We study the effect of gas rarefaction on the interaction of small thermodynamic non-uniformities with a finite body. Considering a two-dimensional set-up, the initial system state is modelled as slight perturbations over its uniform density and temperature fields, prescribed in the vicinity of a thin plate. The problem is analysed in the collisionless limit and complemented by direct simulation Monte Carlo computations to cover the entire range of gas rarefaction rates. The high-Knudsen ‘sink-like’ and ‘source-like’ propagation patterns observed in the density- and temperature-driven set-ups, respectively, are discussed, together with the impact of specular (smooth) and diffuse (isothermal) wall reflections. At highly rarefied conditions, the solid body obstructs part of the gas domain, preventing the propagation of acoustic disturbances therein. With decreasing gas rarefaction, the acoustic field penetrates the obscured area via the effect of molecular collisions. Inspecting the near-field description, the propagation of flow disturbances along the plate surface is examined, and the acoustic force on the body is computed. In the thermally excited case, both normal- and shear-force components change sign at late times, attracting the plate towards the initial perturbation location. With reducing gas rarefaction, the shear force diminishes while the normal force sharply increases due to the decrease in signal decay. Finally, we apply the analysis to study the impact of gas rarefaction on acoustic reciprocity. Notably, acoustic reciprocity does not hold at non-continuum conditions over non-specular surfaces, where boundary reflections propagate in the presence of few molecular collisions, insufficient to retain reciprocal symmetry.

To characterize fluid flow in the slip regime, the use of Navier–Stokes–Fourier (NSF) equations with slip boundary conditions is prevalent. This trend underscores the necessity of developing reliable and accurate slip boundary conditions. According to kinetic theory, slip behaviours are intrinsically linked to the gas scattering processes at the surface. The widely used Maxwell scattering model, which employs a single accommodation coefficient to describe gas scattering processes, reveals its limitations when the difference between accommodation coefficients in the tangential and normal directions becomes significant. In this work, we provide a derivation of velocity slip and temperature jump boundary conditions based on the Cercignani–Lampis–Lord scattering model, which applies two independent accommodation coefficients to describe the gas scattering process. A Knudsen layer correction term is introduced to account for the impact of the surface on the velocity distribution function, which is associated with the scattering model. The governing equation of the correction term is established based on the linearized Boltzmann equation. Additionally, two moments are derived to capture the collision effect in the Knudsen layer: a conserving moment of collision invariants, and an approximate higher-order conserving moment. These moments are then employed to determine the coefficients in the correction term. We demonstrate that the derived slip coefficients align closely with numerical results obtained by solving the Boltzmann equation in the Knudsen layer. Besides, we apply the derived slip boundary conditions within the framework of the NSF equations, yielding numerical results that exhibit excellent consistency with those obtained through molecular-level simulations.

We study the effect of continuum breakdown on the two-dimensional thermoacoustic radiation of a thin plate set in a perfect monatomic gas. The plate is heated harmonically in time and the acoustic field is investigated in the entire range of gas rarefaction rates. Analytical approximations are obtained in the limits of high (free-molecular) and low (continuum limit) gas rarefaction, accompanied by direct simulation Monte Carlo calculations at intermediate flow conditions. While the source acoustic field is of a monopole type in the continuum regime, it turns non-isotropic at non-continuum conditions, exhibiting dipole directivity and exponential decay rate in the collisionless limit. The combined effects of source heating frequency and gas Knudsen number on the far-field acoustic radiation are illustrated and rationalized.

