive transport model for binary reactive fluids by adopting a structure-preserving algorithm to solve the model equations in a rectangular domain.

A phase separation model for binary fluids with hereditary viscosity

We consider a diffuse interface model for an incompressible binary viscoelastic fluid flow. The model consists of the Navier–Stokes–Voigt equations where the instantaneous kinematic viscosity has been replaced by a memory term incorporating hereditary effects coupled with the Cahn–Hilliard equation with Flory–Huggins potential. The resulting system is subject to no‐slip condition for the (volume averaged) fluid velocity and to no‐flux boundary conditions for the order parameter and the chemical potential . We first establish the well‐posedness of the initial and boundary value problem. Then, taking advantage of an Ekman‐type damping, we obtain a dissipative estimate. Thus, we can define a dissipative dynamical system on a suitable phase space. Also, we show that any weak solution is such that regularizes in finite time, and in dimension two, it stays uniformly away from pure phases; that is, the instantaneous strict separation property holds. Finally, we prove the existence of the global attractor, and exploiting the strict separation property, we demonstrate that an exponential attractor exists in dimension two.

Molecular Dynamics Study of Wetting and Adsorption of Binary Mixtures of the Lennard-Jones Truncated and Shifted Fluid on a Planar Wall.

The wetting of surfaces is strongly influenced by adsorbate layers. Therefore, in this work, sessile drops and their interaction with adsorbate layers on surfaces were investigated by molecular dynamics simulations. Binary fluid model mixtures were considered. The two components of the fluid mixture have the same pure component parameters, but one component has a stronger and the other a weaker affinity to the surface. Furthermore, the unlike interactions between both components were varied. All interactions were described by the Lennard-Jones truncated and shifted potential with a cutoff radius of 2.5σ. The simulations were carried out at constant temperature for mixtures of different compositions. The parameters were varied systematically and chosen such that cases with partial wetting as well as cases with total wetting were obtained and the relation between the varied molecular parameters and the phenomenological behavior was elucidated. Data on the contact angle as well as on the mole fraction and thickness of the adsorbate layer were obtained, accompanied by information on liquid and gaseous bulk phases and the corresponding phase equilibrium. Also, the influence of the adsorbate layer on the wetting was studied: for a sufficiently thick adsorbate layer, the wall’s influence on the wetting vanishes, which is then only determined by the adsorbate layer.

Continuous phase transition and microstructure of charged AdS black hole with quintessence

Previously, the Maxwell equal-area law has been used to discuss the conditions satisfied by the phase transition of charged AdS black holes with cloud of string and quintessence, and it was concluded that black holes have phase transition similar to that of vdW system. The phase transition depends on the electric potential of the black hole and is not the one between a large black hole and a small black hole. On the basis of this result, we study the relation between the latent heat of the phase transition and the parameter of dark energy, and use the Landau continuous phase transition theory to discuss the critical phenomenon of the black hole with quintessence and give the critical exponent. By introducing the number density of the black hole molecules, some properties of the microstructure of black holes are studied in terms of a phase transition. It is found that the electric charge of the black hole and the normalization parameter related to the density of quintessence field play a key role in the phase transition. By constructing the binary fluid model of the black hole molecules, we also discuss the microstructure of charged AdS black holes with a cloud of strings and quintessence.

Microstructure and continuous phase transition of a Reissner-Nordstrom-AdS black hole

As is well known that RN-AdS black hole has a phase transition which is similar to that of van der Waals system. The phase transition depends on the electric potential of the black hole and is not the one between a large black hole and a small black hole. On this basis, we introduce a new order parameter and use the Landau continuous phase transition theory to discuss the critical phenomenon of RN-AdS black hole and give the critical exponent. By constructing the binary fluid model of black hole molecules, we investigate the microstructure of black holes. Furthermore, by studying the effect of the spacetime scalar curvature on the phase transition, we find that the charged and uncharged molecules of black holes are with different microstructure red which is like fermion gas and boson gas.

