A computational strategy for the regularized 13 moment equations with enhanced wall-boundary conditions

Gu, Xiaojun; Emerson, David R.

doi:10.1016/j.jcp.2006.11.032

Cited by 108 publications

(97 citation statements)

References 36 publications

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“…There were no explicit wall boundary conditions available for the moment equations, which hamper the use of the moment method in a wide range of practical applications, although the principle for constructing wall boundary conditions was suggested by Grad (1949). The first set of explicit wall boundary conditions (Gu and Emerson 2007) for the regularised moment equations is constructed from Maxwell's kinetic boundary condition (Maxwell 1879). The boundary conditions are further improved for the R13 and R26 moment equations .…”

Section: Lubrication Approximation and Extended Reynolds Equationmentioning

confidence: 99%

A new extended Reynolds equation for gas bearing lubrication based on the method of moments

Zhang

Emerson

2016

Microfluid Nanofluid

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equation has been used to model slider bearings. In the theory of lubrication, the continuum description is assumed and inertial terms are negligible (Reynolds 1886). A pressure equation can then be obtained with one dimension less than the Navier-Stokes (NS) equations. However, as modern read-write heads are typically floating 50 nm or less above the moving disk, the characteristic length scale involved is comparable to the molecular mean free path, , and the continuum assumption is no longer considered to be valid. Any gas in such a narrow spacing will suffer from significant non-equilibrium effects, which can be measured by the Knudsen number, Kn, and relates the ratio of the mean free path, , to the characteristic length scale of the device. For convenience, it is possible to classify four distinct flow regimes based on the Knudsen number (Barber and Emerson 2006): (1) Kn < 10 −3 , represents the continuum flow regime where no-slip boundary conditions can generally be applied; (2) 10 −3 < Kn < 10 −1 , indicates that the flow is in the slip regime and boundary conditions need to account for velocity-slip and temperature-jump conditions; (3) 10 −1 < Kn < 10, represents the transition flow regime and results obtained using continuum-based equations are no longer considered reliable; and (4) Kn > 10, is the free molecular or collisionless flow regime and the continuum hypothesis is not valid.In the case of slider bearings operating in the slip-flow regime, it is usually adequate to modify the Reynolds equation with a velocity-slip boundary condition (Burgdorfer 1959;Hsia and Domoto 1983;Mitsuya 1993;Bahukudumbi and Beskok 2003;Chen and Bogy 2010). However, for many problems, especially those related to disk-drive heads, the operating conditions lie well within the transition regime. This is a region where non-equilibrium effects come not only from the wall but also from the Knudsen layer (Cercignani 2000; Lilley and Sader 2008; Gu et al. AbstractAn extended Reynolds equation based on the regularised 13 moment equations and lubrication theory is derived for gas slider bearings operating in the early to upper transition regime. The new formulation performs well beyond the capability of the conventional Reynolds equation modified with simple velocity-slip models. Both load capacity and pressure distribution can be reliably predicted by the extended Reynolds equation and are in good agreement with available direct simulation Monte Carlo data. In addition, the equations are able to provide both velocity and stress information, which is not conveniently recovered from available kinetic models. Tests indicate that the equations can provide accurate data for Knudsen numbers up to unity but, as expected, begin to deteriorate afterwards.

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Section: Lubrication Approximation and Extended Reynolds Equationmentioning

confidence: 99%

A new extended Reynolds equation for gas bearing lubrication based on the method of moments

Zhang

Emerson

2016

Microfluid Nanofluid

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“…A number of reports have studied the Knudsen layer with DSMC calculations (Bird 1977;Lockerby et al 2005b), with several appearing in the past year (Gu & Emerson 2007;Lilley & Sader 2007;Mizzi et al 2007;Struchtrup & Torrilhon 2007;Torrilhon & Struchtrup 2008). These studies employed Couette flow solutions to capture Kramers' problem, and we have followed suit using the geometry illustrated in figure 5.…”

