The ability to treat non-equilibrium flow problems without evaluating higher than first-order derivatives would prove very advantageous numerically. The method should be insensitive to irregularities in the grid, the straight-forward computation in Discontinuous Galerkin framework, and the easy implementation of boundary conditions.
Due to the low order equations, the less communication between the computational cells small stencil makes its numerical scheme more efficient to implement on parallel computing architecture. Solutions of the generalized gas dynamic equations In this section, we present two flow problems in the near continuum flow regime for the validation of the generalized gas dynamic equations derived in the last section. Both test problems are microchannel flows, but with different flow speeds and non-equilibrium properties.
Analytic solution of force-driven Poiseuille flow in the near continuum flow regime In this subsection, we apply the generalized gas dynamic equations to rarefied gas flows, such as the case of a force-driven Poiseuille flow between two parallel plates.
Both DSMC and kinetic theory have shown that even with a small Kn, the pressure and temperature profiles in this flow exhibit a different qualitative behavior from that predicted by the Navier-Stokes equations.
The flow is assumed to be unidirectional, i. It is difficult to obtain an analytical solution from the above nonlinear system. Here, we try to find an approximate solution using a perturbation method similar to that used in.
From Eqs. First, it is clear that the velocity profile is parabolic, but the profiles of pressure and temperature may be complicated due to the presence of the hyperbolic cosine function. It is interesting to compare the approximate solutions of the present gas dynamic equations to those of the Navier-Stokes-Fourier NSF equations. It is shown that the velocity profile of the NSF equa- tions is also parabolic, which is the same as the that of the generalized hydrodynamic equations.
However, the NSF equations give a constant pressure, which is qualitatively different from the predictions of the new 2 proposed model. It is seen that the temperature and pressure profiles of the present model are in qualitative agreement with the DSMC data. These critical flow behaviors are absent in the profiles of the NSF equations. For example, the temperature minimum only appears in the super-Burnett order if the traditional BGK collision model is used to construct the gas dynamic equations.
Numerical solution for Couette flows For the new gas dynamic equations, a corresponding gas-kinetic scheme can be developed. Given that the distance between walls is less than a mean-free path and the relative wall speed is high, the gas system will be strongly out of equilibrium. Specifically, the velocity distribution for the particles is non-Maxwellian. As in,31 the temperature along the channel is defined as Tx and the temperature perpendicular to walls is Tz.
The new gas dynamics equations are capable of capturing this kind of non-equilibrium flow phenomena. Also, it is necessary for any gas dynamic equations to consider the temperature as a tensor for the non-equilibrium system. Conclusion Based on the multiple stage particle collision BGK model and the Gaussian distribution function as the middle state, the generalized gas dynamic equations have been derived.
Since the gas temperature basically represents the molecular random motion, the direct extension of the temperature concept from a scalar to a second-order symmetric tensor, Tij , is physically reasonable. In the non-equilibrium flow regime, the randomness of the particle distribution indeed depends on the spatial orientation. Pr In the continuum flow regime, the generalized constitutive relationship and the heat flux term are the corre- sponding Navier-Stokes formulations.
They capture the time evolution of the anisotropic non-equilibrium flow variables, as demonstrated in our examples. The traditional temperature concept come from thermodynamics, in which there is no anisotropic par- ticle random motion in space. However, for the non-equilibrium flow transport, due to inadequate particle collisions the random molecule motion can become easily anisotropic. To directly consider the temperature as a tensor rather than a scalar is a reasonable description of a non-equilibrium flow.
For a diluted gas, due to the lack of long-range particle interactions, the randomness of the particle motion is the only source for the dissipation in the system. Under the new definition of the temperature, all dissipative effects in a diluted gas system, such as the viscosity and heat conduction, can be unified under the same concept, Tij.
The current gas dynamic equations can be useful in the study of microflows. Further developments of the gas dynamic equations based on the kinetic model and its scheme in,32 which are valid for compressible shock waves, will be done in the near future.
