Quasi-incompressible Model Research Articles

A Multiscale Poromechanics Model Integrating Myocardial Perfusion and the Epicardial Coronary Vessels

The importance of myocardial perfusion at the outset of cardiac disease remains largely understudied. To address this topic we present a mathematical model that considers the systemic circulation, the coronary vessels, the myocardium, and the interactions among these components. The core of the whole model is the description of the myocardium as a multicompartment poromechanics system. A novel decomposition of the poroelastic Helmholtz potential involved in the poromechanics model allows for a quasi-incompressible model that adequately describes the physical interaction among all components in the porous medium. We further provide a rigorous mathematical analysis that gives guidelines for the choice of the Helmholtz potential. To reduce the computational cost of our integrated model we propose decoupling the deformation of the tissue and systemic circulation from the porous flow in the myocardium and coronary vessels, which allows us to apply the model also in combination with precomputed cardiac displacements, obtained form other models or medical imaging data. We test the methodology through the simulation of a heartbeat in healthy conditions that replicates the systolic impediment phenomenon, which is particularly challenging to capture as it arises from the interaction of several parts of the model.

An extension of the Boussinesq-type models to weakly compressible flows

An extension of the Boussinesq-type models to weakly compressible flows is derived in the fully nonlinear case. The dispersive properties are consistent with the linear theory of compressible fluids at the long-wave limit. The particular case of a vanishing Mach number gives a quasi-incompressible model, intended for coastal wave simulations, which is a hyperbolic version of the Serre–Green–Naghdi equations, with a new treatment of the bathymetric terms. Both the compressible and quasi-incompressible models are hyperbolic four-equation models on an arbitrary bathymetry, with an exact equation of energy conservation. In addition, these models are extended to hyperbolic and fully nonlinear five-equation versions with improved dispersive properties. A remarkable property of the quasi-incompressible model with improved dispersive properties is that it is possible to decrease significantly the sound velocity, and thus the computational time, with the same accuracy or even slightly better. The numerical results show good agreement of the quasi-incompressible model with experimental data and the capability of the compressible model to calculate the decrease of tsunami velocity due to compressible effects.

Phase-field method based on discrete unified gas-kinetic scheme for large-density-ratio two-phase flows.

In this paper, a phase-field method under the framework of discrete unified gas-kinetic scheme (DUGKS) for incompressible multiphase fluid flows is proposed. Two kinetic models are constructed to solve the conservative Allen-Cahn equation that accounts for the interface behavior and the incompressible hydrodynamic equations that govern the flow field, respectively. With a truncated equilibrium distribution function as well as a temporal derivative added to the source term, the macroscopic governing equations can be exactly recovered from the kinetic models through the Chapman-Enskog analysis. Calculation of source terms involving high-order derivatives existed in the quasi-incompressible model is simplified. A series of benchmark cases including four interface-capturing tests and four binary flow tests are carried out. Results compared to that of the lattice Boltzmann method (LBM) have been obtained. A convergence rate of second order can be guaranteed in the test of interface diagonal translation. The capability of the present method to track the interface that undergoes a severe deformation has been verified. Stationary bubble and spinodal decomposition problems, both with a density ratio as high as 1000, are conducted and reliable solutions have been provided. The layered Poiseuille flow with a large viscosity ratio is simulated and numerical results agree well with the analytical solutions. Variation of positions of the bubble front and spike tip during the evolution of Rayleigh-Taylor instability has been predicted precisely. However, the detailed depiction of complicated interface patterns appearing during the evolution process is failed, which is mainly caused by the relatively large numerical dissipation of DUGKS compared to that of LBM. A high-order DUGKS is needed to overcome this problem.

On hydrodynamic phase field models for binary fluid mixtures

Two classes of thermodynamically consistent hydrodynamic phase field models have been developed for binary fluid mixtures of incompressible viscous fluids of possibly different densities and viscosities. One is quasi-incompressible, while the other is incompressible. For the same binary fluid mixture of two incompressible viscous fluid components, which one is more appropriate? To answer this question, we conduct a comparative study in this paper. First, we visit their derivation, conservation and energy dissipation properties and show that the quasi-incompressible model conserves both mass and linear momentum, while the incompressible one does not. We then show that the quasi-incompressible model is sensitive to the density deviation of the fluid components, while the incompressible model is not in a linear stability analysis. Second, we conduct a numerical investigation on coarsening or coalescent dynamics of protuberances using the two models. We find that they can predict quite different transient dynamics depending on the initial conditions and the density difference although they predict essentially the same quasi-steady results in some cases. This study thus cast a doubt on the applicability of the incompressible model to describe dynamics of binary mixtures of two incompressible viscous fluids especially when the two fluid components have a large density deviation.