This work presents an experimental investigation focused on the analysis of aerodynamic properties between two interacting spheres in a supersonic rarefied flow. Atmospheric re-entries of space debris, whether natural or man-made, begin at altitude 120 km, and observations of historical re-entries have shown that fragmentation occurs between 90 and 50 km. The resulting fragments interact with each other, altering their own trajectories while traversing the different flow regimes between the free molecular and continuum regimes. This study focuses on the intermediate slip regime, where viscous effects of varying magnitude can influence the nature of the interactions of the shocks and modify them from the already known behaviour in the continuum regime. Specifically, this study examines how two spheres interact with each other upon re-entry into the atmosphere, focusing particularly on the six types of shock/shock interactions identified by Edney. The experiments were performed in the MARHy wind tunnel, in a steady Mach 4 laminar flow with static pressure 2.67 Pa. To highlight the differences between the six types of interferences, a variety of set-ups and devices were used: flow-field visualization, aerodynamic forces (through two diagnoses, aerodynamic balance and the swinging sphere technique) and wall pressure measurements. Results demonstrate the identification of differences according to the type of interference observed, showing in particular the viscous effect of rarefied flows by making a comparison with the continuum regime.

Two kinetic models are proposed for high-temperature rarefied (or non-equilibrium) gas flows with internal degrees of freedom and radiation. One of the models uses the Boltzmann collision operator to model the translational motion of gas molecules, which has the ability to capture the influence of intermolecular potentials, while the other adopts the relaxation time approximations, which has higher computational efficiency. In our kinetic model equations, not only the transport coefficients such as the shear/bulk viscosity and thermal conductivity but also their underlying relaxation processes are recovered. The non-equilibrium dynamics of gas flow and radiation are tightly coupled, where the transport properties of gas molecules and photons are correlatively dependent. The proposed kinetic models are validated by the direct simulation Monte Carlo method in several non-radiative rarefied gas flows (e.g. the normal shock wave, Fourier flow, Couette flow and the creep flow driven by the Maxwell demon), and the experimental data of planar heat transfer and normal shock waves in nitrogen. Then, the rarefied gas flows with strong radiation are studied based on the kinetic models, not only in the above one-dimensional gas flows, but also in the two-dimensional radiative hypersonic flow passing a cylinder. The characteristics of heat transfer in the tightly coupled fields of gas and radiation are systematically investigated, particularly the influence of the non-equilibrium photon transport and their interactions with gas molecules are revealed. It is found that the radiation makes a profound contribution to the total heat transfer in radiative hypersonic flow at an intermediate photon Knudsen number.

We study the effect of gas rarefaction and wall confinement on the propagation of vibroacoustic disturbances in a microchannel, generated by non-uniform (localized) time-harmonic oscillations of one of the channel walls. The problem is studied in the entire range of gas rarefaction rates, combining continuum and free-molecular limit analyses with direct simulation Monte Carlo calculations. Gas rarefaction is found to strongly increase the signal decay rate, varying between a slowly decaying propagating wave parallel to the channel walls at continuum conditions, to a near-source confined acoustic perturbation in the free-molecular regime. The impact of the stationary scattering wall is examined in detail, and the effect of replacing between fully diffuse and specular boundary reflections is found to slightly reduce the decay rate of the signal. The frequency dependence of the force generated by the gas film on the channel walls is calculated. Here, gas rarefaction smooths the transition between resonance and antiresonance behaviours observed in the continuum regime. A model set-up of a fully specular channel with a point delta source is examined, for which closed-form expressions are found for the effect of the stationary wall on the hydrodynamic perturbations and the acoustic force. These expressions assist in rationalizing the fundamental effect of the scattering wall on the system response.

The thermal conductivity of a molecular gas consists of the translational and internal parts. Although in continuum flows the total thermal conductivity itself is adequate to describe the heat transfer, in rarefied gas flows they need to be modelled separately, according to the relaxation rates of translational and internal heat fluxes in an homogeneous system. This paper is dedicated to quantifying how these relaxation rates affect rarefied gas dynamics. The kinetic model of Wu et al. (J. Fluid Mech., vol. 763, 2015, pp. 24–50) is adapted to recover the relaxation of heat fluxes, which is validated by the direct simulation Monte Carlo method. Then the model of Wu et al., which has the freedom to adjust the relaxation rates, is used to investigate the rate effects of thermal relaxation in problems such as the normal shock wave, creep flow driven by Maxwell's demon and thermal transpiration. It is found that the relaxation rates of heat flux affect rarefied gas flows significantly, even when the total thermal conductivity is fixed.

