Recent years have seen the emergence of new technologies that exploit nanoscale evaporation, ranging from nanoporous membranes for distillation to evaporative cooling in electronics. Despite the increasing depth of fundamental knowledge, there is still a lack of simulation tools capable of capturing the underlying non-equilibrium liquid–vapour phase changes that are critical to these and other such technologies. This work presents a molecular kinetic theory model capable of describing the entire flow field, i.e. the liquid and vapour phases and their interface, while striking a balance between accuracy and computational efficiency. In particular, unlike previous kinetic models based on the isothermal assumption, the proposed model can capture the temperature variations that occur during the evaporation process, yet does not require the computational resources of more complicated mean-field kinetic approaches. We assess the present kinetic model in three test cases: liquid–vapour equilibrium, evaporation into near-vacuum condition, and evaporation into vapour. The results agree well with benchmark solutions, while reducing the simulation time by almost two orders of magnitude on average in the cases studied. The results therefore suggest that this work is a stepping stone towards the development of an accurate and efficient computational approach to optimising the next generation of nanotechnologies based on nanoscale evaporation.

We employ the methods of statistical mechanics to obtain closures for the balance equations of momentum and fluctuation kinetic energy that govern the ballistic motion of grains rebounding at a rigid, bumpy bed that are driven by turbulent or non-turbulent shearing fluids, in the absence of mid-trajectory collisions and fluid velocity fluctuations. We obtain semi-analytical solutions for steady and fully developed saltation over horizontal beds for the vertical profiles of particle concentration and stresses and fluid and particle velocities. These compare favourably with measurements in discrete-element numerical simulations in the wide range of conditions of Earth and other planetary environments. The predictions of the particle horizontal mass flux and its scaling with the amount of particles in the system, the properties of the carrier fluid and the intensity of the shearing also agree with numerical simulations and wind-tunnel experiments.

An open problem of the derivation of the relativistic Vlasov equation for systems of charged particles moving with velocities up to the speed of light and creating the electromagnetic field in accordance with the full set of the Maxwell equations is considered. Moreover, the method of derivation is illustrated on the non-relativistic kinetic model. Independent derivation of the relativistic hydrodynamics is also demonstrated. The key role of these derivations of the hydrodynamic and kinetic equations includes the explicit operator of averaging on the physically infinitesimal volume suggested by L.S. Kuzmenkov.

A thermodynamically consistent kinetic model is proposed for the non-equilibrium transport of confined van der Waals fluids, where the long-range molecular attraction is considered by a mean-field term in the transport equation, and the transport coefficients are tuned to match the experimental data. The equation of state of the van der Waals fluids can be obtained from an appropriate choice of the pair correlation function. By contrast, the modified Enskog theory predicts non-physical negative transport coefficients near the critical temperature and may not be able to recover the Boltzmann equation in the dilute limit. In addition, the shear viscosity and thermal conductivity are predicted more accurately by taking gas molecular attraction into account, while the softened Enskog formula for hard-sphere molecules performs better in predicting the bulk viscosity. The present kinetic model agrees with the Boltzmann model in the dilute limit and with the Navier–Stokes equations in the continuum limit, indicating its capability in modelling dilute-to-dense and continuum-to-non-equilibrium flows. The new model is examined thoroughly and validated by comparing it with the molecular dynamics simulation results. In contrast to the previous studies, our simulation results reveal the importance of molecular attraction even for high temperatures, which holds the molecules to the bulk while the hard-sphere model significantly overestimates the density near the wall. Because the long-range molecular attraction is considered appropriately in the present model, the velocity slip and temperature jump at the surface for the more realistic van der Waals fluids can be predicted accurately.