Lattice Boltzmann method for binary fluids based on mass-conserving quasi-incompressible phase-field theory.

In this paper, a lattice Boltzmann equation (LBE) model is proposed for binary fluids based on a quasi-incompressible phase-field model [J. Shen et al., Commun. Comput. Phys. 13, 1045 (2013)10.4208/cicp.300711.160212a]. Compared with the other incompressible LBE models based on the incompressible phase-field theory, the quasi-incompressible model conserves mass locally. A series of numerical simulations are performed to validate the proposed model, and comparisons with an incompressible LBE model [H. Liang et al., Phys. Rev. E 89, 053320 (2014)PLEEE81539-375510.1103/PhysRevE.89.053320] are also carried out. It is shown that the proposed model can track the interface accurately. As the stationary droplet and rising bubble problems, the quasi-incompressible LBE gives nearly the same predictions as the incompressible model, but the compressible effect in the present model plays a significant role in the phase separation problem. Therefore, in general cases the present mass-conserving model should be adopted.

Dynamics of spiral waves in a cardiac electromechanical model with a local electrical inhomogeneity

Joint effect of electrical heterogeneity (e.g. induced by ischemia) and mechanical deformation is investigated for an anisotropic, quasi–incompressible model of cardiac electromechanical coupling (EMC) using the active strain approach and periodic boundary conditions. Three local inhomogeneities with different geometry are simulated. Under a specific stimulation protocol, the heterogeneities are able to induce spirals. The interplay between the dimension of the electrical inhomogeneity, the EMC and the mechano-electrical feedback provided by the stretch activated current (SAC) determines the dynamics of the spiral waves of excitation, which could extinguish (in the case of low SAC), or be stable (with the tip rotating inside the inhomogeneity), or drift and be annihilated (in the case of high SAC).

Diffuse interface model for incompressible two-phase flows with large density ratios

We investigate the applicability of an incompressible diffuse interface model for two-phase incompressible fluid flows with large viscosity and density contrasts. Diffuse-interface models have been used previously primarily for density-matched fluids, and it remains unclear to what extent such models can be used for fluids of different density, thereby potentially limiting the application of these models. In this paper, 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. 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. The convergence properties of the model are investigated for cases with large density ratio. Quantitative comparisons are made with results from previous studies to validate the model and its numerical implementation. Tests show that the variation in volume during the computation is of the order of machine accuracy, which is consistent with our use of a conservative discretization scheme (finite volume methods) for the Cahn–Hilliard equation. Results of the method are compared with previous work for the change in topology of rising bubbles and Rayleigh–Taylor instability. Additional results are presented for head-on droplet collision and the onset of droplet entrainment in stratified flows.

A lattice Boltzmann algorithm for calculation of the laminar jet diffusion flame

A new two-distribution lattice Boltzmann equation (LBE) algorithm is presented to solve the laminar diffusion flames within the context of Burke–Schumann flame sheet model. One distribution models the transport of the Schvab–Zeldovich coupling function, or the mixture fraction to combine the energy and species equations. The other distribution models the quasi-incompressible Navier–Stokes equations with the low Mach number approximation. In the quasi-incompressible flows, the thermodynamics quantities depend on the coupling function but not on the hydrodynamic pressure, and the fluid components are assumed to be compressible only in the mixing/reaction region. A systematic and consistent approach to deriving LBEs for the general advection–diffusion equation and the quasi-incompressible Navier–Stokes equations are also presented. The streaming step of the LBEs are discretized by the total variation diminishing (TVD) Lax–Wendroff scheme. Numerical simulations are carried out to reproduce the low frequency flame oscillation (or flame flicker) of buoyant jet diffusion flame. Comparison between the quasi-incompressible model and the incompressible model is presented and the role of non-solenoidal velocity is examined.