2.1. Problem and basic assumptions

Consider a slow uniform flow of a monatomic rarefied gas past a sphere with radius $L$. At a great distance from the sphere, the gas is in the equilibrium state with velocity $(v_{\infty },0,0)$, pressure $p_{\infty }$ and temperature $T_{\infty }$. The sphere is kept at the same temperature as the uniform flow and is assumed to be at rest. We investigate the steady behaviour of the gas around the sphere under the following assumptions:

1. (i) the behaviour of the gas is described by the Boltzmann equation;

2. (ii) the gas molecules are reflected from the sphere according to the rule described below;

3. (iii) the Mach number of the flow is small, i.e. $\mathit{Ma}=v_{\infty }/(5RT_{\infty }/3)^{1/2}\ll 1$;

4. (iv) the Knudsen number is finite, i.e. $\mathit{Kn}=\ell _{\infty }/L=O(1)$.

Here, $R$ is the specific gas constant ($R=k_{B}/m$ with $k_{B}$ and $m$ being the Boltzmann constant and the mass of a molecule, respectively) and $\ell _{\infty }$ is the mean free path of the gas molecules at the equilibrium state at rest with pressure $p_{\infty }$ and temperature $T_{\infty }$.

2.2. Basic equations

We first introduce some basic notation. Let $Lx_{i}$ (or $L\boldsymbol{x}$) be the space coordinates and $(2RT_{\infty })^{1/2}{\it\zeta}_{i}$ (or $(2RT_{\infty })^{1/2}{\bf\zeta}$) be the molecular velocity (see figure 1). We also use the spherical coordinate system $(Lr,{\it\theta},{\it\varphi})$ throughout the paper whenever it is convenient. The $r$, ${\it\theta}$ and ${\it\varphi}$ components of ${\bf\zeta}$ are denoted by ${\it\zeta}_{r}$, ${\it\zeta}_{{\it\theta}}$ and ${\it\zeta}_{{\it\varphi}}$, respectively. The relation ${\it\zeta}_{1}={\it\zeta}_{r}\cos {\it\theta}-{\it\zeta}_{{\it\theta}}\sin {\it\theta}$ is frequently used. We denote by ${\it\rho}_{\infty }(2RT_{\infty })^{-3/2}(1+{\it\phi}(\boldsymbol{x},{\bf\zeta}))E$ the velocity distribution function, where ${\it\rho}_{\infty }=p_{\infty }/RT_{\infty }$ is the density of the uniform flow and $E={\rm\pi}^{-3/2}\exp (-|{\bf\zeta}|^{2})$. Further, we introduce the density ${\it\rho}_{\infty }(1+{\it\omega}(\boldsymbol{x}))$, the flow velocity $(2RT_{\infty })^{1/2}u_{i}(\boldsymbol{x})$, the temperature $T_{\infty }(1+{\it\tau}(\boldsymbol{x}))$, the pressure $p_{\infty }(1+P(\boldsymbol{x}))$, the stress tensor $p_{\infty }({\it\delta}_{ij}+\unicode[STIX]{x1D617}_{ij}(\boldsymbol{x}))$ (${\it\delta}_{ij}$ is the Kronecker delta) and the heat-flux vector $p_{\infty }(2RT_{\infty })^{1/2}Q_{i}(\boldsymbol{x})$ of the gas. The components of $\boldsymbol{u}=(u_{i})_{i=1,2,3}$, $\unicode[STIX]{x1D64B}=(\unicode[STIX]{x1D617}_{ij})_{i,j=1,2,3}$ and $\boldsymbol{Q}=(Q_{i})_{i=1,2,3}$ in the spherical coordinate system are denoted by using $r$, ${\it\theta}$ and ${\it\varphi}$ as the subscripts, e.g. $u_{r}$, $u_{{\it\theta}}$, $\unicode[STIX]{x1D617}_{rr}$, etc.

Figure 1.

Problem and coordinate system.

The Boltzmann equation for the present steady problem is written as

(2.1)$$\begin{eqnarray}{\it\zeta}_{i}\frac{\partial {\it\phi}}{\partial x_{i}}=\frac{1}{k}(\mathscr{L}[{\it\phi}]+\mathscr{J}[{\it\phi},{\it\phi}]).\end{eqnarray}$$

Here, $\mathscr{L}[{\it\phi}]$ and $\mathscr{J}[{\it\phi},{\it\phi}]$ are the linear and nonlinear parts of the collision operator (appendix A) and $k=(\sqrt{{\rm\pi}}/2)\mathit{Kn}$. The boundary condition on the sphere is written in the form

(2.2)$$\begin{eqnarray}{\it\phi}=\mathscr{K}[{\it\phi}]\quad \text{for}~{\it\zeta}_{i}n_{i}>0~(r=1),\end{eqnarray}$$

where $\mathscr{K}[{\it\phi}]$ is the (linear) scattering operator whose explicit form depends on the prescribed rule for the molecule–surface interaction and $n_{i}$ is the unit normal vector on the surface pointing to the gas. $\mathscr{K}[{\it\phi}]$ is further expressed by a scattering kernel $\hat{K}$ as

(2.3)$$\begin{eqnarray}\mathscr{K}[{\it\phi}]=\int _{{\it\zeta}_{i\ast }n_{i}<0}\hat{K}({\bf\zeta},{\bf\zeta}_{\ast }){\it\phi}_{\ast }E_{\ast }\,\text{d}{\bf\zeta}_{\ast }\quad ({\it\zeta}_{i}n_{i}>0),\end{eqnarray}$$

where ${\it\phi}_{\ast }={\it\phi}({\bf\zeta}_{\ast })$ and $E_{\ast }={\rm\pi}^{-3/2}\exp (-|{\bf\zeta}_{\ast }|^{2})$. Since the sphere is at rest and its surface temperature is uniform, $\hat{K}$ is independent of the local position on the surface (appendix A). The derivation of (2.3) from a more general scattering kernel is given in appendix A. For the diffuse reflection condition, $\hat{K}=-2\sqrt{{\rm\pi}}{\it\zeta}_{i\ast }n_{i}$. We assume that $\hat{K}$ satisfies the following properties.

1. (i) Positivity: $\hat{K}({\bf\zeta},{\bf\zeta}_{\ast })\geqslant 0$ for ${\it\zeta}_{i}n_{i}>0$ and ${\it\zeta}_{i\ast }n_{i}<0$.

2. (ii) Impermeability: $-\!\int _{{\it\zeta}_{i}n_{i}>0}({\it\zeta}_{k}n_{k}/{\it\zeta}_{j\ast }n_{j})\hat{K}({\bf\zeta},{\bf\zeta}_{\ast })E\,\text{d}{\bf\zeta}=1$ for ${\it\zeta}_{i\ast }n_{i}<0$.

3. (iii) Let

   (2.4)$$\begin{eqnarray}(1+{\it\phi})E=\frac{1+{\hat{c}}_{0}}{{\rm\pi}^{3/2}(1+{\hat{c}}_{4})^{3/2}}\exp \left(-\frac{({\it\zeta}_{i}-{\hat{c}}_{i})^{2}}{1+{\hat{c}}_{4}}\right)\end{eqnarray}$$
   be an arbitrary Maxwellian, where ${\hat{c}}_{0}$, ${\hat{c}}_{i}$, and ${\hat{c}}_{4}$ are independent of ${\bf\zeta}$. Among such ${\it\phi}$, only ${\it\phi}={\hat{c}}_{0}$ satisfies the relation ${\it\phi}=\mathscr{K}[{\it\phi}]$ for ${\it\zeta}_{i}n_{i}>0$.

4. (iv) Local isotropy:

   (2.5)$$\begin{eqnarray}\mathscr{K}[f(\unicode[STIX]{x1D62D}_{ij}{\it\zeta}_{j})]({\bf\zeta})=\mathscr{K}[f({\bf\zeta})](\unicode[STIX]{x1D62D}_{ij}{\it\zeta}_{j}),\end{eqnarray}$$
   where $\unicode[STIX]{x1D62D}_{ij}$ is any orthogonal transformation matrix ($\unicode[STIX]{x1D62D}_{ik}\unicode[STIX]{x1D62D}_{jk}={\it\delta}_{ij}$) that satisfies the condition $\unicode[STIX]{x1D62D}_{ij}n_{j}=n_{i}$.

Finally, the boundary condition at infinity is given by

(2.6)$$\begin{eqnarray}(1+{\it\phi})E\rightarrow (1+{\it\phi}_{\infty })E=\frac{1}{{\rm\pi}^{3/2}}\exp \left(-({\it\zeta}_{i}-\hat{v}_{\infty }{\it\delta}_{i1})^{2}\right)\quad \text{as}~r\rightarrow \infty ,\end{eqnarray}$$

where $\hat{v}_{\infty }=v_{\infty }/(2RT_{\infty })^{1/2}=\sqrt{5/6}\mathit{Ma}$.

The macroscopic quantities of interest, ${\it\omega}$, $u_{i}$, ${\it\tau}$, $P$, $\unicode[STIX]{x1D617}_{ij}$ and $Q_{i}$, are expressed in terms of ${\it\phi}$ as

(2.7a)$$\begin{eqnarray}\displaystyle & {\it\omega}=\langle {\it\phi}\rangle ,\quad (1+{\it\omega})u_{i}=\langle {\it\zeta}_{i}{\it\phi}\rangle , & \displaystyle\end{eqnarray}$$

(2.7b)$$\begin{eqnarray}\displaystyle & \displaystyle {\textstyle \frac{3}{2}}(1+{\it\omega}){\it\tau}=\left\langle \left(|{\bf\zeta}|^{2}-{\textstyle \frac{3}{2}}\right){\it\phi}\right\rangle -(1+{\it\omega})u_{j}^{2}, & \displaystyle\end{eqnarray}$$

(2.7c)$$\begin{eqnarray}\displaystyle & P={\it\omega}+{\it\tau}+{\it\omega}{\it\tau},\quad \unicode[STIX]{x1D617}_{ij}=2\langle {\it\zeta}_{i}{\it\zeta}_{j}{\it\phi}\rangle -2(1+{\it\omega})u_{i}u_{j}, & \displaystyle\end{eqnarray}$$

(2.7d)$$\begin{eqnarray}\displaystyle & \displaystyle Q_{i}=\langle {\it\zeta}_{i}|{\bf\zeta}|^{2}{\it\phi}\rangle -{\textstyle \frac{5}{2}}u_{i}-u_{j}\unicode[STIX]{x1D617}_{ij}-{\textstyle \frac{3}{2}}Pu_{i}-(1+{\it\omega})u_{i}u_{j}^{2}. & \displaystyle\end{eqnarray}$$

${\bf\zeta}$

(2.8)$$\begin{eqnarray}\langle f\rangle =\int f({\bf\zeta})E\,\text{d}{\bf\zeta},\end{eqnarray}$$

where the integration is carried out over the whole space of ${\bf\zeta}$.

To summarise, the present boundary-value problem contains two basic parameters, i.e. $k$ (or $\mathit{Kn}$) and $\hat{v}_{\infty }$ (or $\mathit{Ma}$). Because of the assumptions (iii) and (iv), we assume that

(2.9)$$\begin{eqnarray}\hat{v}_{\infty }\ll 1,\quad k=O(1).\end{eqnarray}$$

The aim of the present study is to investigate the behaviour of the gas in the situation (2.9) by a systematic asymptotic analysis for small $\hat{v}_{\infty }\ll 1$. Finally, we note that the existence of a solution in the present situation was mathematically proved by Ukai & Asano (Reference Ukai and Asano1983).

