Asymptotic far-field behavior of macroscopic quantities in a problem of slow uniform rarefied gas flow past a sphere

The steady behavior of a low Mach number flow of a rarefied gas past a sphere is considered on the basis of the linearized Boltzmann equation with a special interest in the asymptotic behaviors of the flow velocity and temperature in the region far from the sphere. The study is motivated by the previous one [Taguchi, J. Fluid Mech. 774, 363 (2015)], in which an asymptotic analysis of the Boltzmann equation for small Mach numbers was carried out to derive the expression of the drag up to the second order of the Mach number. The derived expression contains two functions of the Knudsen number, corresponding to the leading-order drag obtained from the linearized problem and the second-order correction due to the weak nonlinear effect. This correction is also obtained through the analysis of the linearized problem; it is given by the factor of the term in the flow velocity whose magnitude is inversely proportional to the distance from the sphere and therefore vanishes at infinity. In this study, this factor (and thus the correction) is obtained for a wide range of the Knudsen number for the hard-sphere gas as well as for the ellipsoidal statistical (ES) model of the Boltzmann equation, under the conventional diffuse reflection boundary condition. The construction is based on the universal relation between the linear drag and the factor (correction). With the available data for the linear drag, this allows us to derive the latter from the former. For the ES model with the Prandtl number $\mathrm{Pr}=2/3$, a series of additional numerical computations of the linearized problem is carried out to obtain the linear drag and then the correction. It is also shown that a factor occurring in the temperature, decaying in proportion to the inverse square of the distance from the sphere, is connected to the thermal force exerted on a sphere (thermophoresis), whose numerical values on the basis of the ES model are obtained as well.

Figure 1

Double-log plots of the moments ${\tilde{\omega }}_{\left(1\right)},{\tilde{u}}_{r\left(1\right)}-1,{\tilde{u}}_{t\left(1\right)}+1$, and ${\tilde{\tau }}_{\left(1\right)}$ for $\mathrm{Pr}=2/3$ and 1 on the basis of the ES model under the diffuse reflection boundary condition. The solid line indicates the result for $\mathrm{Pr}=2/3$ and the dash-dot line the result for $\mathrm{Pr}=1$ (or the BGK model). The solution of the free-molecular gas $\left(k=\infty \right)$ is indicated by the dashed line.

Figure 2

$h_D\left(k\right)$ and $c_1\left(k\right)$ under the diffuse reflection condition for the hard-sphere gas $(\diamond )$, for the ES model with $\mathrm{Pr}=2/3$ $(\circ )$, and for the ES model with $\mathrm{Pr}=1$ (or the BGK model) $(•)$. The value of $h_D$ for the hard-sphere gas were taken from Ref. [22] (also shown in Table 2). The solid line indicates the asymptotic formulas (34) and (35) for the hard-sphere gas. The dashed line shows the first two terms of the formulas. In (a), the asymptotic value as $k\to \infty$ is also indicated by dash-dotted line.

Figure 3

$c_2\left(k\right)$ and $c_3\left(k\right)$ under the diffuse reflection condition for the ES model with $\mathrm{Pr}=2/3$ $(\circ )$ and for the ES model with $\mathrm{Pr}=1$ (or the BGK model) $(•)$, and $c_3\left(k\right)$ under the diffuse reflection condition for the hard-sphere gas $(\diamond )$. The solid line indicates the asymptotic formulas (36) and (37) for the hard-sphere gas. In (b), the dash-dotted line indicates the asymptotic value as $k\to \infty$ for the hard-sphere model.

Figure 4

The computational domain. The domains II and III are separated by the characteristic $rsin{\theta }_{\zeta }=1$ $\left(0<{\theta }_{\zeta }\le \pi /2\right)$. The arrow indicates the direction of propagation of the discontinuity in VDF.