Showing papers on "Knudsen number published in 2018"

Anisotropic nonequilibrium hydrodynamic attractor

Abstract: We determine the dynamical attractors associated with anisotropic hydrodynamics (aHydro) and the DNMR equations for a $0+1\mathrm{d}$ conformal system using kinetic theory in the relaxation time approximation. We compare our results to the nonequilibrium attractor obtained from the exact solution of the $0+1\mathrm{d}$ conformal Boltzmann equation, the Navier-Stokes theory, and the second-order Mueller-Israel-Stewart theory. We demonstrate that the aHydro attractor equation resums an infinite number of terms in the inverse Reynolds number. The resulting resummed aHydro attractor possesses a positive longitudinal-to-transverse pressure ratio and is virtually indistinguishable from the exact attractor. This suggests that an optimized hydrodynamic treatment of kinetic theory involves a resummation not only in gradients (Knudsen number) but also in the inverse Reynolds number. We also demonstrate that the DNMR result provides a better approximation of the exact kinetic theory attractor than the Mueller-Israel-Stewart theory. Finally, we introduce a new method for obtaining approximate aHydro equations which relies solely on an expansion in the inverse Reynolds number. We then carry this expansion out to the third order, and compare these third-order results to the exact kinetic theory solution.

Ballistic molecular transport through two-dimensional channels.

Abstract: Gas permeation through nanoscale pores is ubiquitous in nature and has an important role in many technologies1,2. Because the pore size is typically smaller than the mean free path of gas molecules, the flow of the gas molecules is conventionally described by Knudsen theory, which assumes diffuse reflection (random-angle scattering) at confining walls3-7. This assumption holds surprisingly well in experiments, with only a few cases of partially specular (mirror-like) reflection known5,8-11. Here we report gas transport through angstrom-scale channels with atomically flat walls12,13 and show that surface scattering can be either diffuse or specular, depending on the fine details of the atomic landscape of the surface, and that quantum effects contribute to the specularity at room temperature. The channels, made from graphene or boron nitride, allow helium gas flow that is orders of magnitude faster than expected from theory. This is explained by specular surface scattering, which leads to ballistic transport and frictionless gas flow. Similar channels, but with molybdenum disulfide walls, exhibit much slower permeation that remains well described by Knudsen diffusion. We attribute the difference to the larger atomic corrugations at molybdenum disulfide surfaces, which are similar in height to the size of the atoms being transported and their de Broglie wavelength. The importance of this matter-wave contribution is corroborated by the observation of a reversed isotope effect, whereby the mass flow of hydrogen is notably higher than that of deuterium, in contrast to the relation expected for classical flows. Our results provide insights into the atomistic details of molecular permeation, which previously could be accessed only in simulations10,14, and demonstrate the possibility of studying gas transport under controlled confinement comparable in size to the quantum-mechanical size of atoms.

Phonon Hydrodynamic Heat Conduction and Knudsen Minimum in Graphite

Abstract: In the hydrodynamic regime, phonons drift with a nonzero collective velocity under a temperature gradient, reminiscent of viscous gas and fluid flow. The study of hydrodynamic phonon transport has spanned over half a century but has been mostly limited to cryogenic temperatures (∼1 K) and more recently to low-dimensional materials. Here, we identify graphite as a three-dimensional material that supports phonon hydrodynamics at significantly higher temperatures (∼100 K) based on first-principles calculations. In particular, by solving the Boltzmann equation for phonon transport in graphite ribbons, we predict that phonon Poiseuille flow and Knudsen minimum can be experimentally observed above liquid nitrogen temperature. Further, we reveal the microscopic origin of these intriguing phenomena in terms of the dependence of the effective boundary scattering rate on momentum-conserving phonon–phonon scattering processes and the collective motion of phonons. The significant hydrodynamic nature of phonon transpo...