Acknowledgement K. Xu would like to thank Prof. Groth and Mr. McDonald for helpful discussion and Dr. Liu for his help in the numerical testing of Couette flow. References 1 K.
Jiang, Multiple temperature kinetic model for continuum and near continuum flows, Phys. Fluids 19, Levermore, Moment closure hierarchies for kinetic theories, J. Struchtrup and M. Groth, P. Roe, T. Gombosi, S. Brown, On the nonstationary wave structure of a moment closure for rarefied gsa dynamics, paper AIAA McDonald and C.
Groth, Extended fluid-dynamic model for micron-scale flows based on Gaussian moment closure, paper AIAA Bird, Molecular gas dynamics and the direct simulation of gas flows, Clarendon Press, Oxford Xu, Regularization of the Chapman-Enskog expansion and its description of shock structure, Physics Fluids 14 , L Xu and E. Josyula, Continuum formulation for non-equilibrium shock structure calculation, Communications Com- put.
Cai, D. Liu, and K. Fluids 20, Bhatnagar, E. Gross, and M. Krook, A model for collision processes in gases. I: Small amplitude processes in charged and neutral one-component systems, Phys. Chapman and T. Ohwada and K. Xu, The kinetic scheme for full Burnett equations, J. Callaway, Model for lattice thermal conductivity at low temperature, Phy Rev , pp. Gorban and I. Karlin, General approach to constructing model of the Boltzmann equation, Phyica A , pp.
Holway, New statistical models for kinetic theory: methods of construction, Physics of fluids 9 , pp. Levermore and W. Hittinger, Foundations for the generalization of the Godunov method to hyperbolic systems with stiff relxation source terms, PhD Thesis at Aerospace Engineering department, University of Michigan Malek, F. Baras, and A. Garcia, On the validity of hydrodynamics in plane Poiseuille flows, Physica A , The difficulty hitherto experienced in applying the results obtained in the case of the Kinetic Theory of Gases in the well-known form to liquids and intermediary states of matter has been pri marily due to the difficulty of properly interpretating molec ular interaction.
In the case of gases this difficulty is in most part overcome by the introduction of the assumption that a molecule consists of a perfectly elastic sphere not surrounded by any field of force. But since such a state of affairs does not exist, the results obtained in the case of gases hold only in a general way, and the numerical constants involved are therefore of an indefinite nature, while in the case of dense gases and liquids this procedure does not lead to anything that is of use in explaining the facts.
Instead of an atom, or molecule, consisting of a per fectly elastic sphere, it is more likely that each may be regarded simply as a center of forces of attraction and repulsion. If the exact nature of the field of force sur rounding atoms and molecules were known, it would be a definite mathematical problem to determine the resulting properties of matter.
But our knowledge in this connection is at present not sufficiently extensive to permit a develop ment of the subject along these lines. But in whatever way the subject is developed fundamental progress will have been made only if molecular interaction is not, as is usually the case, represented by the collision of elastic spheres.
It will be shown in this book that the subject may be developed to a considerable extent along sound mathe matical lines yielding important results without knowing the exact nature and immediate result of molecular interaction. Thus it will be found, for example, that the definition of the free path of a molecule in connection with viscosity, con duction of heat, diffusion, etc. Since in the gaseous state each kind of path is proportional to the volume of the gas, its interest is then mainly associated with the characteristic factor of the volume which makes the product numerically equal to the path.
A direct physical meaning may be given to this factor. In constructing a general Kinetic Theory the problem that presents itself first for investigation is the dependence of the velocity of translation of a molecule in a substance on its density and temperature.
It is often assumed that this velocity is the same in the liquid as in the gaseous state at the same temperature. It can be shown, however, that this holds only for each molecule at the instant it passes through a point in the substance at which the forces of the surrounding molecules neutralize each other.