3.1. Problem for order ${\it\epsilon}$

At the leading order, the problem is reduced to

(3.8)$$\begin{eqnarray}\displaystyle & \displaystyle {\it\zeta}_{i}\frac{\partial {\it\phi}_{S(1)}}{\partial x_{i}}=\frac{1}{k}\mathscr{L}[{\it\phi}_{S(1)}], & \displaystyle\end{eqnarray}$$

(3.9)$$\begin{eqnarray}\displaystyle & {\it\phi}_{S(1)}=\mathscr{K}[{\it\phi}_{S(1)}]\quad \text{for}~{\it\zeta}_{i}n_{i}>0~(r=1), & \displaystyle\end{eqnarray}$$

(3.10)$$\begin{eqnarray}\displaystyle & {\it\phi}_{S(1)}\rightarrow 2{\it\zeta}_{1}\quad \text{as}~r\rightarrow \infty . & \displaystyle\end{eqnarray}$$

$h_{S(1)}$

$h={\it\omega}$

$u_{i}$

${\it\tau}$

${\it\phi}_{S(1)}$

(3.11a)$$\begin{eqnarray}\displaystyle & {\it\omega}_{S(1)}=\langle {\it\phi}_{S(1)}\rangle ,\quad u_{iS(1)}=\langle {\it\zeta}_{i}{\it\phi}_{S(1)}\rangle , & \displaystyle\end{eqnarray}$$

(3.11b)$$\begin{eqnarray}\displaystyle & \displaystyle {\it\tau}_{S(1)}={\textstyle \frac{2}{3}}\left\langle \left(|{\bf\zeta}|^{2}-{\textstyle \frac{3}{2}}\right){\it\phi}_{S(1)}\right\rangle ,\quad P_{S(1)}={\it\omega}_{S(1)}+{\it\tau}_{S(1)}, & \displaystyle\end{eqnarray}$$

(3.11c)$$\begin{eqnarray}\displaystyle & \displaystyle \unicode[STIX]{x1D617}_{ijS(1)}=2\langle {\it\zeta}_{i}{\it\zeta}_{j}{\it\phi}_{S(1)}\rangle ,\quad Q_{iS(1)}=\langle {\it\zeta}_{i}|{\bf\zeta}|^{2}{\it\phi}_{S(1)}\rangle -{\textstyle \frac{5}{2}}u_{iS(1)}. & \displaystyle\end{eqnarray}$$

3.2. Problem for order ${\it\epsilon}^{2}$

Suppose that ${\it\phi}_{S(1)}$ is known. Then the next-order problem takes the following form:

(3.12)$$\begin{eqnarray}\displaystyle & \displaystyle {\it\zeta}_{i}\frac{\partial {\it\phi}_{S(2)}}{\partial x_{i}}=\frac{1}{k}\mathscr{L}[{\it\phi}_{S(2)}]+\frac{1}{k}\mathscr{J}[{\it\phi}_{S(1)},{\it\phi}_{S(1)}], & \displaystyle\end{eqnarray}$$

(3.13)$$\begin{eqnarray}\displaystyle & {\it\phi}_{S(2)}=\mathscr{K}[{\it\phi}_{S(2)}]\quad \text{for}~{\it\zeta}_{i}n_{i}>0~(r=1), & \displaystyle\end{eqnarray}$$

(3.14)$$\begin{eqnarray}\displaystyle & {\it\phi}_{S(2)}\rightarrow 2{\it\zeta}_{1}^{2}-1\quad \text{as}~r\rightarrow \infty . & \displaystyle\end{eqnarray}$$

$h_{S(2)}$

$h={\it\omega}$

$u_{i}$

${\it\tau}$

${\it\phi}_{S(2)}$

(3.15a)$$\begin{eqnarray}\displaystyle & {\it\omega}_{S(2)}=\langle {\it\phi}_{S(2)}\rangle , & \displaystyle\end{eqnarray}$$

(3.15b)$$\begin{eqnarray}\displaystyle & u_{iS(2)}=\langle {\it\zeta}_{i}{\it\phi}_{S(2)}\rangle -{\it\omega}_{S(1)}u_{iS(1)}, & \displaystyle\end{eqnarray}$$

(3.15c)$$\begin{eqnarray}\displaystyle & \displaystyle {\it\tau}_{S(2)}={\textstyle \frac{2}{3}}\left\langle \left(|{\bf\zeta}|^{2}-{\textstyle \frac{3}{2}}\right){\it\phi}_{S(2)}\right\rangle -{\textstyle \frac{2}{3}}u_{jS(1)}^{2}-{\it\omega}_{S(1)}{\it\tau}_{S(1)}, & \displaystyle\end{eqnarray}$$

(3.15d)$$\begin{eqnarray}\displaystyle & P_{S(2)}={\it\omega}_{S(2)}+{\it\tau}_{S(2)}+{\it\omega}_{S(1)}{\it\tau}_{S(1)}, & \displaystyle\end{eqnarray}$$

(3.15e)$$\begin{eqnarray}\displaystyle & \unicode[STIX]{x1D617}_{ijS(2)}=2\langle {\it\zeta}_{i}{\it\zeta}_{j}{\it\phi}_{S(2)}\rangle -2u_{iS(1)}u_{jS(1)}, & \displaystyle\end{eqnarray}$$

(3.15f)$$\begin{eqnarray}\displaystyle & \displaystyle Q_{iS(2)}=\langle {\it\zeta}_{i}|{\bf\zeta}|^{2}{\it\phi}_{S(2)}\rangle -{\textstyle \frac{5}{2}}u_{iS(2)}-u_{jS(1)}\unicode[STIX]{x1D617}_{ijS(1)}-{\textstyle \frac{3}{2}}P_{S(1)}u_{iS(1)}. & \displaystyle\end{eqnarray}$$

${\it\phi}_{S(2)}$

${\it\phi}_{S(2)}$

$k=\infty$

$k<\infty$

The reason for this ill-posedness can be understood as follows. We are seeking a solution of the Boltzmann equation (2.1) whose magnitude is of the order of ${\it\epsilon}$. If the length scale of variation of the solution is of the order of unity, the nonlinear term of the Boltzmann equation remains much smaller than the other terms. On the other hand, if the length scale of variation is much longer and is of the order of $1/{\it\epsilon}$ (i.e. $\partial {\it\phi}/\partial x_{i}=O({\it\epsilon}{\it\phi})$), the streaming term ${\it\zeta}_{i}\partial {\it\phi}/\partial x_{i}$ becomes comparable to the nonlinear term. In this case, the cumulative effect of the nonlinear term cannot be neglected but plays a decisive role in describing the flow behaviour in the far region, irrespective of the smallness of ${\it\phi}$. In the next section, by taking this into account, we will investigate the behaviour of the gas in the far region as described by the Boltzmann equation.

We end this section by stating the following property of the leading-order inner solution ${\it\phi}_{S(1)}$ concerning its asymptotic behaviour at large $r$.

Proposition 1. Let ${\it\phi}_{S(1)}$ be a solution of the boundary-value problem (3.8)–(3.10), and let ${\it\omega}_{S(1)}$, $u_{iS(1)}$, ${\it\tau}_{S(1)}$ and $P_{S(1)}$ be the associated macroscopic quantities given by (3.11). Then, ${\it\phi}_{S(1)}$, ${\it\omega}_{S(1)}$, $u_{iS(1)}$, ${\it\tau}_{S(1)}$ and $P_{S(1)}$ approach the corresponding values at infinity at the following rate:

(3.16)$$\begin{eqnarray}\displaystyle & {\it\phi}_{S(1)}=2{\it\zeta}_{1}+O(r^{-1}), & \displaystyle\end{eqnarray}$$

(3.17)$$\begin{eqnarray}\displaystyle & u_{iS(1)}={\it\delta}_{i1}+O(r^{-1}), & \displaystyle\end{eqnarray}$$

(3.18a−c)$$\begin{eqnarray}{\it\omega}_{S(1)}=O(r^{-2}),\quad {\it\tau}_{S(1)}=O(r^{-2}),\quad P_{S(1)}=O(r^{-2}).\end{eqnarray}$$

4.0.1. Fluid-dynamic-type equations and velocity distribution function for the outer solution

Since the derivation of the fluid-dynamic-type equations is well established and is expatiated (e.g. Sone Reference Sone2002, Reference Sone2007), we only summarise the results of the analysis here. First, the fluid-dynamic-type equations describing the behaviour of the gas in the far region are summarised as follows:

order ${\it\epsilon}$

(4.8)$$\begin{eqnarray}\frac{\partial P_{O(1)}}{\partial y_{i}}=0,\end{eqnarray}$$

(4.9a)$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\partial u_{iO(1)}}{\partial y_{i}}=0, & \displaystyle\end{eqnarray}$$

(4.9b)$$\begin{eqnarray}\displaystyle & \displaystyle u_{jO(1)}\frac{\partial u_{iO(1)}}{\partial y_{j}}=-\frac{1}{2}\frac{\partial P_{O(2)}}{\partial y_{i}}+\frac{{\it\gamma}_{1}k}{2}\frac{\partial ^{2}u_{iO(1)}}{\partial y_{j}^{2}}, & \displaystyle\end{eqnarray}$$

(4.9c)$$\begin{eqnarray}\displaystyle & \displaystyle u_{jO(1)}\frac{\partial {\it\tau}_{O(1)}}{\partial y_{j}}=\frac{{\it\gamma}_{2}k}{2}\frac{\partial ^{2}{\it\tau}_{O(1)}}{\partial y_{j}^{2}}, & \displaystyle\end{eqnarray}$$

(4.9d)$$\begin{eqnarray}\displaystyle & P_{O(1)}={\it\omega}_{O(1)}+{\it\tau}_{O(1)}; & \displaystyle\end{eqnarray}$$

${\it\epsilon}^{2}$

(4.10a)$$\begin{eqnarray}\displaystyle & & \displaystyle \frac{\partial u_{iO(2)}}{\partial y_{i}}=-u_{jO(1)}\frac{\partial {\it\omega}_{O(1)}}{\partial y_{j}},\end{eqnarray}$$

(4.10b)$$\begin{eqnarray}\displaystyle & & \displaystyle u_{jO(1)}\frac{\partial u_{iO(2)}}{\partial y_{j}}+({\it\omega}_{O(1)}u_{jO(1)}+u_{jO(2)})\frac{\partial u_{iO(1)}}{\partial y_{j}}\nonumber\\ \displaystyle & & \displaystyle \quad =-\frac{1}{2}\frac{\partial }{\partial y_{i}}\left(P_{O(3)}-\frac{{\it\gamma}_{1}{\it\gamma}_{2}-4{\it\gamma}_{3}}{6}k^{2}\frac{\partial ^{2}{\it\tau}_{O(1)}}{\partial y_{j}^{2}}\right)\nonumber\\ \displaystyle & & \displaystyle \qquad +\,\frac{{\it\gamma}_{1}k}{2}\frac{\partial ^{2}u_{iO(2)}}{\partial y_{j}^{2}}+\frac{{\it\gamma}_{4}k}{2}\frac{\partial }{\partial y_{j}}\left[{\it\tau}_{O(1)}\left(\frac{\partial u_{iO(1)}}{\partial y_{j}}+\frac{\partial u_{jO(1)}}{\partial y_{i}}\right)\right],\end{eqnarray}$$

(4.10c)$$\begin{eqnarray}\displaystyle & & \displaystyle u_{jO(1)}\frac{\partial {\it\tau}_{O(2)}}{\partial y_{j}}+({\it\omega}_{O(1)}u_{jO(1)}+u_{jO(2)})\frac{\partial {\it\tau}_{O(1)}}{\partial y_{j}}-\frac{2}{5}u_{jO(1)}\frac{\partial P_{O(2)}}{\partial y_{j}}\nonumber\\ \displaystyle & & \displaystyle \quad =\frac{{\it\gamma}_{1}k}{5}\left(\frac{\partial u_{iO(1)}}{\partial y_{j}}+\frac{\partial u_{jO(1)}}{\partial y_{i}}\right)^{2}+\frac{k}{2}\frac{\partial ^{2}}{\partial y_{j}^{2}}\left({\it\gamma}_{2}{\it\tau}_{O(2)}+\frac{{\it\gamma}_{5}}{2}{\it\tau}_{O(1)}^{2}\right),\end{eqnarray}$$

(4.10d)$$\begin{eqnarray}\displaystyle & & \displaystyle P_{O(2)}={\it\omega}_{O(2)}+{\it\tau}_{O(2)}+{\it\omega}_{O(1)}{\it\tau}_{O(1)},\end{eqnarray}$$

${\it\gamma}_{i}$

$i=1,\dots ,5$

Equation (4.8) shows that the leading-order pressure is constant. On the other hand, (4.9a) and (4.9b) are the continuity equation and the Navier–Stokes equation for an incompressible fluid, respectively, while (4.9c) is an energy equation. Equation (4.9d) is the linearised equation of state. Equations (4.10a–d) are similar to (4.9a–d) but contain some extra terms corresponding to the effect of compressibility and to a non-Navier–Stokes effect (i.e. the thermal stress, the term with ${\it\gamma}_{3}$). They become important when ${\it\tau}_{O(1)}$ (${\it\omega}_{O(1)}$) or $P_{O(2)}$ is not uniform. However, as we will see, they are constant in the present situation, and, therefore, these effects become of the higher order (see § 5.1). It should also be mentioned that the derived equations are essentially the same as the fluid-dynamic-type equations obtained by the S expansion (Sone Reference Sone and Dini1971, Reference Sone2002, Reference Sone2007). The interested reader is referred to Sone (Reference Sone2002, Reference Sone2007) for further discussions on the features of the fluid-dynamic-type equations.