Ballistic molecular transport through two-dimensional channels

Abstract: Gas permeation through nanoscale pores is ubiquitous in nature and plays an important role in a plethora of technologies. Because the pore size is typically smaller than the mean free path of gas molecules, their flow is conventionally described by the Knudsen theory that assumes diffuse reflection (random-angle scattering) at confining walls. This assumption has proven to hold surprisingly well in experiment, and only a few cases of partially specular (mirror-like) reflection are known. Here we report gas transport through angstrom-scale channels with atomically-flat walls and show that surface scattering can be both diffuse or specular, depending on fine details of the surface atomic landscape, and quantum effects contribute to the specularity at room temperature. The channels made from graphene or boron nitride allow a helium gas flow that is orders of magnitude faster than expected from the theory. This is explained by specular surface scattering, which leads to ballistic transport and frictionless gas flow. Similar channels but with molybdenum disulfide walls exhibit much slower permeation that remains well described by Knudsen diffusion. The difference is attributed to stronger atomic corrugations at MoS2 surfaces, which are similar in height to the size of transported atoms and their de Broglie wavelength. The importance of the latter, matter-wave contribution is corroborated by the observation of a reversed isotope effect in which the mass flow of hydrogen is notably higher than that of deuterium, in contrast to the relation expected for classical flows. Our results provide insights into atomistic details of molecular permeation, which so far could be accessed only in simulations, and show a possibility of studying gas transport under a controlled confinement comparable to the quantum-mechanical size of atoms.

Observation of Poiseuille flow of phonons in black phosphorus

Abstract: The travel of heat in insulators is commonly pictured as a flow of phonons scattered along their individual trajectory. In rare circumstances, momentum-conserving collision events dominate, and thermal transport becomes hydrodynamic. One of these cases, dubbed the Poiseuille flow of phonons, can occur in a temperature window just below the peak temperature of thermal conductivity. We report on a study of heat flow in bulk black phosphorus between 0.1 and 80 K. We find a thermal conductivity showing a faster than cubic temperature dependence between 5 and 12 K. Consequently, the effective phonon mean free path shows a nonmonotonic temperature dependence at the onset of the ballistic regime, with a size-dependent Knudsen minimum. These are hallmarks of Poiseuille flow previously observed in a handful of solids. Comparing the phonon dispersion in black phosphorus and silicon, we show that the phase space for normal scattering events in black phosphorus is much larger. Our results imply that the most important requirement for the emergence of Poiseuille flow is the facility of momentum exchange between acoustic phonon branches. Proximity to a structural transition can be beneficial for the emergence of this behavior in clean systems, even when they do not exceed silicon in purity.

Discrete unified gas kinetic scheme for all Knudsen number flows. III. Binary gas mixtures of Maxwell molecules.

Abstract: Recently a discrete unified gas kinetic scheme (DUGKS) in a finite-volume formulation based on the Boltzmann model equation has been developed for gas flows in all flow regimes. The original DUGKS is designed for flows of single-species gases. In this work, we extend the DUGKS to flows of binary gas mixtures of Maxwell molecules based on the Andries-Aoki-Perthame kinetic model [P. Andries et al., J. Stat. Phys. 106, 993 (2002)JSTPBS0022-471510.1023/A:1014033703134. A particular feature of the method is that the flux at each cell interface is evaluated based on the characteristic solution of the kinetic equation itself; thus the numerical dissipation is low in comparison with that using direct reconstruction. Furthermore, the implicit treatment of the collision term enables the time step to be free from the restriction of the relaxation time. Unlike the DUGKS for single-species flows, a nonlinear system must be solved to determine the interaction parameters appearing in the equilibrium distribution function, which can be obtained analytically for Maxwell molecules. Several tests are performed to validate the scheme, including the shock structure problem under different Mach numbers and molar concentrations, the channel flow driven by a small gradient of pressure, temperature, or concentration, the plane Couette flow, and the shear driven cavity flow under different mass ratios and molar concentrations. The results are compared with those from other reliable numerical methods. The results show that the proposed scheme is an effective and reliable method for binary gas mixtures in all flow regimes.

Application of Knudsen thermal force for detection of inert gases

Abstract: Recently, detection and analysis of gas mixtures have become significant for purifications and separation of the natural gas mixture. In the present work, Direct Simulation Monte Carlo (DSMC) method is applied to evaluate the performance of a new micro gas sensor (MIKRA) for mass analysis of three inert gases (Helium, Neon and Argon). This sensor applied the Knudsen force induced by temperature difference at the low-pressure condition to diagnose the main components of the mixture. Since this sensor works in low-pressure condition, Boltzmann equation is used to attain accurate outcomes. To solve these equations, Direct Simulation Monte Carlo (DSMC) approach is used as a robust method for the non-equilibrium flow field. This study performed comprehensive studies to disclose the primary process of force production and applied this for the analysis of the gas mixture. Hence, effects of the main parameter such as temperature gradient and the gap of arms are expansively examined in different ambient pressures. Furthermore, the influence of various mixtures of the (Helium, Neon and Argon) on force generation is also investigated. Our findings show that value of generating Knudsen force significantly varies when the component of the mixture is changed. According to obtained results, the Knudsen force declines as the molecular weight of the gas decreases. In addition, the induced force is highly proportional to the molecular weight rather than other characteristics. Therefore, the Knudsen force is a reliable method for the mass analysis of the mixtures.