A microscale lubrication flow of a gas between eccentric circular cylinders is studied on the basis of kinetic theory. The dimensionless curvature, defined by the mean clearance divided by the radius of the inner cylinder, is small, and the rotation speed of the inner cylinder is also small. The Knudsen number, defined by the mean free path divided by the mean clearance, is arbitrary. The Boltzmann equation is studied analytically using the slowly varying approximation following the method proposed in the author's previous study (Doi, Phys. Rev. Fluids, vol. 7, 2022, 034201). A macroscopic lubrication equation, which is a microscale generalization of the Reynolds lubrication equation, is derived. To assess this, a direct numerical analysis of the Boltzmann equation in a bipolar coordinate system is conducted using the Bhatnagar–Gross–Krook–Welander kinetic equation. It is demonstrated that the solution of the derived lubrication equation approximates that of the Boltzmann equation over a wide range of the eccentricity and the whole range of the Knudsen number. It is also demonstrated that another lubrication equation derived by a formal application of the slowly varying approximation produces a non-negligible error of the order of the square root of the dimensionless curvature for large Knudsen numbers.

Sound wave propagation in rarefied flows of molecular gases confined in micro-channels is investigated numerically. We first validate the employed kinetic model against the experimental results and then systematically study the gas damping and surface force on the transducer as well as the resonance/anti-resonance in confined space. To quantify the impact of the finite relaxation rates of the translational and internal energies on wave propagation, we examine the roles of bulk viscosity and thermal conductivity in depth over a wide range of rarefactions and oscillation frequencies. It is found that the bulk viscosity only exerts influence on the pressure amplitude and its resonance frequency in the slip regime in high oscillations. In addition, the internal degree of freedom is frozen when the bulk viscosity of a molecular gas is large, resulting in the pressure amplitude of sound waves in the molecular gas being the same as in a monatomic gas. Meanwhile, the thermal conductivity has a limited influence on the pressure amplitude in all the simulated flows. In the case of the thermoacoustic wave, we prove that the Onsager–Casimir reciprocal relation also holds for molecular gases, i.e. the pressure deviation induced by the temperature variation is equal to the heat flux induced by the plate oscillation. Our findings enable an enhanced understanding of sound wave propagation in molecular gases, which may facilitate the design of nano-/micro-scale devices.

The present paper deals with the kinetic-theoretic description of the evolution of systems consisting of many particles interacting not only with each other but also with the external world, so that the equation governing their evolution contains an additional term representing such interaction, called the ‘forcing term’. Firstly, the interactions between pairs of particles are both conservative and nonconservative; the latter represents, among others, birth/death rates. The ‘forcing term’ does not express a ‘classical’ force exerted by the external world on the particles, but a more general influence on the effects of mutual interactions of particles, for instance, climate changes, that increase or decrease the different agricultural productions at different times, thus altering the economic relationships between different subsystems, that in turn can be also perturbed by stock market fluctuations, sudden wars, periodic epidemics, and so on. Thus, the interest towards these problems moves the mathematical analysis of the effects of different kinds of forcing terms on solutions to equations governing the collective (that is statistical) behaviour of such nonconservative many-particle systems. In the present paper, we offer a study of the basic mathematical properties of such solutions, along with some numerical simulations to show the effects of forcing terms for a classical prey–predator model in ecology.

Engineering flow systems operating under low pressures and/or at the micro/nano scale generally include a physically adsorbed gas layer next to the surface. In this paper, we develop a scattering kernel that accounts for the effect of adsorption, arising from van der Waals interactions, on the dynamics of molecules impinging on solid smooth surfaces. In the limit of low bulk density, surface adsorption becomes negligible and the scattering kernel recovers consistently the Cercignani–Lampis model, which best describes molecular collisions with a clean, smooth surface. In the limit of high bulk density, a dense adsorbed molecular layer forms next to the surface and its presence is picked up by the Maxwell model with complete diffuse reflection, which better captures the multiple collisions suffered by molecules. A weight coefficient based on the Langmuir adsorption isotherm is incorporated into the modelling to handle the transition between these two limiting conditions of low and high densities. The proposed model is validated against high-fidelity molecular dynamics simulations that are performed for a variety of gas–surface combinations and adsorbed molecular layers with different densities. It is shown that the proposed model very well captures the scattering patterns of beams of gas molecules at different velocities impinging on surfaces, as well as momentum and energy accommodation coefficients in the entire range of explored conditions.