Once the macroscopic quantities are obtained (by solving the fluid-dynamic-type equations), the velocity distribution functions ${\it\phi}_{O(m)}$ are readily obtained from the expressions

(4.11a)$$\begin{eqnarray}\displaystyle & & \displaystyle {\it\phi}_{O(1)}={\it\phi}_{eO(1)},\end{eqnarray}$$

(4.11b)$$\begin{eqnarray}\displaystyle & & \displaystyle {\it\phi}_{O(2)}={\it\phi}_{eO(2)}-k{\it\zeta}_{i}{\it\zeta}_{j}B(|{\bf\zeta}|)\displaystyle \frac{\partial u_{iO(1)}}{\partial y_{j}}-k{\it\zeta}_{i}A(|{\bf\zeta}|)\displaystyle \frac{\partial {\it\tau}_{O(1)}}{\partial y_{i}},\end{eqnarray}$$

$$\begin{eqnarray}\displaystyle \cdots \,. & & \displaystyle \nonumber\end{eqnarray}$$

The functions $A(|{\bf\zeta}|)$ and $B(|{\bf\zeta}|)$ are defined in appendix B and ${\it\phi}_{eO(m)}$ ($m=1,2,\dots$) are the expansion coefficients of the local Maxwellian in ${\it\epsilon}$, i.e.

(4.12)$$\begin{eqnarray}(1+{\it\phi}_{eO})E=\frac{1+{\it\omega}_{O}}{{\rm\pi}^{3/2}(1+{\it\tau}_{O})^{3/2}}\exp \left(-\frac{({\it\zeta}_{i}-u_{iO})^{2}}{1+{\it\tau}_{O}}\right)=E(1+{\it\phi}_{eO(1)}{\it\epsilon}+{\it\phi}_{eO(2)}{\it\epsilon}^{2}+\cdots \!).\end{eqnarray}$$

More specifically,

(4.13a)$$\begin{eqnarray}\displaystyle {\it\phi}_{eO(1)} & = & \displaystyle P_{O(1)}+2{\it\zeta}_{i}u_{iO(1)}+\left(|{\bf\zeta}|^{2}-{\textstyle \frac{5}{2}}\right){\it\tau}_{O(1)},\end{eqnarray}$$

(4.13b)$$\begin{eqnarray}\displaystyle {\it\phi}_{eO(2)} & = & \displaystyle P_{O(2)}+2{\it\zeta}_{i}u_{iO(2)}+\left(|{\bf\zeta}|^{2}-\frac{5}{2}\right){\it\tau}_{O(2)}+2{\it\zeta}_{i}{\it\omega}_{O(1)}u_{iO(1)}+\left(|{\bf\zeta}|^{2}-\frac{3}{2}\right){\it\omega}_{O(1)}{\it\tau}_{O(1)}\nonumber\\ \displaystyle & & \displaystyle +\,2\left({\it\zeta}_{i}{\it\zeta}_{j}-\frac{{\it\delta}_{ij}}{2}\right)u_{iO(1)}u_{jO(1)}+2{\it\zeta}_{i}\left(|{\bf\zeta}|^{2}-\frac{5}{2}\right)u_{iO(1)}{\it\tau}_{O(1)}\nonumber\\ \displaystyle & & \displaystyle +\,\frac{1}{2}\left(|{\bf\zeta}|^{4}-5|{\bf\zeta}|^{2}+\frac{15}{4}\right){\it\tau}_{O(1)}^{2}.\end{eqnarray}$$

4.0.2. Boundary conditions for the outer solution at infinity

Up to now, we have not considered the boundary condition for the outer solution ${\it\phi}_{O}$. At infinity, ${\it\phi}_{O}$ is required to satisfy the condition

(4.14)$$\begin{eqnarray}{\it\phi}_{O}\rightarrow {\it\phi}_{\infty }=\frac{1}{{\rm\pi}^{3/2}}\exp (-({\it\zeta}_{i}-{\it\epsilon}{\it\delta}_{i1})^{2})E^{-1}-1\quad \text{as}~{\it\eta}\rightarrow \infty ,\end{eqnarray}$$

where ${\it\eta}=(y_{i}^{2})^{1/2}(={\it\epsilon}r)$ (cf. (2.6)). If we expand ${\it\phi}_{\infty }$ as ${\it\phi}_{\infty }\!={\it\phi}_{\infty (1)}{\it\epsilon}+{\it\phi}_{\infty (2)}{\it\epsilon}^{2}+\cdots \,$, the boundary condition is satisfied by demanding

(4.15)$$\begin{eqnarray}{\it\phi}_{O(m)}\rightarrow {\it\phi}_{\infty (m)}\quad \text{as}~{\it\eta}\rightarrow \infty\end{eqnarray}$$

($m=1,2,\dots$), where

(4.16a−c)$$\begin{eqnarray}{\it\phi}_{\infty (1)}=2{\it\zeta}_{1},\quad {\it\phi}_{\infty (2)}=2{\it\zeta}_{1}^{2}-1,\quad {\it\phi}_{\infty (3)}={\textstyle \frac{4}{3}}{\it\zeta}_{1}\left({\it\zeta}_{1}^{2}-{\textstyle \frac{3}{2}}\right),\quad \text{etc}.\end{eqnarray}$$

On the other hand, since the functional form of ${\it\phi}_{O(m)}$ with respect to the molecular velocity is specified (cf. (4.11a) and (4.11b)), the above condition is satisfied if the macroscopic quantities contained in ${\it\phi}_{O(m)}$ take special values at ${\it\eta}\rightarrow \infty$. For example for ${\it\phi}_{O(1)}$ and ${\it\phi}_{O(2)}$, the conditions take the following forms:

(4.17a−c)$$\begin{eqnarray}P_{O(1)}\rightarrow 0,\quad u_{iO(1)}\rightarrow {\it\delta}_{i1},\quad {\it\tau}_{O(1)}\rightarrow 0\quad \text{as}~{\it\eta}\rightarrow \infty ,\end{eqnarray}$$

(4.18a−c)$$\begin{eqnarray}P_{O(2)}\rightarrow 0,\quad u_{iO(2)}\rightarrow 0,\quad {\it\tau}_{O(2)}\rightarrow 0\quad \text{as}~{\it\eta}\rightarrow \infty .\end{eqnarray}$$

These conditions will provide natural constraints for the fluid-dynamic-type equations at infinity. The remaining boundary conditions (at ${\it\eta}\rightarrow 0$) will be derived in the course of the analysis by considering the matching with the inner solution in § 5.

5.1. Leading-order problem

In view of the fact that the boundary-value problem (3.8)–(3.10) in § 3 has a solution, we shall start with the following inner problem:

(5.4)$$\begin{eqnarray}\displaystyle & \displaystyle {\it\zeta}_{i}\frac{\partial {\it\phi}_{S(1)}}{\partial x_{i}}=\frac{1}{k}\mathscr{L}[{\it\phi}_{S(1)}], & \displaystyle\end{eqnarray}$$

(5.5)$$\begin{eqnarray}\displaystyle & {\it\phi}_{S(1)}=\mathscr{K}[{\it\phi}_{S(1)}]\quad \text{for}~{\it\zeta}_{i}n_{i}>0~\text{at}~r=1, & \displaystyle\end{eqnarray}$$

(5.6)$$\begin{eqnarray}{\it\phi}_{S(1)}\rightarrow 2{\it\zeta}_{1}\quad \text{as}~r\rightarrow \infty .\end{eqnarray}$$

The problem is the same as (3.8)–(3.10) and therefore has a solution.

Assuming that ${\it\phi}_{S(1)}$ has been obtained, we now consider the leading-order outer solution ${\it\phi}_{O(1)}$. First, we note that the macroscopic quantities $P_{O(1)}$, $u_{iO(1)}$ and ${\it\tau}_{O(1)}$ contained in ${\it\phi}_{O(1)}$ (cf. (4.11a) with (4.13a)) are described by the fluid-dynamic-type equations (4.8) and (4.9a–d). The corresponding boundary condition at infinity is given by (4.17). The remaining condition is provided by the connection (or matching) condition for ${\it\phi}_{O(1)}$ with ${\it\phi}_{S(1)}$ as follows.

To derive the connection condition, we consider a point where $r$ is large but ${\it\epsilon}r$ is small. According to Proposition 1, ${\it\phi}_{S(1)}$ behaves like

(5.7)$$\begin{eqnarray}{\it\phi}_{S(1)}=2{\it\zeta}_{1}+O(r^{-1})\end{eqnarray}$$

at such a point. This means that the inner solution ${\it\phi}_{S(1)}$, if written in terms of the outer variable ${\it\eta}(={\it\epsilon}r)$, has the form

(5.8)$$\begin{eqnarray}{\it\phi}_{S(1)}^{\ast }=2{\it\zeta}_{1}+O({\it\epsilon})\end{eqnarray}$$

at the point under consideration. Here, the asterisk has been added to emphasise that the independent variable of the inner solution has been changed from the original $r$ (or $x_{i}$) to ${\it\eta}$ (or $y_{i}$), i.e. ${\it\phi}_{S(1)}^{\ast }={\it\phi}_{S(1)}|_{r={\it\eta}/{\it\epsilon}}$. From (5.8), we immediately see that ${\it\phi}_{O(1)}$ coincides with ${\it\phi}_{S(1)}^{\ast }$ within an error of $O({\it\epsilon})$, if we impose the following connection condition:

(5.9)$$\begin{eqnarray}{\it\phi}_{O(1)}\rightarrow 2{\it\zeta}_{1}\quad \text{as}~{\it\eta}\rightarrow 0.\end{eqnarray}$$

To proceed further, we note that the functional form of ${\it\phi}_{O(1)}$ is specified as given by (4.11a) (with (4.13a)). Therefore, in order for the above condition to be satisfied, the macroscopic quantities contained in ${\it\phi}_{O(1)}$ should take the following values as ${\it\eta}\rightarrow 0$:

(5.10a−c)$$\begin{eqnarray}P_{O(1)}\rightarrow 0,\quad u_{iO(1)}\rightarrow {\it\delta}_{i1},\quad {\it\tau}_{O(1)}\rightarrow 0\quad \text{as}~{\it\eta}\rightarrow 0.\end{eqnarray}$$

In summary, the matching condition (5.10) and the boundary condition at infinity (4.17) provide appropriate boundary conditions for the leading-order fluid-dynamic-type equations (4.8) and (4.9a–d).

It is easily seen that

(5.11)$$\begin{eqnarray}P_{O(1)}={\it\tau}_{O(1)}={\it\omega}_{O(1)}=0,\end{eqnarray}$$

(5.12a,b)$$\begin{eqnarray}u_{iO(1)}={\it\delta}_{i1},\quad P_{O(2)}=\text{const.,}\end{eqnarray}$$

(trivially) satisfy the equations and the boundary condition (4.17) as well as the matching condition (5.10). Consequently, the leading-order outer solution (4.11a) is reduced to

(5.13)$$\begin{eqnarray}{\it\phi}_{O(1)}=2{\it\zeta}_{1}.\end{eqnarray}$$

That is, ${\it\phi}_{O(1)}$ is just a linearised uniform flow.

In summary, the leading-order solution is given by ${\it\phi}_{O(1)}=2{\it\zeta}_{1}$ (far from the sphere) and by ${\it\phi}_{S(1)}$ (in the vicinity of the sphere), where ${\it\phi}_{S(1)}$ solves the problem (5.4)–(5.6). Incidentally, it is worth noting that the constant pressure $P_{O(2)}$ (cf. (5.12)) is left undetermined at this stage. This constant will be determined later when we consider the next-order outer problem ((5.31) below).