Unified Gas-kinetic Wave-Particle methods I: Continuum and Rarefied Gas Flow

Abstract: The unified gas-kinetic scheme (UGKS) provides a framework for simulating multiscale transport with the updates of both gas distribution function and macroscopic flow variables on the cell size and time step scales. The multiscale dynamics in UGKS is achieved through the coupled particle transport and collision in the particle evolution process within a time step. In this paper, under the UGKS framework, we propose an efficient multiscale unified gas-kinetic wave-particle (UGKWP) method. The gas dynamics in UGKWP method is described by the individual particle movement coupled with the evolution of the probability density function (PDF). During a time step, the trajectories of simulation particles are tracked until collision happens, and the post-collision particles are evolved collectively through the evolution of the corresponding distribution function. The evolution of simulation particles and distribution function is guided by evolution of macroscopic variables. The two descriptions on a gas particle, i.e. wave and particle, switch dynamically with time. A new concept of multiscale multi-efficiency preserving (MMP) method is introduced, and the UGKWP method is shown to be an MMP scheme. The UGKWP method is specially efficient for hypersonic flow simulation in all regimes in comparison with the wave-type discrete ordinate methods, and presents a much lower stochastic noise in the continuum flow regime in comparison with the particle-based Monte Carlo methods. Numerical tests for flows over a wide range of Mach and Knudsen numbers are presented. The examples include mainly the hypersonic flow passing a circular cylinder at Mach numbers $20$ and $30$ and Knudsen numbers $1$ and $10^{-4}$, low speed lid-driven cavity flow, and laminar boundary layer. These results validate the accuracy, efficiency, and multiscale property of UGKWP method.

Direct Visualization of Evaporation in a Two-Dimensional Nanoporous Model for Unconventional Natural Gas

Abstract: Evaporation at the nanoscale is critical to many natural and synthetic systems including rapidly emerging unconventional oil and gas production from nanoporous shale reservoirs. During extraction processes, hydrocarbons confined to nanoscopic pores (ranging from one to a few hundred nanometers in size) can undergo phase change as pressure is reduced. Here, we directly observe evaporation in two-dimensional (2D) nanoporous media at the sub-10 nm scale. Using an experimental procedure that mimics pressure drawdown during shale oil/gas production, our results show that evaporation takes place at pressures significantly lower than predictions from the Kelvin equation (maximum deviation of 11%). We probe evaporation dynamics as a function of superheat and find that vapor transport resistance dominates evaporation rate. The transport resistance is made up of both Knudsen and viscous flow effects, with the magnitude of the Knudsen effect being approximately twice that of the viscous effects here. We also observe...

Flutter and divergence instability of supported piezoelectric nanotubes conveying fluid

Abstract: In this paper, divergence and flutter instabilities of supported piezoelectric nanotubes containing flowing fluid are investigated. To take the size effects into account, the nonlocal elasticity theory is implemented in conjunction with the Euler-Bernoulli beam theory incorporating surface stress effects. The Knudsen number is applied to investigate the slip boundary conditions between the flow and wall of nanotube. The nonlocal governing equations of nanotube are obtained using Newtonian method, including the influence of piezoelectric voltage, surface effects, Knudsen number and nonlocal parameter. Applying Galerkin approach to transform resulting equations into a set of eigenvalue equations under the simple-simple (S-S) and clamped-clamped (C-C) boundary conditions. The effects of the piezoelectric voltage, surface effects, Knudsen number, nonlocal parameter and boundary conditions on the divergence and flutter boundaries of nanotubes are discussed. It is observed that the fluid-conveying nanotubes with both ends supported lose their stability by divergence first and then by flutter with increase in fluid velocity. Results indicate the importance of using piezoelectric voltage, nonlocal parameter and Knudsen number in decrease of critical flow velocities of system. Moreover, the surface effects have a significant role on the eigenfrequencies and critical fluid velocity.