Section: Dsmc Solutions For Hard Sphere and Variable Soft Sphere Molementioning

confidence: 99%

“…While Bird (1977), Lockerby et al (2005b), Gu & Emerson (2007), Mizzi et al (2007), Struchtrup & Torrilhon (2007) and Torrilhon & Struchtrup (2008) do not report the power-law velocity structure of the Knudsen layer, the DSMC solution of Lockerby et al (2005b) does exhibit the power-law dependence with az0.8 for VSS molecules with diffusely reflecting walls (Lilley & Sader 2007). In Bird's study, the walls of the Couette flow simulation were close together, resulting in interference between the Knudsen layers at each wall so that the Kramers' problem was not captured accurately.…”

Section: Dsmc Solutions For Hard Sphere and Variable Soft Sphere Molementioning

confidence: 99%

“…This is exemplified by the work of Reese et al (2003), who report that a two-dimensional DSMC calculation of rarefied air flow around a moving microcantilever required 24 hours on a parallel supercomputer with 3000 processors. Such intensive computational demands have motivated recent interest in the application of high-order hydrodynamic models to simulate rarefied flows (Reese et al 2003;Guo et al 2006;Gu & Emerson 2007;Mizzi et al 2007;Struchtrup & Torrilhon 2007;Torrilhon & Struchtrup 2008), with the expectation that such models may be employed in computational fluid dynamics solvers to accurately capture rarefaction effects with much less computational effort than comparable DSMC calculations.…”

Section: Introductionmentioning

confidence: 99%

“…The structure of the Knudsen layer has been studied extensively (Bardos et al 1986;Cercignani 2000;Sone 2002;Gu & Emerson 2007;Mizzi et al 2007;Struchtrup & Torrilhon 2007;Torrilhon & Struchtrup 2008). Recently, Lilley & Sader (2007) used existing solutions of the linearized Boltzmann equation (LBE) and precise DSMC calculations to examine the structure of the Knudsen layer in detail for shear flow past a solid wall.…”

Section: Introductionmentioning

confidence: 99%

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Velocity profile in the Knudsen layer according to the Boltzmann equation

Lilley

Sader

2008

Proc. R. Soc. A.

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Flow of a dilute gas near a solid surface exhibits non-continuum effects that are manifested in the Knudsen layer. The non-Newtonian nature of the flow in this region has been the subject of a number of recent studies suggesting that the so-called 'effective viscosity' at a solid surface is half that of the standard dynamic viscosity. Using the Boltzmann equation with a diffusely reflecting surface and hard sphere molecules, Lilley & Sader discovered that the flow exhibits a striking power-law dependence on distance from the solid surface where the velocity gradient is singular. Importantly, these findings (i) contradict these recent claims and (ii) are not predicted by existing highorder hydrodynamic flow models. Here, we examine the applicability of these findings to surfaces with arbitrary thermal accommodation and molecules that are more realistic than hard spheres. This study demonstrates that the velocity gradient singularity and power-law dependence arise naturally from the Boltzmann equation, regardless of the degree of thermal accommodation. These results are expected to be of particular value in the development of hydrodynamic models beyond the Boltzmann equation and in the design and characterization of nanoscale flows.

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Computational framework for the regularized 20‐moment equations for non‐equilibrium gas flows

Mizzi

Emerson

et al. 2008

Numerical Methods in Fluids

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This paper presents a framework for solving the regularized 20-moment equations consisting of a set of transport-like governing equations, the required constitutive closure, re-casting of the equations in second-order partial derivative form and derivation of additional wall boundary conditions. Couette flow results reveal that good agreement occurs between the 20-moment equations and direct simulation Monte Carlo data

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A computational strategy for the regularized 13 moment equations with enhanced wall-boundary conditions

Cited by 108 publications

References 36 publications

A new extended Reynolds equation for gas bearing lubrication based on the method of moments

A new extended Reynolds equation for gas bearing lubrication based on the method of moments

Velocity profile in the Knudsen layer according to the Boltzmann equation

Computational framework for the regularized 20‐moment equations for non‐equilibrium gas flows

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