2.1. Problem and basic assumptions

Let us consider a rigid sphere with radius $L$ moving through a monatomic ideal gas with constant translational velocity $\boldsymbol {v}_0$. While translating, the sphere is also rotating around an axis of revolution with a constant angular velocity, $\boldsymbol {\varOmega }_0$. Far from the sphere, the gas is in the equilibrium state at rest with pressure $p_0$ and temperature $T_0$. Further, we assume that the sphere's surface temperature is uniform and is equal to the gas temperature at infinity. For convenience, let us choose a frame of reference translating with the sphere. In this frame, the sphere centre is at rest, and the flow velocity at infinity is $-\boldsymbol {v}_0$. We write this velocity as $\boldsymbol {v}_\infty$ ($=-\boldsymbol {v}_0$). We are now concerned with a steady flow of a rarefied gas past a rotating sphere, as shown in figure 1. We investigate the behaviour of the gas under the following assumptions.

1. (i) The behaviour of the gas is described by the Boltzmann equation (we use the BGK model of the Boltzmann equation for the actual numerical computations).

2. (ii) The gas molecules undergo diffuse reflection on the sphere. More precisely, the velocity distribution of the reflected molecules on the surface constitutes the corresponding part of the Maxwellian distribution characterized by the temperature and (local) surface velocity of the sphere and by the condition that there is no net mass flux across the surface.

3. (iii) The translational speed of the sphere (or the flow speed at infinity in our frame) is small compared with the thermal speed of the gas molecules, i.e. $|\boldsymbol {v}_\infty | \ll (2RT_0)^{1/2}$. Here, $R=k_{B}/m$ is the specific gas constant with $k_{B}$ and $m$ being the Boltzmann constant and the mass of a molecule, respectively. In other words, the Mach number of the flow, $\textit {Ma} = |\boldsymbol {v}_\infty |/(5RT_0/3)^{1/2}$, is small.

4. (iv) The rotational surface velocity of the sphere is of the same order of magnitude as the translational velocity of the sphere, i.e. $L|\boldsymbol {\varOmega }_0|/|\boldsymbol {v}_\infty | = O(1)$.

Figure 1.

Problem.

For the subsequent analysis, we introduce the rectangular coordinate system $L x_i$ ($i=1,2,3$) with its origin at the centre of the sphere (the corresponding position vector is denoted by $L \boldsymbol {x}$). Without loss of generality, we can assume that the $x_1$ axis is parallel to the angular velocity $\boldsymbol {\varOmega }_0$ and that the vector $\boldsymbol {v}_\infty$ lies in the $x_1 x_2$ plane (see figure 1). Then, $\boldsymbol {\varOmega }_0$ and $\boldsymbol {v}_\infty$ are expressed as $\boldsymbol {\varOmega }_0 = (\varOmega _0,0,0)$ and $\boldsymbol {v}_\infty = (v_{\infty 1},v_{\infty 2},0)$ with $v_{\infty 1}$, $v_{\infty 2}$ and $\varOmega _0$ ($=|\boldsymbol {\varOmega }_0|$) being given constants.

2.2. Basic equations

Let us first introduce the following notation. The molecular velocity is denoted by $\xi _i$, $i=1,2,3$ (or by $\boldsymbol {\xi }$) and the velocity distribution function by $f$. Furthermore, we denote by $\rho$ the density, by $v_i$ (or $\boldsymbol {v}$) the flow velocity, by $T$ the temperature, by $p$ the pressure, by $p_{ij}$ the stress tensor and by $q_i$ (or $\boldsymbol {q}$) the heat-flux vector of the gas ($i,j=1,2,3$). Then, we introduce the following dimensionless variables:

(2.1a)$$\begin{gather} \zeta_i = \xi_i/(2RT_0)^{1/2}, \quad \phi(\boldsymbol{x},\boldsymbol{\zeta}) = E^{{-}1} [f /\rho_0(2RT_0)^{{-}3/2}]-1, \end{gather}$$

(2.1b)$$\begin{gather}\omega(\boldsymbol{x}) = \rho/\rho_0 -1, \quad u_i(\boldsymbol{x}) = v_i/(2RT_0)^{1/2}, \quad \tau(\boldsymbol{x}) = T/T_0 - 1, \end{gather}$$

(2.1c)$$\begin{gather}P(\boldsymbol{x}) = p/p_0 - 1, \quad P_{ij}(\boldsymbol{x}) = p_{ij}/p_0 - \delta_{ij}, \quad Q_i(\boldsymbol{x}) = q_i/[p_0 (2RT_0)^{1/2}], \end{gather}$$

where $\rho _0 = p_0/(RT_0)$, $E={\rm \pi} ^{-3/2}\exp (-|\boldsymbol {\zeta }|^{2})$ and $\delta _{ij}$ is the Kronecker delta.

We also use the spherical coordinate system $(Lr, \theta, \varphi )$ related to $x_i$ by $x_1 = r \cos \theta$, $x_2 = r \sin \theta \cos \varphi$ and $x_3 = r \sin \theta \sin \varphi$. Components of vectors and tensors in spherical coordinates are represented by $(r,\theta,\varphi )$ in the subscript, e.g. $(\zeta _r,\zeta _\theta,\zeta _\varphi )$, $P_{r\theta }$, etc. Note that Cartesian components of a vector $a_i$ are related to $(a_r,a_\theta,a_\varphi )$ as

(2.2a)$$\begin{gather} a_1 = a_r \cos \theta - a_\theta \sin \theta, \end{gather}$$

(2.2b)$$\begin{gather}a_2 = a_r \sin \theta \cos \varphi + a_\theta \cos \theta \cos \varphi - a_\varphi \sin \varphi, \end{gather}$$

(2.2c)$$\begin{gather}a_3 = a_r \sin \theta \sin \varphi + a_\theta \cos \theta \sin \varphi + a_\varphi \cos \varphi. \end{gather}$$

Throughout the paper, we write $\zeta$ to denote $|\boldsymbol {\zeta }|=(\zeta _j^{2})^{1/2}$.

The time-independent Boltzmann equation for $\phi$ is written as

(2.3)\begin{equation} \zeta_i \frac{\partial \phi}{\partial x_i} = \frac{1}{k} (\mathscr{L}(\phi) + \mathscr{J}(\phi,\phi)), \end{equation}

where $\mathscr {L}$ and $\mathscr {J}$ represent, respectively, the linearized and nonlinear collision operators, whose explicit forms are given in Appendix A (see also Sone Reference Sone2007). Parameter $k$ is defined by

(2.4)\begin{equation} k = \frac{\sqrt{\rm \pi}}{2} \textit{Kn} = \frac{\sqrt{\rm \pi}}{2} \frac{\ell_0}{L}, \end{equation}

where $\textit {Kn} = \ell _0/L$ is the Knudsen number with $\ell _0$ being the mean free path of the gas molecules in the equilibrium state at rest with density $\rho _0$ and temperature $T_0$. For a hard-sphere gas, $\ell _0$ is given by $\ell _0 = 1/[\sqrt {2} {\rm \pi}d_m^{2} (\rho _0/m)]$ with $d_m$ the diameter of a molecule. On the other hand, for the BGK model introduced below, $\ell _0$ is given by $\ell _0 = (2/\sqrt {{\rm \pi} }) (2RT_0)^{1/2}/A_c \rho _0$ with $A_c$ being a constant ($A_c \rho _0$ is the collision frequency at the reference equilibrium state at rest). Further details concerning (2.3) are given in Appendix A.

The operators $\mathscr {L}$ and $\mathscr {J}$ are spherically symmetric operators, that is, for any functions $F$ and $G$ of $\boldsymbol {\zeta }$, it holds that $\mathscr {L}(F(l_{ij}\zeta _j))(\boldsymbol {\zeta }) = \mathscr {L}(F(\boldsymbol {\zeta }))(l_{ij}\zeta _j)$ and $\mathscr {J}(F(l_{ij}\zeta _j),G(l_{ij}\zeta _j))(\boldsymbol {\zeta }) = \mathscr {J}(F(\boldsymbol {\zeta }),G(\boldsymbol {\zeta }))(l_{ij}\zeta _j)$, where $l_{ij}$ is any orthogonal transformation, i.e. $l_{ij} l_{kj} = \delta _{ik}$. This implies that $\mathscr {L}$ and $\mathscr {J}$ are axially symmetric, meaning that they satisfy the same identities as above for any orthogonal transformation $l_{ij}$ that satisfies $l_{ij} a_j = a_i$, where $a_i$ is a fixed (unit) vector. Since $\mathscr {L}$ and $\mathscr {J}$ are spherically symmetric, $a_i$ can be chosen arbitrarily. The property of axial symmetry of the operators plays a crucial role in the present analysis.

The diffuse reflection boundary condition (e.g. Kogan Reference Kogan1969; Cercignani Reference Cercignani1988; Sone Reference Sone2007) on the sphere is written as

(2.5)\begin{equation} \phi = \frac{1+\sigma_w}{{\rm \pi}^{3/2}} \exp(-\zeta_r^{2} - \zeta_\theta^{2} - (\zeta_\varphi - \hat{\varOmega}_0 \sin \theta)^{2} ) E^{{-}1} - 1, \quad \zeta_r > 0 \quad (r=|\boldsymbol{x}|=1), \end{equation}

with

(2.6)\begin{equation} \sigma_w ={-}2 \sqrt{\rm \pi} \int_{\zeta_r<0} \zeta_r \phi E \,\mathrm{d} \boldsymbol{\zeta}, \end{equation}

where $\mathrm {d} \boldsymbol {\zeta } = \mathrm {d} \zeta _1 \,\mathrm {d} \zeta _2 \,\mathrm {d} \zeta _3$ and $\hat {\varOmega }_0$ is the dimensionless angular velocity defined by

(2.7)\begin{equation} \hat{\varOmega}_0 = \frac{L \varOmega_0}{(2RT_0)^{1/2}}. \end{equation}

The boundary condition at infinity is written as

(2.8)\begin{equation} \phi \to \frac{1}{{\rm \pi}^{3/2}} \exp(-(\zeta_1 - \hat{v}_{\infty 1})^{2} - (\zeta_2 - \hat{v}_{\infty 2})^{2} - \zeta_3^{2}) E^{{-}1} - 1, \quad \text{as} \ r=|\boldsymbol{x}| \to \infty, \end{equation}

with $\hat {v}_{\infty 1}$ and $\hat {v}_{\infty 2}$ given by

(2.9)\begin{equation} \hat{v}_{\infty i} = \frac{v_{\infty i}}{(2RT_0)^{1/2}}, \quad i=1,2. \end{equation}

The macroscopic variables are expressed in terms of $\phi$ as follows:

(2.10a)$$\begin{gather} \omega = \langle \phi \rangle, \quad (1 + \omega) u_i = \langle \zeta_i \phi \rangle, \quad \tfrac{3}{2} (1 + \omega) \tau = \langle (\zeta^{2} - \tfrac{3}{2}) \phi \rangle - (1 + \omega) u_j^{2}, \end{gather}$$

(2.10b)$$\begin{gather}P = \omega + \tau + \omega \tau, \quad P_{ij} = 2 \langle \zeta_i \zeta_j \phi \rangle - 2 (1 + \omega) u_i u_j, \end{gather}$$

(2.10c)$$\begin{gather}Q_i = \langle \zeta_i \zeta^{2} \phi \rangle - \tfrac{5}{2} u_i - u_j P_{ij} - \tfrac{3}{2} P u_i - (1 + \omega) u_i u_j^{2}, \end{gather}$$

where the symbol $\langle \,\, \rangle$ represents the following integral with respect to $\boldsymbol {\zeta }$:

(2.11)\begin{equation} \langle g \rangle = \int_{\mathbb{R}^{3}} g(\boldsymbol{\zeta}) E \,\mathrm{d} \boldsymbol{\zeta}. \end{equation}

In the actual numerical computations, we employ the BGK model of the Boltzmann equation. The BGK model is obtained by replacing $\mathscr {L}(\phi )$ and $\mathscr {J}(\phi,\phi )$ with the following counterparts (see Appendix A):

(2.12)$$\begin{gather} \mathscr{L}^{BGK}(\phi) = g_e(\phi) - \phi, \end{gather}$$

(2.13)$$\begin{gather} \mathscr{J}^{BGK}(\phi) = (1 + \omega) (\phi_e - \phi) - \mathscr{L}^{BGK}(\phi) \nonumber\\\hspace{87pt}= (1 + \omega) (\phi_e - g_e) + \omega \mathscr{L}^{BGK}(\phi), \end{gather}$$

where

(2.14)$$\begin{gather} g_e = \langle \phi \rangle + 2 \zeta_i \langle \zeta_i \phi \rangle + \left( \zeta^{2} - \tfrac{3}{2} \right ) \tfrac{2}{3} \left \langle \left( \zeta^{2} - \tfrac{3}{2} \right ) \phi \right \rangle, \end{gather}$$

(2.15)$$\begin{gather}\phi_e = E^{{-}1}\frac{1+\omega}{{\rm \pi}^{3/2} (1+\tau)^{3/2}} \exp\left(- \frac{(\zeta_i-u_i)^{2}}{1+\tau}\right) -1, \end{gather}$$

and $\omega$, $u_i$ and $\tau$ are given by (2.10a). Note that the operators $\mathscr {L}^{BGK}(\phi )$ and $\mathscr {J}^{BGK}(\phi )$ are spherically symmetric.

2.3. Scaling assumptions

Now we introduce assumptions (iii) and (iv) in § 2.1 and restrict our consideration to slow flows (i.e. low-Mach-number flows). We thus introduce the quantity

(2.16)\begin{equation} \varepsilon := |\hat{\boldsymbol{v}}_{\infty}| = (\hat{v}_{\infty 1}^{2} + \hat{v}_{\infty 2}^{2})^{1/2}, \end{equation}

and assume that $\varepsilon$ is small ($\varepsilon \ll 1$). Note that the Mach number, mentioned in assumption (iii), is related to $\varepsilon$ by $\textit {Ma} = (6/5)^{1/2} \varepsilon$. In the next section, we carry out a perturbative analysis for small $\varepsilon$ in the case $k=O(1)$, $\hat {\varOmega }_0 = O(\varepsilon )$ and $|\phi | = O(\varepsilon )$ (i.e. the weakly nonlinear regime). We thus write $\hat {v}_{\infty 1}$, $\hat {v}_{\infty 2}$ and $\hat {\varOmega }_0$ as

(2.17a–c)\begin{equation} \hat{v}_{\infty 1} = U \varepsilon = \varepsilon \cos \alpha_0, \quad \hat{v}_{\infty 2} = V \varepsilon = \varepsilon \sin \alpha_0, \quad \hat{\varOmega}_0 = S \varepsilon, \end{equation}

where $\alpha _0 \in [\,0,2{\rm \pi} )$ is the azimuth angle of $\boldsymbol {v}_\infty$ (see figure 1). Note that $U^{2}+V^{2}=1$ by definition and that (cf. assumption (iv))

(2.18)\begin{equation} S = \frac{\hat{\varOmega}_0}{\varepsilon} = \frac{L|\boldsymbol{\varOmega}_0|}{|\boldsymbol{v}_\infty|} = O(1). \end{equation}

Then, the boundary conditions are rewritten as

(2.19)$$\begin{gather} \phi = \frac{1+\sigma_w}{{\rm \pi}^{3/2}} \exp(-\zeta_r^{2}-\zeta_\theta^{2} - (\zeta_\varphi - \varepsilon S \sin \theta)^{2} ) E^{{-}1} - 1, \quad \zeta_r > 0 \quad (|\boldsymbol{x}|=1), \end{gather}$$

(2.20)$$\begin{gather}\phi \to \phi_\infty := \frac{1}{{\rm \pi}^{3/2}} \exp(-(\zeta_1- \varepsilon U)^{2} -(\zeta_2- \varepsilon V)^{2} - \zeta_3^{2})E^{{-}1} - 1, \quad \text{as} \ |\boldsymbol{x}| \to \infty, \end{gather}$$

where $\sigma _w$ is given by (2.6).

In summary, the problem to be solved is (2.3), (2.19) with (2.6), and (2.20), where $k$, $U$, $V$ and $S$ are independent of $\varepsilon$.