A Unified Stochastic Particle Bhatnagar-Gross-Krook Method for Multiscale Gas Flows

Abstract: The stochastic particle method based on Bhatnagar-Gross-Krook (BGK) or ellipsoidal statistical BGK (ESBGK) model approximates the pairwise collisions in the Boltzmann equation using a relaxation process. Therefore, it is more efficient to simulate gas flows at small Knudsen numbers than the counterparts based on the original Boltzmann equation, such as the Direct Simulation Monte Carlo (DSMC) method. However, the traditional stochastic particle BGK method decouples the molecular motions and collisions in analogy to the DSMC method, and hence its transport properties deviate from physical values as the time step increases. This defect significantly affects its computational accuracy and efficiency for the simulation of multiscale flows, especially when the transport processes in the continuum regime is important. In the present paper, we propose a unified stochastic particle ESBGK (USP-ESBGK) method by combining the molecular convection and collision effects. In the continuum regime, the proposed method can be applied using large temporal-spatial discretization and approaches to the Navier-Stokes solutions accurately. Furthermore, it is capable to simulate both the small scale non-equilibrium flows and large scale continuum flows within a unified framework efficiently and accurately. The applications of USP-ESBGK method to a variety of benchmark problems, including Couette flow, thermal Couette flow, Poiseuille flow, Sod tube flow, cavity flow, and flow through a slit, demonstrated that it is a promising tool to simulate multiscale gas flows ranging from rarefied to continuum regime.

Numerical Study of Heat Transfer Enhancement for Low-Pressure Flows in a Square Cavity with Two Fins Attached to the Hot Wall Using Al2O3-Air Nanofluid

Abstract: The gaseous low-pressure nanofluid flow of a steady-state two-dimensional laminar natural convection heat transfer in a square cavity of length L with two attached solid fins to the hot wall is numerically investigated. Such flows are found in many engineering applications, such as nuclear reactors and electronic cooling equipment. Physical parameter ranges in this study are as follows: 0 ≤ Kn ≤ 0.1, 103 ≤ Ra ≤106, 0 ≤ ϕ ≤ 0.2, LF/L takes the value of 0.5, HF takes the values of 0.25 to 0.75. Simulation results show that Nusselt number depends directly on the Rayleigh number and inversely on the Knudsen number. In addition, it is found that heat transfer will be enhanced by dispersing the nanoparticles of Al2O3 in the base low-pressure gaseous flow. Moreover, it is found that the Nusselt number of such flows increases as the nano-particle volume fraction increases for the investigated range of volume fractions considered in this study. In addition, a correlation of the Nusselt number among all the investigated parameters in this study is proposed as Nu = 0.2196 Ra0.0829 Kn–0.511 ϕ0.104.

Multiparameter Analysis of Gas Transport Phenomena in Shale Gas Reservoirs: Apparent Permeability Characterization.

Abstract: The large amount of nanoscale pores in shale results in the inability to apply Darcy's law. Moreover, the gas adsorption of shale increases the complexity of pore size characterization and thus decreases the accuracy of flow regime estimation. In this study, an apparent permeability model, which describes the adsorptive gas flow behavior in shale by considering the effects of gas adsorption, stress dependence, and non-Darcy flow, is proposed. The pore size distribution, methane adsorption capacity, pore compressibility, and matrix permeability of the Barnett and Eagle Ford shales are measured in the laboratory to determine the critical parameters of gas transport phenomena. The slip coefficients, tortuosity, and surface diffusivity are predicted via the regression analysis of the permeability data. The results indicate that the apparent permeability model, which considers second-order gas slippage, Knudsen diffusion, and surface diffusion, could describe the gas flow behavior in the transition flow regime for nanoporous shale. Second-order gas slippage and surface diffusion play key roles in the gas flow in nanopores for Knudsen numbers ranging from 0.18 to 0.5. Therefore, the gas adsorption and non-Darcy flow effects, which involve gas slippage, Knudsen diffusion, and surface diffusion, are indispensable parameters of the permeability model for shale.