5.2. Further transformation of the problem for ${\it\phi}_{S(1)}$

Before going into the next-order approximation, here and in the subsequent subsection we shall recall some basic properties of the leading-order inner problem (i.e. the problem (5.4)–(5.6)). Thanks to the axial symmetry of $\mathscr{L}$ and $\mathscr{K}$, we can employ a similarity solution of the form (Sone & Aoki Reference Sone and Aoki1983; Sone Reference Sone2007)

(5.14)$$\begin{eqnarray}{\it\phi}_{S(1)}={\it\Phi}_{a}^{(1)}(r,{\it\zeta}_{r},|{\bf\zeta}|)\cos {\it\theta}+{\it\zeta}_{{\it\theta}}{\it\Phi}_{b}^{(1)}(r,{\it\zeta}_{r},|{\bf\zeta}|)\sin {\it\theta},\end{eqnarray}$$

where ${\it\Phi}_{a}^{(1)}$ and ${\it\Phi}_{b}^{(1)}$ are functions of $r$, ${\it\zeta}_{r}$ and $|{\bf\zeta}|$. Then, the problem is reduced to a (spatially one-dimensional) boundary-value problem for ${\it\Phi}_{a}^{(1)}$ and ${\it\Phi}_{b}^{(1)}$ as follows:

(5.15)$$\begin{eqnarray}\displaystyle & \displaystyle {\it\zeta}_{r}\frac{\partial {\it\Phi}_{a}^{(1)}}{\partial r}+\frac{|{\bf\zeta}|^{2}-{\it\zeta}_{r}^{2}}{r}\frac{\partial {\it\Phi}_{a}^{(1)}}{\partial {\it\zeta}_{r}}+\frac{|{\bf\zeta}|^{2}-{\it\zeta}_{r}^{2}}{r}{\it\Phi}_{b}^{(1)}=\frac{1}{k}\mathscr{L}[{\it\Phi}_{a}^{(1)}], & \displaystyle\end{eqnarray}$$

(5.16)$$\begin{eqnarray}\displaystyle & \displaystyle {\it\zeta}_{r}\frac{\partial {\it\Phi}_{b}^{(1)}}{\partial r}+\frac{|{\bf\zeta}|^{2}-{\it\zeta}_{r}^{2}}{r}\frac{\partial {\it\Phi}_{b}^{(1)}}{\partial {\it\zeta}_{r}}-\frac{{\it\zeta}_{r}}{r}{\it\Phi}_{b}^{(1)}-\frac{{\it\Phi}_{a}^{(1)}}{r}=\frac{1}{k}\mathscr{L}_{1}[{\it\Phi}_{b}^{(1)}], & \displaystyle\end{eqnarray}$$

(5.17)$$\begin{eqnarray}\displaystyle & \displaystyle \left[\begin{array}{@{}c@{}}{\it\Phi}_{a}^{(1)}\\ {\it\Phi}_{b}^{(1)}\end{array}\right]=\left[\begin{array}{@{}c@{}}\mathscr{K}[{\it\Phi}_{a}^{(1)}]\\ \mathscr{K}_{1}[{\it\Phi}_{b}^{(1)}]\end{array}\right]\quad \text{for}~{\it\zeta}_{r}>0~(r=1), & \displaystyle\end{eqnarray}$$

(5.18)$$\begin{eqnarray}\displaystyle & \displaystyle \left[\begin{array}{@{}c@{}}{\it\Phi}_{a}^{(1)}\\ {\it\Phi}_{b}^{(1)}\end{array}\right]\rightarrow \left[\begin{array}{@{}c@{}}2{\it\zeta}_{r}\\ -2\end{array}\right]\quad \text{as}~r\rightarrow \infty , & \displaystyle\end{eqnarray}$$

$\mathscr{L}_{1}$

$\mathscr{K}_{1}$

(5.19a)$$\begin{eqnarray}\displaystyle & \displaystyle \frac{{\it\omega}_{S(1)}}{\cos {\it\theta}}=\widetilde{{\it\omega}}_{S(1)}(r), & \displaystyle\end{eqnarray}$$

(5.19b)$$\begin{eqnarray}\displaystyle & \displaystyle \frac{u_{rS(1)}}{\cos {\it\theta}}=\widetilde{u}_{rS(1)}(r),\quad \frac{u_{{\it\theta}S(1)}}{\sin {\it\theta}}=\widetilde{u}_{{\it\theta}S(1)}(r),\quad u_{{\it\varphi}S(1)}=0, & \displaystyle\end{eqnarray}$$

(5.19c)$$\begin{eqnarray}\displaystyle & \displaystyle \frac{{\it\tau}_{S(1)}}{\cos {\it\theta}}=\widetilde{{\it\tau}}_{S(1)}(r),\quad \frac{P_{S(1)}}{\cos {\it\theta}}=\widetilde{P}_{S(1)}(r), & \displaystyle\end{eqnarray}$$

(5.19d)$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x1D617}_{rrS(1)}}{\cos {\it\theta}}=\widetilde{P}_{rrS(1)}(r),\quad \frac{\unicode[STIX]{x1D617}_{r{\it\theta}S(1)}}{\sin {\it\theta}}=\widetilde{P}_{r{\it\theta}S(1)}(r), & \displaystyle\end{eqnarray}$$

(5.19e)$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\unicode[STIX]{x1D617}_{{\it\theta}{\it\theta}S(1)}}{\cos {\it\theta}}=\widetilde{P}_{{\it\theta}{\it\theta}S(1)}(r),\quad \frac{\unicode[STIX]{x1D617}_{{\it\varphi}{\it\varphi}S(1)}}{\cos {\it\theta}}=\widetilde{P}_{{\it\varphi}{\it\varphi}S(1)}(r),\quad \unicode[STIX]{x1D617}_{r{\it\varphi}S(1)}=\unicode[STIX]{x1D617}_{{\it\theta}{\it\varphi}S(1)}=0, & \displaystyle\end{eqnarray}$$

(5.19f)$$\begin{eqnarray}\displaystyle & \displaystyle \frac{Q_{rS(1)}}{\cos {\it\theta}}=\widetilde{Q}_{rS(1)}(r),\quad \frac{Q_{{\it\theta}S(1)}}{\sin {\it\theta}}=\widetilde{Q}_{{\it\theta}S(1)}(r),\quad Q_{{\it\varphi}S(1)}=0, & \displaystyle\end{eqnarray}$$

$r$

(5.20a)$$\begin{eqnarray}\displaystyle & \displaystyle \widetilde{{\it\omega}}_{S(1)}=2{\rm\pi}\int _{0}^{\infty }\int _{0}^{{\rm\pi}}{\it\zeta}^{2}\sin {\it\theta}_{{\it\zeta}}{\it\Phi}_{a}^{(1)}E\,\text{d}{\it\theta}_{{\it\zeta}}\,\text{d}{\it\zeta}, & \displaystyle\end{eqnarray}$$

(5.20b)$$\begin{eqnarray}\displaystyle & \displaystyle \widetilde{u}_{rS(1)}={\rm\pi}\int _{0}^{\infty }\int _{0}^{{\rm\pi}}{\it\zeta}^{3}\sin 2{\it\theta}_{{\it\zeta}}{\it\Phi}_{a}^{(1)}E\,\text{d}{\it\theta}_{{\it\zeta}}\,\text{d}{\it\zeta}, & \displaystyle\end{eqnarray}$$

(5.20c)$$\begin{eqnarray}\displaystyle & \displaystyle \widetilde{u}_{{\it\theta}S(1)}={\rm\pi}\int _{0}^{\infty }\int _{0}^{{\rm\pi}}{\it\zeta}^{4}\sin ^{3}{\it\theta}_{{\it\zeta}}{\it\Phi}_{b}^{(1)}E\,\text{d}{\it\theta}_{{\it\zeta}}\,\text{d}{\it\zeta}, & \displaystyle\end{eqnarray}$$

(5.20d)$$\begin{eqnarray}\displaystyle & \displaystyle \widetilde{{\it\tau}}_{S(1)}=\frac{4}{3}{\rm\pi}\int _{0}^{\infty }\int _{0}^{{\rm\pi}}{\it\zeta}^{2}\left({\it\zeta}^{2}-\frac{3}{2}\right)\sin {\it\theta}_{{\it\zeta}}{\it\Phi}_{a}^{(1)}E\,\text{d}{\it\theta}_{{\it\zeta}}\,\text{d}{\it\zeta}, & \displaystyle\end{eqnarray}$$

(5.20e)$$\begin{eqnarray}\displaystyle & \displaystyle \widetilde{P}_{S(1)}=\frac{4}{3}{\rm\pi}\int _{0}^{\infty }\int _{0}^{{\rm\pi}}{\it\zeta}^{4}\sin {\it\theta}_{{\it\zeta}}{\it\Phi}_{a}^{(1)}E\,\text{d}{\it\theta}_{{\it\zeta}}\,\text{d}{\it\zeta}=\widetilde{{\it\omega}}_{S(1)}+\widetilde{{\it\tau}}_{S(1)}, & \displaystyle\end{eqnarray}$$

(5.20f)$$\begin{eqnarray}\displaystyle & \displaystyle \widetilde{P}_{rrS(1)}=2{\rm\pi}\int _{0}^{\infty }\int _{0}^{{\rm\pi}}{\it\zeta}^{4}\cos {\it\theta}_{{\it\zeta}}\sin 2{\it\theta}_{{\it\zeta}}{\it\Phi}_{a}^{(1)}E\,\text{d}{\it\theta}_{{\it\zeta}}\,\text{d}{\it\zeta}, & \displaystyle\end{eqnarray}$$

(5.20g)$$\begin{eqnarray}\displaystyle & \displaystyle \widetilde{P}_{r{\it\theta}S(1)}=2{\rm\pi}\int _{0}^{\infty }\int _{0}^{{\rm\pi}}{\it\zeta}^{5}\cos {\it\theta}_{{\it\zeta}}\sin ^{3}{\it\theta}_{{\it\zeta}}{\it\Phi}_{b}^{(1)}E\,\text{d}{\it\theta}_{{\it\zeta}}\,\text{d}{\it\zeta}, & \displaystyle\end{eqnarray}$$

(5.20h)$$\begin{eqnarray}\displaystyle & \displaystyle \widetilde{P}_{{\it\theta}{\it\theta}S(1)}=\widetilde{P}_{{\it\varphi}{\it\varphi}S(1)}=2{\rm\pi}\int _{0}^{\infty }\int _{0}^{{\rm\pi}}{\it\zeta}^{4}\sin ^{3}{\it\theta}_{{\it\zeta}}{\it\Phi}_{a}^{(1)}E\,\text{d}{\it\theta}_{{\it\zeta}}\,\text{d}{\it\zeta}, & \displaystyle\end{eqnarray}$$

(5.20i)$$\begin{eqnarray}\displaystyle & \displaystyle \widetilde{Q}_{rS(1)}={\rm\pi}\int _{0}^{\infty }\int _{0}^{{\rm\pi}}{\it\zeta}^{3}\left({\it\zeta}^{2}-\frac{5}{2}\right)\sin 2{\it\theta}_{{\it\zeta}}{\it\Phi}_{a}^{(1)}E\,\text{d}{\it\theta}_{{\it\zeta}}\,\text{d}{\it\zeta}, & \displaystyle\end{eqnarray}$$

(5.20j)$$\begin{eqnarray}\displaystyle & \displaystyle \widetilde{Q}_{{\it\theta}S(1)}={\rm\pi}\int _{0}^{\infty }\int _{0}^{{\rm\pi}}{\it\zeta}^{4}\left({\it\zeta}^{2}-\frac{5}{2}\right)\sin ^{3}{\it\theta}_{{\it\zeta}}{\it\Phi}_{b}^{(1)}E\,\text{d}{\it\theta}_{{\it\zeta}}\,\text{d}{\it\zeta}. & \displaystyle\end{eqnarray}$$

${\it\zeta}=|{\bf\zeta}|$

${\it\theta}_{{\it\zeta}}$

$0\leqslant {\it\theta}_{{\it\zeta}}\leqslant {\rm\pi}$

${\bf\zeta}$

${\it\zeta}_{r}=|{\bf\zeta}|\cos {\it\theta}_{{\it\zeta}}$

5.3. Asymptotic behaviour of ${\it\phi}_{S(1)}$ at large $r$

Based on the above decomposition, we shall now derive an asymptotic expression for ${\it\phi}_{S(1)}$ at large $r\gg 1$ that will be needed in the subsequent analysis. In this problem, the approach of ${\it\phi}_{S(1)}$ (or (${\it\Phi}_{a}^{(1)},{\it\Phi}_{b}^{(1)}$)) to the (linearised) uniform flow is very slow. If we assume it to be of $r^{-{\it\alpha}}$ (${\it\alpha}>0$), the length scale of variation of ${\it\phi}_{S(1)}$ will be of the order of $r^{\ast }$ for $r>r^{\ast }$, where $r^{\ast }$ is a sufficiently large number. In other words, the local Knudsen number, $k_{loc}=k/r^{\ast }$, is sufficiently small in the region where $r>r_{\ast }$, and, therefore, we can make use of the asymptotic theory of the Boltzmann equation for a slightly rarefied gas (Sone Reference Sone2002, Reference Sone2007) to investigate the asymptotic behaviour of ${\it\phi}_{S(1)}$ at large $r$. Because ${\it\phi}_{S(1)}$ is described by the linearised Boltzmann equation, we can apply the linear theory (or the Grad–Hilbert expansion) (Sone Reference Sone2002). As a result, the flow velocity and the pressure are described by the Stokes equations for an incompressible fluid and the temperature is described by the Laplace equation, i.e. 

