Based on the fast spectral approximation to the Boltzmann collision operator, we present an accurate and efficient deterministic numerical method for solving the Boltzmann equation. First, the linearized Boltzmann equation is solved for Poiseuille and thermal creep flows, where the influence of different molecular models on the mass and heat flow rates is assessed, and the Onsager–Casimir relation at the microscopic level for large Knudsen numbers is demonstrated. Recent experimental measurements of mass flow rates along a rectangular tube with large aspect ratio are compared with numerical results for the linearized Boltzmann equation. Then, a number of two-dimensional microflows in the transition and free-molecular flow regimes are simulated using the nonlinear Boltzmann equation. The influence of the molecular model is discussed, as well as the applicability of the linearized Boltzmann equation. For thermally driven flows in the free-molecular regime, it is found that the magnitudes of the flow velocity are inversely proportional to the Knudsen number. The streamline patterns of thermal creep flow inside a closed rectangular channel are analysed in detail: when the Knudsen number is smaller than a critical value, the flow pattern can be predicted based on a linear superposition of the velocity profiles of linearized Poiseuille and thermal creep flows between parallel plates. For large Knudsen numbers, the flow pattern can be determined using the linearized Poiseuille and thermal creep velocity profiles at the critical Knudsen number. The critical Knudsen number is found to be related to the aspect ratio of the rectangular channel.

Droplet dynamics in microfluidic applications is significantly influenced by surfactants. It remains a research challenge to model and simulate droplet behaviour including deformation, breakup and coalescence, especially in the confined microfluidic environment. Here, we propose a hybrid method to simulate interfacial flows with insoluble surfactants. The immiscible two-phase flow is solved by an improved lattice Boltzmann colour-gradient model which incorporates a Marangoni stress resulting from non-uniform interfacial tension, while the convection–diffusion equation which describes the evolution of surfactant concentration in the entire fluid domain is solved by a finite difference method. The lattice Boltzmann and finite difference simulations are coupled through an equation of state, which describes how surfactant concentration influences interfacial tension. Our method is first validated for the surfactant-laden droplet deformation in a three-dimensional (3D) extensional flow and a 2D shear flow, and then applied to investigate the effect of surfactants on droplet dynamics in a 3D shear flow. Numerical results show that, at low capillary numbers, surfactants increase droplet deformation, due to reduced interfacial tension by the average surfactant concentration, and non-uniform effects from non-uniform capillary pressure and Marangoni stresses. The role of surfactants on the critical capillary number ( $Ca_{cr}$ ) of droplet breakup is investigated for various confinements (defined as the ratio of droplet diameter to wall separation) and Reynolds numbers. For clean droplets, $Ca_{cr}$ first decreases and then increases with confinement, and the minimum value of $Ca_{cr}$ is reached at a confinement of 0.5; for surfactant-laden droplets, $Ca_{cr}$ exhibits the same variation in trend for confinements lower than 0.7, but, for higher confinements, $Ca_{cr}$ is almost a constant. The presence of surfactants decreases $Ca_{cr}$ for each confinement, and the decrease is also attributed to the reduction in average interfacial tension and non-uniform effects, which are found to prevent droplet breakup at low confinements but promote breakup at high confinements. In either clean or surfactant-laden cases, $Ca_{cr}$ first remains almost unchanged and then decreases with increasing Reynolds number, and a higher confinement or Reynolds number favours ternary breakup. Finally, we study the collision of two equal-sized droplets in a shear flow in both surfactant-free and surfactant-contaminated systems with the same effective capillary numbers. It is identified that the non-uniform effects in the near-contact interfacial region immobilize the interfaces when two droplets are approaching each other and thus inhibit their coalescence.

In the study of rarefied gas dynamics, the discrete velocity method (DVM) has been widely employed to solve the gas kinetic equations. Although various versions of DVM have been developed, their performance, in terms of modeling accuracy and computational efficiency, is yet to be comprehensively studied in all the flow regimes. Here, the traditional third-order time-implicit Godunov DVM (GDVM) and the recently developed discrete unified gas-kinetic scheme (DUGKS) are analysed in finding steady-state solutions of the low-speed force-driven Poiseuille and lid-driven cavity flows. With the molecular collision and free streaming being treated simultaneously, the DUGKS preserves the second-order accuracy in the spatial and temporal discretizations in all flow regimes. Towards the hydrodynamic flow regime, not only is the DUGKS faster than the GDVM when using the same spatial mesh, but also requires less spatial resolution than that of the GDVM to achieve the same numerical accuracy. From the slip to free molecular flow regimes, however, the DUGKS is slower than the GDVM, due to the complicated flux evaluation and the restrictive time step which is smaller than the maximum effective time step of the GDVM. Therefore, the DUGKS is preferable for problems involving different flow regimes, particularly when the hydrodynamic flow regime is dominant. For highly rarefied gas flows, if the steady-state solution is mainly concerned, the implicit GDVM, which can boost the convergence significantly, is a better choice

A kinetic model of the Boltzmann equation for non-vibrating polyatomic gases is proposed, based on the Rykov model for diatomic gases. We adopt two velocity distribution functions (VDFs) to describe the system state; inelastic collisions are the same as in the Rykov model, but elastic collisions are modelled by the Boltzmann collision operator (BCO) for monatomic gases, so that the overall kinetic model equation reduces to the Boltzmann equation for monatomic gases in the limit of no translational–rotational energy exchange. The free parameters in the model are determined by comparing the transport coefficients, obtained by a Chapman–Enskog expansion, to values from experiment and kinetic theory. The kinetic model equations are solved numerically using the fast spectral method for elastic collision operators and the discrete velocity method for inelastic ones. The numerical results for normal shock waves and planar Fourier/Couette flows are in good agreement with both conventional direct simulation Monte Carlo (DSMC) results and experimental data. Poiseuille and thermal creep flows of polyatomic gases between two parallel plates are also investigated. Finally, we find that the spectra of both spontaneous and coherent Rayleigh–Brillouin scattering (RBS) compare well with DSMC results, and the computational speed of our model is approximately 300 times faster. Compared to the Rykov model, our model greatly improves prediction accuracy, and reveals the significant influence of molecular models. For coherent RBS, we find that the Rykov model could overpredict the bulk viscosity by a factor of two.