Finally, we make the following comment. According to the von Kármán relation (Sone Reference Sone2007), the Reynolds number ${{Re}} = \rho _0 |\boldsymbol {v}_\infty | L/\mu _0$, the Mach number $\textit {Ma}$ and the Knudsen number $\textit {Kn}$ are not independent but are related to each other by the relation ${{Re}} \sim \textit {Ma}/\textit {Kn}$. Here, $\mu _0$ is the viscosity of the gas at the reference state. Thus, the present analysis corresponds physically to a situation in which the Reynolds number is small, i.e.

(2.21)\begin{equation} {{Re}} \sim \frac{\varepsilon}{k} \ll 1. \end{equation}

3.1. Inner solution

We first consider a solution to the problem whose length scale of variation is of the order of unity (or of the order of $L$ in dimensional space). We call this length scale the inner scale for the reason to be clarified below. Accordingly, the solution with this length scale is called the inner solution and hereafter is designated by attaching the subscript $F$, i.e. $\partial \phi _{F}/\partial x_i = O(\phi _{F})$. We assume that the inner solution can be expanded in $\varepsilon$ as

(3.1)\begin{equation} \phi_{F} = \varepsilon \phi_{F1} + \varepsilon^{2} \phi_{F2} + o(\varepsilon^{2}). \end{equation}

It may be mentioned that the remainder may not be a simple power series of $\varepsilon$ but contains terms like $\varepsilon ^{3} \ln \varepsilon$, as in the Navier–Stokes theory (Proudman & Pearson Reference Proudman and Pearson1957; Chester, Breach & Proudman Reference Chester, Breach and Proudman1969). In the present study, we will not address this issue further and concentrate on the force exerted on a particle to $\varepsilon ^{2}$ order.

Corresponding to the expansion (3.1), the macroscopic variables are also expanded in $\varepsilon$ as

(3.2)\begin{equation} h_{F} = \varepsilon h_{F1} + \varepsilon^{2} h_{F2} + o(\varepsilon^{2}) \quad (h = \omega,u_i, \tau, P, P_{ij}, Q_i). \end{equation}

The relations between $h_{Fm}$ and $\phi _{Fm}$ ($m = 1,2, \ldots$) are obtained by substituting the expansions of $h_{F}$ and $\phi _{F}$ into the definitions of the macroscopic variables (2.10) with $h=h_{F}$ and $\phi =\phi _{F}$ and by equating terms with the same power of $\varepsilon$. We thus obtain, for the first two orders in $\varepsilon$,

(3.3a)$$\begin{gather} \omega_{F1} = \langle \phi_{F1} \rangle, \quad u_{i F1} = \langle \zeta_i \phi_{F1} \rangle, \quad \tau_{F1} = \tfrac{2}{3} \langle (\zeta^{2} - \tfrac{3}{2}) \phi_{F1} \rangle, \end{gather}$$

(3.3b)$$\begin{gather}P_{F1} = \omega_{F1} + \tau_{F1}, \quad P_{ij F1} = 2 \langle \zeta_i \zeta_j \phi_{F1} \rangle, \quad Q_{i F1} = \langle \zeta_i \zeta^{2} \phi_{F1} \rangle - \tfrac{5}{2} u_{i F1}, \end{gather}$$

(3.4a)$$\begin{gather} \omega_{F2} = \langle \phi_{F2} \rangle, \quad u_{i F2} = \langle \zeta_i \phi_{F2} \rangle - \omega_{F1} u_{i F1}, \end{gather}$$

(3.4b)$$\begin{gather}\tau_{F2} = \tfrac{2}{3} \langle (\zeta^{2} - \tfrac{3}{2} ) \phi_{F2} \rangle - \tfrac{2}{3} (u_{j F1})^{2} - \omega_{F1} \tau_{F1}, \end{gather}$$

(3.4c)$$\begin{gather}P_{F2} = \omega_{F2} + \tau_{F2} + \omega_{F1} \tau_{F1}, \quad P_{ij F2} = 2 \langle \zeta_i \zeta_j \phi_{F2} \rangle -2 u_{i F1} u_{j F1}, \end{gather}$$

(3.4d)$$\begin{gather}Q_{i F2} = \langle \zeta_i \zeta^{2} \phi_{F2} \rangle - \tfrac{5}{2} u_{i F2} - u_{j F1} P_{ij F1} -\tfrac{3}{2} P_{F1} u_{i F1}. \end{gather}$$

Note that the nonlinearity enters the relations in the form of a product of lower-order terms in (3.4).

If we substitute the expansion (3.1) into (2.3) and collect terms with the same power of $\varepsilon$, we obtain a sequence of linearized Boltzmann equations for $\phi _{Fm}$ ($m=1,2$), i.e.

(3.5a)$$\begin{gather} \zeta_i \frac{\partial \phi_{F1}}{\partial x_i} = \frac{1}{k} \mathscr{L}(\phi_{F1}), \end{gather}$$

(3.5b)$$\begin{gather}\zeta_i \frac{\partial \phi_{F2}}{\partial x_i} = \frac{1}{k} \mathscr{L}(\phi_{F2}) + \frac{1}{k} \mathscr{J}(\phi_{F1},\phi_{F1}). \end{gather}$$

Equation (3.5a) is the linearized Boltzmann equation for $\phi _{F1}$, while (3.5b) is the linearized Boltzmann equation for $\phi _{F2}$ with an inhomogeneous term. Similarly, if we insert the expansion into the diffuse reflection condition on the sphere, i.e. (2.19) with (2.6), we obtain a sequence of boundary conditions for $\phi _{Fm}$ ($m=1,2$) on the sphere (see (3.7) and (3.65)). Provided appropriate boundary conditions at infinity are given, they form a sequence of boundary-value problems for $\phi _{Fm}$, which can be solved successively from the lowest order.

3.2. Leading order

The equation and the boundary condition on the sphere for $\phi _{F1}$ are given by

(3.6)$$\begin{gather} \zeta_i \frac{\partial \phi_{F1}}{\partial x_i} = \frac{1}{k} \mathscr{L}(\phi_{F1}), \end{gather}$$

(3.7)$$\begin{gather}\phi_{F1} = \mathscr{K}(\phi_{F1}) + 2 S \zeta_\varphi \sin \theta, \quad \zeta_r > 0 \quad (|\boldsymbol{x}|=1), \end{gather}$$

where $\mathscr {K}(\cdot )$ represents

(3.8)\begin{equation} \mathscr{K}(g) ={-} 2 \sqrt{\rm \pi} \int_{\zeta_r < 0} \zeta_r g(\boldsymbol{\zeta}) E \,\mathrm{d} \boldsymbol{\zeta}. \end{equation}

To derive the corresponding boundary condition at infinity, we expand $\phi _\infty$ in (2.20) as $\phi _\infty = \varepsilon \phi _{\infty 1} + \varepsilon ^{2} \phi _{\infty 2} + \cdots$ and retain the leading-order term. This yields

(3.9)\begin{equation} \phi_{F1} \to 2(\zeta_1 U + \zeta_2 V), \quad \text{as} \ |\boldsymbol{x}| \to \infty. \end{equation}

Equations (3.6)–(3.9) form a boundary-value problem of the linearized Boltzmann equation for unknown $\phi _{F1}$.

In view of the linearity of the problem, we seek the solution in the form

(3.10)\begin{equation} \phi_{F1} = \varPhi_{\mathrm{U}}^{(1)} + \varPhi_{\mathrm{S}}^{(1)}, \end{equation}

where $\varPhi _{\mathrm {U}}^{(1)}$ and $\varPhi _{\mathrm {S}}^{(1)}$ solve the following problems:

(3.11)$$\begin{gather} \zeta_i \frac{\partial \varPhi_J^{(1)}}{\partial x_i} = \frac{1}{k} \mathscr{L}(\varPhi_J^{(1)}) \quad (J=\textrm{U},\textrm{S}), \end{gather}$$

(3.12)$$\begin{gather}\varPhi_J^{(1)} = \mathscr{K}(\varPhi_J^{(1)}) + I_{{w},J}^{(1)}, \quad \zeta_r > 0, \ |\boldsymbol{x}| = 1, \end{gather}$$

(3.13)$$\begin{gather}\varPhi_J^{(1)} \to I_{\infty,J}^{(1)}, \quad \text{as} \ |\boldsymbol{x}| \to \infty, \end{gather}$$

with

(3.14a,b)$$\begin{gather} I_{{w},\textrm{U}}^{(1)} = 0, \quad I_{{w},\textrm{S}}^{(1)} = 2 \zeta_\varphi S \sin \theta, \end{gather}$$

(3.15a,b)$$\begin{gather}I_{\infty,\textrm{U}}^{(1)} = 2U\zeta_1 + 2V\zeta_2, \quad I_{\infty,\textrm{S}}^{(1)} = 0. \end{gather}$$

The problem for $\varPhi _{\mathrm {S}}^{(1)}$ (hereafter referred to as problem S) describes the steady flow of a rarefied gas around a rotating sphere without any flows at infinity (Loyalka Reference Loyalka1992; Andreev & Popov Reference Andreev and Popov2010; Taguchi et al. Reference Taguchi, Saito and Takata2019). The problem for $\varPhi _{\mathrm {U}}^{(1)}$ (hereafter referred to as problem U) is equivalent to the boundary-value problem describing a uniform flow of rarefied gas past a sphere in the absence of sphere rotation, which has been extensively studied in the literature (e.g. Cercignani et al. Reference Cercignani, Pagani and Bassanini1968; Sone & Aoki Reference Sone and Aoki1977a; Takata et al. Reference Takata, Sone and Aoki1993; Kalempa & Sharipov Reference Kalempa and Sharipov2020; see also Sone Reference Sone2007).

Using the property of axial symmetry of the operator $\mathscr {L}$ given in Appendix B, we seek $\varPhi _{\mathrm {U}}^{(1)}$ and $\varPhi _{\mathrm {S}}^{(1)}$ in the forms (i.e. similarity solutions)

(3.16a)\begin{align} \varPhi_{\mathrm{U}}^{(1)} &= (U \cos \theta + V \sin \theta \cos \varphi) \varphi_{\mathrm{U} a}^{(1)}(r,\zeta_r,\zeta) \nonumber\\ &\quad + [\zeta_\theta (U \sin \theta - V \cos \theta \cos \varphi) + V \zeta_\varphi \sin \varphi] \varphi_{\mathrm{U} b}^{(1)}(r,\zeta_r,\zeta), \end{align}

(3.16b)\begin{gather} \varPhi_{\mathrm{S}}^{(1)} = S \zeta_\varphi \sin \theta \varphi_{\mathrm{S}}^{(1)}(r,\zeta_r,\zeta), \end{gather}

where the functions $\varphi _{\mathrm {U} a}^{(1)}(r,\zeta _r,\zeta )$, $\varphi _{\mathrm {U} b}^{(1)}(r,\zeta _r,\zeta )$ and $\varphi _{\mathrm {S}}^{(1)}(r,\zeta _r,\zeta )$ solve the following boundary-value problems in space in one dimension (in spherical coordinates):

1. (a) Problem U

   (3.17a)$$\begin{gather} \zeta_r \frac{\partial \varphi_{\mathrm{U} a}^{(1)}}{\partial r} + \frac{\zeta^{2} - \zeta_r^{2}}{r} \frac{\partial \varphi_{\mathrm{U} a}^{(1)}}{\partial \zeta_r} + \frac{\zeta^{2} - \zeta_r^{2}}{r} \varphi_{\mathrm{U} b}^{(1)} = \frac{1}{k} \mathscr{L}_0(\varphi_{\mathrm{U} a}^{(1)}), \end{gather}$$
   (3.17b)$$\begin{gather}\zeta_r \frac{\partial \varphi_{\mathrm{U} b}^{(1)}}{\partial r} + \frac{\zeta^{2} - \zeta_r^{2}}{r} \frac{\partial \varphi_{\mathrm{U} b}^{(1)}}{\partial \zeta_r} - \frac{\zeta_r}{r} \varphi_{\mathrm{U} b}^{(1)} - \frac{\varphi_{\mathrm{U} a}^{(1)}}{r} = \frac{1}{k} \mathscr{L}_1(\varphi_{\mathrm{U} b}^{(1)}), \end{gather}$$
   (3.17c)$$\begin{gather}\varphi_{\mathrm{U} a}^{(1)} = \mathscr{K}(\varphi_{\mathrm{U} a}^{(1)}), \quad \varphi_{\mathrm{U} b}^{(1)} = 0 , \quad \zeta_r > 0, \quad \text{at} \ r = 1, \end{gather}$$
   (3.17d)$$\begin{gather}\varphi_{\mathrm{U} a}^{(1)} \to 2 \zeta_r, \quad \varphi_{\mathrm{U} b}^{(1)} \to -2, \quad \text{as} \ r \to \infty. \end{gather}$$

2. (b) Problem S

   (3.18a)$$\begin{gather} \zeta_r \frac{\partial \varphi_{\mathrm{S}}^{(1)}}{\partial r} + \frac{\zeta^{2} - \zeta_r^{2}}{r} \frac{\partial \varphi_{\mathrm{S}}^{(1)}}{\partial \zeta_r} - \frac{\zeta_r}{r} \varphi_{\mathrm{S}}^{(1)} = \frac{1}{k} \mathscr{L}_1(\varphi_{\mathrm{S}}^{(1)}), \end{gather}$$
   (3.18b)$$\begin{gather}\varphi_{\mathrm{S}}^{(1)} = 2, \quad \zeta_r > 0, \quad \text{at} \ r = 1, \end{gather}$$
   (3.18c)$$\begin{gather}\varphi_{\mathrm{S}}^{(1)} \to 0, \quad \text{as} \ r \to \infty. \end{gather}$$

Here, the operators $\mathscr {L}_0$ and $\mathscr {L}_1$ are defined in Appendix B. Note that $\mathscr {L}_0(\varphi _{\mathrm {U} a}^{(1)})$, $\mathscr {L}_1(\varphi _{\mathrm {U} b}^{(1)})$ and $\mathscr {L}_1(\varphi _{\mathrm {S}}^{(1)})$ appearing on the right-hand sides of (3.17a), (3.17b) and (3.18a) are functions of $r$, $\zeta _r$ and $\zeta$. For clarity, table 1 summarizes the notation for the similarity solutions.

Table 1.

Functions appearing in the similarity solutions.