Following the stability analysis method in classic fluid dynamics, a linear stability equation (LSE) suitable for rarefied flows is derived based on the Bhatnagar–Gross–Krook (BGK) equation. The global method and singular value decomposition method are used for modal and non-modal analysis, respectively. This approach is validated by results obtained from Navier–Stokes (NS) equations. The modal analysis shows that LSEs based on NS equations (NS-LSEs) begin to fail when the Knudsen number ($Kn$) increases past $\sim$0.01, regardless of whether a slip model is used. When $Kn\geq 0.01$, the growth rate of the least stable mode is generally underestimated by the NS-LSEs. Under a fixed wavenumber, the pattern (travelling or standing wave) of the least stable mode changes with $Kn$; when the mode presents the same pattern, the growth rate decreases almost linearly with increasing $Kn$; otherwise, rarefaction effects may not stabilize the flow. The characteristic lengths of the different modes are different, and the single-scale classic stability analysis method cannot predict multiple modes accurately, even when combined with a slip model and even for continuum flow. However, non-modal analysis shows that this error does not affect the transient growth because modes with small growth rates offer little contribution to the transient growth. In rarefied flow, as long as the Mach number ($Ma$) is large enough, transient growth will occur in some wavenumber ranges. The rarefaction effect plays a stabilizing role in transient growth. The NS-LSEs-based method always overestimates the maximum transient growth.

Kinetic models of polyatomic gas typically account for the internal degrees of freedom at the level of the two-particle distribution function. However, close to the hydrodynamic limit, the internal (rotational) degrees of freedom tend to be well represented just by rotational kinetic energy density. We account for the rotational energy by augmenting the ellipsoidal statistical Bhatnagar–Gross–Krook (ES–BGK) model, an extension of the BGK model, at the level of the single-particle distribution function with an advection–diffusion–relaxation equation for the rotational energy. This reduced model respects the $H$ theorem and recovers the compressible hydrodynamics for polyatomic gases as its macroscopic limit. As required for a polyatomic gas model, this extension of the ES–BGK model not only has the correct specific heat ratio but also allows for three independent tunable transport coefficients: thermal conductivity, shear viscosity and bulk viscosity. We illustrate the effectiveness of the model via a lattice Boltzmann method implementation.

The Poiseuille and thermal transpiration flows of a dense gas between two parallel plates are investigated on the basis of the Enskog kinetic equation under the diffuse reflection boundary condition. In contrast to the case of an ideal gas, the density and the gradients of pressure and the normal stress component in the flow direction are not uniform in the direction normal to the plates for a dense gas. The non-uniform normal stress gradient contributes also to the acceleration or deceleration of the thermal transpiration flow for small Knudsen numbers. The profiles of mass and heat flows as well as the net mass flows are obtained for various Knudsen numbers and ratios of the molecular diameter to the distance of plates. In the analysis of the Poiseuille flow, most characteristics of a force-driven flow with a small force are recovered. However, for the case of a dense gas, differences between the force-driven and the present pressure-driven flows are observed even within the linearized regime for small force and pressure gradient, especially at the microscopic level. The behaviour of the velocity distribution functions, in particular, the way in which they approach the ones for the Boltzmann equation as the molecular diameter becomes smaller, is clarified.

