New First and Second Order Slip Models for the Compressible Reynolds Equation

2003 ◽  
Vol 125 (3) ◽  
pp. 558-561 ◽  
Author(s):  
Lin Wu ◽  
D. B. Bogy
Keyword(s):  
Free Path ◽  
Reynolds Equation ◽  
Mean Free Path ◽  
Second Order ◽  
Surface Element ◽  
Length Scale ◽  
Mean Velocity ◽  
First Order ◽  
Lubrication Equation ◽  
The Mean

In the original derivations of the first order and the second order slip models of the generalized Reynolds equation in the literature [3,4], a length scale equal to the mean free path of the gas molecules was used in a Taylor series expansion of the mean velocity field. The coefficients of the correction terms in the derived lubrication equation depend on that length scale. This choice of the length scale is arbitrary to some extent. In this paper, new first order and the second order slip models are derived using a somewhat more physical approach, in which the requirement that the expansion length scale be the mean free path is relaxed. In this approach the momentum transfer rate across each surface element is obtained by summing up the contributions from each group of molecules impinging on the surface at an angle θ to the surface normal within a solid angle dω. The new second order slip lubrication equation appears to be preferable to the original one when the inverse Knudsen number is small, and it is free of any contact pressure singularity, whereas the new first order slip model continues to contain the unacceptable pressure singularity in the limit as the spacing approaches zero, as does the original first order model.

scholarly journals Determination of the coefficient of inter-diffusion of gases and the velocity of ions under an electric force, in terms of mean free paths

1912 ◽  
Vol 86 (586) ◽  
pp. 197-206 ◽  
Keyword(s):  
Free Path ◽  
Equations Of Motion ◽  
Mean Free Path ◽  
Electric Force ◽  
Mean Velocity ◽  
The Mean ◽  
Definition Of ◽  
Good Agreement ◽  
Diffusion Of Gases

1. The properties of gases which depend on the velocity of agitation of molecules and the lengths of their free paths may easily be expressed in terms of the mean velocity of agitation and the mean free path when certain assumptions are made in order to simplify the investigations. The expressions thus found on the principles of the kinetic theory are in good agreement with the experimental results in most cases, but the formulæ that have been obtained for the coefficient of inter-diffusion of gases and the velocity of particles acted on by an external force are not so satisfactory. The equations of motion of two inter-diffusing gases have been given by Maxwell, and it may be shown from these that the exact value of the ratio of the coefficient of diffusion of ions to the velocity under unit electric force is N e /II, where N is the number of molecules per cubic centimetre of a gas at pressure II, and e the charge on an ion. The method adopted by Maxwell is perfectly general, there are no assumptions made as to the distribution of the velocities of agitation, and no particular definition of a collision of a free path is involved, so that there can be little doubt as to the accuracy of the result.

scholarly journals The velocity of diffusion in a mixed gas; the second approximation

1941 ◽  
Vol 179 (977) ◽  
pp. 159-169 ◽  
Keyword(s):  
Capillary Tube ◽  
Mean Free Path ◽  
Uniform Motion ◽  
Mixed Gas ◽  
Mean Velocity ◽  
Diffusion Term ◽  
First Order ◽  
Mean Motion ◽  
Pressure Diffusion ◽  
The Mean

The general equation of diffusion in a mixed gas in non-uniform motion, in the presence of forces imparting differential accelerations to the two constituents, and of gradients of composition, pressure, and temperature, has been carried to a second approximation. This adds nine new terms to the expression for the velocity of diffusion, and each term involves as a factor a new ‘second order’ diffusion coefficient. All the new terms depend on the gradients of the mean motion of the gas, and vanish if this is uniform . One of the new terms is proportional to the space gradient of the tensor that expresses the rate of distortion of the gas; two other terms include parts involving the second space differential coefficients of the mean motion. The other portions of the two latter terms, and the remaining six terms, depend on products of the velocity gradients and the factors that appear in the first-order diffusion equation, namely the gradients of composition, pressure, and temperature, and the relative accelerations of the two types of molecule. The expressions for the nine new diffusion coefficients are extremely complicated, and have been evaluated only approximately. The new terms in the velocity of diffusion are with one exception negligible com pared with the first-order terms, at pressures above 10 -6 atm., except possibly in shock waves where the mean velocity of the gas alters by an appreciable fraction in a distance equal to the mean free path. The excepted term is the one which depends on the gradient of the rate-of-distortion tensor; in certain circumstances this is comparable with, though smaller than, the first-order pressure-diffusion term. It materially reduces the rate of diffusion in a stream of mixed gas flowing under pressure along a fine capillary tube.

scholarly journals On the calculation of the velocity and temperature distributions for flow along a flate plate

1936 ◽  
Vol 154 (882) ◽  
pp. 364-377 ◽  
Keyword(s):  
Kinetic Theory ◽  
Turbulent Flow ◽  
Free Path ◽  
Mean Free Path ◽  
Mixing Length ◽  
Shearing Stress ◽  
Mean Velocity ◽  
Temperature Distributions ◽  
The Mean