Suppose that $\varphi _{\mathrm {U} a}^{(1)}$ and $\varphi _{\mathrm {U} b}^{(1)}$ (or $\varPhi _{\mathrm {U}}^{(1)}$) and $\varphi _{\mathrm {S}}^{(1)}$ (or $\varPhi _{\mathrm {S}}^{(1)}$) are known. Then, the leading-order macroscopic quantities $\omega _{F1}$, $u_{iF1}$, etc., are obtained by substituting (3.10) with (3.16) into (3.3). To this end, we first introduce the following notation:

(3.19a)$$\begin{gather} \tilde{\omega}[g] = \langle g \rangle, \quad \tilde{u}_i[g] = \langle \zeta_i g \rangle, \quad \tilde{\tau}[g] = \tfrac{2}{3} \langle (\zeta^{2} - \tfrac{3}{2} ) g \rangle, \end{gather}$$

(3.19b)$$\begin{gather}\tilde{P}[g] = \tfrac{2}{3} \langle \zeta^{2} g \rangle = \tilde{\omega}[g] + \tilde{\tau}[g], \quad \tilde{P}_{ij}[g] = 2 \langle \zeta_i \zeta_j g \rangle, \quad \tilde{Q}_{i}[g] = \langle \zeta_i (\zeta^{2} - \tfrac{5}{2} ) g \rangle, \end{gather}$$

where $g=g(\zeta _i)$. Note that, for every position $\boldsymbol {x}$ in the gas, $\varPhi _{\mathrm {U}}^{(1)}$ and $\varPhi _{\mathrm {S}}^{(1)}$ (and thus $\phi _{F1}$) are of the form $g=a_0 g_0(\zeta _r,\zeta ) + b_i \zeta _j (\delta _{ij} - \hat {x}_i \hat {x}_j) g_1(\zeta _r,\zeta )$, where $\hat {x}_i = x_i/r = x_i/|\boldsymbol {x}|$ ($\hat {x}_i^{2}=1$) and $a_0$ and $b_i$ are independent of $\zeta _i$. For such a function, the components of $\tilde {u}_i^{g} = \tilde {u}_i[g]$, $\tilde {P}_{ij}^{g} = \tilde {P}_{ij}[g]$ and $\tilde {Q}_i^{g} = \tilde {Q}_i[g]$ in spherical coordinates are calculated as

(3.20) \begin{equation} \left.\begin{array}{c@{}} \tilde{u}_r^{g} = a_0 \langle \zeta_r g_0 \rangle, \quad \tilde{u}_i^{g} t_i = \dfrac{b_i t_i}{2} \langle (\zeta^{2}-\zeta_r^{2}) g_1 \rangle, \quad \tilde{P}_{rr}^{g} = 2 a_0 \langle \zeta_r^{2} g_0 \rangle,\\ \tilde{P}_{ij}^{g} t_i t_j = a_0 \langle (\zeta^{2} - \zeta_r^{2}) g_0 \rangle, \quad \tilde{P}_{rj}^{g} t_j = b_j t_j \langle \zeta_r (\zeta^{2} - \zeta_r^{2}) g_1 \rangle,\\ \tilde{Q}_r^{g} = a_0 \left \langle \zeta_r \left(\zeta^{2} - \dfrac{5}{2}\right) g_0 \right \rangle, \quad \tilde{Q}_i^{g} t_i = \dfrac{b_i t_i}{2} \left\langle (\zeta^{2}-\zeta_r^{2}) \left(\zeta^{2} - \dfrac{5}{2}\right) g_1 \right\rangle, \end{array}\right\} \end{equation}

where $t_i$ is an arbitrary unit vector perpendicular to $\hat {x}_i$. Therefore, if we further introduce the notation

(3.21a)$$\begin{gather} \tilde{u}_{r}[\varphi] = \langle \zeta_r \varphi \rangle, \quad \tilde{u}_{t}[\varphi] = \tfrac{1}{2} \langle (\zeta^{2} - \zeta_r^{2}) \varphi \rangle, \end{gather}$$

(3.21b)$$\begin{gather}\left.\begin{array}{c@{}} \tilde{P}_{rr}[\varphi] = 2 \langle \zeta_r^{2} \varphi \rangle,\\ \tilde{P}_{tt}[\varphi] = \langle (\zeta^{2} - \zeta_r^{2}) \varphi \rangle (= 2 \tilde{u}_t[\varphi]),\\ \tilde{P}_{rt}[\varphi] = \langle \zeta_r (\zeta^{2} - \zeta_r^{2}) \varphi \rangle, \end{array}\right\} \end{gather}$$

(3.21c)$$\begin{gather}\tilde{Q}_{r}[\varphi] = \langle \zeta_r (\zeta^{2} - \tfrac{5}{2}) \varphi \rangle, \quad \tilde{Q}_{t}[\varphi] = \tfrac{1}{2} \langle (\zeta^{2} - \zeta_r^{2}) (\zeta^{2} - \tfrac{5}{2} ) \varphi \rangle, \end{gather}$$

where $\varphi =\varphi (\zeta _r,\zeta )$, the leading-order macroscopic quantities (as functions of $(r,\theta,\varphi )$) are expressed in the forms

(3.22a)$$\begin{gather} \omega_{F1} = (U \cos \theta + V \sin \theta \cos \varphi) \tilde{\omega}_{\mathrm{U} a}^{(1)}(r), \end{gather}$$

(3.22b)$$\begin{gather}u_{r F1} = (U \cos \theta + V \sin \theta \cos \varphi) \tilde{u}_{r,\mathrm{U} a}^{(1)}(r), \end{gather}$$

(3.22c)$$\begin{gather}u_{\theta F1} = (U \sin \theta - V \cos \theta \cos \varphi) \tilde{u}_{t,\mathrm{U} b}^{(1)}(r), \end{gather}$$

(3.22d)$$\begin{gather}u_{\varphi F1} = (V \sin \varphi) \tilde{u}_{t,\mathrm{U} b}^{(1)}(r) + (S \sin \theta) \tilde{u}_{t,\mathrm{S}}^{(1)}(r), \end{gather}$$

(3.22e)$$\begin{gather}\tau_{F1} = (U \cos \theta + V \sin \theta \cos \varphi) \tilde{\tau}_{\mathrm{U} a}^{(1)}(r), \end{gather}$$

(3.22f)$$\begin{gather}P_{F1} = (U \cos \theta + V \sin \theta \cos \varphi) \tilde{P}_{\mathrm{U} a}^{(1)}(r), \end{gather}$$

(3.22g)$$\begin{gather}P_{rr F1} = (U \cos \theta + V \sin \theta \cos \varphi) \tilde{P}_{rr,\mathrm{U} a}^{(1)}(r), \end{gather}$$

(3.22h)$$\begin{gather}P_{r\theta F1} = (U \sin \theta - V \cos \theta \cos \varphi) \tilde{P}_{rt,\mathrm{U} b}^{(1)}(r), \end{gather}$$

(3.22i)$$\begin{gather}P_{r\varphi F1} = (V \sin \varphi) \tilde{P}_{rt,\mathrm{U} b}^{(1)}(r) + (S \sin \theta) \tilde{P}_{rt,\mathrm{S}}^{(1)}(r), \end{gather}$$

(3.22j)$$\begin{gather}P_{\theta \theta F1} = P_{\varphi \varphi F1} = (U \cos \theta + V \sin \theta \cos \varphi) \tilde{P}_{tt,\mathrm{U} a}^{(1)}(r), \end{gather}$$

(3.22k)$$\begin{gather}P_{\theta \varphi F1} = 0, \end{gather}$$

(3.22l)$$\begin{gather}Q_{r F1} = (U \cos \theta + V \sin \theta \cos \varphi)\tilde{Q}_{r,\mathrm{U} a}^{(1)}(r), \end{gather}$$

(3.22m)$$\begin{gather}Q_{\theta F1} = (U \sin \theta - V \cos \theta \cos \varphi)\tilde{Q}_{t,\mathrm{U} b}^{(1)}(r), \end{gather}$$

(3.22n)$$\begin{gather}Q_{\varphi F1} = (V \sin \varphi) \tilde{Q}_{t,\mathrm{U} b}^{(1)}(r) + (S \sin \theta) \tilde{Q}_{t,\mathrm{S}}^{(1)}(r), \end{gather}$$

where

(3.23) \begin{equation} \left.\begin{array}{c@{}} \tilde{\omega}_{\mathrm{U} a}^{(1)} = \tilde{\omega}[\varphi_{\mathrm{U} a}^{(1)}], \quad \tilde{u}_{r,\mathrm{U} a}^{(1)} = \tilde{u}_{r}[\varphi_{\mathrm{U} a}^{(1)}], \quad \tilde{\tau}_{\mathrm{U} a}^{(1)} = \tilde{\tau}[\varphi_{\mathrm{U} a}^{(1)}], \quad \tilde{P}_{\mathrm{U} a}^{(1)} = \tilde{P}[\varphi_{\mathrm{U} a}^{(1)}],\\ \tilde{P}_{rr,\mathrm{U} a}^{(1)} = \tilde{P}_{rr}[\varphi_{\mathrm{U} a}^{(1)}], \quad \tilde{P}_{tt,\mathrm{U} a}^{(1)} = \tilde{P}_{tt}[\varphi_{\mathrm{U} a}^{(1)}], \quad \tilde{Q}_{r,\mathrm{U} a}^{(1)} = \tilde{Q}_{r}[\varphi_{\mathrm{U} a}^{(1)}],\\ \tilde{u}_{t,\mathrm{U} b}^{(1)} = \tilde{u}_{t}[\varphi_{\mathrm{U} b}^{(1)}], \quad \tilde{P}_{rt,\mathrm{U} b}^{(1)} = \tilde{P}_{rt}[\varphi_{\mathrm{U} b}^{(1)}], \quad \tilde{Q}_{t,\mathrm{U} b}^{(1)} = \tilde{Q}_{t}[\varphi_{\mathrm{U} b}^{(1)}],\\ \tilde{u}_{t,\mathrm{S}}^{(1)} = \tilde{u}_{t}[\varphi_{\mathrm{S}}^{(1)}], \quad \tilde{P}_{rt,\mathrm{S}}^{(1)} = \tilde{P}_{rt}[\varphi_{\mathrm{S}}^{(1)}], \quad \tilde{Q}_{t,\mathrm{S}}^{(1)} = \tilde{Q}_{t}[\varphi_{\mathrm{S}}^{(1)}]. \end{array}\right\} \end{equation}

Note that $\tilde {\omega }_{\mathrm {U} a}^{(1)}$, $\tilde {u}_{r,\mathrm {U} a}^{(1)}$, etc., depend on $r$ through $\varphi _{\mathrm {U} a}^{(1)}$, $\varphi _{\mathrm {U} b}^{(1)}$ and $\varphi _{\mathrm {S}}^{(1)}$, as shown explicitly in (3.22). We will not repeat similar comments in what follows.

Finally, let us consider the force and the torque acting on the sphere. Let $p_0 L^{2} \mathcal {F}_i$ and $p_0 L^{3} \mathcal {M}_i$ denote, respectively, the force and moment of force (about the origin) acting on the sphere. Then, $\mathcal {F}_i$ and $\mathcal {M}_i$ are given in terms of the stress tensor as

(3.24a,b)\begin{equation} \mathcal{F}_i ={-} \int_{|\boldsymbol{x}|=1} P_{ij F} n_j \,\mathrm{d}S, \quad \mathcal{M}_i ={-} \int_{|\boldsymbol{x}|=1} \epsilon_{ijk} x_j P_{kl F} n_l \,\mathrm{d}S, \end{equation}

where $\mathrm {d}S$ is the surface element, $n_i$ is the unit normal vector on the sphere pointing to the gas, $\epsilon _{ijk}$ ($i,j,k=1,2,3$) is the Eddington epsilon (the permutation symbol) and the integration is carried out over the whole surface $|\boldsymbol {x}|=1$. We expand the (dimensionless) force and torque in $\varepsilon$ as

(3.25a,b)\begin{equation} \mathcal{F}_i = \varepsilon \mathcal{F}_i^{(1)} + \varepsilon^{2} \mathcal{F}_i^{(2)} + o(\varepsilon^{2}), \quad \mathcal{M}_i = \varepsilon \mathcal{M}_i^{(1)} + \varepsilon^{2} \mathcal{M}_i^{(2)} + o(\varepsilon^{2}), \end{equation}

where $\mathcal {F}_i^{(m)}$ and $\mathcal {M}_i^{(m)}$, $m=1,2$, are given by

(3.26a,b)\begin{equation} \mathcal{F}_i^{(m)} ={-} \int_{|\boldsymbol{x}|=1} P_{ij Fm} n_j \,\mathrm{d}S, \quad \mathcal{M}_i^{(m)} ={-} \int_{|\boldsymbol{x}|=1} \epsilon_{ijk} x_j P_{kl Fm} n_l \,\mathrm{d}S. \end{equation}

Substituting (3.22g)–(3.22i) into (3.26a,b) with $m=1$, the force $\mathcal {F}_i^{(1)}$ and the moment of force $\mathcal {M}_i^{(1)}$ are obtained as

(3.27a)$$\begin{gather} \mathcal{F}_1^{(1)} = U h_D, \quad \mathcal{F}_2^{(1)} = V h_D, \quad \mathcal{F}_3^{(1)} = 0, \end{gather}$$

(3.27b)$$\begin{gather}\mathcal{M}_1^{(1)} = S h_M, \quad \mathcal{M}_2^{(1)} = 0, \quad \mathcal{M}_3^{(1)} = 0, \end{gather}$$

where

(3.28)$$\begin{gather} h_D ={-} \tfrac{4}{3} {\rm \pi}(\tilde{P}_{rr,\mathrm{U} a}^{(1)} - 2 \tilde{P}_{rt,\mathrm{U} b}^{(1)})|_{r=1}, \end{gather}$$

(3.29)$$\begin{gather}h_M ={-} \tfrac{8}{3} {\rm \pi}\tilde{P}_{rt,\mathrm{S}}^{(1)}|_{r=1}. \end{gather}$$

Introducing the two unit vectors

(3.30a,b)\begin{equation} (e_i)_{i=1,2,3} = (U,V,0), \quad (\hat{e}_i)_{i=1,2,3} = (1,0,0), \end{equation}

the force and the torque acting on the sphere are summarized as

(3.31a,b)\begin{equation} \mathcal{F}_i = \varepsilon h_D e_i + O(\varepsilon^{2}), \quad \mathcal{M}_i = \varepsilon S h_M \hat{e}_i + O(\varepsilon^{2}). \end{equation}

Thus, no transverse force acts on the sphere at leading order, i.e. $\mathcal {F}_3=O(\varepsilon ^{2})$. It should be noted that $h_D$ and $h_M$ depend on $k$ through $\varphi _{\mathrm {U} a}^{(1)}$, $\varphi _{\mathrm {U} b}^{(1)}$ and $\varphi _{\mathrm {S}}^{(1)}$. Therefore, we write $h_D=h_D(k)$ and $h_M=h_M(k)$. In other words, the magnitudes of the force and torque vary with $k$.

To summarize, the sphere is subject to a drag force but no transverse force acts on the sphere at the order $\varepsilon$. The drag and torque are modulated by the Knudsen number through the functions $h_D(k)$ and $h_M(k)$ given by (3.28) and (3.29), respectively.

3.3. Slowly varying solution

In the preceding section, we considered the leading-order problem under the condition that the length scale of variation of the solution is of the order of unity. The solution does not support a transverse force on the sphere. Thus, we are motivated to proceed to the next-order problem in $\varepsilon$.

We note that the linearized Boltzmann equation considered in the preceding subsection may not provide an approximate solution to the original (nonlinear) problem for small $\varepsilon$ uniformly in space. Indeed, using the asymptotic representation of $\phi _{F1}=\varPhi _{\mathrm {U}}^{(1)}+\varPhi _{\mathrm {S}}^{(1)}$ for $r = |\boldsymbol {x}| \gg 1$ (see Proposition 3.1 given below), it can be shown that the linearization is valid in the region $|\boldsymbol {x}| \ll \varepsilon ^{-1}$ for $\varPhi _{\mathrm {U}}$, although no such restriction is found for $\varPhi _{\mathrm {S}}$. In the region beyond this range, the nonlinear term $\mathscr {J}(\varepsilon \varPhi _{\mathrm {U}},\varepsilon \varPhi _{\mathrm {U}})$, integrated over a long distance, gives a non-negligible contribution to the behaviour of $\varepsilon \varPhi _{\mathrm {U}}$. (Note that the term $\zeta _i \partial (\varepsilon \varPhi _{\mathrm {U}})/\partial x_i$ is comparable with the nonlinear term $\mathscr {J}(\varepsilon \varPhi _{\mathrm {U}},\varepsilon \varPhi _{\mathrm {U}})$ when the length scale of variation of $\varPhi _{\mathrm {U}}$ is of the order of $\varepsilon ^{-1}$.) In other words, we encounter a situation analogous to Whitehead's paradox in the Navier–Stokes theory (Van Dyke Reference Van Dyke1975; Taguchi Reference Taguchi2015).

Given this observation, we introduce another length scale to describe the solution in the far region. More specifically, from now on, we assume a solution whose length scale of variation is of the order of $1/\varepsilon$ in the far region. We call this solution the slowly varying solution (or the outer solution) and designate it by attaching the subscript $H$, i.e. $\partial \phi _{H}/\partial x_i = O(\varepsilon \phi _{H})$. The situation is schematically shown in figure 2.

Figure 2.

Schematic of the solution structure. The gas region is divided into two, the inner region $1 < r \ll \varepsilon ^{-1}$ and the outer region $1 \ll r < \infty$, overlapping each other in the crossover region $1 \ll r \ll \varepsilon ^{-1}$.

To analyse the slowly varying solution, it is convenient to introduce a new spatial variable (called the outer or slow variable) by

(3.32)\begin{equation} y_i = \varepsilon x_i, \end{equation}

and assume that $\phi _{H} = \phi _{H}(y_i,\zeta _i)$. Then, the Boltzmann equation for $\phi _{H}$ is recast as

(3.33)\begin{equation} \zeta_i \frac{\partial \phi_{H}}{\partial y_i} = \frac{1}{k\varepsilon} \mathscr{L}(\phi_{H}) + \frac{1}{k\varepsilon} \mathscr{J}(\phi_{H},\phi_{H}). \end{equation}

We seek a solution to (3.33) in the form of a power series in $\varepsilon$, i.e.

