Oscillatory flows of a slightly rarefied gas: a kinetic theory investigation

© 2013 Dr. Jason NassiosClassical continuum theory provides a rigorous framework for studying a diverse range of flow phenomena, with solutions of the Navier-Stokes equations subject to the usual no-slip boundary condition at a solid wall being widely reported. Flows exhibited by miniaturised devices, such as as those found in nanoelectromechanical systems (NEMS), can occur in a regime that obviates the use of the Navier-Stokes equations. More advanced approaches that rigorously describe the kinetic nature of gas flows are thus required to contend with the intrinsic gas rarefaction. The Boltzmann equation describes the evolution of the mass distribution function of a dilute gas resulting from interparticle collisions and phase space advection. This enables the study of gas flows at arbitrary degrees of rarefaction. Practical use of this formalism in the study of non-equilibrium flows is complicated by the nature of the collision integral in the Boltzmann equation, which typically exhibits a quadratic nonlinearity in the mass distribution function. In addition, several useful models for the interparticle force, such as the hard sphere model and the variable hard sphere model, yield particle collision frequencies that are dependent on the relative velocity of colliding particles. This restricts the scope for direct analysis. For this reason, model equations for the collision integral have been proposed. In particular, the relaxation approximation that was independently proposed by Bhatnagar, Gross and Krook (1954) and Welander (1954), has been widely applied to model dilute gas flows. This is commonly referred to as the BGK relaxation time approximation, and dramatically simplifies the Boltzmann equation. While not an exact formulation, this model displays many of the characteristics of real gas flows, and has been solved both numerically and analytically for a wide range of flows. Analytical formulations often utilise matched asymptotic expansions. This approach facilitates a rigorous mathematical treatment of localised non-equilibrium effects that arise for non-vanishing Knudsen number Kn ≡ λ / L, where the mean free path is defined as λ and L is a characteristic geometric length scale of the flow. Pioneering asymptotic studies of steady flows in a slightly rarefied gas where Kn<<1 were presented by Cercignani (1964) and Sone (1969, 1971, 1974). These studies showed that the Stokes Equations of creeping flow and the usual no-slip boundary condition can be derived from the linearised Boltzmann-BGK equation in the continuum limit, i.e., where Kn=0. Analytical slip boundary conditions and Knudsen layer corrections were also derived at higher-order in the Knudsen number Kn. Importantly, the Stokes equations were proven to describe the bulk flow (away from any wall) for all Knudsen number. In contrast, general asymptotic analyses for unsteady flows typically assume large Knudsen number, i.e., λ / L >> 1. Modern NEMS can generate highly oscillatory flows at length scales that are comparable in order to the mean free path of many gases at standard laboratory conditions. Consequently, the primary aim of this thesis is to investigate the effect of oscillatory (time-varying) unsteady motion on the flow of a slightly rarefied gas. This is performed within the framework of the Boltzmann-BGK equation. All walls are solid and of arbitrary smooth shape. In the near-continuum limit where the characteristic oscillation frequency of the flow ω is much smaller than the collision frequency of gas particles ν, we generalise existing steady theory to the unsteady (oscillatory) case. This formally elucidates the effect of unsteadiness on all classical (steady) hydrodynamic equations, slip models and Knudsen layer corrections, up to second-order in the Knudsen number Kn. 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. This latter finding is consistent with the numerical observation of Hadjiconstantinou (2005). We also present a complementary analysis in the high oscillation frequency limit, where the oscillation frequency of the body ω greatly exceeds the collision frequency of the gas particles ν, i.e., ω >> ν. A matched asymptotic expansion in the small parameter ν / ω is performed. 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. These expression are defined 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, the oscillatory (time-varying) thermal creep flow generated by temperature gradients imposed along two parallel plane walls is investigated. We also explore the flow that arises when unsteady temperature fields of constant amplitude are imposed along two parallel walls, where numerical solutions of the linearised Boltzmann-BGK equation validate our asymptotic formulae in both limits