Hydrodynamic Phase Field Model Research Articles

Thermodynamically consistent phase-field modelling of activated solute transport in binary solvent fluids

Active diffusion of substances in binary immiscible and incompressible fluids with different densities occurs universally in nature and industry, but relevant mathematical models and numerical simulation have been studied scarcely so far. In this paper, a thermodynamically consistent phase-field model is established to describe the activated solute transport in binary fluids with different densities. A mixed free-energy function for multiple solutes is proposed, which leads to different solute chemical potentials in binary solvent fluids, thus it has the ability to characterize the solubility difference of the solutes in two solvents. The two-phase flow is governed by a general hydrodynamic phase-field model that can account for general average velocity and different densities. The proposed model is derived rigorously using the second law of thermodynamics. Moreover, a general multi-component solute diffusion model is established using the Maxwell–Stefan approach, which involves the crossing influences between different solutes. To solve the model effectively, an efficient, linearized and decoupled numerical method is proposed for the model as well. The proposed numerical method can preserve the thermodynamical consistency, i.e. obeying an energy dissipation law at the discrete level, as well as guarantee the mass conservation law for the solutes and solvent fluids. Numerical experiments are carried out to show that the proposed model and numerical method can simulate various processes of the solute active diffusion in two-phase solvent fluids.

Quasi-incompressible models for binary fluid flows in porous media

We derive a quasi-incompressible hydrodynamic phase field model and consistent physical boundary conditions for flows of binary incompressible fluids of distinct intrinsic densities in porous media subject to an external force using the generalized Onsager principle (GOP). When the external force is conservative, the model not only conserves mass and volume fraction but also dissipates the total mechanical energy with respect to the consistent boundary conditions in a fixed domain. In the case of gravity, an extended family of hydrodynamical phase field models parametrized by a specific volume fraction parameter, ϕˆ, is derived via the GOP in the context of the buoyancy force, which reduces to the previous model at ϕˆ=0. In the case of a constant mobility and equal intrinsic densities in the binary fluid system, the extended model is identical to the original one for any ϕˆ. While ϕˆ is chosen as the spatially averaged volume fraction in the general case, the extended model may not be dissipative and thus different from the model at ϕˆ=0. Numerical examples are presented to highlight the difference between the two models and interfacial instability with respect to distinct densities when the two models are distinct.

Energy stable and mass conservative numerical method for a generalized hydrodynamic phase-field model with different densities

A general hydrodynamic phase-field model for two-phase flows with general average velocity and variable densities is rigorously derived using thermodynamic laws and Onsager’s variational principle. The pressure is naturally involved in both the Cahn–Hilliard equation and Navier–Stokes equation. The proposed model includes two famous phase-field models as its special cases. The model admits a natural energy dissipation law. A semi-implicit, totally linear, and energy stable numerical scheme is proposed for the model, which uses an intermediate velocity involving all driving forces, including surface tension, pressure, and gravity. The tight coupling relationship between pressure and velocity is decoupled. Another advantage of the proposed scheme is that the intermediate velocity allows us to preserve the mass conservation, and consequently, there is no need to impose any mass balance equation in the Navier–Stokes equation as usual. The discrete energy dissipation law is proved rigorously. Several numerical examples are simulated to demonstrate that the proposed method can preserve the energy stability and total mass conservation for complex hydrodynamical flow problems with large density contrasts and gravity.

A second order fully-discrete linear energy stable scheme for a binary compressible viscous fluid model

We present a linear, second order fully discrete numerical scheme on a staggered grid for a thermodynamically consistent hydrodynamic phase field model of binary compressible fluid flows, derived from the generalized Onsager Principle. The hydrodynamic model possesses not only the variational structure in its constitutive equation, but also warrants the mass, linear momentum conservation as well as energy dissipation. We first reformulate the model using the energy quadratization method into an equivalent form and then discretize the reformulated model to obtain a semidiscrete partial differential equation system using the Crank-Nicolson method in time. The semi-discrete numerical scheme preserves the mass conservation and energy dissipation law in time. Then, we discretize the semi-discrete PDE system on a staggered grid in space to arrive at a fully discrete scheme using 2nd order finite difference methods, which respects a discrete energy dissipation law. We prove the unique solvability of the linear system resulting from the fully discrete scheme. Mesh refinements and numerical examples on phase separation due to spinodal decomposition in binary polymeric fluids and interface evolution in the gas-liquid mixture are presented to show the convergence property and the usefulness of the new scheme in applications, respectively.