(3.34)\begin{equation} \phi_{H} = \varepsilon \phi_{H1} + \varepsilon^{2} \phi_{H2} + \cdots. \end{equation}

Likewise, the macroscopic quantities $h_{H}$ ($h=\omega$, $u_i$, $\tau$, etc.) are expanded in $\varepsilon$ as

(3.35)\begin{equation} h_{H} = \varepsilon h_{H1} + \varepsilon^{2} h_{H2} + \cdots. \end{equation}

The relations between $h_{Hm}$ and $\phi _{Hm}$ are the same as those between $h_{Fm}$ and $\phi _{Fm}$ except that the subscript should be changed from $F$ to $H$ (see (3.3) and (3.4)).

The above expansion for $\phi _{H}$ is a Hilbert-type expansion starting from the order $\varepsilon$, which is equivalent to the S expansion (Sone Reference Sone1971, Reference Sone2002, Reference Sone2007). Since a detailed description of the expansion is given in Sone (Reference Sone2002, Reference Sone2007), we only give the results necessary for the subsequent analysis, omitting the derivation.

3.3.1. Fluid-dynamic-type equations

First, we summarize fluid-dynamic-type equations describing the behaviour of the gas in the far region. That is, the macroscopic quantities $h_{Hm}$, $m=1,2,\ldots$, are described by the following (incompressible) Navier–Stokes-type equations (hereafter, we call them the Navier–Stokes equations).

Order $\varepsilon$:

(3.36)\begin{gather} \frac{\partial P_{H1}}{\partial y_i} = 0, \end{gather}

(3.37a)$$\begin{gather} \frac{\partial u_{j H1}}{\partial y_j} = 0, \end{gather}$$

(3.37b)$$\begin{gather}u_{j H1} \frac{\partial u_{i H1}}{\partial y_j} ={-} \frac{1}{2} \frac{\partial P_{H2}}{\partial y_i} + \frac{\gamma_1 k}{2} \Delta u_{i H1}, \end{gather}$$

(3.37c)$$\begin{gather}u_{j H1} \frac{\partial \tau_{H1}}{\partial y_j} = \frac{\gamma_2 k}{2} \Delta \tau_{H1}, \end{gather}$$

(3.37d)$$\begin{gather}\omega_{H1} = P_{H1} - \tau_{H1}. \end{gather}$$

Order $\varepsilon ^{2}$:

(3.38a)\begin{align} &\hspace{6pc}\frac{\partial u_{j H2}}{\partial y_j} ={-} u_{j H1} \frac{\partial \omega_{H1}}{\partial y_j},\end{align}

(3.38b)\begin{align} &u_{j H1} \frac{\partial u_{i H2}}{\partial y_j} + (\omega_{H1} u_{j H1} + u_{j H2}) \frac{\partial u_{i H1}}{\partial y_j} \nonumber\\ &\quad ={-} \frac{1}{2} \frac{\partial}{\partial y_i} \left( P_{H3} - \frac{\gamma_1 \gamma_2 - 4 \gamma_3}{6} k^{2} \Delta \tau_{H1} \right) \nonumber\\ &\qquad + \frac{\gamma_1 k}{2} \Delta u_{i H2} + \frac{\gamma_4 k}{2} \frac{\partial}{\partial y_j} \left[\tau_{H1} \left(\frac{\partial u_{i H1}}{\partial y_j} + \frac{\partial u_{j H1}}{\partial y_i}\right) \right], \end{align}

(3.38c)\begin{align} &u_{j H1} \frac{\partial \tau_{H2}}{\partial y_j} + (\omega_{H1} u_{j H1} + u_{j H2}) \frac{\partial \tau_{H1}}{\partial y_j} - \frac{2}{5} u_{j H1} \frac{\partial P_{H2}}{\partial y_j} \nonumber\\ &\quad = \frac{\gamma_1 k}{5} \left(\frac{\partial u_{i H1}}{\partial y_j} + \frac{\partial u_{j H1}}{\partial y_i}\right)^{2} + \frac{k}{2} \Delta \left( \gamma_2 \tau_{H2} + \frac{\gamma_5}{2} (\tau_{H1})^{2} \right), \end{align}

(3.38d)\begin{align} \omega_{H2} &= P_{H2} - \tau_{H2} - \omega_{H1} \tau_{H1}. \end{align}

Here, $\Delta = \partial ^{2}/\partial y_j^{2}$ is the Laplacian operator and $\gamma _i$, $i=1,\ldots,5$, are constants defined by

(3.39a)$$\begin{gather} \gamma_1 = \tfrac{2}{15} \langle \zeta^{4} B \rangle, \quad \gamma_2 = \tfrac{4}{15} \langle \zeta^{4} A \rangle, \end{gather}$$

(3.39b)$$\begin{gather}\gamma_3 = \tfrac{2}{15} \langle \zeta^{4} AB \rangle = \tfrac{2}{3} \langle \zeta^{4} D_1 \rangle + \tfrac{2}{15} \langle \zeta^{6} D_2 \rangle ={-} \tfrac{4}{15} \langle \zeta^{4} F \rangle, \end{gather}$$

(3.39c)$$\begin{gather}\gamma_4 ={-}\tfrac{5}{2} \gamma_1 + \tfrac{2}{15} \langle \zeta^{6} B \rangle + \tfrac{1}{15} \langle \zeta^{4} B C \rangle, \end{gather}$$

(3.39d)$$\begin{gather}\gamma_5 ={-}6 \gamma_2 + \tfrac{4}{15} \langle \zeta^{6} A\rangle + \tfrac{4}{15} \langle \zeta^{2} A G\rangle, \end{gather}$$

where $A(\zeta )$, $B(\zeta )$, $C(\zeta )$, $D_1(\zeta )$, $D_2(\zeta )$, $F(\zeta )$ and $G(\zeta )$ are functions defined in Appendix C. Physically, $\gamma _i$ are dimensionless transport coefficients. For example, the viscosity $\mu _0$ and the thermal conductivity $\lambda _0$ at the reference equilibrium state at rest are expressed as

(3.40a,b)\begin{equation} \mu_0 = \frac{\sqrt{\rm \pi}}{2} \gamma_1 \frac{p_0 \ell_0}{(2RT_0)^{1/2}}, \quad \lambda_0 = \frac{5}{4} \sqrt{\rm \pi} \gamma_2 \frac{R p_0 \ell_0}{(2RT_0)^{1/2}}. \end{equation}

The numerical value of $\gamma _i$ depends on the molecular model. For the BGK model, $\gamma _i=1$. The values of $\gamma _i$ for the hard-sphere model are summarized in table 2.

Table 2.

The numerical values of $\gamma _i$ for a hard-sphere gas (Sone Reference Sone2002, Reference Sone2007). The values for the BGK model are also shown.

3.3.2. Velocity distribution functions and boundary conditions for the Navier–Stokes system at infinity

Suppose that the macroscopic variables $P_{Hm}$, $u_{iHm}$ and $\tau _{Hm}$ ($m \ge 1$) satisfy the fluid-dynamic-type equations  (3.36)–(3.38). Then, $\phi _{Hm}$, $m=1,2$, are expressed using $P_{Hn}$, $u_{iHn}$ and $\tau _{Hn}$ ($n=1, 2$) as

(3.41a)$$\begin{gather} \phi_{H1} = \phi_{e H1}, \end{gather}$$

(3.41b)$$\begin{gather}\phi_{H2} = \phi_{e H2} - \frac{1}{2}k \zeta_i \zeta_j B(\zeta) \left(\frac{\partial u_{i H1}}{\partial y_j} + \frac{\partial u_{j H1}}{\partial y_i} \right) - k \zeta_i A(\zeta) \frac{\partial \tau_{H1}}{\partial y_i}, \end{gather}$$

where

(3.42a)\begin{equation} \phi_{e H1} = P_{H1} + 2 \zeta_i u_{i H1} + \left(\zeta^{2}- \frac{5}{2}\right) \tau_{H1},\end{equation}

(3.42b)\begin{align} \phi_{e H2} &= P_{H2} + 2 \zeta_i u_{i H2} + \left(\zeta^{2}- \frac{5}{2}\right) \tau_{H2} + 2\zeta_i u_{i H1} P_{H1} + \left(\zeta^{2}-\frac{5}{2} \right) \tau_{H1} P_{H1} \nonumber\\ &\quad + \frac{2}{3} \left(\zeta^{2} - \frac{3}{2}\right) (u_{j H1})^{2} + 2 \left(\zeta_i \zeta_j - \frac{\zeta^{2}}{3} \delta_{ij} \right) u_{i H1} u_{jH1} + 2 \zeta_i \left(\zeta^{2}-\frac{7}{2} \right) u_{i H1} \tau_{H1} \nonumber\\ &\quad + \frac{1}{2} \left(\zeta^{4}-7\zeta^{2}+\frac{35}{4} \right) (\tau_{H1})^{2}, \end{align}

and the functions $A(\zeta )$ and $B(\zeta )$ are defined in Appendix C. It should be noted that $\phi _{e H1}$ and $\phi _{e H2}$ are the first two terms of the expansion $\phi _{eH} = \varepsilon \phi _{e H1} + \varepsilon ^{2} \phi _{e H2} + \cdots$ of the Maxwellian

(3.43)\begin{equation} (1 + \phi_{e H})E = \frac{1+P_{H}}{{\rm \pi}^{3/2} (1+\tau_{H})^{5/2}} \exp \left( - \frac{(\zeta_i-u_{i H})^{2}}{1+\tau_{H}}\right), \end{equation}

obtained by inserting the expansions of $P_H$, $u_{iH}$ and $\tau _H$ in $\varepsilon$ (see (3.35)). Therefore, for $\phi _H$ to satisfy the boundary condition at infinity, i.e. (2.20) with $\phi =\phi _H$, the macroscopic variables contained in $\phi _{H1}$ and $\phi _{H2}$ should satisfy the following conditions:

(3.44)$$\begin{gather} P_{H1} \to 0, \quad u_{iH1} \to e_i = (U,V,0), \quad \tau_{H1} \to 0, \quad \text{as} \ \eta \to \infty, \end{gather}$$

(3.45)$$\begin{gather}P_{H2} \to 0, \quad u_{iH2} \to 0, \quad \tau_{H2} \to 0, \quad \text{as} \ \eta \to \infty, \end{gather}$$

where

(3.46)\begin{equation} \eta = \varepsilon r = |\boldsymbol{y}| \quad \text{(outer variable)}. \end{equation}

These conditions serve as a part of the boundary conditions for the Navier–Stokes equations. The remaining conditions are derived by matching the outer solution with the inner solution, as shown next.

3.4. Outer problem

In the preceding subsection, we introduced a slowly varying solution characterized by the longer length scale of variation. This solution is meaningful only if it can be matched with the inner solution while meeting the boundary condition at infinity. This subsection shows that it is indeed the case and we determine the first two terms of the $\varepsilon$ expansion of $\phi _H$.

3.4.1. Preliminary

We begin with the following results.

Proposition 3.1 Let $(\varphi _{\mathrm {U} a}^{(1)},\varphi _{\mathrm {U} b}^{(1)})$ be a solution to the boundary-value problem ( 3.17) for $k < \infty$ and let $\varphi _{\mathrm {S}}^{(1)}$ be a solution to the boundary-value problem ( 3.18) for $k < \infty$.Then, they have the following asymptotic representations as $r \to \infty$:

(3.47)\begin{align} \varphi_{\mathrm{U} a}^{(1)} & = 2 \zeta_r \left(1 + \frac{c_1}{r} + \frac{c_2}{r^{3}}\right) + \left(\zeta^{2} - \frac{5}{2}\right) \frac{c_3}{r^{2}} \nonumber\\ &\quad + \frac{k}{r^{2}} \left[\gamma_1 c_1 + \frac{2 c_3}{r} \zeta_r A(\zeta) - \frac{1}{2} \left(c_1 + \frac{3 c_2}{r^{2}}\right)(\zeta^{2}-3\zeta_r^{2})B(\zeta) \right]\nonumber\\ &\quad - \frac{k^{2}}{r^{3}} \left\{ 2 c_1 \zeta_r D_1(\zeta) + 2 \left[c_1(2\zeta^{2}-3\zeta_r^{2}) + \frac{3 c_2}{r^{2}} (3\zeta^{2}-5\zeta_r^{2})\right] \zeta_r D_2(\zeta) \right. \nonumber\\ &\quad \left. -\, \frac{3 c_3}{r} (\zeta^{2}-3\zeta_r^{2})F(\zeta)\right\}, \end{align}

(3.48)\begin{align} \varphi_{\mathrm{U} b}^{(1)} & ={-}2 -\frac{c_1}{r} + \frac{c_2}{r^{3}} + \frac{k}{r^{3}} \left(c_3 A(\zeta) + \frac{3 c_2}{r} \zeta_r B(\zeta) \right) \nonumber\\ &\quad - \frac{k^{2}}{r^{3}} \left\{ c_1 D_1(\zeta) + \left[ \frac{c_1}{2} (\zeta^{2} - 3 \zeta_r^{2}) + \frac{9 c_2}{2 r^{2}} (\zeta^{2} - 5 \zeta_r^{2}) \right] D_2(\zeta)\right. \nonumber\\ &\quad \left.+ \frac{6 c_3}{r} \zeta_r F(\zeta)\right\}, \end{align}

(3.49)\begin{align} \varphi_{\mathrm{S}}^{(1)} &= \frac{c_4}{r^{2}} \left[ 2 + \frac{3 k}{r} \zeta_r B(\zeta) - \frac{3 k^{2}}{r^{2}} (\zeta^{2} - 5 \zeta_r^{2}) D_2(\zeta) \right], \end{align}

where $c_i$, $i=1,2,3,4$, are constants independent of $r$, $\zeta _r$ and $\zeta$, and $A(\zeta )$, $B(\zeta )$, $D_1(\zeta )$, $D_2(\zeta )$ and $F(\zeta )$ are the solutions to the integral equations ( C1) with the subsidiary conditions ( C2), given in Appendix C. The constant $\gamma _1$ is defined in (3.39a).

It should be noted that $c_i$ depends on $k$. The next proposition relates $c_1$ and $c_4$ with $h_D$ and $h_M$.

Proposition 3.2 Constants $c_1$ and $c_4$ are related to $h_D$ and $h_M$, defined in (3.28) and (3.29), by

(3.50a,b)\begin{equation} c_1 ={-} \frac{h_D}{4{\rm \pi} \gamma_1 k}, \quad c_4 ={-}\frac{h_M}{8{\rm \pi}\gamma_1 k} \quad (0 < k < \infty). \end{equation}

The proofs of Propositions 3.1 and 3.2 are given in Appendix D. We also mention that the constant $c_3$ is related to the thermophoretic force acting on a single sphere (Taguchi & Suzuki Reference Taguchi and Suzuki2017).