The lattice Boltzmann method (LBM), which was originally designed for near-incompressible Navier–Stokes flows, has been extended to rarefied gas flows with high-order quadrature in recent years. Although the ability of the high-order LBM to capture rarefaction effects has been demonstrated by many authors, its accuracy and efficiency are often undermined by numerical dissipation introduced by the off-lattice abscissas in Gauss–Hermite quadrature. Here, using the spontaneous Rayleigh–Brillouin scattering problem as the benchmark, we assess the accuracy and efficiency of the high-order LBM with on-lattice quadrature rules up to 39th order. The numerical error comprises two parts, one due to the rarefaction effect and the other due to temporal-spatial discretization, and we find that the former depends not only on the number of discrete velocities, but also on their distribution in velocity space. With a quadrature of 29th order, the error between the LBM and the discrete velocity method is found to be below 1 % up to $Kn=2.0$. Compared with a finite-volume Bhatnagar–Gross–Krook solver using Gauss–Hermite quadrature, the on-lattice LBM has a numerical dissipation several orders of magnitude lower, and achieves the same accuracy with fewer discrete velocities.

Based on an accurate numerical solution of the kinetic equation using well-resolved spatial and velocity grids, the separation of rarefied gas flow in a microchannel with double rectangular bends is investigated over a wide range of Knudsen and Reynolds numbers. Rarefaction effects are found to play different roles in flow separation (vortex formation) at the concave and convex corners. Flow separations near the concave and convex corners are only observed for a Knudsen number up to $0.04$ and $0.01$, respectively. With further increase of the Knudsen number, flow separation disappears. Due to the velocity slip at the solid walls, the concave (convex) vortex is suppressed (enhanced), which leads to the late (early) onset of separation of rarefied gas flows with respect to the Reynolds number. The critical Reynolds numbers for the emergence of concave and convex vortices are found to be as low as $0.32\times 10^{-3}$ and $30.8$, respectively. The slip velocity near the concave (convex) corner is found to increase (decrease) when the Knudsen number increases. An adverse pressure gradient appears near the concave corner for all the examined Knudsen numbers, while for the convex corner it only occurs when the Knudsen number is less than $0.1$. Due to the secondary flow and adverse pressure gradient near the rectangular bends, the mass flow rate ratio between the bent and straight channels of the same length is a non-monotonic function of the Knudsen number. Our results clarify the diversified and often contradictory observations reported in the literature about flow rate enhancement and vortex formation in bent microchannels.

Although the thermal conductivity of molecular gases can be measured straightforwardly and accurately, it is difficult to experimentally determine its separate contributions from the translational and internal motions of gas molecules. Yet, this information is critical in rarefied gas dynamics as the rarefaction effects corresponding to these motions are different. In this paper, we propose a novel methodology to extract the translational thermal conductivity (or equivalently, the translational Eucken factor) of molecular gases from the Rayleigh–Brillouin scattering (RBS) experimental data. From the numerical simulation of the Wu et al. (J. Fluid Mech., vol. 763, 2015, pp. 24–50) model we find that, in the kinetic regime, in addition to bulk viscosity, the RBS spectrum is sensitive to the translational Eucken factor, even when the total thermal conductivity is fixed. Thus it is not only possible to extract the bulk viscosity, but also the translational Eucken factor of molecular gases from RBS light scattering spectra measurements. Such experiments bear the additional advantage that gas–surface interactions do not affect the measurements. By using the Wu et al. model, bulk viscosities (due to the rotational relaxation of gas molecules only) and translational Eucken factors of $\textrm {N}_2$, $\textrm {CO}_2$ and $\textrm {SF}_6$ are simultaneously extracted from RBS experiments.