Linear second order in time energy stable schemes for hydrodynamic models of binary mixtures based on a spatially pseudospectral approximation

We develop two linear, second order energy stable schemes for solving the governing system of partial differential equations of a hydrodynamic phase field model of binary fluid mixtures. We first apply the Fourier pseudo-spectral approximation to the partial differential equations in space to obtain a semi-discrete, time-dependent, ordinary differential and algebraic equation (DAE) system, which preserves the energy dissipation law at the semi-discrete level. Then, we discretize the DAE system by the Crank-Nicolson (CN) and the second-order backward differentiation/extrapolation (BDF/EP) method in time, respectively, to obtain two fully discrete systems. We show that the CN method preserves the energy dissipation law while the BDF/EP method does not preserve it exactly but respects the energy dissipation property of the hydrodynamic model. The two new fully discrete schemes are linear, unconditional stable, second order accurate in time and high order in space, and uniquely solvable as linear systems. Numerical examples are presented to show the convergence property as well as the efficiency and accuracy of the new schemes in simulating mixing dynamics of binary polymeric solutions.

Second Order Fully Discrete Energy Stable Methods on Staggered Grids for Hydrodynamic Phase Field Models of Binary Viscous Fluids

We present second order, fully discrete, energy stable methods on spatially staggered grids for a hydrodynamic phase field model of binary viscous fluid mixtures in a confined geometry subject to both physical and periodic boundary conditions. We apply the energy quadratization strategy to develop a linear-implicit scheme. We then extend it to a decoupled, linear scheme by introducing an intermediate velocity term so that the phase variable, velocity field, and pressure can be solved sequentially. The two new, fully discrete linear schemes are then shown to be unconditionally energy stable, and the linear systems resulting from the schemes are proved uniquely solvable. Rates of convergence of the two linear schemes in both space and time are verified numerically. The decoupled scheme tends to introduce excessive dissipation compared to the coupled one. The coupled scheme is then used to simulate fluid drops of one fluid in the matrix of another fluid as well as mixing dynamics of binary polymeric, viscous solutions. The numerical results in mixing dynamics reveals the dramatic difference between the morphology in the simulations obtained using the two different boundary conditions (physical vs. periodic), demonstrating the importance of using proper boundary conditions in fluid dynamics simulations.

An energy stable algorithm for a quasi-incompressible hydrodynamic phase-field model of viscous fluid mixtures with variable densities and viscosities

A quasi-incompressible hydrodynamic phase field model for flows of fluid mixtures of two incompressible viscous fluids of distinct densities and viscosities is derived by using the generalized Onsager principle, which warrants the variational structure, the mass conservation and energy dissipation law. We recast the model in an equivalent form and discretize the equivalent system in space firstly to arrive at a time-dependent ordinary differential and algebraic equation (DAE) system, which preserves the mass conservation and energy dissipation law at the semi-discrete level. Then, we develop a temporal discretization scheme for the DAE system, where the mass conservation and the energy dissipation law are once again preserved at the fully discretized level. We prove that the fully discretized algorithm is unconditionally energy stable. Several numerical examples, including drop dynamics of viscous fluid drops immersed in another viscous fluid matrix and mixing dynamics of binary polymeric solutions, are presented to show the convergence property as well as the accuracy and efficiency of the new scheme.

Decoupled Energy Stable Schemes for a Phase Field Model of Three-Phase Incompressible Viscous Fluid Flow

We develop a numerical approximation for a hydrodynamic phase field model of three immiscible, incompressible viscous fluid phases. The model is derived from a generalized Onsager principle following an energetic variational formulation and is consisted of the momentum transport equation and coupled phase transport equations. It conserves the volume of each phase and warrants the total energy dissipation in time. Its numerical approximation is given by a set of easy-to-implement, semi-discrete, linear, decoupled elliptic equations at each time step, which can be solved efficiently using fast solvers. We prove that the scheme is energy stable. Mesh refinement tests and three numerical examples of three-phase viscous fluid flows in 3D are presented to benchmark the effectiveness of the model as well as the efficiency of the numerical scheme.