3.4.2. Leading- and second-order outer solutions

Let us consider an arbitrary point $\boldsymbol {x}$ in the domain such that $r=|\boldsymbol {x}| \gg 1$ and $r=|\boldsymbol {x}| \ll 1/\varepsilon$. From Proposition 3.1, the leading-order inner solution at this point is approximated by

(3.51a)$$\begin{gather} \varphi_{\mathrm{U} a}^{(1)}(r,\zeta_r,\zeta) = 2 \zeta_r \left(1 + \frac{c_1}{r} \right) + O(r^{{-}2}), \enspace \varphi_{\mathrm{U} b}^{(1)}(r,\zeta_r,\zeta) ={-}2 -\frac{c_1}{r} + O(r^{{-}3}), \end{gather}$$

(3.51b)$$\begin{gather}\varphi_{\mathrm{S}}^{(1)}(r,\zeta_r,\zeta) = O(r^{{-}2}), \end{gather}$$

or equivalently by

(3.52)\begin{align} \phi_{F1} & = 2 \zeta_1 U + 2 \zeta_2 V \nonumber\\ &\quad + 2\zeta_r (U \cos \theta + V \sin \theta \cos \varphi) \frac{c_1}{r} - \zeta_\theta (U \sin \theta - V \cos \theta \cos \varphi) \frac{c_1}{r} \nonumber\\ &\quad- \zeta_\varphi V \sin \varphi \,\frac{c_1}{r} + O(r^{{-}2}), \end{align}

using (3.10) and (3.16). Thus, the inner solution has a far-field asymptotic representation of the form

(3.53)\begin{equation} \phi_F = \varepsilon \left(2 \zeta_1 U + 2 \zeta_2 V + \frac{1}{r}(\cdots) + O(r^{{-}2}) \right) + \cdots, \quad r=|\boldsymbol{x}| \gg 1, \end{equation}

where the part ‘$(\cdots )$’ in the parentheses is independent of $r$ and $\varepsilon$. Now we consider the limit $\varepsilon \searrow 0$ (and $r \nearrow \infty$) with $\varepsilon r \ (=\eta )$ fixed. Substituting $r=\eta /\varepsilon$ into the above expression and arranging the resulting terms in increasing order of $\varepsilon$, we obtain an $\varepsilon$ expansion of $\phi _F$ of the form

(3.54)\begin{align} \phi_F^{*} & = \varepsilon (2\zeta_1 U + 2\zeta_2 V) \nonumber\\ &\quad + \varepsilon^{2}[2\zeta_r (U \cos \theta + V \sin \theta \cos \varphi) - \zeta_\theta (U \sin \theta - V \cos \theta \cos \varphi) - \zeta_\varphi V \sin \varphi] \frac{c_1}{\eta} \nonumber\\ &\quad + \cdots, \quad \text{as } \varepsilon \searrow 0 \text{ with } \varepsilon r\,(=\eta) \text{ fixed}. \end{align}

Here, we have attached $^{*}$ to indicate that $\phi _F$ is expressed using the outer variable $\eta$ (or $y_i$). Thus, a term-by-term comparison between (3.41) (with (3.42)) and (3.54) shows that $\phi _H$ can be made to match $\phi _F^{*}$ by imposing the following conditions:

(3.55)\begin{gather} P_{H1} \to 0, \quad u_{iH1} \to (U,V,0), \quad \tau_{H1} \to 0, \end{gather}

(3.56a)$$\begin{gather} P_{H2} \to 0, \end{gather}$$

(3.56b)$$\begin{gather}u_{rH2} \to (U \cos \theta + V \sin \theta \cos \varphi) \frac{c_1}{\eta}, \end{gather}$$

(3.56c)$$\begin{gather}u_{\theta H2} \to - (U \sin \theta - V \cos \theta \cos \varphi) \frac{c_1}{2\eta}, \end{gather}$$

(3.56d)$$\begin{gather}u_{\varphi H2} \to - V \sin \varphi \frac{c_1}{2\eta}, \end{gather}$$

(3.56e)$$\begin{gather}\tau_{H2} \to 0, \end{gather}$$

as $\eta \to 0_+$. Together with the conditions (3.44) and (3.45), these conditions serve as appropriate boundary conditions for the Navier–Stokes equations (3.36)–(3.38).

The leading-order problem, (3.36), (3.37), (3.44) and (3.55), is trivially solved by

(3.57a–c)\begin{equation} u_{iH1} = e_i, \quad P_{H1} = \tau_{H1} =\omega_{H1} = 0, \quad P_{H2} = 0. \end{equation}

(Note that $P_{H2}$ is determined up to an additive constant but this constant can be made to vanish by using the condition (3.56a).) Then, the next-order set of equations (3.38) reduce to the following system of linear partial differential equations:

(3.58a)$$\begin{gather} \frac{\partial u_{j H2}}{\partial y_j} = 0, \end{gather}$$

(3.58b)$$\begin{gather}e_j \frac{\partial u_{i H2}}{\partial y_j} ={-} \frac{1}{2} \frac{\partial P_{H3}}{\partial y_i} + \frac{\gamma_1 k}{2} \Delta u_{i H2}, \end{gather}$$

(3.58c)$$\begin{gather}e_j \frac{\partial \tau_{H2}}{\partial y_j} = \frac{\gamma_2 k}{2} \Delta \tau_{H2}, \end{gather}$$

(3.58d)$$\begin{gather}\omega_{H2} ={-} \tau_{H2}, \end{gather}$$

which, equipped with the boundary conditions (3.45) and (3.56b)–(3.56e), describes the behaviour of $u_{iH2}$, $P_{H3}$, $\tau _{H2}$ and $\omega _{H2}$ in the far region. We first note that $\tau _{H2} = 0$ is the trivial solution of (3.58c) satisfying (3.56e) and the last condition of (3.45). Next, (3.58b) with (3.58a) is the Oseen equation for incompressible flow, and the solution subject to the boundary conditions (3.56b)–(3.56d) and (3.45) is easily obtained (e.g. Rubinow & Keller Reference Rubinow and Keller1961).

Here, we summarize the second-order macroscopic quantities in the outer region. They are given by

(3.59a)\begin{gather} u_{rH2} = \frac{c_1 \gamma_1 k}{2} \frac{1}{\eta^{2}}\left[{-}1 + \left(1 + \frac{\eta}{\gamma_1 k}(1+U \cos \theta + V \sin \theta \cos \varphi) \right)\right. \nonumber\\ \quad \left.\times \exp\left({-\frac{\eta}{\gamma_1 k}(1-U\cos \theta - V \sin \theta \cos \varphi)}\right) \right], \end{gather}

(3.59b)$$\begin{gather} u_{\theta H2} ={-} \frac{c_1}{2 \eta} (U \sin \theta - V \cos \theta \cos \varphi) \exp\left(- \frac{\eta}{\gamma_1 k}(1-U\cos \theta - V \sin \theta \cos \varphi)\right), \end{gather}$$

(3.59c)$$\begin{gather}u_{\varphi H2} ={-} \frac{c_1}{2 \eta} V \sin \varphi \exp\left({- \frac{\eta}{\gamma_1 k}(1-U\cos \theta - V \sin \theta \cos \varphi)}\right), \end{gather}$$

(3.59d)$$\begin{gather}\omega_{H2} = \tau_{H2} = P_{H2} = 0, \end{gather}$$

(3.59e)$$\begin{gather}P_{H3} = \frac{c_1 \gamma_1 k}{\eta^{2}} (U \cos \theta + V \sin \theta \cos \varphi). \end{gather}$$

Note that $(u_{rH2},u_{\theta H2},u_{\varphi H2})$ are proportional to the $k$-dependent constant $c_1$.

Finally, we can readily obtain the explicit forms of $\phi _{H1}$ and $\phi _{H2}$ from (3.41a) and (3.41b) by substituting (3.57a–c) and (3.59). In particular, the outer solution is a Maxwellian to order $\varepsilon ^{2}$, i.e.

(3.60)\begin{equation} (1 + \phi_H)E = \frac{1}{{\rm \pi}^{3/2}} \exp(-(\zeta_i - \varepsilon u_{iH1} - \varepsilon^{2} u_{iH2})^{2} ) (1 + o(\varepsilon^{2})). \end{equation}

3.5. Second-order inner problem

Now we return to the near region and consider the second-order approximation, i.e. $\phi _{F2}$. As before, we consider an arbitrary point $\boldsymbol {x}$ such that $|\boldsymbol {x}| \gg 1$ and $|\boldsymbol {x}| \ll 1/\varepsilon$. Because $(u_{rH2},u_{\theta H2},u_{\varphi H2})$ are expanded for small $\eta \ll 1$ as

(3.61a)\begin{align} u_{r H2}(\eta,\theta,\varphi) & = \frac{c_1}{\eta} (U \cos \theta + V \sin \theta \cos \varphi) \nonumber\\ &\quad + \frac{c_1}{4 \gamma_1 k} (U \cos \theta + V \sin \theta \cos \varphi - 1)[3 (U \cos \theta + V \sin \theta \cos \varphi) + 1 ] \nonumber\\ &\quad + O(\eta),\\ u_{\theta H2}(\eta,\theta,\varphi) & ={-}\frac{c_1}{2 \eta} (U \sin \theta - V \cos \theta \cos \varphi) \nonumber\\ &\quad - \frac{c_1}{2 \gamma_1 k}(U \sin \theta - V \cos \theta \cos \varphi) \,(U \cos \theta + V \sin \theta \cos \varphi - 1) \nonumber\\ &\quad + O(\eta),\end{align}

(3.61b)\begin{align} u_{\varphi H2}(\eta,\theta,\varphi) & ={-} \frac{c_1}{2 \eta} V \sin \varphi - \frac{c_1}{2 \gamma_1 k}V \sin \varphi(U \cos \theta + V \sin \theta \cos \varphi - 1) \nonumber\\ &\quad + O(\eta), \end{align}

the outer solution $\phi _H = \varepsilon \phi _{H1} + \varepsilon ^{2} \phi _{H2} + \cdots$ has the following expansion in terms of the inner variable:

(3.62)\begin{align} \phi_H^{*} & = \varepsilon \left[ 2 (\zeta_1 U + \zeta_2 V) + 2 \zeta_r (U \cos \theta + V \sin \theta \cos \varphi) \frac{c_1}{r} \right. \nonumber\\ & \quad \left. -\, \zeta_\theta (U \sin \theta - V \cos \theta \cos \varphi) \frac{c_1}{r} - \zeta_\varphi V \sin \varphi \frac{c_1}{r} \right] \nonumber\\ &\quad + \varepsilon^{2} \left[ \zeta_r (U \cos \theta + V \sin \theta \cos \varphi - 1) (3(U \cos \theta + V \sin \theta \cos \varphi) + 1) \frac{c_1}{2 \gamma_1 k} \right. \nonumber\\ &\quad - \zeta_\theta (U \sin \theta - V \cos \theta \cos \varphi) (U \cos \theta + V \sin \theta \cos \varphi - 1) \frac{c_1}{\gamma_1 k} \nonumber\\ & \quad \left. -\, \zeta_\varphi V \sin \varphi (U \cos \theta + V \sin \theta \cos \varphi - 1) \frac{c_1}{\gamma_1 k} + 2 (e_j \zeta_j)^{2} - 1 \right] \nonumber\\ &\quad + \cdots, \quad \text{as } \varepsilon \searrow 0 \text{ with } \eta/\varepsilon\,({=}r) \text{ fixed}. \end{align}

Here, $^{*}$ has been attached to indicate that $\phi _H$ is expressed using the inner variable $r$ (or $x_i$). We observe that the $\varepsilon$-order term coincides with the far-field expansion of $\varepsilon \phi _{F1}$ for $r \gg 1$ (see (3.52)). Hence, for the inner solution to match the outer solution to order $\varepsilon ^{2}$, we should impose the following matching condition:

(3.63)\begin{align} \phi_{F2} & \to \zeta_r (U \cos \theta + V \sin \theta \cos \varphi - 1) [3(U \cos \theta + V \sin \theta \cos \varphi) + 1] \frac{c_1}{2 \gamma_1 k} \nonumber\\ &\quad - \zeta_\theta (U \sin \theta - V \cos \theta \cos \varphi) \,(U \cos \theta + V \sin \theta \cos \varphi - 1) \frac{c_1}{\gamma_1 k} \nonumber\\ &\quad - \zeta_\varphi V \sin \varphi \,(U \cos \theta + V \sin \theta \cos \varphi - 1) \frac{c_1}{\gamma_1 k} + 2 (e_j \zeta_j)^{2} - 1, \quad \text{as} \ |\boldsymbol{x}| \to \infty. \end{align}

In summary, the equation and boundary conditions for $\phi _{F2}$ are given by

(3.64)$$\begin{gather} \zeta_j \frac{\partial \phi_{F2}}{\partial x_j} = \frac{1}{k} \mathscr{L}(\phi_{F2}) + \frac{1}{k} \mathscr{J}(\phi_{F1},\phi_{F1}), \end{gather}$$

(3.65)$$\begin{gather}\phi_{F2} = \mathscr{K}(\phi_{F2}) + (2 \zeta_\varphi^{2} - 1) S^{2} \sin^{2} \theta + 2 \mathscr{K}(\phi_{F1}) \zeta_\varphi S \sin \theta, \quad \zeta_r > 0, \ |\boldsymbol{x}| = 1, \end{gather}$$

(3.66)\begin{align} \phi_{F2} &\to \zeta_r (U \cos \theta + V \sin \theta \cos \varphi - 1) (3(U \cos \theta + V \sin \theta \cos \varphi) + 1) \frac{c_1}{2 \gamma_1 k} \nonumber\\ &\quad - \zeta_\theta (U \sin \theta - V \cos \theta \cos \varphi) \,(U \cos \theta + V \sin \theta \cos \varphi - 1) \frac{c_1}{\gamma_1 k} \nonumber\\ & \quad - \zeta_\varphi V \sin \varphi \,(U \cos \theta + V \sin \theta \cos \varphi - 1) \frac{c_1}{\gamma_1 k} \nonumber\\ & \quad + 2 (e_j \zeta_j)^{2} - 1, \quad \text{as} \ |\boldsymbol{x}| \to \infty. \end{align}

Note that $\phi _{F2}$ depends on the leading-order solution $\phi _{F1}$ through $\mathscr {J}(\phi _{F1},\phi _{F1})$, $\mathscr {K}(\phi _{F1})$ and $c_1(=-h_D/4 {\rm \pi}\gamma _1 k)$. In the remaining part, assuming the existence of a solution for the above problem, we derive expressions for the force and torque acting on the sphere.

3.6. Similarity solutions for the second-order problem and the second-order force and torque acting on the sphere

We begin by noting the following.

1. (i) The right-hand side of (3.66) is arranged in the form

   (3.67)\begin{equation} \text{(right-hand side)} ={-} \frac{c_1}{2 \gamma_1 k} (2 e_j \zeta_j) + (\text{other terms}). \end{equation}
   Note that the first term is a constant multiple of $\varPhi _{\mathrm {U}}^{(1)}$ at infinity (see (3.13) with $J=\mathrm {U}$).

2. (ii) Since $\phi _{F1} = \varPhi _{\mathrm {U}}^{(1)} + \varPhi _{\mathrm {S}}^{(1)}$, the inhomogeneous term in (3.64) is decomposed as

   (3.68)\begin{equation} \mathscr{J}(\phi_{F1},\phi_{F1}) = \mathscr{J}(\varPhi_{\mathrm{U}}^{(1)},\varPhi_{\mathrm{U}}^{(1)}) + \mathscr{J}(\varPhi_{\mathrm{S}}^{(1)},\varPhi_{\mathrm{S}}^{(1)}) + 2 \mathscr{J}(\varPhi_{\mathrm{U}}^{(1)},\varPhi_{\mathrm{S}}^{(1)}). \end{equation}

Then, noting the linearity of the problem, we seek $\phi _{F2}$ in the form

(3.69)\begin{equation} \phi_{F2} ={-}\frac{c_1}{2\gamma_1 k} \varPhi_{\mathrm{U}}^{(1)} + \varPhi_{\mathrm{UU}}^{(2)} + \varPhi_{\mathrm{SS}}^{(2)} + \varPhi_{\mathrm{US}}^{(2)}. \end{equation}

Here, $\varPhi _{\mathrm {UU}}^{(2)}$, $\varPhi _{\mathrm {SS}}^{(2)}$ and $\varPhi _{\mathrm {US}}^{(2)}$ solve the following problems:

(3.70)$$\begin{gather} \zeta_i \frac{\partial \varPhi_J^{(2)}}{\partial x_i} = \frac{1}{k} \mathscr{L}(\varPhi_J^{(2)}) + \frac{1}{k} I_J^{(2)} \quad (J = \mathrm{UU}, \,\mathrm{SS}, \,\mathrm{US}), \end{gather}$$

(3.71)$$\begin{gather}\varPhi_J^{(2)} = \mathscr{K}(\varPhi_J^{(2)}) + I_{w,J}^{(2)}, \quad \zeta_r > 0, \ r = |\boldsymbol{x}|=1, \end{gather}$$

(3.72)$$\begin{gather}\varPhi_J^{(2)} \to I_{\infty,J}^{(2)} \quad \text{as} \ r = |\boldsymbol{x}| \to \infty, \end{gather}$$

where

(3.73a)\begin{gather} I_{\mathrm{UU}}^{(2)} = \mathscr{J}(\varPhi_{\mathrm{U}}^{(1)},\varPhi_{\mathrm{U}}^{(1)}), \quad I_{\mathrm{SS}}^{(2)} = \mathscr{J}(\varPhi_{\mathrm{S}}^{(1)},\varPhi_{\mathrm{S}}^{(1)}), \quad I_{\mathrm{US}}^{(2)} = 2\mathscr{J}(\varPhi_{\mathrm{U}}^{(1)},\varPhi_{\mathrm{S}}^{(1)}), \end{gather}

(3.73b)\begin{gather} I_{{w},\mathrm{UU}}^{(2)} = 0, \quad I_{{w},\mathrm{SS}}^{(2)} = (2 \zeta_\varphi^{2} - 1) S^{2} \sin^{2} \theta, \quad I_{{w},\mathrm{US}}^{(2)} = 2 \mathscr{K}(\varPhi_{\mathrm{U}}^{(1)}|_{r=1}) \zeta_\varphi S \sin \theta, \end{gather}

(3.73c)\begin{align} I_{\infty,\mathrm{UU}}^{(2)} &= (U \cos \theta + V \sin \theta \cos \varphi)^{2} \left(\frac{3}{2} \frac{c_1}{\gamma_1 k} \zeta_r + 3 \,\zeta_r^{2} - \zeta^{2} \right) \nonumber\\ &\quad + (U \cos \theta + V \sin \theta \cos \varphi) [(U \sin \theta - V \cos \theta \cos \varphi) \zeta_\theta + V \sin \varphi\ \zeta_\varphi ]\nonumber\\ &\quad \times \left(- \frac{c_1}{\gamma_1 k} - 4 \zeta_r \right) + 2 \left\{[(U \sin \theta - V \cos \theta \cos \varphi)^{2} - V^{2} \sin^{2} \varphi ] \frac{\zeta_\theta^{2} - \zeta_\varphi^{2}}{2}\right. \nonumber\\ &\quad \left.\vphantom{\frac{\zeta_\theta^{2} - \zeta_\varphi^{2}}{2}} + 2 V \sin \varphi (U \sin \theta - V \cos \theta \cos \varphi) \zeta_\theta \zeta_\varphi \right\} - \frac{c_1}{2\gamma_1 k} \zeta_r + \zeta^{2} - \zeta_r^{2} - 1, \end{align}

(3.73d)\begin{gather} I_{\infty,\mathrm{SS}}^{(2)} = I_{\infty,\mathrm{US}}^{(2)} = 0. \end{gather}

Note that $\mathscr {K}(\varPhi _{\mathrm {S}}^{(1)}|_{r=1}) = 0$ has been used in the second condition of (3.73b).