This paper is dedicated to the development of particle charge and velocity second-order moment transport equations for monodisperse particles in gas flow with a tribocharging effect. The full transport equations for the particle charge–velocity covariance and the charge variance are derived in the framework of the kinetic theory of granular flow assuming that the electrostatic interaction does not modify the collision dynamics. The collision integrals are solved without presuming the form of the electric part for the particle probability density function. The full second-order transport equation model is tested in a one-dimensional periodic domain. The results show that this model is able to capture more important physical mechanisms that are neglected by simple algebraic models proposed in the past. An in-depth analysis of the transport equations is also performed. This study reveals that, for sufficiently small covariance characteristic destruction time scales, the transient and third-order moments terms can be safely neglected. In addition, two different reduced-order models are proposed: a more general algebraic model that takes into account the variance effect and a semi-algebraic model that only resolves a transport equation for the charge variance coupled with an algebraic model for the covariance. The former could, however, lead to non-physical predictions in many cases, while the latter can be a suitable alternative only for a sufficiently small interparticle collision time. Finally, a simple chart based on test case simulations is proposed to show under which conditions a semi-algebraic model could be considered as a suitable alternative.

We revisit the construction of discrete kinetic models for single-component isothermal two-phase flows. Starting from a kinetic model for a non-ideal fluid, we show that, under conventional scaling, the Navier–Stokes equations with a non-ideal equation of state are recovered in the hydrodynamic limit. A scaling based on the smallness of velocity increments is then introduced, which recovers the full Navier–Stokes–Korteweg equations. The proposed model is realized on a standard lattice and validated on a variety of benchmarks. Through a detailed study of thermodynamic properties including co-existence densities, surface tension, Tolman length and sound speed, we show thermodynamic consistency, well-posedness and convergence of the proposed model. Furthermore, hydrodynamic consistency is demonstrated by verification of Galilean invariance of the dissipation rate of shear and normal modes and the study of visco-capillary coupling effects. Finally, the model is validated on dynamic test cases in three dimensions with complex geometries and large density ratios such as drop impact on textured surfaces and mercury drops coalescence.

Particle-resolved direct numerical simulations are employed to investigate the particle–particle drag force in the bidisperse gas–particle suspensions where the particles are smooth and the single-particle velocity distribution function is Maxwellian. The particle Reynolds number ranges from 6.7 to 123.8, and in this range the particle inertia is high enough that the lubrication force is not essential. It is found that the relation derived by the kinetic theory of granular flow (KTGF) highly overestimates the particle–particle drag force. This is because the pre-collision velocities of colliding particles are not completely uncorrelated with each other. From the time sequence of collision events, it is observed that the particle pair that has just collided will probably collide again after a short time due to the restriction of the particle motion in dense suspensions. Since the post-collision velocities of the first collision cannot relax entirely in such a short time, the relative velocity before the subsequent collision is statistically smaller than the domain-averaged relative velocity. Consequently, the particle–particle drag force is over-predicted when the domain-averaged relative velocity is used. For this reason, this work assumes that the particle–particle drag force is determined by the relative velocity within a local region near large particles. When the local region is set to be the spherical shells centred on the centres of large particles and with an outer radius of a mean free path of small particles, the KTGF-based relation can reasonably predict the particle–particle drag force.

The aim of this paper is twofold: the first aim is to formulate and validate a multi-scale discrete Boltzmann method (DBM) based on density functional kinetic theory for thermal multiphase flow systems, ranging from continuum to transition flow regime; the second aim is to present some new insights into the thermo-hydrodynamic non-equilibrium (THNE) effects in the phase separation process. Methodologically, for bulk flow, DBM includes three main pillars: (i) the determination of the fewest kinetic moment relations, which are required by the description of significant THNE effects beyond the realm of continuum fluid mechanics; (ii) the construction of an appropriate discrete equilibrium distribution function recovering all the desired kinetic moments; (iii) the detection, description, presentation and analysis of THNE based on the moments of the non-equilibrium distribution ( $f-f^{(eq)}$). The incorporation of appropriate additional higher-order thermodynamic kinetic moments considerably extends the DBM's capability of handling larger values of the liquid–vapour density ratio, curbing spurious currents, and ensuring mass/momentum/energy conservation. Compared with the DBM with only first-order THNE (Gan et al., Soft Matt., vol. 11 (26), 2015, pp. 5336–5345), the model retrieves kinetic moments beyond the third-order super-Burnett level, and is accurate for weak, moderate and strong THNE cases even when the local Knudsen number exceeds $1/3$. Physically, the ending point of the linear relation between THNE and the concerned physical parameter provides a distinct criterion to identify whether the system is near or far from equilibrium. Besides, the surface tension suppresses the local THNE around the interface, but expands the THNE range and strengthens the THNE intensity away from the interface through interface smoothing and widening.