(5.21a−d)$$\begin{eqnarray}\frac{\partial u_{iS(1)}}{\partial x_{i}}=0,\quad \frac{\partial P_{S(1)}}{\partial x_{i}}-{\it\gamma}_{1}k\frac{\partial ^{2}u_{iS(1)}}{\partial x_{j}^{2}}=0,\quad \frac{\partial ^{2}{\it\tau}_{S(1)}}{\partial x_{j}^{2}}=0,\quad P_{S(1)}={\it\omega}_{S(1)}+{\it\tau}_{S(1)}.\end{eqnarray}$$

In addition, the velocity distribution function is expressed in terms of $P_{S(1)}$, $u_{iS(1)}$ and ${\it\tau}_{S(1)}$ as follows (Sone Reference Sone2002):

(5.22)$$\begin{eqnarray}\displaystyle {\it\phi}_{S(1)}\! & = & \displaystyle \!P_{S(1)}+2{\it\zeta}_{i}u_{iS(1)}+\left(|{\bf\zeta}|^{2}-\frac{5}{2}\right){\it\tau}_{S(1)}\nonumber\\ \displaystyle & & \displaystyle -\,k{\it\zeta}_{i}{\it\zeta}_{j}B(|{\bf\zeta}|)\frac{\partial u_{iS(1)}}{\partial x_{j}}-k{\it\zeta}_{i}A(|{\bf\zeta}|)\frac{\partial {\it\tau}_{S(1)}}{\partial x_{i}}+\frac{k}{{\it\gamma}_{1}}{\it\zeta}_{i}D_{1}(|{\bf\zeta}|)\frac{\partial P_{S(1)}}{\partial x_{i}}\nonumber\\ \displaystyle & & \displaystyle +\,k^{2}{\it\zeta}_{i}{\it\zeta}_{j}{\it\zeta}_{k}D_{2}(|{\bf\zeta}|)\frac{\partial ^{2}u_{iS(1)}}{\partial x_{j}\partial x_{k}}-k^{2}{\it\zeta}_{i}{\it\zeta}_{j}F(|{\bf\zeta}|)\frac{\partial ^{2}{\it\tau}_{S(1)}}{\partial x_{i}\partial x_{j}}+\cdots \,,\end{eqnarray}$$

where the functions $A(|{\bf\zeta}|)$, $B(|{\bf\zeta}|),\dots ,F(|{\bf\zeta}|)$ as well as the constant ${\it\gamma}_{1}$ are defined in appendix B.

Since the angular dependence of the macroscopic quantities is explicit in (5.19), its substitution into (5.21) yields a set of ordinary differential equations for $\widetilde{u}_{rS(1)}(r)$, $\widetilde{u}_{{\it\theta}S(1)}(r)$, $\widetilde{P}_{S(1)}(r)$ and $\widetilde{{\it\tau}}_{S(1)}(r)$, which is easily integrated under the condition $\widetilde{u}_{rS(1)}\rightarrow 1$, $\widetilde{u}_{{\it\theta}S(1)}\rightarrow -1$, $\widetilde{P}_{S(1)}\rightarrow 0$, $\widetilde{{\it\tau}}_{S(1)}\rightarrow 0$ at $r\rightarrow \infty$, corresponding to the condition ${\it\phi}_{S(1)}\rightarrow 2{\it\zeta}_{1}$. As the result, we obtain

(5.23a)$$\begin{eqnarray}\displaystyle & \displaystyle \widetilde{u}_{rS(1)}=1+\frac{c_{1}}{r}+\frac{c_{2}}{r^{3}},\quad \widetilde{u}_{{\it\theta}S(1)}=-\left(1+\frac{c_{1}}{2r}-\frac{c_{2}}{2r^{3}}\right), & \displaystyle\end{eqnarray}$$

(5.23b)$$\begin{eqnarray}\displaystyle & \displaystyle \widetilde{{\it\omega}}_{S(1)}=\frac{{\it\gamma}_{1}kc_{1}-c_{3}}{r^{2}},\quad \widetilde{{\it\tau}}_{S(1)}=\frac{c_{3}}{r^{2}},\quad \widetilde{P}_{S(1)}=\frac{{\it\gamma}_{1}kc_{1}}{r^{2}}, & \displaystyle\end{eqnarray}$$

$c_{1}$

$c_{2}$

$c_{3}$

$r=1$

${\it\Phi}_{a}^{(1)}$

${\it\Phi}_{b}^{(1)}$

$r$

(5.24a)$$\begin{eqnarray}\displaystyle {\it\Phi}_{a}^{(1)} & = & \displaystyle 2\left(1+\frac{c_{1}}{r}+\frac{c_{2}}{r^{3}}\right){\it\zeta}_{r}+\frac{c_{3}}{r^{2}}\left(|{\bf\zeta}|^{2}-\frac{5}{2}\right)\nonumber\\ \displaystyle & & \displaystyle +\,\frac{k}{r^{2}}\left[{\it\gamma}_{1}c_{1}+\frac{2c_{3}}{r}{\it\zeta}_{r}A-\frac{1}{2}\left(c_{1}+\frac{3c_{2}}{r^{2}}\right)(|{\bf\zeta}|^{2}-3{\it\zeta}_{r}^{2})B\right]\nonumber\\ \displaystyle & & \displaystyle -\,\frac{k^{2}}{r^{3}}\left\{2c_{1}{\it\zeta}_{r}D_{1}-\frac{3c_{3}}{r}(|{\bf\zeta}|^{2}-3{\it\zeta}_{r}^{2})F\right.\nonumber\\ \displaystyle & & \displaystyle \left.+\,2\left[c_{1}(2|{\bf\zeta}|^{2}-3{\it\zeta}_{r}^{2})+\frac{3c_{2}}{r^{2}}(3|{\bf\zeta}|^{2}-5{\it\zeta}_{r}^{2})\right]{\it\zeta}_{r}D_{2}\right\},\end{eqnarray}$$

(5.24b)$$\begin{eqnarray}\displaystyle {\it\Phi}_{b}^{(1)} & = & \displaystyle -2-\frac{c_{1}}{r}+\frac{c_{2}}{r^{3}}+\frac{k}{r^{3}}\left(c_{3}A+\frac{3c_{2}}{r}{\it\zeta}_{r}B\right)\nonumber\\ \displaystyle & & \displaystyle -\,\frac{k^{2}}{r^{3}}\left\{c_{1}D_{1}+\frac{6c_{3}}{r}{\it\zeta}_{r}F+\frac{1}{2}\left[c_{1}(|{\bf\zeta}|^{2}-3{\it\zeta}_{r}^{2})+\frac{9c_{2}}{r^{2}}(|{\bf\zeta}|^{2}-5{\it\zeta}_{r}^{2})\right]D_{2}\right\},\qquad\end{eqnarray}$$

$A(|{\bf\zeta}|)$

$B(|{\bf\zeta}|)$

$c_{1}$

$c_{2}$

$c_{3}$

${\it\Phi}_{a}^{(1)}$

${\it\Phi}_{b}^{(1)}$

$c_{1}$

$c_{2}$

$c_{3}$

5.4. Second-order outer problem for ${\it\phi}_{O(2)}$

Now we proceed to the second-order approximation in ${\it\epsilon}$. We first consider the outer problem for ${\it\phi}_{O(2)}$ here, and subsequently ${\it\phi}_{S(2)}$ in the following subsection. To begin with, we repeat the fluid-dynamic-type equations describing the overall behaviour of the macroscopic quantities associated with the second-order outer solution. That is, owing to (5.11) and (5.12), the pressure $P_{O(2)}$ is constant and the fluid-dynamic-type equations (4.10a–d) are simplified to

(5.25a)$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\partial u_{iO(2)}}{\partial y_{i}}=0, & \displaystyle\end{eqnarray}$$

(5.25b)$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\partial u_{iO(2)}}{\partial y_{1}}=-\frac{1}{2}\frac{\partial P_{O(3)}}{\partial y_{i}}+\frac{{\it\gamma}_{1}k}{2}\frac{\partial ^{2}u_{iO(2)}}{\partial y_{j}^{2}}, & \displaystyle\end{eqnarray}$$

(5.25c)$$\begin{eqnarray}\displaystyle & \displaystyle \frac{\partial {\it\tau}_{O(2)}}{\partial y_{1}}=\frac{{\it\gamma}_{2}k}{2}\frac{\partial ^{2}{\it\tau}_{O(2)}}{\partial y_{j}^{2}}, & \displaystyle\end{eqnarray}$$

(5.25d)$$\begin{eqnarray}\displaystyle & P_{O(2)}={\it\omega}_{O(2)}+{\it\tau}_{O(2)}. & \displaystyle\end{eqnarray}$$

(5.26a−c)$$\begin{eqnarray}P_{O(2)}\rightarrow 0,\quad u_{iO(2)}\rightarrow 0,\quad {\it\tau}_{O(2)}\rightarrow 0\quad \text{as}~{\it\eta}\rightarrow \infty ,\end{eqnarray}$$

as well as by an appropriate matching condition at ${\it\eta}\rightarrow 0$ derived below.

As we have seen in § 5.1, ${\it\phi}_{O(1)}$ coincides with ${\it\phi}_{S(1)}^{\ast }$ if we allow an error of the order of ${\it\epsilon}$ (cf. (5.8) and (5.13)). The leading-order term in the difference can be made precise if we refine ${\it\phi}_{S(1)}^{\ast }$ with the aid of (5.24) and (5.14), that is,

(5.27)$$\begin{eqnarray}{\it\phi}_{S(1)}^{\ast }-{\it\phi}_{O(1)}={\it\epsilon}\frac{c_{1}}{{\it\eta}}(2{\it\zeta}_{r}\cos {\it\theta}-{\it\zeta}_{{\it\theta}}\sin {\it\theta})+O({\it\epsilon}^{2}).\end{eqnarray}$$

Thus, it follows that, in order for ${\it\phi}_{O(1)}+{\it\phi}_{O(2)}{\it\epsilon}$ to coincide with ${\it\phi}_{S(1)}^{\ast }$ within an error of $O({\it\epsilon}^{2})$, the following condition should be imposed:

(5.28)$$\begin{eqnarray}{\it\phi}_{O(2)}\rightarrow \frac{c_{1}}{{\it\eta}}(2{\it\zeta}_{r}\cos {\it\theta}-{\it\zeta}_{{\it\theta}}\sin {\it\theta})\quad \text{as}~{\it\eta}\rightarrow 0.\end{eqnarray}$$

To further derive the corresponding condition for the macroscopic quantities, we recall that the functional form of ${\it\phi}_{O(2)}$ (with respect to ${\bf\zeta}$) is specified and is given by (4.11b), which, with the aid of (5.11) and (5.12), reduces to

(5.29)$$\begin{eqnarray}{\it\phi}_{O(2)}=P_{O(2)}+2{\it\zeta}_{i}u_{iO(2)}+\left(|{\bf\zeta}|^{2}-{\textstyle \frac{5}{2}}\right){\it\tau}_{O(2)}+2{\it\zeta}_{1}^{2}-1.\end{eqnarray}$$

Therefore, in order for (5.28) to be satisfied, the macroscopic quantities contained in (5.29) should take the following values as ${\it\eta}\rightarrow 0$:

(5.30a−d)$$\begin{eqnarray}{\it\eta}u_{rO(2)}\rightarrow c_{1}\cos {\it\theta},\quad {\it\eta}u_{{\it\theta}O(2)}\rightarrow -\frac{c_{1}}{2}\sin {\it\theta},\quad {\it\eta}u_{{\it\varphi}O(2)}\rightarrow 0,\quad {\it\eta}{\it\tau}_{O(2)}\rightarrow 0\quad \text{as}~{\it\eta}\rightarrow 0.\end{eqnarray}$$

Here, we have used the fact that $P_{O(2)}$ and $2{\it\zeta}_{1}^{2}-1$ are independent of ${\it\eta}$. In summary, the second and third conditions of (5.26) as well as (5.30) provide the boundary conditions for the fluid-dynamic-type equations (5.25a–c).