Physically, the problem for $\varPhi _{\mathrm {UU}}^{(2)}$ (or for $\varPhi _{\mathrm {SS}}^{(2)}$) describes the second-order Mach number effect in the uniform flow over a sphere (or in the swirling flow around a rotating sphere). On the other hand, the problem for $\varphi _{\mathrm {US}}^{(2)}$ describes the cross effect between the flow past a sphere and the flow induced by a sphere rotation. We call them problems UU, SS and US, respectively. As we see below, only problem US is essential for the transverse force acting on the sphere.

Using the similarity solutions for $\varPhi _{\mathrm {U}}^{(1)}$ and $\varPhi _{\mathrm {S}}^{(1)}$ shown in (3.16) and the formulas in Appendix B, we can transform the inhomogeneous term $I_{J}^{(2)}$ ($J=\mathrm {UU},\mathrm {SS},\mathrm {US}$) into

(3.74)\begin{align} I_{\mathrm{UU}}^{(2)} & = (U \cos \theta + V \sin \theta \cos \varphi)^{2} \nonumber\\ &\quad \times \left[\mathscr{J}_0(\varphi_{\mathrm{U} a}^{(1)},\varphi_{\mathrm{U} a}^{(1)}) - \frac{\zeta^{2} - \zeta_r^{2}}{2} \mathscr{J}_2(\varphi_{\mathrm{U} b}^{(1)},\varphi_{\mathrm{U} b}^{(1)}) - \mathscr{J}_3(\varphi_{\mathrm{U} b}^{(1)},\varphi_{\mathrm{U} b}^{(1)}) \right] \nonumber\\ &\quad + 2 (U \cos \theta + V \sin \theta \cos \varphi) [(U \sin \theta - V \cos \theta \cos \varphi) \zeta_\theta + V \sin \varphi\ \zeta_\varphi] \mathscr{J}_1(\varphi_{\mathrm{U} a}^{(1)},\varphi_{\mathrm{U} b}^{(1)}) \nonumber\\ &\quad + \left\{[ (U \sin \theta - V \cos \theta \cos \varphi)^{2} - V^{2} \sin^{2} \varphi] \frac{\zeta_\theta^{2} - \zeta_\varphi^{2}}{2}\right. \nonumber\\ &\quad + 2V \left.\sin \varphi (U \sin \theta - V \cos \theta \cos \varphi) \zeta_\theta \zeta_\varphi \vphantom{\frac{\zeta_\theta^{2} - \zeta_\varphi^{2}}{2}}\right\} \mathscr{J}_2(\varphi_{\mathrm{U} b}^{(1)},\varphi_{\mathrm{U} b}^{(1)}) \nonumber\\ &\quad + \frac{\zeta^{2} - \zeta_r^{2}}{2} \mathscr{J}_2(\varphi_{\mathrm{U} b}^{(1)},\varphi_{\mathrm{U} b}^{(1)}) + \mathscr{J}_3(\varphi_{\mathrm{U} b}^{(1)},\varphi_{\mathrm{U} b}^{(1)}), \end{align}

(3.75)\begin{align} I_{\mathrm{SS}}^{(2)} & = S^{2} \sin^{2} \theta \left[\frac{\zeta^{2}-\zeta_r^{2}}{2} \mathscr{J}_2(\varphi_{\mathrm{S}}^{(1)},\varphi_{\mathrm{S}}^{(1)}) + \mathscr{J}_3(\varphi_{\mathrm{S}}^{(1)},\varphi_{\mathrm{S}}^{(1)}) \right] \nonumber\\ &\quad - S^{2} \sin^{2} \theta \frac{\zeta_\theta^{2} - \zeta_\varphi^{2}}{2} \mathscr{J}_2(\varphi_{\mathrm{S}}^{(1)},\varphi_{\mathrm{S}}^{(1)}), \end{align}

(3.76)\begin{equation} I_{\mathrm{US}}^{(2)} = I_{\mathrm{US}}^{(2)\sharp} + I_{\mathrm{US}}^{(2)\flat}, \end{equation}

(3.77)\begin{align} I_{\mathrm{US}}^{(2)\sharp} & = 2 S \sin \theta (U \cos \theta + V \sin \theta \cos \varphi) \zeta_\varphi \mathscr{J}_1(\varphi_{\mathrm{U} a}^{(1)},\varphi_{\mathrm{S}}^{(1)}) \nonumber\\ &\quad + 2 S \sin \theta \left[ - V \sin \varphi \frac{\zeta_\theta^{2} - \zeta_\varphi^{2}}{2} + (U \sin \theta - V \cos \theta \cos \varphi) \zeta_\theta \zeta_\varphi \right] \mathscr{J}_2(\varphi_{\mathrm{U} b}^{(1)},\varphi_{\mathrm{S}}^{(1)}) \nonumber\\ &\quad + 2 S V \sin \theta \sin \varphi \left[\frac{\zeta^{2} - \zeta_r^{2}}{2} \mathscr{J}_2(\varphi_{\mathrm{U} b}^{(1)},\varphi_{\mathrm{S}}^{(1)}) + \mathscr{J}_3(\varphi_{\mathrm{U} b}^{(1)},\varphi_{\mathrm{S}}^{(1)}) \right], \end{align}

(3.78)\begin{equation} I_{\mathrm{US}}^{(2)\flat} = 2 S \sin \theta \,(U \sin \theta - V \cos \theta \cos \varphi)\, \zeta_r \mathscr{J}_4(\varphi_{\mathrm{U} b}^{(1)},\varphi_{\mathrm{S}}^{(1)}). \end{equation}

Here, the operators $\mathscr {J}_i$, $i=0,1,2,3,4$, are defined in Appendix B. Note also that $\mathscr {J}_i(\cdot,\cdot )$ appearing above are functions of $r$, $\zeta _r$ and $\zeta$ through $\varphi _{\mathrm {U} a}^{(1)}$, $\varphi _{\mathrm {U} b}^{(1)}$ and $\varphi _{\mathrm {S}}^{(1)}$. Given these forms, we consider the following similarity solutions for $\varPhi _{\mathrm {UU}}^{(2)}$, $\varPhi _{\mathrm {US}}^{(2)}$ and $\varPhi _{\mathrm {SS}}^{(2)}$ (see table 1):

(3.79)\begin{align} \varPhi_{\mathrm{UU}}^{(2)} & = (U \cos \theta + V \sin \theta \cos \varphi)^{2} \,\varphi_{\mathrm{UU} a}^{(2)}(r,\zeta_r,\zeta) \nonumber\\ &\quad + (U \cos \theta + V \sin \theta \cos \varphi) [(U \sin \theta - V \cos \theta \cos \varphi ) \zeta_\theta + V \sin \varphi\ \zeta_\varphi] \varphi_{\mathrm{UU} b}^{(2)}(r,\zeta_r,\zeta) \nonumber\\ &\quad + \left[((U \sin \theta - V \cos \theta \cos \varphi)^{2} - V^{2} \sin^{2} \varphi ) \frac{\zeta_\theta^{2} - \zeta_\varphi^{2}}{2}\right. \nonumber\\ &\quad \left.\vphantom{\frac{\zeta_\theta^{2} - \zeta_\varphi^{2}}{2}}+ 2 V \sin \varphi (U \sin \theta - V \cos \theta \cos \varphi) \zeta_\theta \zeta_\varphi \right] \varphi_{\mathrm{UU} c}^{(2)}(r,\zeta_r,\zeta) + \varphi_{\mathrm{UU} d}^{(2)}(r,\zeta_r,\zeta), \end{align}

(3.80)\begin{align} \varPhi_{\mathrm{SS}}^{(2)} & = S^{2} \cos^{2} \theta \varphi_{\mathrm{SS} a}^{(2)}(r,\zeta_r,\zeta) + S^{2} \cos \theta \sin \theta\, \zeta_\theta \varphi_{\mathrm{SS} b}^{(2)}(r,\zeta_r,\zeta) \nonumber\\ &\quad + S^{2} \frac{\zeta_\theta^{2} - \zeta_\varphi^{2}}{2} \sin^{2} \theta \varphi_{\mathrm{SS} c}^{(2)}(r,\zeta_r,\zeta) + S^{2} \varphi_{\mathrm{SS} d}^{(2)}(r,\zeta_r,\zeta), \end{align}

(3.81)\begin{equation} \varPhi_{\mathrm{US}}^{(2)} = \varPhi_{\mathrm{US}}^{(2)\sharp} + \varPhi_{\mathrm{US}}^{(2)\flat}, \end{equation}

(3.82)\begin{align} \varPhi_{\mathrm{US}}^{(2)\sharp} & = S \sin \theta (U \cos \theta + V \sin \theta \cos \varphi) \zeta_\varphi \varphi_{\mathrm{US} a}^{(2)\sharp}(r,\zeta_r,\zeta) \nonumber\\ &\quad + S \cos \theta [ (U \sin \theta - V \cos \theta \cos \varphi) \zeta_\varphi - V \sin \varphi\, \zeta_\theta ] \varphi_{\mathrm{US} b}^{(2)\sharp}(r,\zeta_r,\zeta) \nonumber\\ &\quad + S \sin \theta \left[{-}V \sin \varphi \frac{\zeta_\theta^{2} - \zeta_\varphi^{2}}{2} + (U \sin \theta - V \cos \theta \cos \varphi) \zeta_\theta \zeta_\varphi \right] \varphi_{\mathrm{US} c}^{(2)\sharp}(r,\zeta_r,\zeta) \nonumber\\ &\quad + SV \sin \theta \sin \varphi \,\varphi_{\mathrm{US} d}^{(2)\sharp}(r,\zeta_r,\zeta), \end{align}

(3.83)\begin{align} \varPhi_{\mathrm{US}}^{(2)\flat} & = S \sin \theta (U \sin \theta - V \cos \theta \cos \varphi) \varphi_{\mathrm{US} a}^{(2)\flat}(r,\zeta_r,\zeta) \nonumber\\ &\quad + S [ (U \sin 2 \theta - V \cos 2\theta \cos \varphi) \zeta_\theta + V \cos \theta \sin \varphi\ \zeta_\varphi] \varphi_{\mathrm{US} b}^{(2)\flat}(r,\zeta_r,\zeta) \nonumber\\ &\quad + S \sin \theta \left[ (U \sin \theta - V \cos \theta \cos \varphi) \frac{\zeta_\theta^{2} - \zeta_\varphi^{2}}{2} + V \sin \varphi\, \zeta_\theta \zeta_\varphi \right] \varphi_{\mathrm{US} c}^{(2)\flat}(r,\zeta_r,\zeta) \nonumber\\ &\quad + SU \varphi_{\mathrm{US} d}^{(2)\flat}(r,\zeta_r,\zeta), \end{align}

where $(\varphi _{\mathrm {UU} \alpha }^{(2)},\varphi _{\mathrm {SS} \alpha }^{(2)},\varphi _{\mathrm {US} \alpha }^{(2)\sharp },\varphi _{\mathrm {US} \alpha }^{(2)\flat })$, $\alpha =a,b,c,d$, which are functions of $r$, $\zeta _r$ and $\zeta$, are the solutions to the boundary-value problems shown in Appendix E. Note that they also depend on the parameter $k$.

Because of the similarity solutions, we can obtain the explicit dependency of the second-order macroscopic quantities on $\theta$ and $\varphi$. Below, we only give the results for $P_{rr F2}$, $P_{r\theta F2}$ and $P_{r\varphi F2}$ for conciseness. That is, substituting (3.69) with (3.79)–(3.83) into the definition of $P_{ijF2}$ (see (3.4c)), we obtain

(3.84a)\begin{align} P_{rr F2}(r,\theta,\varphi) & ={-} \frac{c_1}{2 \gamma_1 k} (U \cos \theta + V \sin \theta \cos \varphi) \underline{\tilde{P}_{rr,\mathrm{U} a}^{(1)}} \nonumber\\ &\quad + (U \cos \theta + V \sin \theta \cos \varphi)^{2} \underline{(\tilde{P}_{rr}[\varphi_{\mathrm{UU} a}^{(2)}] - 2 \,(\,\tilde{u}_{r,\mathrm{U} a}^{(1)})^{2})} \nonumber\\ &\quad + \underline{\tilde{P}_{rr}[\varphi_{\mathrm{UU} d}^{(2)}]} + S^{2} \cos^{2} \theta \underline{\tilde{P}_{rr}[\varphi_{\mathrm{SS} a}^{(2)}]} + S^{2} \underline{\tilde{P}_{rr}[\varphi_{\mathrm{SS} d}^{(2)}]} \nonumber\\ &\quad + S V \sin \theta \sin \varphi \underline{\tilde{P}_{rr}[\varphi_{\mathrm{US} d}^{(2)\sharp}]} + S \sin \theta (U \sin \theta - V \cos \theta \cos \varphi) \underline{\tilde{P}_{rr}[\varphi_{\mathrm{US} a}^{(2)\flat}]} \nonumber\\ &\quad + S U \underline{\tilde{P}_{rr}[\varphi_{\mathrm{US} d}^{(2)\flat}]}, \end{align}

(3.84b)\begin{align} P_{r{\theta F2}}(r,\theta,\varphi) & ={-} \frac{c_1}{2 \gamma_1 k} (U \sin \theta - V \cos \theta \cos \varphi) \underline{\tilde{P}_{rt,\mathrm{U} b}^{(1)}} \nonumber\\ &\quad + (U \cos \theta + V \sin \theta \cos \varphi) (U \sin \theta - V \cos \theta \cos \varphi) \nonumber\\ &\quad \times \underline{(\tilde{P}_{rt}[\varphi_{\mathrm{UU} b}^{(2)}] - 2 \,\tilde{u}_{r,\mathrm{U} a}^{(1)} \,\tilde{u}_{t,\mathrm{U} b}^{(1)} )} \nonumber\\ &\quad + S^{2} \cos \theta \sin \theta \underline{\tilde{P}_{rt}[\varphi_{\mathrm{SS} b}^{(2)}]} - S V \cos \theta \sin \varphi\underline{\tilde{P}_{rt}[\varphi_{\mathrm{US} b}^{(2)\sharp}]} \nonumber\\ &\quad + S (U \sin 2 \theta - V \cos 2 \theta \cos \varphi) \underline{\tilde{P}_{rt}[\varphi_{\mathrm{US} b}^{(2)\flat}]}, \end{align}