Rarefaction throttling effect: Influence of the bend in micro-channel gaseous flow

Abstract: Micro-bends are frequently encountered in micro-electro-mechanical systems as a basic unit of complex geometry. It is essential for a deep understanding of the rarefied gas flow through bent channels. In this paper, a two-dimensional pressure-driven gas flow in a micro-channel with two bends is investigated by solving the Bhatnagar-Gross-Krook kinetic equation via the discrete velocity method in the slip and transition flow regimes. The results show that the mass flow rate (MFR) through the bent channel is slightly higher than that in the straight channel in the slip flow regime but drops significantly as the Knudsen number increases further. It is demonstrated that the increase in MFR is not due to the rarefaction effect but due to the increase in cross section of the bent corners. As the rarefaction effect becomes more prominent, the low-velocity zones at the corners expand and the gas flow is “squeezed” into the inner corner. The narrowed flow section is similar to the throttling effect caused by the valve, and both the changes in MFRs and the pressure distribution also confirm this effect. The classical Knudsen minimum changes due to this “rarefaction throttling effect.” The Knudsen number at which the minimum MFR occurs gradually increases with the bend angle and finally disappears in the transition flow regime. In addition, the onset of rarefaction throttling effect shifts to a smaller Knudsen number with a lower tangential momentum accommodation coefficient.Micro-bends are frequently encountered in micro-electro-mechanical systems as a basic unit of complex geometry. It is essential for a deep understanding of the rarefied gas flow through bent channels. In this paper, a two-dimensional pressure-driven gas flow in a micro-channel with two bends is investigated by solving the Bhatnagar-Gross-Krook kinetic equation via the discrete velocity method in the slip and transition flow regimes. The results show that the mass flow rate (MFR) through the bent channel is slightly higher than that in the straight channel in the slip flow regime but drops significantly as the Knudsen number increases further. It is demonstrated that the increase in MFR is not due to the rarefaction effect but due to the increase in cross section of the bent corners. As the rarefaction effect becomes more prominent, the low-velocity zones at the corners expand and the gas flow is “squeezed” into the inner corner. The narrowed flow section is similar to the throttling effect caused by the v...

Hydrodynamic Heat Transport Regime in Bismuth: A Theoretical Viewpoint.

Abstract: Bismuth is one of the rare materials in which second sound has been experimentally observed. Our exact calculations of thermal transport with the Boltzmann equation predict the occurrence of this Poiseuille phonon flow between $\ensuremath{\approx}1.5$ and $\ensuremath{\approx}3.5\text{ }\text{ }\mathrm{K}$, in a sample size of 3.86 and 9.06 mm, consistent with the experimental observations. Hydrodynamic heat flow characteristics are given for any temperature: heat wave propagation length, drift velocity, and Knudsen number. We discuss a gedanken experiment allowing us to assess the presence of a hydrodynamic regime in any bulk material.

Coupling of high Knudsen number and non-ideal gas effects in microporous media

Abstract: High Knudsen number non-ideal gas flows in porous media are important and fundamental in various applications including shale gas exploitation and carbon dioxide sequestration. Because of the small pore size in tight rocks, the Knudsen number (Kn) may be high (i.e. much higher than 0.01) even though the gas is really dense. In fact, due to the high pressure and temperature underground, the gas usually manifests a strong non-ideal gas effect. Understanding the coupling mechanism of the high Kn effect and non-ideal gas effect is a premise to accurately model deep-seated underground gas exploitation or carbon dioxide sequestration. In this work, we theoretically analyse the high Kn non-ideal gas flows in microporous media. Based on the relative importance of the non-ideal gas effect and high Kn effect, the coupling is divided into four types. The analysis is subsequently validated by multiscale numerical simulations, in which the four types of coupling are clearly demonstrated. After applying the analysis to laboratory measurements, we propose a characteristic pressure model to calculate the gas permeability of tight rocks with better precision. The new model incorporates the non-ideal gas effect with the high Kn effect accurately and better bridges the laboratory measurements with the reservoir engineering.

Incompressible Navier-Stokes-Fourier Limit from The Boltzmann Equation: Classical Solutions

Abstract: The global classical solution to the incompressible Navier-Stokes-Fourier equa- tion with small initial data in the whole space is constructed through a zero Knudsen number limit from the solutions to the Boltzmann equation with general collision kernels. The key point is the uniform estimate of the Sobolev norm on the global solutions to the Boltzmann equation.