We consider the two-dimensional steady channel flow of a rarefied gas over a backward facing step in the limit of large Knudsen numbers. The free-molecular problem is solved analytically for both diffuse and specular-reflecting channel boundaries, and the solutions are validated through comparison with direct simulation Monte Carlo calculations. Prescribing the density and temperature differences between the inlet and outlet external equilibrium conditions, the results for the density- and temperature-drop-driven flows are analysed and contrasted, revealing higher flow velocities and mass flow rates in the former. While the flow rate is unaffected by the step geometry in the specular case, it increases with the step size in the diffuse-reflecting set-up. At conditions where small flow velocities occur, flow detachment is observed in the form of streamlines connecting the step edge stagnation points. Considering the problem at finite Knudsen numbers, the collisionless-flow regime breaks down at higher Knudsen numbers for lower gas speed flows, followed by the occurrence of step flow separation and recirculation.

A comprehensive study of the fundamental characteristics of leading-edge separation in rarefied hypersonic flows is undertaken and its salient features are elucidated. Separation of a boundary layer undergoing strong expansion is typical in many practical hypersonic applications such as base flows of re-entry vehicles and flows over deflected control surfaces. Boundary layer growth under such conditions is influenced by effects of rarefaction and thermal non-equilibrium, thereby differing significantly from the conventional no-slip Blasius type. A leading-edge separation configuration presents a fundamental case for studying the characteristics of such a flow separation but with minimal influence from a pre-existing boundary layer. In this work, direct simulation Monte Carlo computations have been performed to investigate flow separation and reattachment in a low-density hypersonic flow over such a configuration. Distinct features of leading-edge flow, limited boundary layer growth, separation, shear layer, flow structure in the recirculation region and reattachment are all explained in detail. The fully numerical shear layer profile after separation is compared against a semi-theoretical profile, which is obtained using the numerical separation profile as the initial condition on existing theoretical concepts of shear layer analysis based on continuum flow separation. Experimental studies have been carried out to determine the surface heat flux using thin-film gauges and computations showed good agreement with the experimental data. Flow visualisation experiments using the non-intrusive planar laser-induced fluorescence technique have been performed to image the fluorescence of nitric oxide, from which velocity and rotational temperature distributions of the separated flow region are determined.

The pressure field of a pulsating sphere is a canonical problem in classical acoustics, used to illustrate the acoustic efficiency of a monopole source at continuum conditions. We consider the counterpart vibroacoustic and thermoacoustic problems in a rarefied gas, to investigate the effect of continuum breakdown on monopole radiation. Focusing on small-amplitude normal-to-boundary mechanical and heat-flux excitations, the perturbation field is analysed in the entire range of gas rarefaction and input frequencies. Numerical calculations are carried out via the direct simulation Monte Carlo method, and are used to validate analytical predictions in the free-molecular and near-continuum regimes. In the latter, the regularized thirteen moments model (R13) is applied, to capture the system response at states where the Navier–Stokes–Fourier (NSF) description breaks down. Comparing with the continuum inviscid solution, the results quantitate the dampening effect of gas rarefaction on source point-wise strength and acoustic power. At near-continuum conditions, the acoustic field is composed of exponentially decaying ‘compression’, ‘thermal’ and ‘Knudsen-layer’ modes, reflecting thermoviscous and higher-order rarefaction effects. With reducing rarefaction, the contributions of the latter two modes vanish, and the former degenerates into the ideal-flow inverse-to-distance decaying wave. Stronger attenuation is obtained with increasing rarefaction, where boundary sphericity results in a ‘geometric reduction’ of the molecular layer affected by the source. Notably, while the R13 model at low frequencies appears valid up to moderate gas rarefaction rates, both the NSF and R13 descriptions break down at common low Knudsen numbers at higher frequencies. Further study should therefore be carried out to extend the applicability of moment models to unsteady flows with short time scales.