From the first condition of (5.26), $P_{O(2)}(=\text{const}.)$ must vanish identically, i.e.

(5.31)$$\begin{eqnarray}P_{O(2)}=0.\end{eqnarray}$$

On the other hand, (5.25a) and (5.25b) are, respectively, the continuity equation and the Oseen equation for an incompressible fluid, and (5.25c) is an energy equation with a linear convection term. We first observe that ${\it\tau}_{O(2)}=0$ is a trivial solution of (5.25c) satisfying the boundary condition (5.26) (the third condition) as well as the matching condition (5.30) (the fourth condition). Consequently, the density ${\it\omega}_{O(2)}$ also vanishes owing to (5.31) and (5.25d). On the other hand, the axisymmetric solution of the Oseen equation, vanishing at infinity and having the proper singularity (see (5.30)) at the origin, is given by the Oseenlet oriented along the $x_{1}$ axis (Imai Reference Imai1973), that is,

(5.32a)$$\begin{eqnarray}\displaystyle & \displaystyle \frac{u_{rO(2)}}{{\it\alpha}}=-\frac{1}{{\it\eta}^{2}}+\frac{1}{{\it\eta}^{2}}\left[1+\frac{{\it\eta}}{{\it\gamma}_{1}k}(1+\cos {\it\theta})\right]\exp \left(-\frac{{\it\eta}}{{\it\gamma}_{1}k}(1-\cos {\it\theta})\right), & \displaystyle\end{eqnarray}$$

(5.32b)$$\begin{eqnarray}\displaystyle & \displaystyle \frac{u_{{\it\theta}O(2)}}{{\it\alpha}}=-\frac{1}{{\it\gamma}_{1}k}\frac{\sin {\it\theta}}{{\it\eta}}\exp \left(-\frac{{\it\eta}}{{\it\gamma}_{1}k}(1-\cos {\it\theta})\right), & \displaystyle\end{eqnarray}$$

(5.32c)$$\begin{eqnarray}\displaystyle & u_{{\it\varphi}O(2)}=0, & \displaystyle\end{eqnarray}$$

(5.32d)$$\begin{eqnarray}\displaystyle & \displaystyle \frac{P_{O(3)}}{{\it\alpha}}=\frac{2\cos {\it\theta}}{{\it\eta}^{2}}, & \displaystyle\end{eqnarray}$$

${\it\alpha}$

$P_{O(3)}$

${\it\phi}_{O(3)}$

To determine ${\it\alpha}$, we shall make use of the matching condition (5.30). For this purpose, we expand (5.32a) and (5.32b) in terms of (small) ${\it\eta}$ to obtain

(5.33a)$$\begin{eqnarray}\displaystyle & \displaystyle \frac{u_{rO(2)}}{{\it\alpha}}=\frac{2}{{\it\gamma}_{1}k}\frac{\cos {\it\theta}}{{\it\eta}}-\frac{(1-\cos {\it\theta})(1+3\cos {\it\theta})}{2({\it\gamma}_{1}k)^{2}}+O({\it\eta}), & \displaystyle\end{eqnarray}$$

(5.33b)$$\begin{eqnarray}\displaystyle & \displaystyle \frac{u_{{\it\theta}O(2)}}{{\it\alpha}}=-\frac{1}{{\it\gamma}_{1}k}\frac{\sin {\it\theta}}{{\it\eta}}+\frac{\sin {\it\theta}(1-\cos {\it\theta})}{({\it\gamma}_{1}k)^{2}}+O({\it\eta}). & \displaystyle\end{eqnarray}$$

${\it\eta}$

${\it\eta}\rightarrow 0$

(5.34a,b)$$\begin{eqnarray}{\it\eta}u_{rO(2)}\rightarrow \frac{2{\it\alpha}}{{\it\gamma}_{1}k}\cos {\it\theta},\quad {\it\eta}u_{{\it\theta}O(2)}\rightarrow -\frac{{\it\alpha}}{{\it\gamma}_{1}k}\sin {\it\theta}\quad ({\it\eta}\rightarrow 0).\end{eqnarray}$$

Thus, a comparison with the first two conditions in (5.30) immediately yields

(5.35)$$\begin{eqnarray}{\it\alpha}=\frac{{\it\gamma}_{1}kc_{1}}{2}.\end{eqnarray}$$

To summarise, the second-order outer solution has been obtained as

(5.36a)$$\begin{eqnarray}\displaystyle & \displaystyle u_{rO(2)}=\frac{{\it\gamma}_{1}kc_{1}}{2{\it\eta}^{2}}\left[-1+\left(1+\frac{{\it\eta}}{{\it\gamma}_{1}k}(1+\cos {\it\theta})\right)\exp \left(-\frac{{\it\eta}}{{\it\gamma}_{1}k}(1-\cos {\it\theta})\right)\right], & \displaystyle\end{eqnarray}$$

(5.36b)$$\begin{eqnarray}\displaystyle & \displaystyle u_{{\it\theta}O(2)}=-c_{1}\frac{\sin {\it\theta}}{2{\it\eta}}\exp \left(-\frac{{\it\eta}}{{\it\gamma}_{1}k}(1-\cos {\it\theta})\right), & \displaystyle\end{eqnarray}$$

(5.36c)$$\begin{eqnarray}\displaystyle & u_{{\it\varphi}O(2)}={\it\omega}_{O(2)}={\it\tau}_{O(2)}=P_{O(2)}=0, & \displaystyle\end{eqnarray}$$

(5.37)$$\begin{eqnarray}{\it\phi}_{O(2)}=2{\it\zeta}_{r}u_{rO(2)}+2{\it\zeta}_{{\it\theta}}u_{{\it\theta}O(2)}+2{\it\zeta}_{1}^{2}-1.\end{eqnarray}$$

Note that ${\it\phi}_{O(2)}$ depends on the constant $c_{1}$ through $u_{rO(2)}$ and $u_{{\it\theta}O(2)}$. It may also be noted that, owing to (5.13) and (5.37), the outer solution is a Maxwellian up to the order ${\it\epsilon}^{2}$, i.e. 

(5.38)$$\begin{eqnarray}(1+{\it\phi}_{O})E=\frac{1}{{\rm\pi}^{3/2}}\text{exp}(-({\it\zeta}_{i}-{\it\epsilon}{\it\delta}_{i1}-{\it\epsilon}^{2}u_{iO(2)})^{2})+O({\it\epsilon}^{3}).\end{eqnarray}$$

5.5. Second-order inner problem for ${\it\phi}_{S(2)}$: connection condition

Finally, we consider the second-order inner problem. The equation and the boundary condition on the sphere are the same as (3.12) and (3.13), i.e.

(5.39)$$\begin{eqnarray}\displaystyle & \displaystyle {\it\zeta}_{i}\frac{\partial {\it\phi}_{S(2)}}{\partial x_{i}}=\frac{1}{k}\mathscr{L}[{\it\phi}_{S(2)}]+\frac{1}{k}\mathscr{J}[{\it\phi}_{S(1)},{\it\phi}_{S(1)}], & \displaystyle\end{eqnarray}$$

(5.40)$$\begin{eqnarray}\displaystyle & \displaystyle {\it\phi}_{S(2)}=\mathscr{K}[{\it\phi}_{S(2)}],\quad {\it\zeta}_{i}n_{i}>0\quad (r=1), & \displaystyle\end{eqnarray}$$

To derive the connection condition, we consider again a point where $r$ is large but ${\it\epsilon}r$ is small. Owing to (5.33) (with (5.35)) and (5.37), ${\it\phi}_{O(2)}$ has the following asymptotic representation at the point:

(5.41)$$\begin{eqnarray}\displaystyle {\it\phi}_{O(2)} & = & \displaystyle \frac{2c_{1}}{{\it\eta}}\left({\it\zeta}_{r}\cos {\it\theta}-\frac{{\it\zeta}_{{\it\theta}}}{2}\sin {\it\theta}\right)+\frac{c_{1}}{2{\it\gamma}_{1}k}\left[-(1-\cos {\it\theta})(1+3\cos {\it\theta}){\it\zeta}_{r}\right.\nonumber\\ \displaystyle & & \displaystyle \left.+\,2\sin {\it\theta}(1-\cos {\it\theta}){\it\zeta}_{{\it\theta}}\right]+2{\it\zeta}_{1}^{2}-1+O({\it\eta}).\end{eqnarray}$$

Thus, noting the relation ${\it\eta}={\it\epsilon}r$ (see also (5.13)), the outer solution ${\it\phi}_{O}\,(={\it\epsilon}{\it\phi}_{O(1)}+{\it\epsilon}^{2}{\it\phi}_{O(2)}+O({\it\epsilon}^{3}))$ is written in terms of the inner variable $r$ as

(5.42)$$\begin{eqnarray}\displaystyle {\it\phi}_{O}^{\ast } & = & \displaystyle {\it\epsilon}\left[2{\it\zeta}_{1}+\frac{2c_{1}}{r}\left({\it\zeta}_{r}\cos {\it\theta}-\frac{{\it\zeta}_{{\it\theta}}}{2}\sin {\it\theta}\right)\right]+{\it\epsilon}^{2}\left\{\frac{c_{1}}{2{\it\gamma}_{1}k}\left[-(1-\cos {\it\theta})(1+3\cos {\it\theta}){\it\zeta}_{r}\right.\right.\nonumber\\ \displaystyle & & \displaystyle \left.\left.+\,2\sin {\it\theta}(1-\cos {\it\theta}){\it\zeta}_{{\it\theta}}\right]+2{\it\zeta}_{1}^{2}-1\right\}+O({\it\epsilon}^{3}).\end{eqnarray}$$

Here, the asterisk has been attached to indicate that the outer solution is expressed in terms of the inner variable, i.e. ${\it\phi}_{O}^{\ast }={\it\phi}_{O}|_{{\it\eta}={\it\epsilon}r}$. It confirms that the terms of order ${\it\epsilon}$ coincide with ${\it\phi}_{S(1)}$ (at large $r\gg 1$) up to the terms of $r^{-1}$. On the other hand, the terms of ${\it\epsilon}^{2}$ order cannot be represented by ${\it\phi}_{S(1)}$. Therefore, in order for ${\it\phi}_{S}$ to match ${\it\phi}_{O}^{\ast }$, ${\it\phi}_{S(2)}$ should satisfy the following connection condition as $r\rightarrow \infty$:

(5.43)$$\begin{eqnarray}{\it\phi}_{S(2)}\rightarrow \frac{c_{1}}{2{\it\gamma}_{1}k}\left[-(1-\cos {\it\theta})(1+3\cos {\it\theta}){\it\zeta}_{r}+2\sin {\it\theta}(1-\cos {\it\theta}){\it\zeta}_{{\it\theta}}\right]+2{\it\zeta}_{1}^{2}-1\quad \text{as}~r\rightarrow \infty .\end{eqnarray}$$

In summary, the boundary-value problem for ${\it\phi}_{S(2)}$ consists of (5.39) and (5.40), supplemented by the connection condition (5.43). It may be noted that the term $2{\it\zeta}_{1}^{2}-1$ in (5.43) corresponds to the uniform flow (cf. (3.7) or (4.16)) and the rest is the contribution from the slowly varying outer solution.