(3.84c)\begin{align} P_{r \varphi F2}(r,\theta,\varphi) & ={-} \frac{c_1}{2 \gamma_1 k} V \sin \varphi \underline{\tilde{P}_{rt,\mathrm{U} b}^{(1)}} + (U \cos \theta + V \sin \theta \cos \varphi) V \sin \varphi \nonumber\\ & \quad \times \underline{(\tilde{P}_{rt}[\varphi_{\mathrm{UU} b}^{(2)}] - 2 \tilde{u}_{r,\mathrm{U} a}^{(1)} \tilde{u}_{t,\mathrm{U} b}^{(1)} )} \nonumber\\ &\quad + S \sin \theta \,(U \sin \theta + V \sin \theta \cos \varphi) \underline{(\tilde{P}_{rt}[\varphi_{\mathrm{US} a}^{(2)\sharp}] - 2 \tilde{u}_{r,\mathrm{U} a}^{(1)} \tilde{u}_{t,\mathrm{S}}^{(1)})} \nonumber\\ &\quad + S \cos \theta \,(U \sin \theta - V \cos \theta \cos \varphi) \underline{\tilde{P}_{rt}[\varphi_{\mathrm{US} b}^{(2)\sharp}]} \nonumber\\ &\quad + S V \cos \theta \sin \varphi \underline{\tilde{P}_{rt}[\varphi_{\mathrm{US} b}^{(2)\flat}]}, \end{align}

where $\tilde {P}_{rr}[\,\cdot \,]$ and $\tilde {P}_{rt}[\,\cdot \,]$ are defined in (3.21b) and the underlined quantities are functions of $r$. Finally, substituting (3.84) into (3.26a,b) (with $m=2$) and carrying out the surface integral, we obtain the second-order force and torque as follows:

(3.85a)$$\begin{gather} \mathcal{F}_1^{(2)} ={-} \frac{c_1}{2 \gamma_1 k} U h_D, \quad \mathcal{F}_2^{(2)} ={-} \frac{c_1}{2 \gamma_1 k} V h_D, \quad \mathcal{F}_3^{(2)} ={-} SV h_L, \end{gather}$$

(3.85b)$$\begin{gather}\mathcal{M}_1^{(2)} = \mathcal{M}_2^{(2)} = \mathcal{M}_3^{(2)} = 0, \end{gather}$$

where the dimensionless transverse force $h_L$ is defined by

(3.86)\begin{equation} h_L = \frac{4}{3} {\rm \pi}(\tilde{P}_{rt}[\varphi_{\mathrm{US} a}^{(2)\sharp}] - \tilde{P}_{rt}[\varphi_{\mathrm{US} b}^{(2)\sharp}] + \tilde{P}_{rr}[\varphi_{\mathrm{US} d}^{(2)\sharp}])|_{r=1}. \end{equation}

Note that $h_L$ depends on $k$ through $\varphi _{\mathrm {US} a}^{(2)\sharp }$, $\varphi _{\mathrm {US} b}^{(2)\sharp }$ and $\varphi _{\mathrm {US} d}^{(2)\sharp }$, and hence we write $h_L=h_L(k)$. Also, it is interesting to note that the transverse force is determined by $\varPhi _{\mathrm {US}}^{(2)}$ describing the cross effect between the uniform and rotational flows, in particular by the part $\varPhi _{\mathrm {US}}^{(2)\sharp }$.

3.7. Summary of the force and torque acting on the sphere

Here, we summarize the expressions of the force and torque acting on the sphere derived from the asymptotic analysis. Let us consider a slightly general situation in which the flow velocity at infinity and the angular velocity of the sphere are given by $\boldsymbol {v}_{\infty } = (v_{\infty 1},v_{\infty 2},v_{\infty 3})$ and $\boldsymbol {\varOmega}_0 = (\varOmega _{0,1},\varOmega _{0,2},\varOmega _{0,3})$, respectively. We also define the small parameter $\varepsilon = |\boldsymbol {v}_\infty |/(2RT_0)^{1/2}$ as before, and denote the dimensionless angular velocity by $S_i = L \varOmega _{0,i}/|\boldsymbol {v}_\infty |$. Then, the force $p_0 L^{2} \mathcal {F}_i$ and the torque $p_0 L^{3} \mathcal {M}_i$ acting on the sphere are written as

(3.87)$$\begin{gather} \mathcal{F}_i = \varepsilon \left(1 + \frac{h_D(k)}{8{\rm \pi} (\gamma_1 k)^{2}} \varepsilon \right) e_i h_D(k) + \varepsilon^{2} (\epsilon_{ijl} e_j S_l) \,h_L(k) + o(\varepsilon^{2}), \end{gather}$$

(3.88)$$\begin{gather}\mathcal{M}_i = \varepsilon S_i h_M(k) + o(\varepsilon^{2}), \end{gather}$$

where $e_i = v_{\infty i}/|\boldsymbol {v}_\infty |$ and $h_D$, $h_L$ and $h_M$ are given by (3.28), (3.86) and (3.29), respectively. In (3.87), we have used (3.50a) to eliminate $c_1$. Note also that the functional dependencies of $h_D$, $h_L$ and $h_M$ on $k$ are explicitly shown in the above formulas.

The formulas (3.87) and (3.88) give the force and torque acting on a rotating sphere as functions of $k$ once the functions $h_D(k)$, $h_M(k)$ and $h_L(k)$ are known. The numerical values of $h_D(k)$ were obtained in Takata et al. (Reference Takata, Sone and Aoki1993) (see also Sone Reference Sone2007), in which the steady flow of a rarefied gas past a sphere (problem U with $U=1$ and $V=0$) was investigated based on the linearized Boltzmann equation for a hard-sphere gas under the diffuse reflection boundary condition. Later, Taguchi & Suzuki (Reference Taguchi and Suzuki2017) obtained the numerical values of $h_D(k)$ for the ellipsoidal statistical model (Holway Reference Holway1966; Andries et al. Reference Andries, Le Tallec, Perlat and Perthame2000; Brull & Schneider Reference Brull and Schneider2008). Concerning $h_M$, Taguchi et al. (Reference Taguchi, Saito and Takata2019) investigated the steady flow around a rotating sphere (problem S) based on the ellipsoidal statistical model (and the BGK model), in which the numerical values of $h_M(k)$ were obtained for both the BGK and the ellipsoidal statistical model. We also mention that the time-dependent behaviour of a gas caused by the impulsive rotation of a sphere has been recently studied (Taguchi, Tsuji & Kotera Reference Taguchi, Tsuji and Kotera2021). For readers’ convenience, we summarize the values of $h_D(k)$ and $h_M(k)$ obtained in those studies in tables 3 and 4, respectively, together with new data obtained in the present study. On the other hand, the function $h_L(k)$, which describes the effect of gas rarefaction on the transverse force, is unknown. Therefore, in the next section, we compute $h_L(k)$ based on the BGK model.

Table 3.

Values of $h_D$ for various $k$ for a hard-sphere gas (HS) (Takata et al. Reference Takata, Sone and Aoki1993) and for the BGK model (Taguchi & Suzuki Reference Taguchi and Suzuki2017) under the diffuse refection boundary condition. Here $k=\infty$ shows the value of $h_D$ in the free molecular limit ($h_D(\infty )=2\sqrt {{\rm \pi} }({\rm \pi} +8)/3$). For the BGK model, the values of $h_D$ for $k<0.1$ and $k>10$ have been newly obtained in this study.

Table 4.

Values of $-h_M$ for various $k$ for the BGK model under the diffuse refection boundary condition (Taguchi et al. Reference Taguchi, Saito and Takata2019). Here $k=\infty$ shows the value of $-h_M$ in the free molecular limit ($h_M(\infty )=-8\sqrt {{\rm \pi} }/3$). The values of $h_M$ for $k<0.1$ and $k>10$ have been newly obtained in this study.

In table 3, the values of $h_D$ for a hard-sphere gas and for the BGK model are shown against $k$. The viscosity is not adjusted between these models (cf. (3.40a)). If we regard the viscosity given by (3.40a) as the common basic quantity, we obtain the relation $k^{BGK} = 1.2700042427 \times k^\textrm {HS}$. With this conversion, a better agreement of $h_D$ between the two models can be obtained (Takata et al. Reference Takata, Sone and Aoki1993; see also figure 4a below). The drag problem for a rarefied gas has also been investigated by Kalempa & Sharipov (Reference Kalempa and Sharipov2020) based on the Shakhov kinetic model. We can derive the values of $h_D$ for the Shakhov model from their results.

For subsequent discussions, let us summarize the asymptotic expressions of $h_D(k)$, $h_L(k)$ and $h_M(k)$ for small $k \ll 1$ (for the general molecular model). They are given by

(3.89a)$$\begin{gather} h_D = 6 {\rm \pi}\gamma_1 k (1 + k_0 k + O(k^{2})), \quad k \ll 1, \end{gather}$$

(3.89b)$$\begin{gather}h_L = 2 {\rm \pi}(1 + 3 k_0 k + O(k^{2})), \quad k \ll 1, \end{gather}$$

(3.89c)$$\begin{gather}h_M ={-}8 {\rm \pi}\gamma_1 k (1 + 3 k_0 k + O(k^{2})), \quad k \ll 1. \end{gather}$$

Here, $k_0$ represents the slip coefficient (shear slip). The value of $k_0$ under the diffuse reflection boundary condition is known as (Sone Reference Sone2002, Reference Sone2007)

(3.90)\begin{equation} k_0 =\begin{cases} -1.25395 & (\text{hard-sphere gas}), \\ -1.01619 & (\text{BGK model}). \end{cases} \end{equation}

The next-order term in the formulas (3.89a) and (3.89c), namely the $k^{2}$-order term in the parentheses, can be found in Sone (Reference Sone2007) for $h_D$ and in Taguchi et al. (Reference Taguchi, Saito and Takata2019) for $h_M$. Note that the expression (3.89b) is new; we give more details on its derivation in § 4.3.

To conclude this section, we present the force and torque using the dimensional quantities initially introduced at the beginning of § 2.1 for a moving sphere. Let $\boldsymbol {F}$ and $\boldsymbol {M}$ denote the force and the moment of force (around the centre) acting on the sphere, respectively. Then, they are given by

(3.91)$$\begin{gather} \boldsymbol{F} ={-}6 {\rm \pi}\mu_0 L \boldsymbol{v}_0 (1 + \tfrac{3}{8} \,{{Re}} \,\bar{h}_D(k)) \bar{h}_D(k) + {\rm \pi}\rho_0 L^{3} (\boldsymbol{\varOmega}_0 \times \boldsymbol{v}_0) \bar{h}_L(k), \end{gather}$$

(3.92)$$\begin{gather}\boldsymbol{M} ={-}8 {\rm \pi}\mu_0 L^{3} \boldsymbol{\varOmega}_0 \bar{h}_M(k) (1 + o({{Re}})), \end{gather}$$

where

(3.93a–c)\begin{equation} \bar{h}_D(k) = \frac{h_D(k)}{6{\rm \pi} \gamma_1 k}, \quad \bar{h}_L(k) = \frac{h_L(k)}{2{\rm \pi}}, \quad \bar{h}_M(k) ={-} \frac{h_M(k)}{8{\rm \pi} \gamma_1 k}, \end{equation}

and ${{Re}}$ is the Reynolds number (see the last paragraph of § 2). The drag $\boldsymbol {F}_D$ and the lift $\boldsymbol {F}_L$ (transverse force) are therefore written as

(3.94)$$\begin{gather} \boldsymbol{F}_D ={-}6 {\rm \pi}\mu_0 L \boldsymbol{v}_0 (1 + \tfrac{3}{8} \,{{Re}} \,\bar{h}_D(k)) \bar{h}_D(k), \end{gather}$$

(3.95)$$\begin{gather}\boldsymbol{F}_L = {\rm \pi}\rho_0 L^{3} (\boldsymbol{\varOmega}_0 \times \boldsymbol{v}_0) \bar{h}_L(k). \end{gather}$$

We remark that although the asymptotic analysis has been carried out under the assumption $k = O(1)$, the formulas are applicable to arbitrary $k$ as long as the Reynolds number ${{Re}}$ is small. In particular, using the asymptotic expressions (3.89) in (3.94), (3.95), and (3.92) (with (3.93a–c)), we obtain for small $k$ ($=({\sqrt {{\rm \pi} }}/{2}) \textit {Kn}$) the following expressions:

(3.96)$$\begin{gather} \boldsymbol{F}_D ={-}6 {\rm \pi}\mu_0 L \boldsymbol{v}_0 (1 + \kappa_0 \textit{Kn} + \tfrac{3}{8} {{Re}} + \cdots ), \quad \textit{Kn} \ll 1, \end{gather}$$

(3.97)$$\begin{gather}\boldsymbol{F}_L = {\rm \pi}\rho_0 L^{3} (\boldsymbol{\varOmega}_0 \times \boldsymbol{v}_0) (1 + 3 \kappa_0 \textit{Kn} + \cdots), \quad \textit{Kn} \ll 1, \end{gather}$$

(3.98)$$\begin{gather}\boldsymbol{M} ={-} 8 {\rm \pi}\mu_0 L^{3} \boldsymbol{\varOmega}_0 (1 + 3 \kappa_0 \textit{Kn} + \cdots), \quad \textit{Kn} \ll 1, \end{gather}$$

where $\kappa _0 = (\sqrt {{\rm \pi} }/2) k_0$ and $\textit {Kn}$ has been used instead of $k$ ($\kappa _0 \approx -1.11128$ for the hard-sphere model and $\kappa _0 \approx -0.90057$ for the BGK model; see (3.90)).

4.1. Remarks on the asymptotic analysis based on the BGK model

The asymptotic analysis described in § 3 can be carried out also for the BGK model in a similar way, and essentially the same results are derived. To present the results for the BGK model in a unified way, we expand the operator $\mathscr {J}^{BGK}(\phi )$ (see (2.13)) as

(4.1)\begin{equation} \mathscr{J}^{BGK}(\phi) = \mathop {\mathscr{J}}\limits{^\circ}^{BGK}(\phi,\phi) + O(\phi^{3}), \end{equation}

where

(4.2)\begin{align} \mathop {\mathscr{J}}\limits{^\circ}^{BGK}(\phi,\phi) & = 2 \left(\zeta_i \zeta_j - \frac{\zeta^{2}}{3} \delta_{ij}\right) \tilde{u}_i[\phi] \tilde{u}_j[\phi] + 2 \zeta_i \left(\zeta^{2} - \frac{5}{2} \right) \tilde{u}_i[\phi] \tilde{\tau}[\phi] \nonumber\\ &\quad + \frac{1}{2} \left(\zeta^{4} - 5 \zeta^{2} + \frac{15}{4} \right) \tilde{\tau}[\phi]^{2} + \tilde{\omega}[\phi] \mathscr{L}^{BGK}(\phi), \end{align}

and $O(\phi ^{3})$ is the remainder. Here, $\tilde {\omega }[\,\cdot \,]$, $\tilde {u}_i[\,\cdot \,]$ and $\tilde {\tau }[\,\cdot \,]$ are defined in (3.19a). Operator $\mathop {\mathscr {J}}\limits{^\circ}^{BGK}(\phi,\phi )$ is a nonlinear operator and quadratic in $\phi$. We can therefore define the symmetric bilinear form associated with $\mathop {\mathscr {J}}\limits{^\circ}^{BGK}(\phi,\phi )$ by

(4.3)\begin{align} \mathop {\mathscr{J}}\limits{^\circ}^{BGK}(\psi,\phi) & = \frac{1}{2} (\mathop {\mathscr{J}}\limits{^\circ}^{BGK}(\psi+\phi,\psi+\phi) - \mathop {\mathscr{J}}\limits{^\circ}^{BGK}(\psi,\psi) - \mathop {\mathscr{J}}\limits{^\circ}^{BGK}(\phi,\phi)) \nonumber\\ & = \left(\zeta_i \zeta_j - \frac{\zeta^{2}}{3} \delta_{ij}\right) ( \tilde{u}_i[\psi]\tilde{u}_j[\phi] + \tilde{u}_i[\phi]\tilde{u}_j[\psi] ) \nonumber\\ &\quad + \zeta_i \left(\zeta^{2} - \frac{5}{2}\right) (\tilde{u}_i[\psi]\tilde{\tau}[\phi] + \tilde{u}_i[\phi]\tilde{\tau}[\psi]) \nonumber\\ &\quad + \frac{1}{2} \left(\zeta^{4} - 5 \zeta^{2} + \frac{15}{4} \right) \tilde{\tau}[\psi]\,\tilde{\tau}[\phi] \nonumber\\ &\quad + \frac{1}{2} (\tilde{\omega}[\psi]\, \mathscr{L}^{BGK}(\phi) + \tilde{\omega}[\phi]\, \mathscr{L}^{BGK}(\psi)). \end{align}

Note that $\mathscr {L}^{BGK}(\phi ) = 2 \mathop {\mathscr {J}}\limits{^\circ}^{BGK}(1,\phi )$ holds (cf. the sentence below (A8)). With these preparations, the results in § 3 apply to the case of the BGK model by taking into account the following correspondence:

(4.4) \begin{equation} \left.\begin{array}{c@{}} \mathscr{L} \to \mathscr{L}^{BGK}, \quad \mathscr{J} \to \mathop {\mathscr{J}}\limits{^\circ}^{BGK} \quad \text{(except for (3.33))},\\ \mathscr{L}_i \to \mathscr{L}_i^{BGK} \quad (i=0,1,2), \quad \mathscr{J}_i \to \mathop {\mathscr{J}}\limits{^\circ}_i^{BGK} \quad (i=0,1,2,3,4), \end{array}\right\} \end{equation}

where $\mathscr {L}_i^{BGK}$ and $\mathop {\mathscr {J}}\limits{^\circ}_i^{BGK}$ are the operators derived from $\mathscr {L}^{BGK}$ and $\mathop {\mathscr {J}}\limits{^\circ}^{BGK}$, whose explicit forms are given in Appendix B (see (B9) and (B10)). Note that in (3.33), the term $\mathscr {J}(\phi _H,\phi _H)$ should be replaced by $\mathscr {J}^{BGK}(\phi _H)$.