We consider the linearized form of the regularized 13-moment equations (R13) to model the slow, steady gas dynamics surrounding a rigid, heat-conducting sphere when a uniform temperature gradient is imposed far from the sphere and the gas is in a state of rarefaction. Under these conditions, the phenomenon of thermophoresis, characterized by forces on the solid surfaces, occurs. The R13 equations, derived from the Boltzmann equation using the moment method, provide closure to the mass, momentum and energy conservation laws in the form of constitutive, transport equations for the stress and heat flux that extend the Navier–Stokes–Fourier model to include non-equilibrium effects. We obtain analytical solutions for the field variables that characterize the gas dynamics and a closed-form expression for the thermophoretic force on the sphere. We also consider the slow, streaming flow of gas past a sphere using the same model resulting in a drag force on the body. The thermophoretic velocity of the sphere is then determined from the balance between thermophoretic force and drag. The thermophoretic force is compared with predictions from other theories, including Grad’s 13-moment equations (G13), variants of the Boltzmann equation commonly used in kinetic theory, and with recently published experimental data. The new results from R13 agree well with results from kinetic theory up to a Knudsen number (based on the sphere’s radius) of approximately 0.1 for the values of solid-to-gas heat conductivity ratios considered. However, in this range of Knudsen numbers, where for a very high thermal conductivity of the solid the experiments show reversed thermophoretic forces, the R13 solution, which does result in a reversal of the force, as well as the other theories predict significantly smaller forces than the experimental values. For Knudsen numbers between 0.1 and 1 approximately, the R13 model of thermophoretic force qualitatively shows the trend exhibited by the measurements and, among the various models considered, results in the least discrepancy.

The steady behaviour of a rarefied gas around a rotating sphere is studied numerically on the basis of the linearised ellipsoidal statistical model of the Boltzmann equation, also known as the ES model, and the Maxwell diffuse–specular boundary condition. It is demonstrated numerically that the normal derivative of the circumferential component of the flow velocity and that of the heat flux diverge on the boundary with a rate $s^{-1/2}$, where $s$ is the normal distance from the boundary. Further, it is demonstrated that the diverging term is proportional to the magnitude of the jump discontinuity of the velocity distribution function on the boundary, which originates from the mismatch of the incoming and outgoing data on the boundary. The moment of force exerted on the sphere is also obtained for a wide range of the Knudsen number and for various values of the accommodation coefficient.

This paper presents a comprehensive investigation into flow past a circular cylinder where compressibility and rarefaction effects play an important role. The study focuses on steady subsonic flow in the Reynolds-number range 0.1–45. Rarefaction, or non-equilibrium, effects in the slip and early transition regime are accounted for using the method of moments and results are compared to data from kinetic theory obtained from the direct simulation Monte Carlo method. Solutions obtained for incompressible continuum flow serve as a baseline to examine non-equilibrium effects on the flow features. For creeping flow, where the Reynolds number is less than unity, the drag coefficient predicted by the moment equations is in good agreement with kinetic theory for Knudsen numbers less than one. When flow separation occurs, we show that the effects of rarefaction and velocity slip delay flow separation and will reduce the size of the vortices downstream of the cylinder. When the Knudsen number is above 0.028, the vortex length shows an initial increase with the Reynolds number, as observed in the standard no-slip continuum regime. However, once the Reynolds number exceeds a critical value, the size of the downstream vortices decreases with increasing Reynolds number until they disappear. An existence criterion, which identifies the limits for the presence of the vortices, is proposed. The flow physics around the cylinder is further analysed in terms of velocity slip, pressure and skin friction coefficients, which highlights that viscous, rarefaction and compressibility effects all play a complex role. We also show that the local Knudsen number, which indicates the state of the gas around the cylinder, can differ significantly from its free-stream value and it is essential that computational studies of subsonic gas flows in the slip and early transition regime are able to account for these strong non-equilibrium effects.

