Second-Order Knudsen-Layer Analysis for the Generalized Slip-Flow Theory II: Curvature Effects

Hattori, Masanari; Takata, Shigeru

doi:10.1007/s10955-015-1364-0

Cited by 13 publications

(34 citation statements)

References 22 publications

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“…We analyze the Boltzmann equation (15) with conditions (25) and (27) when the Knudsen number Kn is small, that is, when ε 1. Since our reference time is t 0 = L/(2RT 0 ) 1/2 = 1/εν c , it is much larger than the mean free time 1/ν c .…”

Section: Dimensionless Formmentioning

confidence: 99%

“…where ζ * = |ζ ζ ζ * | = (ζ 2 * i ) 1/2 , E indicates the function defined by (26), ϕ(ζ ζ ζ ) is an arbitrary function of ζ ζ ζ , and the same convention as in (16) is used, i.e., ϕ * = ϕ(ζ ζ ζ * ), ϕ = ϕ(ζ ζ ζ ), etc. Then, we define the following extended linearized collision operator…”

Section: Linearized Collision Operatormentioning

confidence: 99%

“…K /∂ ζ wi diverges there [17,25,26]. This means that we cannot neglect this term in O(εR) in (73) at the point ζ wi n i = 0, η = 0.…”

Section: Knudsen-layer Equationmentioning

confidence: 99%

“…As mentioned in Sect. 5.1, the Chapman-Enskog solution can be made to satisfy the kinetic initial condition (25) with the choice (51). From the form of (55) and (57), it follows thatf…”

Section: Summary and Remarksmentioning

confidence: 99%

“…then the Chapman-Enskog solution (39) satisfies (25) up to O(ε), since ∂v i /∂ x j = 0 and ∂T /∂ x i = 0. Therefore, under assumption (iv) in Sect.…”

Section: Introduction Of the Knudsen Layermentioning

confidence: 99%

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Slip Boundary Conditions for the Compressible Navier–Stokes Equations

et al. 2017

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Abstract. The slip boundary conditions for the compressible Navier-Stokes equations are derived systematically from the Boltzmann equation on the basis of the ChapmanEnskog solution of the Boltzmann equation and the analysis of the Knudsen layer adjacent to the boundary. The resulting formulas of the slip boundary conditions are summarized with explicit values of the slip coefficients for hard-sphere molecules as well as the Bhatnagar-Gross-Krook (BGK) model. These formulas, which can be applied to specific problems immediately, help to prevent the use of often used slip boundary conditions that are either incorrect or without theoretical basis.

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Section: Dimensionless Formmentioning

confidence: 99%

Section: Linearized Collision Operatormentioning

confidence: 99%

“…K /∂ ζ wi diverges there [17,25,26]. This means that we cannot neglect this term in O(εR) in (73) at the point ζ wi n i = 0, η = 0.…”

Section: Knudsen-layer Equationmentioning

confidence: 99%

“…As mentioned in Sect. 5.1, the Chapman-Enskog solution can be made to satisfy the kinetic initial condition (25) with the choice (51). From the form of (55) and (57), it follows thatf…”

Section: Summary and Remarksmentioning

confidence: 99%

“…then the Chapman-Enskog solution (39) satisfies (25) up to O(ε), since ∂v i /∂ x j = 0 and ∂T /∂ x i = 0. Therefore, under assumption (iv) in Sect.…”

Section: Introduction Of the Knudsen Layermentioning

confidence: 99%

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Slip Boundary Conditions for the Compressible Navier–Stokes Equations

et al. 2017

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Gradient Divergence of Fluid-Dynamic Quantities in Rarefied Gases on Smooth Boundaries

Takata

Taguchi

2017

J Stat Phys

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The behavior of fluid-dynamic (or macroscopic) quantities of rarefied gases is studied, with a special interest in its non-analytic feature near boundaries. It is shown that their gradients normal to the boundary diverge even if the boundary is smooth, irrespective of the value of the (nonzero) Knudsen number. The boundary geometry determines the diverging rate. On a planar or concave boundary, the logarithmic divergence ln s should be observed, where s is the normal distance from the boundary. In other cases, the diverging rate is enhanced to be the inverse-power s −1/n , where n(≥ 2) is the degree of the dominant terms of the polynomial which locally represents the boundary. Some numerical demonstrations are given as well.

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Singular Behavior of the Macroscopic Quantity Near the Boundary for a Lorentz-Gas Model with the Infinite-Range Potential

Takata

Hattori

2022

J Stat Phys

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Possibility of the diverging gradient of the macroscopic quantity near the boundary is investigated by a mono-speed Lorentz-gas model, with a special attention to the regularizing effect of the grazing collision for the infinite-range potential on the velocity distribution function (VDF) and its influence on the macroscopic quantity. By careful numerical analyses of the steady one-dimensional boundary-value problem, it is confirmed that the grazing collision suppresses the occurrence of a jump discontinuity of the VDF on the boundary. However, as the price for that regularization, the collision integral becomes no longer finite in the direction of the molecular velocity parallel to the boundary. Consequently, the gradient of the macroscopic quantity diverges, even stronger than the case of the finite-range potential. A conjecture about the diverging rate in approaching the boundary is made as well for a wide range of the infinite-range potentials, accompanied by the numerical evidence.

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Second-Order Knudsen-Layer Analysis for the Generalized Slip-Flow Theory II: Curvature Effects

Cited by 13 publications

References 22 publications

Slip Boundary Conditions for the Compressible Navier–Stokes Equations

Slip Boundary Conditions for the Compressible Navier–Stokes Equations

Gradient Divergence of Fluid-Dynamic Quantities in Rarefied Gases on Smooth Boundaries

Singular Behavior of the Macroscopic Quantity Near the Boundary for a Lorentz-Gas Model with the Infinite-Range Potential

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