The classical Mott-Smith solution for one-dimensional normal shock wave structure is extended to the two-dimensional regular shock reflection problem. The solution for the non-equilibrium molecular velocity distribution function along the symmetry-plane streamline is obtained as a weighted sum of four Maxwellians. An analysis of applicability of the solution has been performed using the results of direct simulation Monte Carlo calculations for a range of incident shock wave intensities. Accuracy of the solution improves with increasing $Ma_n$, the Mach number normal to the shock front, so that the solution becomes rather accurate for strong shocks with $Ma_n>8$.

It is well known that the Poiseuille mass flow rate along microchannels shows a stationary point as the fluid density decreases, referred to as the Knudsen minimum. Surprisingly, if the flow characteristic length is comparable to the molecular size, the Knudsen minimum disappears, as reported for the first time by Wu et al. (J. Fluid Mech., vol. 794, 2016, pp. 252–266). However, there is still no fundamental understanding why the mass flow rate monotonically increases throughout the entire range of flow regimes. Although diffusion is believed to dominate the fluid transport at the nanoscale, here we show that the Fick's first law fails in capturing this behaviour, and so diffusion alone is insufficient to explain this confined flow phenomenon. Rather, we show that the Knudsen minimum disappears in tight confinements because the decay of the mass flow rate due to the decreasing density effects is overcome by the enhancing contribution to the flow provided by the fluid velocity slip at the wall.

The transport coefficients of a dilute gas of inelastic hard spheres immersed in a gas of elastic hard spheres (molecular gas) are determined. We assume that the number density of the granular gas is much smaller than that of the surrounding molecular gas, so that the latter is not affected by the presence of the granular particles. In this situation, the molecular gas may be treated as a thermostat (or bath) of elastic hard spheres at a fixed temperature. The Boltzmann kinetic equation is the starting point of the present work. The first step is to characterise the reference state in the perturbation scheme, namely the homogeneous state. Theoretical results for the granular temperature and kurtosis obtained in the homogeneous steady state are compared against Monte Carlo simulations showing a good agreement. Then, the Chapman–Enskog method is employed to solve the Boltzmann equation to first order in spatial gradients. In dimensionless form, the Navier–Stokes–Fourier transport coefficients of the granular gas are given in terms of the mass ratio $m/m_g$ ($m$ and $m_g$ being the masses of a granular and a gas particle, respectively), the (reduced) bath temperature and the coefficient of restitution. Interestingly, previous results derived from a suspension model based on an effective fluid–solid interaction force are recovered in the Brownian limit ($m/m_g \to \infty$). Finally, as an application of the theory, a linear stability analysis of the homogeneous steady state is performed showing that this state is always linearly stable.

A kinetic model called the $\nu$-model is proposed to replace the complicated Boltzmann collision operator in the simulation of rarefied flows of monatomic gas. The model follows the relaxation-time approximation, but the collision frequency (i.e. inverse relaxation time) is a function of the molecular velocity to reflect part of the collision details of the Boltzmann equation, and the target velocity distribution function (VDF) to which the VDF relaxes is close to that used in the Shakhov model. Based on the numerical simulation of strong non-equilibrium shock waves, a half-theoretical and half-empirical collision frequency is designed for different intermolecular potentials: the $\nu$-model shows significantly improved accuracy, and the underlying mechanism is analysed. The $\nu$-model also performs well in canonical rarefied microflows, especially in thermal transpiration, where kinetic models with velocity-independent collision frequency lack the capability to distinguish the influence of intermolecular potentials.