Simplified kinetic modelling of polyatomic gases and liquid-vapour flows

Abstract

The relentless miniaturization of technology has pushed fluid dynamics into previously unexplored territories at the micro- and nanoscale. In these domains, micro- and nano-electromechanical systems (MEMS and NEMS) encounter unique fluid behaviors, particularly rarefaction effects, which fundamentally affect their performance. Traditional modelling approaches face a critical dilemma: continuum models, such as the Navier-Stokes equations, fail to capture these phenomena, while molecular dynamics (MD) simulations, though accurate, remain computationally intractable for practical applications. Kinetic theory bridges this gap by providing an efficient framework for capturing these effects. It offers a statistical framework for describing the behavior of large numbers of particles in terms of a distribution function, i.e. the probability of particles having particular velocities and positions at a given time. The cornerstone of kinetic theory is the Boltzmann equation, a fundamental tool that captures the evolution of distribution functions through its two components: a free-stream term, which describes the motion of particles under external forces, and a collision term, which accounts for interactions between molecules. While the Boltzmann equation is well-suited for rarefied monatomic gases, extensions such as the Rykov equation for polyatomic gases and the Enskog equation for dense gases address more complex conditions by incorporating additional effects in the collision term. However, the original Boltzmann equation is a highly complex nonlinear integro-differential equation, and its extensions are even more intricate, making them challenging to apply in practical engineering contexts. This thesis focuses on developing simplified kinetic models for polyatomic gases and liquid-vapour flows, validating these models against experimental and MD simulation results, and exploring their applicability across a variety of engineering scenarios.

In the first part of this thesis, a simplified kinetic model for polyatomic gases is developed and applied to the numerical study of sound waves generated by a vibrating plate in a rarefied polyatomic gas environment confined within microchannels. The goal is to accurately capture gas damping, particularly the surface forces acting on the plates of MEMS devices. Unlike monatomic gases, polyatomic gases exhibit bulk viscosity, and both translational and rotational degrees of freedom contribute to their thermal conductivity. To understand the effects of these two transport coefficients, sound wave propagation is studied over a wide range of rarefaction conditions and plate vibration frequencies. The results reveal that bulk viscosity significantly influences the pressure amplitude (equivalent to the surface force) and the resonance frequency, primarily in the slip flow regime at high oscillation frequencies. Additionally, at high bulk viscosity, the internal degrees of freedom are effectively frozen, causing the pressure amplitude of sound waves in a polyatomic gas to resemble that of a monatomic gas. In contrast, thermal conductivity has a limited effect on the pressure amplitude across all simulated conditions. For thermoacoustic waves, the study confirms the validity of the Onsager-Casimir reciprocal relation (OCRR) for polyatomic gases, demonstrating that the pressure deviation induced by temperature changes equals the heat flux generated by plate vibrations. Validating this relation enhances simulation efficiency by reducing the required number of kinetic coefficients and provides an additional criterion for numerical accuracy.

Building on the work involving polyatomic gases, this thesis transitions to the kinetic modelling of liquid-vapour flows, which are critical for NEMS applications such as cooling systems and water desalination membranes. A simplified kinetic model is developed to simulate nanoscale evaporation processes, striking a balance between computational efficiency and accuracy. This model captures the entire flow field, encompassing both the liquid and vapour phases as well as their interface. Unlike previous models that predominantly assumed isothermal conditions, this model accounts for temperature variations during evaporation. The model’s validity is demonstrated through three benchmark cases: liquid-vapour equilibrium, evaporation into near-vacuum conditions, and evaporation into a vapour environment. The results show excellent agreement with benchmark solutions, with a significant reduction in computation time of almost two orders of magnitude, without compromising accuracy.

The final part of the thesis extends the one-dimensional modelling of liquid-vapour flows to two dimensions and examines droplet ripening—a near-equilibrium process in which larger droplets grow at the expense of smaller ones. This process, which is poorly understood at the micro- and nanoscale, where interfacial-bulk interactions dominate, has significant implications for natural phenomena and technological applications, such as cloud formation and nanomaterial synthesis. The results reveal that the primary driving force behind droplet growth is the pressure difference between the vapour phase and the droplets. Notably, the rate of change in droplet size is found to be linear with the Laplace pressure when the small droplet has a bulk region, while non-linear effects emerge when the bulk region disappears. Furthermore, the slope of the rate of change in the linear regime increases exponentially with temperature, independent of droplet number or arrangement.

This thesis contributes to the development of simplified kinetic models for polyatomic gases and liquid-vapour flows, providing new insights into sound wave propagation, one-dimensional non-equilibrium evaporation, and two-dimensional droplet ripening. The findings offer a foundation for advancing kinetic modelling techniques, with important implications for both fundamental research and engineering applications.