Search Results (7)

The Euler–Boltzmann equations are an important class of mathematical models that describe the coupling between particle transport and macroscopic fluid dynamics. They find broad applications in plasma physics, rarefied gas dynamics, and astrophysics. In these fields, incorporating a time-dependent damping term is crucial for modeling real-world scenarios, as opposed to idealized inviscid conditions. In recent years, there has been growing interest in the long-time behavior of their solutions. This paper focuses on the initial value problem for the three-dimensional Euler–Boltzmann equations with time-dependent damping, aiming to investigate the finite-time blowup behavior of classical solutions. We use an integration method with general test function f and show that if the initial data are sufficiently large, classical solutions of the Euler–Boltzmann equations with time-dependent damping in ${\mathbb{R}}^3$ will blowup on or before the finite time $T^{*}>0$. Full article

Accurate representations of slip and transitional flow regimes present a challenge in the simulation of rarefied gas flow in confined systems with complex geometries. In these regimes, continuum-based formulations may not capture the physics correctly. This work considers a regularized multi-relaxation time lattice Boltzmann (LB) method with mixed Maxwellian diffusive and halfway bounce-back wall boundary treatments to capture flow at high Kn. The simulation results are validated against atomistic simulation results from the literature. We examine the convergence behavior of LB for confined systems as a function of inlet and outlet treatments, complexity of the geometry, and magnitude of pressure gradient and show that convergence is sensitive to all three. The inlet and outlet boundary treatments considered in this work include periodic, pressure, and a generalized periodic boundary condition. Compared to periodic and pressure treatments, simulations of complex domains using a generalized boundary treatment conserve mass but require more iterations to converge. Convergence behavior in complex domains improves at higher magnitudes of pressure gradient across the computational domain, and lowering the porosity deteriorates the convergence behavior for complex domains. Full article

The gas-kinetic scheme (GKS) and the unified gas-kinetic scheme (UGKS) are numerical methods based on the gas-kinetic theory, which have been widely used in the numerical simulations of high-speed and non-equilibrium flows. Both methods employ a multiscale flux function constructed from the integral solutions of kinetic equations to describe the local evolution process of particles’ free transport and collision. The accumulating effect of particles’ collision during transport process within a time step is used in the construction of the schemes, and the intrinsic simulating flow physics in the schemes depends on the ratio of the particle collision time and the time step, i.e., the so-called cell’s Knudsen number. With the initial distribution function reconstructed from the Chapman–Enskog expansion, the GKS can recover the Navier–Stokes solutions in the continuum regime at a small Knudsen number, and gain multi-dimensional properties by taking into account both normal and tangential flow variations in the flux function. By employing a discrete velocity distribution function, the UGKS can capture highly non-equilibrium physics, and is capable of simulating continuum and rarefied flow in all Knudsen number regimes. For high-speed non-equilibrium flow simulation, the real gas effects should be considered, and the computational efficiency and robustness of the schemes are the great challenges. Therefore, many efforts have been made to improve the validity and reliability of the GKS and UGKS in both the physical modeling and numerical techniques. In this paper, we give a review of the development of the GKS and UGKS in the past decades, such as physical modeling of a diatomic gas with molecular rotation and vibration at high temperature, plasma physics, computational techniques including implicit and multigrid acceleration, memory reduction methods, and wave–particle adaptation. Full article

The Knudsen paradox—the non-monotonous variation of mass-flow rate with the Knudsen number—is a unique and well-established signature of micro-channel rarefied flows. A particle which is not of insignificant size in relation to the duct geometry can significantly alter the flow behavior when introduced in such a system. In this work, we investigate the effects of a stationary particle on a micro-channel Poiseuille flow, from continuum to free-molecular conditions, using the direct simulation Monte-Carlo (DSMC) method. We establish a hydrodynamic basis for such an investigation by evaluating the flow around the particle and study the blockage effect on the Knudsen paradox. Our results show that with the presence of a particle this paradoxical behavior is altered. The effect is more significant as the particle becomes large and results from a shift towards relatively more ballistic molecular motion at shorter geometrical distances. The need to account for combinations of local and non-local transport effects in modeling reactive gas–solid flows in confined geometries at the nano-scale and in nanofabrication of model pore systems is discussed in relation to these results. Full article