Large Eddy Simulation of liquid sheet breakup using a two-phase lattice Boltzmann method

A two-phase flow Large Eddy Simulation (LES) model is developed within the framework of phase-field lattice Boltzmann methods (LBM). The proposed model is based on a standard Smagorinsky model for subgrid closure and uses the Multiple-Relaxation-Time (MRT) collision operator. Relevant test cases are considered to demonstrate the capability and accuracy of the new model in dealing with two-phase flows at wide range of density contrasts. These include the Rayleigh–Taylor instability at two distinct density ratios, the breakup of a turbulent liquid column, and assisted breakup of a liquid layer. The validated model is then utilized to examine a liquid sheet emerging from a nozzle into a stagnant gas at different Reynolds numbers ranging from 1000 to 6000. The influence of jet injection velocity on the flow structure, variation of the jet centerline velocity, and jet area is also investigated. This work reports for the first time the extension of the Smagorinsky model for LES to the binary fluid model of Fakhari & Lee [1] and its application in simulation of turbulent liquid sheet breakup at the density ratios as high as 250.

The roles of wettability and surface tension in droplet formation during inkjet printing

This paper describes a lattice Boltzmann-based binary fluid model for inkjet printing. In this model, a time-dependent driving force is applied to actuate the droplet ejection. As a result, the actuation can be accurately controlled by adjusting the intensity and duration of the positive and negative forces, as well as the idle time. The present model was verified by reproducing the actual single droplet ejection process captured by fast imaging. This model was subsequently used to investigate droplet formation in piezoelectric inkjet printing. It was determined that the wettability of the nozzle inner wall and the surface tension of the ink are vital factors controlling the print quality and speed. Increasing the contact angle of the nozzle inner delays the droplet breakup time and reduces the droplet velocity. In contrast, higher surface tension values promote earlier droplet breakup and faster drop velocity. These results indicate that the hydrophilic modification of the nozzle inner wall and the choice of inks with high surface tensions will improve printing quality.

Mesoscopic model for binary fluids

We propose a model for studying binary fluids based on the mesoscopic molecular simulation technique known as multiparticle collision, where the space and state variables are continuous, and time is discrete. We include a repulsion rule to simulate segregation processes that does not require calculation of the interaction forces between particles, so binary fluids can be described on a mesoscopic scale. The model is conceptually simple and computationally efficient; it maintains Galilean invariance and conserves the mass and energy in the system at the micro- and macro-scale, whereas momentum is conserved globally. For a wide range of temperatures and densities, the model yields results in good agreement with the known properties of binary fluids, such as the density profile, interface width, phase separation, and phase growth. We also apply the model to the study of binary fluids in crowded environments with consistent results.

Model-based Analysis of Ammonia Generation in SCR-systems

The selective catalytic reduction (SCR) of nitrogenous oxides is the most promising technique to meet prospective emission regulations of cars. Due to its toxic potential the reducing agent ammonia cannot be stored in a car but it can be carried in the form of an urea-water-solution (UWS) which is injected into the hot exhaust gas and reacts to ammonia. This formation of ammonia upstream the catalytic converter is responsible for the efficiency of the complete SCR-process-chain. This paper deals with the numerical analysis and description of interactions between droplets of the urea-water-solution and the exhaust gas. For the description of the droplets' trajectories, a numeric model was developed, which considers, in contrast to previous surveys, the loss of the droplets' mass due to evaporation of water and thermal decomposition of urea. The model for the droplets' motion was extended by an evaporation model for binary fluids and additionally on a kinetic approach to describe the thermal decomposition. This model is able to determine the exact position of flying droplets, the droplets' velocity, the composition and its current temperature in a gas flow. These parameters are substantial for further considerations of the interactions between droplets and hot surfaces, e.g. wall wetting.

The sharp-interface limit of the Cahn–Hilliard/Navier–Stokes model for binary fluids

AbstractThe Cahn–Hilliard model is increasingly often being used in combination with the incompressible Navier–Stokes equation to describe unsteady binary fluids in a variety of applications ranging from turbulent two-phase flows to microfluidics. The thickness of the interface between the two bulk fluids and the mobility are the main parameters of the model. For real fluids they are usually too small to be directly used in numerical simulations. Several authors proposed criteria for the proper choice of interface thickness and mobility in order to reach the so-called ‘sharp-interface limit’. In this paper the problem is approached by a formal asymptotic expansion of the governing equations. It is shown that the mobility is an effective parameter to be chosen proportional to the square of the interface thickness. The theoretical results are confirmed by numerical simulations for two prototypal flows, namely capillary waves riding the interface and droplets coalescence. The numerical analysis of two different physical problems confirms the theoretical findings and establishes an optimal relationship between the effective parameters of the model.

