Slow uniform flow of a rarefied gas past an infinitely thin circular disk

TL;DR

The paper solves the slow, axisymmetric flow of a rarefied gas past an infinitesimally thin circular disk using the linearized BGK Boltzmann model with diffuse reflection. By formulating an integral equation along characteristics, it accurately captures discontinuities in the velocity distribution function at the disk edge and reveals a kinetic edge layer with magnitude scaling as $\mathrm{Kn}^{1/2}$. It also uncovers a thermal polarization near the edge that shares the same $\mathrm{Kn}^{1/2}$ scaling, and computes the drag force across the Knudsen-number spectrum, recovering known limits in both the free-molecular and near-continuum regimes. The approach demonstrates a robust method for three-dimensional, edge-affected rarefied flows and provides insights into edge-layer structure relevant for micro/nano gas dynamics.

Abstract

The classical problem of steady rarefied gas flow past an infinitely thin circular disk is revisited, with particular emphasis on the gas behavior near the disk edge. The uniform flow is assumed to be perpendicular to the disk surface. An integral equation for the velocity distribution function, derived from the linearized Bhatnagar-Gross-Krook (BGK) model of the Boltzmann equation and subject to diffuse reflection boundary conditions, is solved numerically. The numerical method fully accounts for the discontinuity in the velocity distribution function that arises due to the presence of the edge. It is found that a kinetic boundary layer forms near the disk edge, extending over several mean free paths, and that its magnitude scales as $\mathrm{Kn}^{1/2}$ as the Knudsen number $\mathrm{Kn}$ (defined with respect to the disk radius) tends to zero. A thermal polarization effect, previously studied for spherical geometries, is also observed in the disk case, with a more pronounced manifestation near the edge that exhibits the same $\mathrm{Kn}^{1/2}$ scaling. The drag force acting on the disk is computed over a wide range of Knudsen numbers and shows good agreement with existing results for a hard-sphere gas and in the near-free-molecular regime.

Figures (15)

• Figure 1: Problem: a flow past a circular disk.
• Figure 2: (a) Backward characteristics (dashed line) from the point $\bm{x}$ in the direction of $-\bm{\zeta}$ in the case of $0 \leq r < 1$. The thick solid arrow indicates the molecular velocity $\bm{\zeta}$. (b) shows a projected view of from the positive side of the $x_1$ axis.
• Figure 3: (a) Backward characteristics (dashed line) from the point $\bm{x}$ in the direction of $-\bm{\zeta}$ in the case of $r > 1$. See the caption of Fig. \ref{['fig:trajectory_0r1']}.
• Figure 4: Cross sections of the boundary $\partial \varOmega$ in the $\theta_\zeta\,\varphi_\zeta$ plane, where the VDF is discontinuous, for various values of $r$ in the cases of (a) $x=1$ and (b) $x=0.2$. For a given $x$, the solid red curves represent $\theta_\zeta = \theta_{\zeta *}^+(x,r,\varphi_\zeta)$ as a function of $\varphi_\zeta$ ($r<1$); the solid (dash-dotted) blue curves represent $\theta_\zeta = \theta_{\zeta *}^+(x,r,\varphi_\zeta)$ ($\theta_\zeta = \theta_{\zeta *}^-(x,r,\varphi_\zeta)$) as a function of $\varphi_\zeta$ ($r>1$); the solid black curves represent $\theta_\zeta = \theta_{\zeta *}^+(x,r,\varphi_\zeta)$ as a function of $\varphi_\zeta$ ($r=1$). The values of $r \in [0.8, 1.2]$ not shown in the panels are $r=0.8+0.05 m$ ($m=0,1,\ldots,8$). When $r > 1$, the curve $\theta_{\zeta} = \theta_{\zeta *}^{+}$ (solid blue curves) and $\theta_{\zeta} = \theta_{\zeta *}^{-}$ (dash-dotted blue curves) are joined at $\varphi_\zeta = \varphi_{\zeta *}$ indicated by open circles. The black dashed line indicates $\theta_\zeta = \text{arctan}(\cot \varphi_\zeta/x)$, which gives the trajectory of $\varphi_\zeta = \varphi_{\zeta *}(r)$ for $r \ge 1$.
• Figure 5: The geometrical interpretations of $\tilde{x}$, $\tilde{r}$, $\tilde{\varphi}_{\zeta}$, and $r_{\mathrm{w}}$ [(a)] and a view from the positive side of the $x_1$ axis [(b)]. Suppose that we move along the characteristics from $\bm{x}$ to $\tilde{\bm{x}} = \bm{x} - \bm{\ell} s$ for a given $\bm{\ell}=\bm{\zeta}/\zeta$. Then, the cylindrical coordinates $(x,r)$ of $\bm{x}$ change to $(\tilde{x},\tilde{r})$ at $\tilde{\bm{x}}$. Furthermore, at $\tilde{\bm{x}}$, the azimuth angle $\varphi_{\zeta}$ of $\bm{\zeta}$ changes to $\tilde{\varphi}_{\zeta}$. If we project the trajectory onto the plane $x_1=0$ and call the resulting segment PS, the length of the segment OS gives $\tilde{r}$, and the angle between the two lines SP and OS gives $\tilde{\varphi}_{\zeta}$. In the case where $(x,r,\theta_{\zeta},\varphi_{\zeta}) \in \varOmega$, if the intersection of the characteristic with the disk is denoted by T, the length of the segment OT gives $r_{\mathrm{w}}$.