With the aid of direct simulation Monte Carlo (DSMC), we conduct a detailed investigation pertaining to the fluid and thermal characteristics of rarefied gas flow with regard to various arrangements for radiometric pumps featuring vane and ratchet structures. For the same, we consider three categories of radiometric pumps consisting of channels with their bottom or top surfaces periodically patterned with different structures. The structures in the design of the first category are assumed to be on the bottom wall and consist of either a simple vane, a right-angled triangular fin or an isosceles triangular fin. The arrangements on the second category of radiometric pumps consist of an alternating diffuse–specular right-angled fin and an alternating diffuse–specular isosceles fin on the bottom wall. The third category contains either a channel with double isosceles triangular fins on its lowermost surface or a zigzag channel with double isosceles triangular fins on both walls. In the first and the third categories, the surfaces of the channel and its structures are considered as diffuse reflectors. The temperature is kept steady on the horizontal walls of the channel; thus, radiometric flow is created by subjecting the adjacent sides of the vane/ratchet to constant but unequal temperatures. On the other hand, for the second category, radiometric flow is introduced by specifying different top/bottom channel wall temperatures. The DSMC simulations are performed at a Knudsen number based on the vane/ratchet height of approximately one. The dominant mechanism in the radiometric force production is clarified for the examined configurations. Our results demonstrate that, at the investigated Knudsen number, the zigzag channel experiences maximum induced velocity with a parabolic velocity profile, whereas its net radiometric force vanishes. In the case of all other configurations, the flow pattern resembles a Couette flow in the open section of the channel situated above the vane/ratchet. To further enhance the simulations, the predictions of the finite volume discretization of the Boltzmann Bhatnagar–Gross–Krook (BGK)–Shakhov equation for the mass flux dependence on temperature difference and Knudsen number are also reported for typical test cases.

We present a method for calculating the hydrodynamic interactions between particles in the kinetic (or transition regime), characterized by non-negligible particle Knudsen numbers. Such particles are often present in aerosol systems. The method is based on our extended Kirkwood–Riseman theory (Corson et al., Phys. Rev. E, vol. 95 (1), 2017c, 013103), which accounts for interactions between spheres using the velocity field around a translating sphere as a function of Knudsen number. Results for the two-sphere problem at small Knudsen numbers are in good agreement with those obtained using Felderhof’s interaction actions for mixed slip-stick boundary conditions, which are accurate to order $r^{-7}$ (Felderhof, Physica A, vol. 89 (2), 1977, pp. 373–384). The strength of the interactions decreases with increasing Knudsen number. Results for two fractal aggregates demonstrate that one can apply a point force approach for interactions between particles in the transition regime; the interaction tensor is similar to the Oseen tensor for continuum flow. Using this point force approach, we present an analysis for the settling of an unbounded cloud of particles. Our analysis shows that for sufficiently high volume fractions and cloud radii, the cloud behaves as a gas droplet in continuum flow even when the individual particles are small relative to the mean free path of the gas. The method presented here can be applied in a Brownian dynamics simulation analogous to Stokesian dynamics to study the behaviour of a dense aerosol system.

In the present paper, analytical formulae for the shear lift forces on nanocylinders moving in linear shear flows in the free molecule regime are derived on the basis of the gas kinetic theory. The model takes into account the intermolecular interactions between the nanocylinders and gas molecules, i.e., the non-rigid-body effect. It is shown that the resulting formulae are consistent with the previous theory in the limit of rigid-body collisions. The lift forces acting on carbon nanotubes and long-chain $n$ -alkanes are evaluated as examples. It is found that the non-rigid-body effect is of great importance for small nanocylinders at low temperatures.

In this paper, an analytical investigation of two-dimensional conventional Burnett equations has been undertaken for gaseous flow through a long microchannel. The analytical solution is obtained by using perturbation analysis around the classical Navier–Stokes solution with appropriate boundary conditions. The perturbation expansion is employed with the smallness parameter $\unicode[STIX]{x1D716}$, taken as the ratio of height to length of the microchannel. The solution for pressure is obtained by solving the cross-stream momentum equation while the velocity distribution is obtained from the streamwise momentum equation. The resulting ordinary differential equations in pressure and velocity are third-order and second-order, respectively. The required boundary conditions for pressure are obtained from direct simulation Monte Carlo (DSMC) data. The obtained analytical solution matches the available DSMC solution well. This is perhaps the first analytical solution of the Burnett equations using the perturbation approach.