Kármán and Prandtl were the first investigators to publish theoretical ults for problems of turbulent flow involving plane boundaries. Before nsidering any particular problem the general considerations of these iters will be outlined. Prandtl's is, perhaps, the easier method to follow. He considered a bulent motion in which the mean velocity u remains parallel to a tain direction—O x , say,—and is a function of y only, O y being perpendicular to O x , and he arrived at the result τ = ρ l 2 | du / dy | du / dy (1) the shearing stress, where ρ is the density of the fluid and l is a length, led the mixing length; it is the analogue of the mean free path in the etic theory of gases. The conception of the mixing length of the sent problem is physically much less surely grounded than the mean e path of the kinetic theory.

scholarly journals High frequency oscillatory flows in a slightly rarefied gas according to the Boltzmann–BGK equation

2013 ◽  
Vol 729 ◽  
pp. 1-46 ◽  
Author(s):  
Jason Nassios ◽  
John E. Sader
Keyword(s):  
Boundary Layer ◽  
Free Path ◽  
Frequency Ratio ◽  
Oscillation Frequency ◽  
Mean Free Path ◽  
Mean Velocity ◽  
Oscillatory Flows ◽  
First Order ◽  
Bgk Equation ◽  
High Oscillation Frequency

AbstractThe 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 $\nu / \omega $, where $\nu $ is the collision frequency of gas particles and $\omega $ 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 $\nu / \omega $. 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.

scholarly journals Ballistic Heat Transport in Nanocomposite: The Role of the Shape and Interconnection of Nanoinclusions

Nanomaterials ◽  
2021 ◽  
Vol 11 (8) ◽  
pp. 1982
Author(s):  
Paul Desmarchelier ◽  
Alice Carré ◽  
Konstantinos Termentzidis ◽  
Anne Tanguy
Keyword(s):  
Thermal Conductivity ◽  
Molecular Dynamics ◽  
Molecular Dynamics Simulations ◽  
Free Path ◽  
Mean Free Path ◽  
Strong Increase ◽  
Moderate Increase ◽  
Energy Propagation ◽  
The Mean ◽  
Dynamics Simulations

In this article, the effect on the vibrational and thermal properties of gradually interconnected nanoinclusions embedded in an amorphous silicon matrix is studied using molecular dynamics simulations. The nanoinclusion arrangement ranges from an aligned sphere array to an interconnected mesh of nanowires. Wave-packet simulations scanning different polarizations and frequencies reveal that the interconnection of the nanoinclusions at constant volume fraction induces a strong increase of the mean free path of high frequency phonons, but does not affect the energy diffusivity. The mean free path and energy diffusivity are then used to estimate the thermal conductivity, showing an enhancement of the effective thermal conductivity due to the existence of crystalline structural interconnections. This enhancement is dominated by the ballistic transport of phonons. Equilibrium molecular dynamics simulations confirm the tendency, although less markedly. This leads to the observation that coherent energy propagation with a moderate increase of the thermal conductivity is possible. These findings could be useful for energy harvesting applications, thermal management or for mechanical information processing.

Experiments on the flow of heat in liquid helium below 0.7 °K

1958 ◽  
Vol 246 (1246) ◽  
pp. 390-405 ◽  
Keyword(s):  
Thermal Conductivity ◽  
Liquid Helium ◽  
Free Path ◽  
Empirical Relation ◽  
Mean Free Path ◽  
Series Of Experiments ◽  
The Mean ◽  
Set Up ◽  
Low Pressures ◽  
Measured Thermal Conductivity

A series of experiments has been performed to study the steady flow of heat in liquid helium in tubes of diameter 0.05 to 1.0 cm at temperatures between 0.25 and 0.7 °K. The results are interpreted in terms of the flow of a gas of phonons, in which the mean free path λ varies with temperature, and may be either greater or less than the diameter of the tube d . When λ ≫ d the flow is limited by the scattering of the phonons at the walls, and the effect of the surface has been studied, but when λ ≪ d viscous flow is set up in which the measured thermal conductivity is increased above that for wall scattering. This behaviour is very similar to that observed in the flow of gases at low pressures, and by applying kinetic theory to the problem it can be shown that the mean free path of the phonons characterizing viscosity can be expressed by the empirical relation λ = 3.8 x 10 -3 T -4.3 cm. This result is inconsistent with the temperature dependence of λ as T -9 predicted theoretically by Landau & Khalatnikov (1949).

Light scattering by phonons in the presence of a heat flow

1968 ◽  
Vol 46 (24) ◽  
pp. 2843-2845 ◽  
Author(s):  
Allan Griffin
Keyword(s):  
Light Scattering ◽  
Heat Flow ◽  
Temperature Gradient ◽  
Free Path ◽  
Mean Free Path ◽  
Photon Scattering ◽  
The Mean ◽  
Anti Stokes

If the temperature in an insulating crystal decreases in the z-direction, there are more phonons with momentum qz > 0 than with qz < 0. The resulting difference between the Stokes and anti-Stokes Brillouin intensities is proportional to the mean free path of the phonon involved and to the temperature gradient. The effect should be observable by either neutron or photon scattering.

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