Numerical investigation of head-on droplet collision with lattice Boltzmann method

In the present work, head-on droplet collision is numerically investigated with the lattice Boltzmann method (LBM). The simulation is based on the high density ratio LBM multiphase model for incompressible binary fluids proposed by Lee et al., and the model is implemented in the axisymmetric form. Compared with previous LBM-based numerical studies on droplet collision, density ratio is substantially increased. The investigation is conducted from aspects of effect of Weber number, effect of Ohnesorge number, effect of colliding velocity and unequal-size droplet collision. The evolution of total kinetic energy and cumulative dissipated energy are studied, and a very detailed picture of the collision process is presented and discussed with time-resolved evolution of the flow field, the kinetic energy distribution, and the viscous dissipation rate distribution, which provides a broader and more in-depth insight into the droplet dynamics and the energy transfer and dissipation process.

Modeling the Transport of Nanoparticle-Filled Binary Fluids through Micropores

Understanding the transport of multicomponent fluids through porous medium is of great importance for a number of technological applications, ranging from ink jet printing and the production of textiles to enhanced oil recovery. The process of capillary filling is relatively well understood for a single-component fluid; much less attention, however, has been devoted to investigating capillary filling processes that involve multiphase fluids, and especially nanoparticle-filled fluids. Here, we examine the behavior of binary fluids containing nanoparticles that are driven by capillary forces to fill well-defined pores or microchannels. To carry out these studies, we use a hybrid computational approach that combines the lattice Boltzmann model for binary fluids with a Brownian dynamics model for the nanoparticles. This hybrid approach allows us to capture the interactions among the fluids, nanoparticles, and pore walls. We show that the nanoparticles can dynamically alter the interfacial tension between the two fluids and the contact angle at the pore walls; this, in turn, strongly affects the dynamics of the capillary filling. We demonstrate that by tailoring the wetting properties of the nanoparticles, one can effectively control the filling velocities. Our findings provide fundamental insights into the dynamics of this complex multicomponent system, as well as potential guidelines for a number of technological processes that involve capillary filling with nanoparticles in porous media.

Three-dimensional lattice Boltzmann model for immiscible two-phase flow simulations

We present an improved three-dimensional 19-velocity lattice Boltzmann model for immisicible binary fluids with variable viscosity and density ratios. This model uses a perturbation step to generate the interfacial tension and a recoloring step to promote phase segregation and maintain surfaces. A generalized perturbation operator is derived using the concept of a continuum surface force together with the constraints of mass and momentum conservation. A theoretical expression for the interfacial tension is determined directly without any additional analysis and assumptions. The recoloring algorithm proposed by Latva-Kokko and Rothman is applied for phase segregation, which minimizes the spurious velocities and removes lattice pinning. This model is first validated against the Laplace law for a stationary bubble. It is found that the interfacial tension is predicted well for density ratios up to 1000. The model is then used to simulate droplet deformation and breakup in simple shear flow. We compute droplet deformation at small capillary numbers in the Stokes regime and find excellent agreement with the theoretical Taylor relation for the segregation parameter β=0.7. In the limit of creeping flow, droplet breakup occurs at a critical capillary number 0.35<Ca(c)<0.4 for the viscosity ratio of unity, consistent with previous numerical simulations and experiments. Droplet breakup can also be promoted by increasing the Reynolds number. Finally, we numerically investigate a single bubble rising under buoyancy force in viscous fluids for a wide range of Eötvös and Morton numbers. Numerical results are compared with theoretical predictions and experimental results, and satisfactory agreement is shown.