The quasi-molecular regime coincides with Knudsen numbers in the range of 0. The dominant damping mechanisms in this regime are structure-molecular and intermolecular collisions.
Analytical solutions of the Boltzmann equation in the quasi-molecular regime do not exist, but several approximations like the Burnett equation are available [12]. One possibility to get a better insight into this transition regime is to use the assumptions of the free molecular flow and to compare the systematic deviations of the measurement results with the analytical solutions of the free molecular flow.
As an important precondition for the experimental evaluation surrounding surfaces from the bulk material, the mounting and the measurement chamber are in sufficiently large distance to the cantilever so that their influence on the damping of the fundamental mode of the device can be neglected.
Device Fabrication The fabrication process for the device is based on the silicon-on-insulator SOI technology. Both the current- carrying leads and the bond pads consists of a nm thick gold layer and a 70 nm chromium layer acting as bonding agent.
In the resulting multi-layered structure the stresses of the chromium and the gold layer compensate each other and the structure remains undeflected. The structure, schematically shown in figure 1, consists of a clamped-free cantilever with a length l of 2 mm, a width b of 1. A Top View 1. Figure 1: Schematic top view and cross-section of the test device with the Au-lead on the top of the Lorentz-force actuated cantilever. Experimental Setup A scanning laser Doppler-vibrometer Polytec MSA Micromotion Analyzer, shown in figure 2 com- prising displacement and velocity decoders is used to characterize the out-of-plane vibrations of the cantilever structures.
Because of its considerable broadband noise, the vibrometer output signal is mea- sured with a lock-in amplifier SR, Stanford Research Instruments.
A waveform generator excites the vibrating structure around its natural frequency in a static magnetic field. The sinusoidal current in the lead on the top of the cantilever causes a harmonic oscillation due to the Lorentz force.
Owing to high intrinsic quality factors, the cantilever vibration may take several seconds to settle. The amplitude of the cantilever vibra- tion due to the excitation is always much smaller than its length l to ensure linearity. The nitrogen pressure in the vacuum-chamber is measured with a high vacuum gauge Pfeiffer IMR and maintained by a flow controller MKS 50 sccm at the high vacuum port of a turbomolecular pump, enabling a dynamic equilibrium in the pressure range of 20 mPa to Pa.
As a result a fully automated control of the nitrogen atmosphere is established for interactions with the vibrating test structure. Helmholtz coil. In the case of a weakly damped system it is feasible to determine the quality factor using the linewidth of the resonance.
At the lower boundary of the investigated pressure range intrinsic and quasi-molecular damping governs the quality factor. The intrinsic quality factor is independent of the ambient pressure and can therefore be found experimentally at the ultimate pressure of the vacuum system 20 mPa. In Blom et al. The idea of his derivation is based on a net difference of the particle density between the top and the bottom side of the vibrating cantilever.
In consequence, a nonzero molecular pressure difference acts on the cantilever, which counteracts the propulsive forces. Several publications have laid emphasis on the determination of the damping coefficient, but Martin et al. Thermal effects are completely neglected, but the result of equation 6 is valid for a wide range of MEMS cantilevers, where the height h is much smaller than the width b of the beam.
Viscous squeeze-film damping is well known and exactly analyzed [16, 17]. Due to flexural vibrations of the cantilever perpendicular to a fixed wall the gas in between is compressed [16].
The gas can partially escape through air slits that surround the moving cantilever opposite to the chip carrier, which induces friction losses since compression work is not fully recovered. Rarefaction effects were considered by Veijola et al. The Reynolds equation is a simplification of the Navier-Stokes equation.
Hence, it stands for a continuum approach, that is adapted to the transition between the macroscopic and microscopic description of rarefied gas flows [19]. But the Navier-Stokes equation in turn applies only to a subset of problems covered by the Boltzmann equation. This work presents a model of squeezed-film damped beams in the molecular regime.
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