Due to the failure of the continuum hypothesis for higher Knudsen numbers, rarefied gases and microflows of gases are particularly difficult to model. Macroscopic transport equations compete with particle methods, such as the Direct Simulation Monte Carlo method (DSMC), to find accurate solutions in the rarefied gas regime. Due to growing interest in micro flow applications, such as micro fuel cells, it is important to model and understand evaporation in this flow regime. Here, evaporation boundary conditions for the R13 equations, which are macroscopic transport equations with applicability in the rarefied gas regime, are derived. The new equations utilize Onsager relations, linear relations between thermodynamic fluxes and forces, with constant coefficients, that need to be determined. For this, the boundary conditions are fitted to DSMC data and compared to other R13 boundary conditions from kinetic theory and Navier–Stokes–Fourier (NSF) solutions for two one-dimensional steady-state problems. Overall, the suggested fittings of the new phenomenological boundary conditions show better agreement with DSMC than the alternative kinetic theory evaporation boundary conditions for R13. Furthermore, the new evaporation boundary conditions for R13 are implemented in a code for the numerical solution of complex, two-dimensional geometries and compared to NSF solutions. Different flow patterns between R13 and NSF for higher Knudsen numbers are observed. Full article

In this study, the Boltzmann equation with electric acceleration term is discretized and solved by the unified gas-kinetic scheme (UGKS). The charged particle transport driven by electric field is included in the electric acceleration term. To capture non-equilibrium distribution function, the probability distribution functions of gas is discretized in a discrete velocity space. After discretization, the numerical flux for distribution function is computed to update the microscopic and macroscopic states. The flux is decided by an integral solution of Boltzmann equation based on characteristic problem. An electron-ion collision model is introduced in the Boltzmann Bhatnagar-Gross-Krook (BGK) equation. This finite volume method for the UGKS couples the free transport and long-range interaction between particles. For simplicity, the electric field induced by charged particles is controlled by the Poisson’s equation, which is solved using the Green’s function for two dimensional plasma system subjected to the symmetry or periodic boundary conditions. Two numerical cases, linear Landau damping and Gaussian beam, are carried out to validate the proposed method. The linear electron plasma wave damping is simulated based on electron-ion collision operator. Comparison results show good accuracy and higher efficiency than particle based methods. Difference between Poisson’s equation and complete electromagnetic Maxwell equation is presented by numerical results based on the two models. Highly non-equilibrium and rarefied plasma flows, such as electron flows driven by electromagnetic field, can be simulated easily. The UGKS-Poisson model is proved to be promising in plasma flow simulation. Full article

Nanopores are extremely developed and randomly distributed in shale gas reservoirs. Due to the rarefied conditions in shale strata, multiple gas transport mechanisms coexist and need further understanding. The commonly used slip models are mostly based on Maxwell slip boundary condition, which assumes elastic collisions between gas molecules and solid surfaces. However, gas molecules do not rebound from solid surfaces elastically, but rather are adsorbed on them and then re-emitted after some time lag. A Langmuir slip permeability model was established by introducing Langmuir slip BC. Knudsen diffusion of bulk phase gas and surface diffusion of adsorbed gas were also coupled into our nanopore transport model. Considering the effects of real gas, stress dependence, thermodynamic phase changes due to pore confinement, surface roughness, gas molecular volume, and pore enlargement due to gas desorption during depressurization, a unified gas transport model in organic shale nanopores was established, which was then upscaled by coupling effective porosity and tortuosity to describe practical SGR properties. The bulk phase transport model, single capillary model, and upscaled porous media model were validated by data from experimental data, lattice Boltzmann method or model comparisons. Based on the new gas transport model, the equivalent permeability of different flow mechanisms as well as the flux proportion of each mechanism to total flow rate was investigated in different pore radius and pressure conditions. The study in this paper revealed special gas transport characteristics in shale nonopores and provided a robust foundation for accurate simulation of shale gas production. Full article