5.6. Further transformation of the problem for ${\it\phi}_{S(2)}$

With the aid of the relation ${\it\zeta}_{1}={\it\zeta}_{r}\cos {\it\theta}-{\it\zeta}_{{\it\theta}}\sin {\it\theta}$, the matching condition (5.43) is transformed into

(5.44)$$\begin{eqnarray}\displaystyle {\it\phi}_{S(2)} & \rightarrow & \displaystyle -\frac{c_{1}}{{\it\gamma}_{1}k}{\it\zeta}_{1}-\frac{c_{1}}{2{\it\gamma}_{1}k}\left[(1-3\cos ^{2}{\it\theta}){\it\zeta}_{r}+2{\it\zeta}_{{\it\theta}}\cos {\it\theta}\sin {\it\theta}\right]\nonumber\\ \displaystyle & & \displaystyle +\,2({\it\zeta}_{r}\cos {\it\theta}-{\it\zeta}_{{\it\theta}}\sin {\it\theta})^{2}-1\quad \text{as}~r\rightarrow \infty .\end{eqnarray}$$

Then, by the linearity of the problem, we can set the solution in the form

(5.45)$$\begin{eqnarray}{\it\phi}_{S(2)}=-\frac{c_{1}}{2{\it\gamma}_{1}k}{\it\phi}_{S(1)}+{\it\Phi}^{(2)},\end{eqnarray}$$

where the first term satisfies the homogeneous equation and the (homogeneous) boundary condition on the sphere as well as the first term of the matching condition (see also (5.6)), while the second term ${\it\Phi}^{(2)}$ solves the following problem:

(5.46)$$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle {\it\zeta}_{i}\frac{\partial {\it\Phi}^{(2)}}{\partial x_{i}}=\frac{1}{k}\mathscr{L}[{\it\Phi}^{(2)}]+\frac{1}{k}\mathscr{J}[{\it\phi}_{S(1)},{\it\phi}_{S(1)}],\end{eqnarray}$$

(5.47)$$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle {\it\Phi}^{(2)}=\mathscr{K}[{\it\Phi}^{(2)}]\quad \text{for}~{\it\zeta}_{i}n_{i}>0~(r=1),\end{eqnarray}$$

(5.48)$$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle {\it\Phi}^{(2)}\rightarrow -\frac{c_{1}}{2{\it\gamma}_{1}k}\left[(1-3\cos ^{2}{\it\theta}){\it\zeta}_{r}+2{\it\zeta}_{{\it\theta}}\cos {\it\theta}\sin {\it\theta}\right]\nonumber\\ \displaystyle & & \displaystyle \quad +\,2({\it\zeta}_{r}\cos {\it\theta}-{\it\zeta}_{{\it\theta}}\sin {\it\theta})^{2}-1\quad (r\rightarrow \infty ).\end{eqnarray}$$

$\mathscr{L}$

$\mathscr{J}$

$\mathscr{K}$

${\it\Phi}^{(2)}$

(5.49)$$\begin{eqnarray}{\it\Phi}^{(2)}={\it\Phi}_{a}^{(2)}(r,{\it\zeta}_{r},|{\bf\zeta}|)\cos ^{2}{\it\theta}+{\it\zeta}_{{\it\theta}}{\it\Phi}_{b}^{(2)}\cos {\it\theta}\sin {\it\theta}+\frac{{\it\zeta}_{{\it\theta}}^{2}-{\it\zeta}_{{\it\varphi}}^{2}}{2}{\it\Phi}_{c}^{(2)}\sin ^{2}{\it\theta}+{\it\Phi}_{d}^{(2)},\end{eqnarray}$$

where ${\it\Phi}_{J}^{(2)}$ ($J=a,b,c$ or $d$) are functions of $r$, ${\it\zeta}_{r}$ and $|{\bf\zeta}|$. Then, the problem is reduced to a (spatially one-dimensional) boundary-value problem for (${\it\Phi}_{a}^{(2)},{\it\Phi}_{b}^{(2)},{\it\Phi}_{c}^{(2)}$,${\it\Phi}_{d}^{(2)}$) as follows:

(5.50a)$$\begin{eqnarray}\displaystyle & \displaystyle {\it\zeta}_{r}\frac{\partial {\it\Phi}_{a}^{(2)}}{\partial r}+\frac{|{\bf\zeta}|^{2}-{\it\zeta}_{r}^{2}}{r}\frac{\partial {\it\Phi}_{a}^{(2)}}{\partial {\it\zeta}_{r}}+\frac{3}{2}\frac{|{\bf\zeta}|\sin {\it\theta}_{{\it\zeta}}}{r}{\it\Phi}_{b}^{(2)}=\frac{1}{k}\mathscr{L}[{\it\Phi}_{a}^{(2)}]+\frac{1}{k}I_{a}, & \displaystyle\end{eqnarray}$$

(5.50b)$$\begin{eqnarray}\displaystyle & \displaystyle {\it\zeta}_{r}\frac{\partial {\it\Phi}_{b}^{(2)}}{\partial r}+\frac{|{\bf\zeta}|^{2}-{\it\zeta}_{r}^{2}}{r}\frac{\partial {\it\Phi}_{b}^{(2)}}{\partial {\it\zeta}_{r}}-\frac{{\it\zeta}_{r}}{r}{\it\Phi}_{b}^{(2)}-\frac{2}{r}{\it\Phi}_{a}^{(2)}+\frac{|{\bf\zeta}|^{2}-{\it\zeta}_{r}^{2}}{r}{\it\Phi}_{c}^{(2)}=\frac{1}{k}\mathscr{L}_{1}[{\it\Phi}_{b}^{(2)}]+\frac{1}{k}I_{b}, & \displaystyle \nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

(5.50c)$$\begin{eqnarray}\displaystyle & \displaystyle {\it\zeta}_{r}\frac{\partial {\it\Phi}_{c}^{(2)}}{\partial r}+\frac{|{\bf\zeta}|^{2}-{\it\zeta}_{r}^{2}}{r}\frac{\partial {\it\Phi}_{c}^{(2)}}{\partial {\it\zeta}_{r}}-2\frac{{\it\zeta}_{r}}{r}{\it\Phi}_{c}^{(2)}-\frac{{\it\Phi}_{b}^{(2)}}{r}=\frac{1}{k}\mathscr{L}_{2}[{\it\Phi}_{c}^{(2)}]+\frac{1}{k}I_{c}, & \displaystyle\end{eqnarray}$$

(5.50d)$$\begin{eqnarray}\displaystyle & \displaystyle {\it\zeta}_{r}\frac{\partial {\it\Phi}_{d}^{(2)}}{\partial r}+\frac{|{\bf\zeta}|^{2}-{\it\zeta}_{r}^{2}}{r}\frac{\partial {\it\Phi}_{d}^{(2)}}{\partial {\it\zeta}_{r}}-\frac{|{\bf\zeta}|^{2}-{\it\zeta}_{r}^{2}}{2r}{\it\Phi}_{b}^{(2)}=\frac{1}{k}\mathscr{L}[{\it\Phi}_{d}^{(2)}]+\frac{1}{k}I_{d}, & \displaystyle\end{eqnarray}$$

(5.51)$$\begin{eqnarray}\displaystyle \left[\begin{array}{@{}c@{}}{\it\Phi}_{a}^{(2)}\\ {\it\Phi}_{b}^{(2)}\\ {\it\Phi}_{c}^{(2)}\\ {\it\Phi}_{d}^{(2)}\end{array}\right]=\left[\begin{array}{@{}c@{}}\mathscr{K}[{\it\Phi}_{a}^{(2)}]\\ \mathscr{K}_{1}[{\it\Phi}_{b}^{(2)}]\\ \mathscr{K}_{2}[{\it\Phi}_{c}^{(2)}]\\ \mathscr{ K}[{\it\Phi}_{d}^{(2)}]\end{array}\right]\quad \text{for}~{\it\zeta}_{r}>0~(r=1), & & \displaystyle\end{eqnarray}$$

(5.52)$$\begin{eqnarray}\displaystyle \left[\begin{array}{@{}c@{}}{\it\Phi}_{a}^{(2)}\\ {\it\Phi}_{b}^{(2)}\\ {\it\Phi}_{c}^{(2)}\\ {\it\Phi}_{d}^{(2)}\end{array}\right]\rightarrow \left[\begin{array}{@{}c@{}}\displaystyle \frac{3}{2}\frac{c_{1}}{{\it\gamma}_{1}k}{\it\zeta}_{r}+3{\it\zeta}_{r}^{2}-|{\bf\zeta}|^{2}\\ \displaystyle -\frac{c_{1}}{{\it\gamma}_{1}k}-4{\it\zeta}_{r}\\ 2\\ \displaystyle -\frac{c_{1}}{2{\it\gamma}_{1}k}{\it\zeta}_{r}+|{\bf\zeta}|^{2}-{\it\zeta}_{r}^{2}-1\end{array}\right]\quad (r\rightarrow \infty ), & & \displaystyle\end{eqnarray}$$

$\mathscr{L}_{i}$

$\mathscr{K}_{i}$

$i=1$

$\mathscr{J}_{i}$

$i=1,2$

$(I_{a},I_{b},I_{c},I_{d})$

(5.53a)$$\begin{eqnarray}\displaystyle \displaystyle I_{a} & = & \displaystyle \mathscr{J}[{\it\Phi}_{a}^{(1)},{\it\Phi}_{a}^{(1)}]-\frac{|{\bf\zeta}|^{2}-{\it\zeta}_{r}^{2}}{2}\mathscr{J}_{2}[{\it\Phi}_{b}^{(1)},{\it\Phi}_{b}^{(1)}]-\mathscr{J}_{3}[{\it\Phi}_{b}^{(1)},{\it\Phi}_{b}^{(1)}],\end{eqnarray}$$

(5.53b)$$\begin{eqnarray}\displaystyle I_{b} & = & \displaystyle 2\mathscr{J}_{1}[{\it\Phi}_{a}^{(1)},{\it\Phi}_{b}^{(1)}],\end{eqnarray}$$

(5.53c)$$\begin{eqnarray}\displaystyle I_{c} & = & \displaystyle \mathscr{J}_{2}[{\it\Phi}_{b}^{(1)},{\it\Phi}_{b}^{(1)}],\end{eqnarray}$$

(5.53d)$$\begin{eqnarray}\displaystyle \displaystyle I_{d} & = & \displaystyle \frac{|{\bf\zeta}|^{2}-{\it\zeta}_{r}^{2}}{2}\mathscr{J}_{2}[{\it\Phi}_{b}^{(1)},{\it\Phi}_{b}^{(1)}]+\mathscr{J}_{3}[{\it\Phi}_{b}^{(1)},{\it\Phi}_{b}^{(1)}].\end{eqnarray}$$

(5.54a)$$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle {\it\omega}_{S(2)}=-\frac{c_{1}}{2k}\widetilde{{\it\omega}}_{S(1)}\cos {\it\theta}+\widetilde{{\it\omega}}_{S(2)}^{a}\cos ^{2}{\it\theta}+\widetilde{{\it\omega}}_{S(2)}^{d},\end{eqnarray}$$

(5.54b)$$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle u_{rS(2)}=-\frac{c_{1}}{2k}\widetilde{u}_{rS(1)}\cos {\it\theta}+\left(\widetilde{u}_{rS(2)}^{a}-\widetilde{{\it\omega}}_{S(1)}\widetilde{u}_{rS(1)}\right)\cos ^{2}{\it\theta}+\widetilde{u}_{rS(2)}^{d},\end{eqnarray}$$

(5.54c)$$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle u_{{\it\theta}S(2)}=-\frac{c_{1}}{2k}\widetilde{u}_{{\it\theta}S(1)}\sin {\it\theta}+\left(\widetilde{u}_{{\it\theta}S(2)}^{b}-\widetilde{{\it\omega}}_{S(1)}\widetilde{u}_{{\it\theta}S(1)}\right)\cos {\it\theta}\sin {\it\theta},\end{eqnarray}$$

(5.54d)$$\begin{eqnarray}\displaystyle & & \displaystyle u_{{\it\varphi}S(2)}=0,\end{eqnarray}$$

(5.54e)$$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle {\it\tau}_{S(2)}=-\frac{c_{1}}{2k}\widetilde{{\it\tau}}_{S(1)}\cos {\it\theta}+\left(\widetilde{{\it\tau}}_{S(2)}^{a}-\frac{2}{3}(\widetilde{u}_{rS(1)})^{2}-\widetilde{{\it\omega}}_{S(1)}\widetilde{{\it\tau}}_{S(1)}\right)\cos ^{2}{\it\theta}+\widetilde{{\it\tau}}_{S(2)}^{d}\nonumber\\ \displaystyle & & \displaystyle \qquad \quad -\,\frac{2}{3}(\widetilde{u}_{{\it\theta}S(1)})^{2}\sin ^{2}{\it\theta},\end{eqnarray}$$