Phase-Field Model of Long-Time Glasslike Relaxation in Binary Fluid Mixtures

We present a new phase-field model for binary fluids, exhibiting typical signatures of soft-glassy behavior, such as long-time relaxation, aging, and long-term dynamical arrest. The present model allows the cost of building an interface to vanish locally within the interface, while preserving positivity of the overall surface tension. A crucial consequence of this property, which we prove analytically, is the emergence of free-energy minimizing density configurations, hereafter named "compactons," to denote their property of being localized to a finite-size region of space and strictly zero elsewhere (no tails). Thanks to compactness, any arbitrary superposition of compactons still is a free-energy minimizer, which provides a direct link between the complexity of the free-energy landscape and the morphological complexity of configurational space.

Derivation and thermodynamics of a lattice Boltzmann model with soluble amphiphilic surfactant

We derive a lattice Boltzmann model of a ternary fluid mixture, one component of which consists of amphiphilic molecules with a rotational degree of freedom. The model was first introduced elsewhere where it was constructed on the basis of the earlier Shan-Chen type lattice Boltzmann model for binary fluids. We provide a rigorous derivation of the model from an underlying continuum kinetic theory. In particular, we show how the model can be interpreted as a discretization of a Vlasov-Boltzmann type kinetic theory for a fluid composed of two species of oppositely charged monopoles and a composite, dipolar molecule. We also derive a free-energy functional for the model, including the contribution from the amphiphiles, and confirm that the free energy is lowered during phase separation and during the formation of a stable emulsion.

Rayleigh–Taylor instability in a viscoelastic binary fluid

The effects of polymer additives on Rayleigh–Taylor (RT) instability of immiscible fluids is investigated using the Oldroyd-B viscoelastic model. Analytic results obtained exploiting the phase-field approach show that in polymer solution the growth rate of the instability speeds up with elasticity (but remains slower than in the pure solvent case). Numerical simulations of the viscoelastic binary fluid model confirm this picture.

Acoustic lattice Boltzmann model for immiscible binary fluids with a species-dependent impedance

A lattice Boltzmann model is considered for immiscible binary fluids with species-dependent acoustic impedance. Acoustic waves and pulses are simulated within the model, and the reflection and transmission components at the fluid interface are found to compare well with theory. The model is then applied to simulate the acoustic field due to plane-wave propagation through a bubble.

Lattice Boltzmann model for simulating immiscible two-phase flows

The lattice Boltzmann equation is often promoted as a numerical simulation tool that is particularly suitable for predicting the flow of complex fluids. This paper develops a two-dimensional 9-velocity (D2Q9) lattice Boltzmann model for immiscible binary fluids with variable viscosities and density ratio using a single relaxation time for each fluid. In the macroscopic limit, this model is shown to recover the Navier–Stokes equations for two-phase flows. This is achieved by constructing a two-phase component of the collision operator that induces the appropriate surface tension term in the macroscopic equations. A theoretical expression for surface tension is determined. The validity of this analysis is confirmed by comparing numerical and theoretical predictions of surface tension as a function of density. The model is also shown to predict Laplace's law for surface tension and Poiseuille flow of layered immiscible binary fluids. The spinodal decomposition of two fluids of equal density but different viscosity is then studied. At equilibrium, the system comprises one large low viscosity bubble enclosed by the more viscous fluid in agreement with theoretical arguments of Renardy and Joseph (1993 Fundamentals of Two-Fluid Dynamics (New York: Springer)). Two other simulations, namely the non-equilibrium rod rest and the coalescence of two bubbles, are performed to show that this model can be used to simulate two fluids with a large density ratio.

Simulation of break-up behavior of immiscible droplet under shear field by means of continuous-velocity lattice gas cellular automata

Continuous-velocity lattice gas cellular automata method is applied to the simulation of the motion of a single droplet under a three-dimensional shear field. Immiscible binary fluid model with segregation intensity parameter is employed. A simple collision rule is introduced in order to describe the momentum transport from moving boundaries to the fluid. The calculations well simulate two types of the deformation and break-up behaviors, fracture and tipstreaming, of the droplet. The segregation intensity parameter of the fluid particles reasonably controls the surface tension of the droplet. The break-up behavior of the droplet is well correlated with the ratio of the shear strength and the segregation intensity parameter.