Fully Discretized Energy Stable Schemes for Hydrodynamic Equations Governing Two-Phase Viscous Fluid Flows

We develop systematically a numerical approximation strategy to discretize a hydrodynamic phase field model for a binary fluid mixture of two immiscible viscous fluids, derived using the generalized Onsager principle that warrants not only the variational structure but also the energy dissipation property. We first discretize the governing equations in space to arrive at a semi-discretized, time-dependent ordinary differential and algebraic equation (DAE) system in which a corresponding discrete energy dissipation law is preserved. Then, we discretize the DAE system in time to obtain a fully discretized system using a structure preserving finite difference method like the Crank---Nicolson method, which satisfies a fully discretized energy dissipation law. Alternatively, we solve the first order DAE system using the integration factor method after the algebraic equation is solved firstly. The integration factor method, which treats the linear, spatial derivative terms explicitly. Finally, two numerical examples are presented to compare the efficiency and accuracy of the two proposed methods.

Margination of white blood cells: a computational approach by a hydrodynamic phase field model

We numerically investigate margination of white blood cells and demonstrate the dependency on a number of conditions including haematocrit, the deformability of the cells and the Reynolds number. The approach, which is based on a mesoscopic hydrodynamic Helfrich-type model, reproduces previous results, e.g. a decreasing tendency for margination with increasing deformability and a non-monotonic dependency on haematocrit. The consideration of inertia effects, which may be of relevance in various parts of the cardiovascular system, indicates a decreasing tendency for margination with increasing Reynolds number. The effect is discussed by analysing inertial and non-inertial lift forces for single cells under different flow conditions and large-scale two-dimensional simulations of interacting red blood cells and white blood cells in an idealized blood vessel.

A decoupled energy stable scheme for a hydrodynamic phase-field model of mixtures of nematic liquid crystals and viscous fluids

We develop a linear, first-order, decoupled, energy-stable scheme for a binary hydrodynamic phase field model of mixtures of nematic liquid crystals and viscous fluids that satisfies an energy dissipation law. We show that the semi-discrete scheme in time satisfies an analogous, semi-discrete energy-dissipation law for any time-step and is therefore unconditionally stable. We then discretize the spatial operators in the scheme by a finite-difference method and implement the fully discrete scheme in a simplified version using CUDA on GPUs in 3 dimensions in space and time. Two numerical examples for rupture of nematic liquid crystal filaments immersed in a viscous fluid matrix are given, illustrating the effectiveness of this new scheme in resolving complex interfacial phenomena in free surface flows of nematic liquid crystals.

Disjoining potential and spreading of thin liquid layers in the diffuse-interface model coupled to hydrodynamics

The hydrodynamic phase field model is applied to the problem of film spreading on a solid surface. The disjoining potential, responsible for modification of the fluid properties near a three-phase contact line, is computed from the solvability conditions of the density field equation with appropriate boundary conditions imposed on the solid support. The equations describing the motion of a spreading film are derived in the lubrication approximation (in the limit of small contact angles). In the case of quasiequilibrium spreading, it is shown that the correct sharp-interface limit is obtained, and sample solutions are obtained by numerical integration. It is further shown that evaporation or condensation may strongly affect the dynamics near the contact line, and that it is necessary to account for kinetic retardation of the interphase transport to build up a consistent theory.

Quasi–incompressible Cahn–Hilliard fluids and topological transitions

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 non–locality (dispersion) associated with an internal length scale and localized dissipation due to mixing. The non–locality introduces a dimensional surface energy; dissipation is added to handle the loss of regularity of solutions to the sharp interface equations and to provide a mechanism for topological changes. In particular, we study a non–trivial limit when both components are incompressible, the pressure is kinematic but the velocity field is non–solenoidal (quasi–incompressibility). To demonstrate the effects of quasi–incompressibility, we analyse the linear stage of spinodal decomposition in one dimension. We show that when the densities of the fluids are not perfectly matched, the evolution of the concentration field causes fluid motion even if the fluids are inviscid. In the limit of infinitely thin and well–separated interfacial layers, an appropriately scaled quasi–incompressible Euler–Cahn–Hilliard system converges to the classical sharp interface model. In order to investigate the behaviour of the model outside the range of parameters where the sharp interface approximation is sufficient, we consider a simple example of a change of topology and show that the model permits the transition to occur without an associated singularity.