(5.54f)$$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle P_{S(2)}=-\frac{c_{1}}{2k}\widetilde{P}_{S(1)}\cos {\it\theta}+\!\left(\widetilde{P}_{S(2)}^{a}-\frac{2}{3}(\widetilde{u}_{rS(1)})^{2}\right)\cos ^{2}{\it\theta}\!+\!\widetilde{P}_{S(2)}^{d}\!-\!\frac{2}{3}(\widetilde{u}_{{\it\theta}S(1)})^{2}\sin ^{2}{\it\theta},\nonumber\\ \displaystyle & & \displaystyle\end{eqnarray}$$

(5.54g)$$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle \unicode[STIX]{x1D617}_{rrS(2)}=-\frac{c_{1}}{2k}\widetilde{P}_{rrS(1)}\cos {\it\theta}+\left(\widetilde{P}_{rrS(2)}^{a}-2(\widetilde{u}_{rS(1)})^{2}\right)\cos ^{2}{\it\theta}+\widetilde{P}_{rrS(2)}^{d},\end{eqnarray}$$

(5.54h)$$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle \unicode[STIX]{x1D617}_{r{\it\theta}S(2)}=-\frac{c_{1}}{2k}\widetilde{P}_{r{\it\theta}S(1)}\sin {\it\theta}+\left(\widetilde{P}_{r{\it\theta}S(2)}^{b}-2\widetilde{u}_{rS(1)}\widetilde{u}_{{\it\theta}S(1)}\right)\cos {\it\theta}\sin {\it\theta},\end{eqnarray}$$

(5.54i)$$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D617}_{r{\it\varphi}S(2)}=0,\end{eqnarray}$$

(5.54j)$$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle \unicode[STIX]{x1D617}_{{\it\theta}{\it\theta}S(2)}=-\frac{c_{1}}{2k}\widetilde{P}_{{\it\theta}{\it\theta}(1)}\cos {\it\theta}+\widetilde{P}_{{\it\theta}{\it\theta}S(2)}^{a}\cos ^{2}{\it\theta}+\left(\frac{1}{4}\widetilde{P}_{{\it\theta}{\it\theta}S(2)}^{c}-2(\widetilde{u}_{{\it\theta}S(1)})^{2}\right)\sin ^{2}{\it\theta}\nonumber\\ \displaystyle & & \displaystyle \qquad \quad +\,\widetilde{P}_{{\it\theta}{\it\theta}S(2)}^{d},\end{eqnarray}$$

(5.54k)$$\begin{eqnarray}\displaystyle & & \displaystyle \unicode[STIX]{x1D617}_{{\it\theta}{\it\varphi}S(2)}=0,\end{eqnarray}$$

(5.54l)$$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle \unicode[STIX]{x1D617}_{{\it\varphi}{\it\varphi}S(2)}=-\frac{c_{1}}{2k}\widetilde{P}_{{\it\varphi}{\it\varphi}S(1)}\cos {\it\theta}+\widetilde{P}_{{\it\varphi}{\it\varphi}S(2)}^{a}\cos ^{2}{\it\theta}-\frac{1}{4}\widetilde{P}_{{\it\varphi}{\it\varphi}S(2)}^{c}\sin ^{2}{\it\theta}+\widetilde{P}_{{\it\varphi}{\it\varphi}S(2)}^{d},\end{eqnarray}$$

(5.54m)$$\begin{eqnarray}\displaystyle & & \displaystyle \displaystyle Q_{rS(2)}=-\frac{c_{1}}{2k}\widetilde{Q}_{rS(1)}\cos {\it\theta}+\left(\widetilde{Q}_{rS(2)}^{a}+\frac{5}{2}\widetilde{{\it\omega}}_{S(1)}\widetilde{u}_{rS(1)}-\widetilde{u}_{rS(1)}\widetilde{P}_{rrS(1)}\right.\nonumber\\ \displaystyle & & \displaystyle \qquad \quad \left.-\,\frac{3}{2}\widetilde{P}_{S(1)}\widetilde{u}_{rS(1)}\right)\cos ^{2}{\it\theta}+\widetilde{Q}_{rS(2)}^{d}-\widetilde{u}_{{\it\theta}S(1)}\widetilde{P}_{r{\it\theta}S(1)}\sin ^{2}{\it\theta},\end{eqnarray}$$

(5.54n)$$\begin{eqnarray}\displaystyle & & \displaystyle Q_{{\it\theta}S(2)}=-\frac{c_{1}}{2k}\widetilde{Q}_{{\it\theta}S(1)}\sin {\it\theta}+\left(\widetilde{Q}_{{\it\theta}S(2)}^{b}+\frac{5}{2}\widetilde{{\it\omega}}_{S(1)}\widetilde{u}_{{\it\theta}S(1)}-\widetilde{u}_{rS(1)}\widetilde{P}_{r{\it\theta}S(1)}\right.\nonumber\\ \displaystyle & & \displaystyle \qquad \quad \left.-\,\widetilde{u}_{{\it\theta}S(1)}\widetilde{P}_{{\it\theta}{\it\theta}(1)}^{a}-\frac{3}{2}\widetilde{P}_{S(1)}\widetilde{u}_{{\it\theta}S(1)}\right)\cos {\it\theta}\sin {\it\theta},\end{eqnarray}$$

(5.54o)$$\begin{eqnarray}\displaystyle & & \displaystyle Q_{{\it\varphi}S(2)}=0,\end{eqnarray}$$

$r$

$J=a$

$d$

(5.55a)$$\begin{eqnarray}\displaystyle & \displaystyle \widetilde{{\it\omega}}_{S(2)}^{J}=2{\rm\pi}\int _{0}^{\infty }\int _{0}^{{\rm\pi}}{\it\zeta}^{2}\sin {\it\theta}_{{\it\zeta}}{\it\Phi}_{J}^{(2)}E\,\text{d}{\it\theta}_{{\it\zeta}}\,\text{d}{\it\zeta}, & \displaystyle\end{eqnarray}$$

(5.55b)$$\begin{eqnarray}\displaystyle & \displaystyle \widetilde{u}_{rS(2)}^{J}={\rm\pi}\int _{0}^{\infty }\int _{0}^{{\rm\pi}}{\it\zeta}^{3}\sin 2{\it\theta}_{{\it\zeta}}{\it\Phi}_{J}^{(2)}E\,\text{d}{\it\theta}_{{\it\zeta}}\,\text{d}{\it\zeta}, & \displaystyle\end{eqnarray}$$

(5.55c)$$\begin{eqnarray}\displaystyle & \displaystyle \widetilde{u}_{{\it\theta}S(2)}^{b}={\rm\pi}\int _{0}^{\infty }\int _{0}^{{\rm\pi}}{\it\zeta}^{4}\sin ^{3}{\it\theta}_{{\it\zeta}}{\it\Phi}_{b}^{(2)}E\,\text{d}{\it\theta}_{{\it\zeta}}\,\text{d}{\it\zeta}, & \displaystyle\end{eqnarray}$$

(5.55d)$$\begin{eqnarray}\displaystyle & \displaystyle \widetilde{{\it\tau}}_{S(2)}^{J}=\frac{4}{3}{\rm\pi}\int _{0}^{\infty }\int _{0}^{{\rm\pi}}{\it\zeta}^{2}\left({\it\zeta}^{2}-\frac{3}{2}\right)\sin {\it\theta}_{{\it\zeta}}{\it\Phi}_{J}^{(2)}E\,\text{d}{\it\theta}_{{\it\zeta}}\,\text{d}{\it\zeta}, & \displaystyle\end{eqnarray}$$

(5.55e)$$\begin{eqnarray}\displaystyle & \displaystyle \widetilde{P}_{S(2)}^{J}=\frac{4}{3}{\rm\pi}\int _{0}^{\infty }\int _{0}^{{\rm\pi}}{\it\zeta}^{4}\sin {\it\theta}_{{\it\zeta}}{\it\Phi}_{J}^{(2)}E\,\text{d}{\it\theta}_{{\it\zeta}}\,\text{d}{\it\zeta}=\widetilde{{\it\omega}}_{S(2)}^{J}+\widetilde{{\it\tau}}_{S(2)}^{J}, & \displaystyle\end{eqnarray}$$

(5.55f)$$\begin{eqnarray}\displaystyle & \displaystyle \widetilde{P}_{rrS(2)}^{J}=2{\rm\pi}\int _{0}^{\infty }\int _{0}^{{\rm\pi}}{\it\zeta}^{4}\cos {\it\theta}_{{\it\zeta}}\sin 2{\it\theta}_{{\it\zeta}}{\it\Phi}_{J}^{(2)}E\,\text{d}{\it\theta}_{{\it\zeta}}\,\text{d}{\it\zeta}, & \displaystyle\end{eqnarray}$$

(5.55g)$$\begin{eqnarray}\displaystyle & \displaystyle \widetilde{P}_{r{\it\theta}S(2)}^{b}=2{\rm\pi}\int _{0}^{\infty }\int _{0}^{{\rm\pi}}{\it\zeta}^{5}\cos {\it\theta}_{{\it\zeta}}\sin ^{3}{\it\theta}_{{\it\zeta}}{\it\Phi}_{b}^{(2)}E\,\text{d}{\it\theta}_{{\it\zeta}}\,\text{d}{\it\zeta}, & \displaystyle\end{eqnarray}$$

(5.55h)$$\begin{eqnarray}\displaystyle & \left[\begin{array}{@{}c@{}}\widetilde{P}_{{\it\theta}{\it\theta}S(2)}^{J}\\ \widetilde{P}_{{\it\theta}{\it\theta}S(2)}^{c}\end{array}\right]=\left[\begin{array}{@{}c@{}}\widetilde{P}_{{\it\varphi}{\it\varphi}S(2)}^{J}\\ \widetilde{P}_{{\it\varphi}{\it\varphi}S(2)}^{c}\end{array}\right]=2{\rm\pi}\displaystyle \int _{0}^{\infty }\displaystyle \int _{0}^{{\rm\pi}}{\it\zeta}^{4}\sin ^{3}{\it\theta}_{{\it\zeta}}\left[\begin{array}{@{}c@{}}{\it\Phi}_{J}^{(2)}\\ ({\it\zeta}\sin {\it\theta}_{{\it\zeta}})^{2}{\it\Phi}_{c}^{(2)}\end{array}\right]E\,\text{d}{\it\theta}_{{\it\zeta}}\,\text{d}{\it\zeta}, & \displaystyle\end{eqnarray}$$

(5.55i)$$\begin{eqnarray}\displaystyle & \displaystyle \widetilde{Q}_{rS(2)}^{J}={\rm\pi}\int _{0}^{\infty }\int _{0}^{{\rm\pi}}{\it\zeta}^{3}\left({\it\zeta}^{2}-\frac{5}{2}\right)\sin 2{\it\theta}_{{\it\zeta}}{\it\Phi}_{J}^{(2)}E\,\text{d}{\it\theta}_{{\it\zeta}}\,\text{d}{\it\zeta}, & \displaystyle\end{eqnarray}$$

(5.55j)$$\begin{eqnarray}\displaystyle & \displaystyle \widetilde{Q}_{{\it\theta}S(2)}^{b}={\rm\pi}\int _{0}^{\infty }\int _{0}^{{\rm\pi}}{\it\zeta}^{4}\left({\it\zeta}^{2}-\frac{5}{2}\right)\sin ^{3}{\it\theta}_{{\it\zeta}}{\it\Phi}_{b}^{(2)}E\,\text{d}{\it\theta}_{{\it\zeta}}\,\text{d}{\it\zeta}, & \displaystyle\end{eqnarray}$$

${\it\zeta}=|{\bf\zeta}|$

${\it\theta}_{{\it\zeta}}=\cos ^{-1}({\it\zeta}_{r}/|{\bf\zeta}|)$

In summary, to obtain the flow field in the vicinity of the sphere to the second order in ${\it\epsilon}$, it suffices to solve one more spatially one-dimensional problem, in addition to the leading-order linearised problem. More precisely, we solve (5.50)–(5.52) to obtain ${\it\Phi}_{J}^{(2)}$ ($J=a,b,c,d$) and compute $\widetilde{{\it\omega}}_{S(2)}^{J}$, $\widetilde{u}_{rS(2)}^{J}$, $\widetilde{u}_{{\it\theta}S(2)}^{b}$, etc. from (5.55). Then, the second-order macroscopic quantities are readily obtained from (5.54). In this paper, however, we do not carry out this analysis, leaving it to a subsequent study. Instead, we derive the expression for the drag to the second order in ${\it\epsilon}$ in the remaining part, by assuming the existence of the solution to the problem.