Kinetic theory provides a rigorous foundation for calculating the dynamics of gas flow at arbitrary degrees of rarefaction, with solutions of the Boltzmann equation requiring numerical methods in many cases of practical interest. Importantly, the near-continuum regime can be examined analytically using asymptotic techniques. These asymptotic analyses often assume steady flow, for which analytical slip models have been derived. Recently, developments in nanoscale fabrication have stimulated research into the study of oscillatory non-equilibrium flows, drawing into question the applicability of the steady flow assumption. In this article, we present a formal asymptotic analysis of the unsteady linearized Boltzmann-BGK equation, generalizing existing theory to the oscillatory (time-varying) case. We consider the near-continuum limit where the mean free path and oscillation frequency are small. The complete set of hydrodynamic equations and associated boundary conditions are derived for arbitrary Stokes number and to second order in the Knudsen number. The first-order steady boundary conditions for the velocity and temperature are found to be unaffected by oscillatory flow. In contrast, the second-order boundary conditions are modified relative to the steady case, except for the velocity component tangential to the solid wall. Application of this general asymptotic theory is explored for the oscillatory thermal creep problem, for which unsteady effects manifest themselves at leading order.

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Kinetic theory provides a rigorous foundation to explore the unsteady (oscillatory) flow of a dilute gas, which is often generated by nanomechanical devices. Recently, formal asymptotic analyses of unsteady (oscillatory) flows at small Knudsen numbers have been derived from the linearised Boltzmann–Bhatnagar–Gross–Krook (Boltzmann–BGK) equation, in both the low- and high-frequency limits (Nassios & Sader, J. Fluid Mech., vol. 708, 2012, pp. 197–249 and vol. 729, 2013, pp. 1–46; Takata & Hattori, J. Stat. Phys., vol. 147, 2012, pp. 1182–1215). These asymptotic theories predict that unsteadiness can couple strongly with heat transport to dramatically modify the overall gas flow. Here, we study the gas flow generated between two parallel plane walls whose temperatures vary sinusoidally in time. Predictions of the asymptotic theories are compared to direct numerical solutions, which are valid for all Knudsen numbers and normalised frequencies. Excellent agreement is observed, providing the first numerical validation of the asymptotic theories. The asymptotic analyses also provide critical insight into the physical mechanisms underlying these flow phenomena, establishing that mass conservation (not momentum or energy) drives the flows – this explains the identical results obtained using different previous theoretical treatments of these linear thermal flows. This study highlights the unique gas flows that can be generated under oscillatory non-isothermal conditions and the importance of both numerical and asymptotic analyses in explaining the underlying mechanisms.

The Boltzmann equation provides a rigorous theoretical framework to study dilute gas flows at arbitrary degrees of rarefaction. Asymptotic methods have been applied to steady flows, enabling the development of analytical formulae. For unsteady (oscillatory) flows, two important limits have been studied: (i) at low oscillation frequency and small mean free path, slip models have been derived; and (ii) at high oscillation frequency and large mean free path, the leading-order dynamics are free-molecular. In this article, the complementary case of small mean free path and high oscillation frequency is examined in detail. All walls are solid and of arbitrary smooth shape. We perform a matched asymptotic expansion of the unsteady linearized Boltzmann-BGK equation in the small parameter ν/ω, where ν is the collision frequency of gas particles and ω is the characteristic oscillation frequency of the flow. Critically, an algebraic expression is derived for the perturbed mass distribution function throughout the bulk of the gas away from any walls, at all orders in the frequency ratio ν/ω. This is supplemented by a boundary layer correction defined by a set of first-order differential equations. This system is solved explicitly and in complete generality. We thus provide analytical expressions up to first order in the frequency ratio, for the density, temperature, mean velocity and stress tensor of the gas, in terms of the temperature and mean velocity of the wall, and the applied body force. In stark contrast to other asymptotic regimes, these explicit formulae eliminate the need to solve a differential equation for a body of arbitrary geometry. To illustrate the utility of these results, we study the oscillatory thermal creep problem for which we find a tangential boundary layer flow arises at first order in the frequency ratio.

Economic consequence analysis is one of many inputs to terrorism contingency planning. Computable general equilibrium (CGE) models are being used more frequently in these analyses, in part because of their capacity to accommodate high levels of event-specific detail. In modeling the potential economic effects of a hypothetical terrorist event, two broad sets of shocks are required: (1) physical impacts on observable variables (e.g., asset damage); (2) behavioral impacts on unobservable variables (e.g., investor uncertainty). Assembling shocks describing the physical impacts of a terrorist incident is relatively straightforward, since estimates are either readily available or plausibly inferred. However, assembling shocks describing behavioral impacts is more difficult. Values for behavioral variables (e.g., required rates of return) are typically inferred or estimated by indirect means. Generally, this has been achieved via reference to extraneous literature or ex ante surveys. This article explores a new method. We elucidate the magnitude of CGE-relevant structural shifts implicit in econometric evidence on terrorist incidents, with a view to informing future ex ante event assessments. Ex post econometric studies of terrorism by Blomberg et al. yield macro econometric equations that describe the response of observable economic variables (e.g., GDP growth) to terrorist incidents. We use these equations to determine estimates for relevant (unobservable) structural and policy variables impacted by terrorist incidents, using a CGE model of the United States. This allows us to: (i) compare values for these shifts with input assumptions in earlier ex ante CGE studies; and (ii) discuss how future ex ante studies can be informed by our analysis.