Session A27: Rarefied Gases

The Effect of Rotational Non-equilibrium on Chemical Reaction Rates Predicted by the Quantum-Kinetic (Q-K) Model for Direct Simulation Monte Carlo (DSMC) Simulations

The effect of non-equilibrium rotational excitation on dissociation and exchange reaction rates predicted by Bird's Q-K model is analyzed. The effect of rotational non-equilibrium is introduced as a perturbation to the effect of vibrational non-equilibrium. For dissociation reactions, a small but measurable improvement in the rates is observed. For exchange reactions, the change is negligible. These findings agree with experimental observations and theoretical predictions. The results from one-dimensional stagnation-streamline and two-dimensional axi-symmetric DSMC implementations of the original and modified Q-K models are compared for a typical re-entry flow. The influence of rotational non-equilibrium in promoting chemical reactions is seen to be small for this type of flow. Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.   

Asymptotic analysis of the Boltzmann-BGK equation for oscillatory gas flows with application to thermal creep

Kinetic theory provides a rigorous foundation for calculating the dynamics of gas flow at arbitrary degrees of rarefaction. Solutions to the Boltzmann equation require numerical methods in many cases of practical interest. However, the near-continuum regime has been analyzed analytically using asymptotic techniques. These asymptotic analyses often assume steady flow, for which analytical slip models have been derived. Recent developments in nanofabrication have stimulated research into the study of oscillatory flows, drawing into question the applicability of the steady flow assumption. In this talk, I will discuss some key findings of a formal asymptotic analysis of the unsteady linearized Boltzmann-BGK equation, which generalizes existing theory to the unsteady case. The near-continuum limit is considered where the mean free path and oscillation frequency are small. A brief exploration of the implications of this theory for the oscillatory thermal creep problem will be presented, where temperature gradients along adjacent walls generate a flow.   

Numerical study of oscillatory Couette flow in rarefied gas

Gas flows generated by nanoscale devices can achieve oscillation frequencies comparable to the intermolecular collision frequency. Modeling these flows often requires the use of kinetic theory, because such operation invalidates the use of standard continuum treatments. The Bhatnagar-Gross-Krook (BGK) kinetic model approximates the effect of molecular collisions as a relaxation process, and is often used to describe non-equilibrium gas flows. In this talk, I will present a numerical study of oscillatory Couette flow to obtain benchmark solutions using the BGK model. Applicability of the lattice Boltzmann method to such flow is also assessed using these benchmark solutions.   

Modeling fluid flows in micro devices: the challenge of Knudsen-layer behavior

Microscale gas flows often display non-standard fluid behavior and near a solid bounding surface. This leads to the formation of a Knudsen layer (KL): a local non-equilibrium region of thickness of few mean free paths (MFP) from the surface. Linear constitutive relations for shear stress and heat flux are no longer necessarily valid in the KL. To account this, we investigate a power-law (PL) form of the probability distribution function for free paths of rarefied gas molecules in arbitrary wall confinements. PL based geometry dependent MFP models are derived for complex geometry systems by taking into account the boundary limiting effects on the molecular free paths. Molecular dynamics numerical experiments are carried out to rigorously validate the PL model, under wide range of rarefaction conditions. As gas transport properties can be related to the MFP through kinetic theory, the Navier-Stokes-Fourier (N-S-F) constitutive relations are then modified in order to better capture the flow behavior in the KL. The new modeling technique tested for isothermal and non-isothermal gas flows in both planar and non-planar confinements. The results show that our approach greatly improves the near-wall accuracy of the N-S-F equations, well beyond the slip-flow regime.   

Exploiting coupled heat and momentum transfer in nanostructured gas-filled channels

The velocity distribution of a gas confined between surfaces held at different temperatures shows a significant deviation from the Maxwell distribution as long as the mean free path of the molecules is comparable to the channel dimensions. When one of the surfaces is suitably structured this non-equilibrium distribution can be exploited to transfer momentum in tangential direction between the two surfaces. Hence the walls experience a net force parallel to their surface which opens the possibility to extract work from the system. Since both surfaces are held at constant temperature this mode of momentum transfer is different from thermal creep flow that has gained more attention so far. We investigate this situation in the limit of free molecular flow for the case that the unstructured surface is allowed to move tangentially with respect to the structured surface. By this the possibility to operate the system as a thermal engine is investigated and its efficiency assessed.   

Flow of a rarefied gas around moving vanes in Crookes radiometer: Numerical analysis of a model problem

A model for flows around moving vanes in a Crookes radiometer is proposed. More precisely, a series of uniformly spaced parallel flat plates heated on their single sides, moving in a channel at a constant speed, is considered in the case where the direction of motion is perpendicular to the plates. The flow around the plates is investigated numerically on the basis of the ellipsoidal statistical (ES) model of the Boltzmann equation and the diffuse reflection boundary condition by means of an accurate finite-difference method. Special attention is paid to the discontinuity contained in the velocity distribution function. The pressure distribution around the edge is found to be different from that reported previously [S. Taguchi and K. Aoki, J. Fluid Mech. \textbf{694}, 191--224 (2012)], in which the plate is assumed to have a stationary position in a closed container. The reason for the difference is also discussed.   

Thermal transpiration of a slightly rarefied gas through a horizontal straight pipe in the presence of weak gravitation

Thermal transpiration of a slightly rarefied gas in the presence of weak gravitation is studied based on kinetic theory. We consider the situation where the Knudsen number (the mean free path divided by the characteristic length of the cross section) is small and the dimensionless gravity (the characteristic length divided by the ascent height of the molecules against gravity) is of the order of the square of the Knudsen number. The behavior of the gas is studied analytically based on the the fluid-dynamic-type equation derived from the Boltzmann equation. When the temperature gradient is imposed along the pipe, the pressure gradient is produced not only in the vertical direction but also in the horizontal direction due to the effect of gravity. This horizontal pressure gradient, which is of the higher order of the Knudsen number, induces a flow of the order of the Knudsen number, and produces a relatively finite effect on thermal transpiration. The velocity profile is considerably different from that of the conventional thermal transpiration due to the effect of weak gravitation. A direct numerical analysis of a flow through a long channel is conducted based on the model Boltzmann equation, and the mechanism of this phenomenon is demonstrated.   

Lattice Boltzmann simulations of genuinely multidimensional rarefied flows in microchannels

We present lattice Boltzmann simulations of rarefied slip flows driven by applied pressure differences across microchannels of finite length. We correctly capture the nonlinear streamwise pressure variation and the cross-channel velocity component, as well as the streamwise velocity and volume flux. The former effects are both absent from almost all previous work that approximated the pressure difference using a uniform body force. We demonstrate second-order convergence of both velocity components towards the asymptotic solution for long microchannels, and slower convergence of the pressure. We use the standard lattice Boltzmann formulation that reduces to a second-order recurrence relation for the streamwise velocity in uniform shear, and whose analytical solution gives a parabolic profile from wall to wall. We therefore cannot capture Knudsen boundary layers, but instead implement Maxwell--Navier slip boundary conditions directly on the hydrodynamic moments of our discrete velocity model. Our only parameter is the tangential momentum accommodation coefficient, so we require no fitting to known solutions. Our moment-based approach shows that existing boundary conditions impose conditions on higher non-hydrodynamic moments rather than on the tangential fluid velocity itself.