Since $\mathop {\mathscr {J}}\limits{^\circ}_4^{BGK}(\cdot,\cdot )=0$ holds identically (see (B10e)), the inhomogeneous terms for the problem (E10)–(E12) are absent for the BGK model (see also (E16)). Consequently, the term $\varPhi _{\mathrm {US}}^{(2)\flat }$ (or $\varphi _{\mathrm {US} \alpha }^{(2)\flat }$, $\alpha =a,b,c,d$) in $\varPhi _{\mathrm {US}}^{(2)}$ happens to be identically zero for this model.

4.2. Cross-coupling formula

We now derive an alternative formula for $h_L$ for the general collision operator. We can immediately obtain the corresponding result for the BGK model by adopting the corresponding BGK collision operators as shown in (4.4). The following discussion is based on the symmetry relation that relates two problems described by the linearized Boltzmann equation (see, e.g. Sharipov Reference Sharipov1994; Takata Reference Takata2009a,Reference Takatab).

We consider the following auxiliary problem:

(Problem $\text {U}'$)  Consider a sphere (radius $L$) at rest with uniform surface temperature $T_0$ immersed in a rarefied gas. Let $L x_i$ denote Cartesian coordinates whose origin is at the sphere centre. We denote by $\rho _0$, $T_0$, $p_0 (= \rho_0 RT_0)$ and $\boldsymbol {v}_\infty '$ the (constant) density, temperature, pressure and flow velocity of the gas at infinity and let $\boldsymbol {v}_\infty ' = (0,0,(2RT_0)^{1/2} u_\infty )$ ($u_\infty >0$). We investigate the steady behaviour of the gas under the same assumptions as (i) and (ii) in § 2 and

1. (iii)' the flow velocity at infinity $\boldsymbol {v}_\infty '$ (or $u_\infty$) is so small that the equation and boundary conditions can be linearized around the reference state at rest with density $\rho _0$ and temperature $T_0$.

In other words, we consider the same sphere as in the original problem in § 2, but the sphere is at rest, and the flow over it is perpendicular to the $x_1x_2$ plane.

Let $\rho _0(2RT_0)^{-3/2}(1+ u_\infty \varPhi _{\mathrm {U}*}^{(1)}(\boldsymbol {x},\boldsymbol {\zeta })) E$ denote the velocity distribution function of the gas molecules in problem $\text {U}'$. Then, using the same notation as before, $\varPhi _{\mathrm {U}*}^{(1)}$ satisfies the following equation and boundary conditions:

(4.5a)$$\begin{gather} \zeta_i \frac{\partial \varPhi_{\mathrm{U}*}^{(1)}}{\partial x_i} = \frac{1}{k} \mathscr{L}(\varPhi_{\mathrm{U}*}^{(1)}), \quad |\boldsymbol{x}|>1, \end{gather}$$

(4.5b)$$\begin{gather}\varPhi_{\mathrm{U}*}^{(1)} = \mathscr{K}(\varPhi_{\mathrm{U}*}^{(1)}), \quad \zeta_r > 0, \ |\boldsymbol{x}| = 1, \end{gather}$$

(4.5c)$$\begin{gather}\varPhi_{\mathrm{U}*}^{(1)} \to 2 \zeta_3 \quad \text{as} \ |\boldsymbol{x}| \to \infty. \end{gather}$$

Note that by assumption (i), we implicitly assume that the molecular model is common for problem $\text {U}'$ and the original one in § 2.

The problem $\text {U}'$ is essentially problem U that arose in § 3.2 (see (3.11)–(3.13) with $J=\mathrm {U}$), but there is a difference in the flow direction at infinity. We chose this flow direction to be consistent with the direction of the transverse force in the original problem. Moreover, the equivalence between problem $\text {U}'$ and problem U suggests that we can write $\varPhi _{\mathrm {U}*}^{(1)}$ in terms of the similarity solutions. Indeed, straightforward computations show that $\varPhi _{\mathrm {U}*}^{(1)}$ is given by

(4.6)\begin{equation} \varPhi_{\mathrm{U}*}^{(1)} = (\sin \theta \sin \varphi) \varphi_{\mathrm{U} a}^{(1)}(r,\zeta_r,\zeta) - (\zeta_\theta \cos \theta \sin \varphi + \zeta_\varphi \cos \varphi) \varphi_{\mathrm{U} b}^{(1)}(r,\zeta_r,\zeta), \end{equation}

where $\varphi _{\mathrm {U} a}^{(1)}$ and $\varphi _{\mathrm {U} b}^{(1)}$ are the solution to (3.17). Therefore, if we know the solution to problem U, we essentially know the solution to problem $\text {U}'$.

We now return to the boundary-value problem for $\varPhi _{\mathrm {US}}^{(2)}$ arising in the second-order problem in the asymptotic analysis, that is, (3.70)–(3.72) with $J=\mathrm {US}$. The equations for $\varPhi _{\mathrm {U}*}^{(1)}$ and $\varPhi _{\mathrm {US}}^{(2)}$ are both linearized Boltzmann equations, and the application of the symmetry relation described in Takata (Reference Takata2009a) (Proposition 2 there) leads to the following identity, valid for the entire range of $k$ ($0 < k < \infty$):

(4.7)\begin{equation} SV h_L = \int_{|\boldsymbol{x}|=1} \langle \zeta_r \varPhi_{\mathrm{U}*}^{(1)} I_{{w},\mathrm{US}}^{\textrm{(2)}-} \rangle \,\mathrm{d}S - \frac{1}{k} \int_{|\boldsymbol{x}|>1} \langle \varPhi_{\mathrm{U}*}^{(1)} I_{\mathrm{US}}^{(2)-} \rangle \,\mathrm{d} \boldsymbol{x}, \end{equation}

where $I_{{w},\mathrm {US}}^{(2)}$ and $I_{\mathrm {US}}^{(2)}$ are given in (3.73b) and (3.73a), $\mathrm {d} \boldsymbol {x} = \mathrm {d} x_1 \,\mathrm {d} x_2 \,\mathrm {d} x_3$ and $^{-}$ indicates the reflection operator acting on the variable $\zeta _i$, that is, $F^{-}(\zeta _i) = F(-\zeta _i)$ for any function $F(\zeta _i)$. The range of the first (second) integral is the whole surface of the sphere (the whole gas region). Note that $I_{{w},\mathrm {US}}^{(2)}$ and $I_{\mathrm {US}}^{(2)}$ are determined by $\varPhi _{\mathrm {U}}^{(1)}$ and $\varPhi _{\mathrm {S}}^{(1)}$. Therefore, if we know $\varPhi _{\mathrm {U}}^{(1)}$ and $\varPhi _{\mathrm {S}}^{(1)}$ (and thus $\varPhi _{\mathrm {U}*}^{(1)}$), the above identity gives a closed expression for $h_L$, which can be evaluated without knowledge of $\varPhi _{\mathrm {US}}^{(2)}$. Moreover, substituting (3.16) and (4.6) and integrating the result with respect to $\theta$ and $\varphi$, we can further simplify (4.7) to

(4.8)\begin{equation} h_L = \frac{4}{3} {\rm \pi}\left[I_{{w}} - \frac{1}{k} (I_1 + I_2)\right], \end{equation}

where

(4.9a)$$\begin{gather} I_{{w}} = \mathscr{K}(\varphi_{\mathrm{U} a}^{(1)}|_{r=1}) \tilde{P}_{rt,\mathrm{U} b}^{(1)}(r=1), \end{gather}$$

(4.9b)$$\begin{gather}I_1 = 2 \int_1^{\infty} r^{2} \left \langle \varphi_{\mathrm{U} a}^{(1)} \left[ \frac{\zeta^{2} - \zeta_r^{2}}{2} \mathscr{J}_2(\varphi_{\mathrm{U} b}^{(1)},\varphi_{\mathrm{S}}^{(1)})^{-} + \mathscr{J}_3(\varphi_{\mathrm{U} b}^{(1)},\varphi_{\mathrm{S}}^{(1)})^{-} \right] \right \rangle \mathrm{d} r, \end{gather}$$

(4.9c)$$\begin{gather}I_2 = 2 \int_1^{\infty} r^{2} \left \langle \frac{\zeta^{2} - \zeta_r^{2}}{2} \varphi_{\mathrm{U} b}^{(1)} \mathscr{J}_1(\varphi_{\mathrm{U} a}^{(1)},\varphi_{\mathrm{S}}^{(1)})^{-} \right \rangle \mathrm{d} r. \end{gather}$$

Here, $\mathscr {J}_l(\cdot,\cdot )^{-}(r,\zeta _r,\zeta ) = \mathscr {J}_l(\cdot,\cdot )(r,-\zeta _r,\zeta )$, $l=1,2,3$. In this formula, $h_L$ is expressed in terms of $(\varphi _{\mathrm {U} a}^{(1)},\varphi _{\mathrm {U} b}^{(1)})$ and $\varphi _{\mathrm {S}}^{(1)}$ only, namely the solutions of the elementary problems U and S. In this way, we can bypass the difficulty to solve the second-order problem as far as $h_L$ is concerned.

The formula (4.8) (or (4.7)) underlines that the transverse force is a cross effect between the translational and swirling motion. In this sense, we may refer to (4.8) as a cross-coupling formula. Sharipov (Reference Sharipov2011) discussed a similar expression for a general weakly perturbed Boltzmann system. In this connection, we stress that, although the transverse force in the present problem is of second order in $\varepsilon$, it is a leading-order effect since the first-order transverse force degenerates. We also remark that the asymptotic properties of $\varPhi _{\mathrm {U}*}^{(1)}$ and $\varPhi _{\mathrm {US}}^{(2)}$ as $|\boldsymbol {x}| \to \infty$ play a crucial role in deriving the formula (4.7), as discussed in Takata (Reference Takata2009a). To see this point more clearly, we give a direct derivation of the formula in Appendix F.

4.3. The function $h_L(k)$ for $k \to \infty$ and for $k \ll 1$

As an application of the formula (4.8), let us first consider two limiting cases, the case of $k \to \infty$ and the case of $k \ll 1$.

First, we explain the case of $k \to \infty$ (i.e. the free molecular gas or collisionless gas). In this case, the term $k^{-1}(I_1 + I_2)$ is negligibly small compared to $I_{{w}}$ and therefore is ignored. On the other hand, the term $I_{w}$ can be easily calculated with the aid of the free molecular solution for problem U (see Appendix G for a summary of the free molecular solution). Since $\mathscr {K}(\varphi _{\mathrm {U} a}^{(1)}|_{r=1}) = - \sqrt {{\rm \pi} }$ and $\tilde {P}_{rt,\mathrm {U} b}^{(1)}(r=1) = 1/\sqrt {{\rm \pi} }$ (cf. (G3e)), we obtain

(4.10)\begin{equation} h_L \to - \tfrac{4}{3} {\rm \pi}\quad \text{as} \ k \to \infty \quad (\text{collisionless gas}). \end{equation}

Meanwhile, we can also compute $h_L$ directly from (3.86) using the second-order macroscopic quantities for the free molecular flow. According to (G6e), (G7c) and (G7e), we have $\tilde {P}_{rr}[\varphi _{\mathrm {US} d}^{(2)\sharp }](r=1) = 0$, $\tilde {P}_{rt}[\varphi _{\mathrm {US} b}^{(2)\sharp }](r=1) = 0$ and $\tilde {P}_{rt}[\varphi _{\mathrm {US} a}^{(2)\sharp }](r=1) = -1$. Therefore, (3.86) gives $h_L=- (4/3) {\rm \pi}$, which coincides with (4.10).

Equation (4.10) implies that a sphere moving in a collisionless gas with translational and angular velocities $\boldsymbol {v}_0$ and $\boldsymbol {\varOmega }_0$ is subject to the (dimensional) transverse force $-\tfrac {2}{3} {\rm \pi}\rho _0 L^{3} (\boldsymbol {\varOmega }_0 \times \boldsymbol {v}_0)$. The direction of the force is therefore opposite to the corresponding force in the continuum flow (e.g. Rubinow & Keller Reference Rubinow and Keller1961). This result, pointed out by Wang (Reference Wang1972), Ivanov & Yanshin (Reference Ivanov and Yanshin1980) and Borg et al. (Reference Borg, Söderholm and Essén2003), is known as the inverse Magnus effect. It may be worth noting that Ivanov & Yanshin (Reference Ivanov and Yanshin1980) arrived at this conclusion without the smallness assumptions for $\boldsymbol {\varOmega }_0$ and $\boldsymbol {v}_0$.

Next, we consider the case of $k \ll 1$. To explicitly evaluate the terms $I_{w}$, $I_1$ and $I_2$ in (4.8), we need the explicit forms of the velocity distribution function for problems U and S (i.e. $\varphi _{\mathrm {U} a}^{(1)}$, $\varphi _{\mathrm {U} b}^{(1)}$ and $\varphi _{\mathrm {S}}^{(1)}$) for small $k$. Such expressions can be obtained with the aid of the asymptotic theory of the Boltzmann equation with small Knudsen numbers (Sone Reference Sone2002, Reference Sone2007). More precisely, we first apply the theory to the system (3.11)–(3.13) to obtain asymptotic expressions for $\varphi _{\mathrm {U} a}^{(1)}$, $\varphi _{\mathrm {U} b}^{(1)}$ and $\varphi _{\mathrm {S}}^{(1)}$ for small $k$. Then, we substitute them into (4.9) to compute $I_{w}$, $I_1$ and $I_2$ explicitly.

In this study, we intend to derive an expression for $h_L(k)$ for small $k$ up to order $k$. This means that we need to obtain $I_1$ and $I_2$ (or $I_{w}$) to order $k^{2}$ (or $k$). The derivation is straightforward but lengthy. Therefore, we omit it here and only give the final results. That is, $I_{w}$, $I_1$ and $I_2$ are expressed for small $k$ as

(4.11a)$$\begin{gather} I_{w} ={-} \tfrac{9}{4} \gamma_1^{2} k^{2} + O(k^{3}), \quad k \ll 1, \end{gather}$$

(4.11b)$$\begin{gather}I_1 ={-}\tfrac{11}{8} k (1 + 3 k_0 k + O(k^{2})), \quad k \ll 1, \end{gather}$$

(4.11c)$$\begin{gather}I_2 ={-}\tfrac{1}{8} k ( 1 + 3 k_0 k + O(k^{2})), \quad k \ll 1. \end{gather}$$

Substituting them into (4.8), we obtain (3.89b). Note that the leading-order term of (3.89b) coincides with the result obtained by Rubinow & Keller (Reference Rubinow and Keller1961) based on the incompressible Navier–Stokes equation (with no-slip boundary conditions). Since $k_0$ is negative (see (3.90)), the magnitude of the transverse force decreases with an increase of $k$ when $k \ll 1$.

4.4. $h_L(k)$ for intermediate values of $k$

In this subsection, we compute the function $h_L(k)$ numerically utilizing the formula (4.8). We employ the BGK model of the Boltzmann equation to reduce the complexity in the numerical integration. We thus replace the operators $\mathscr {J}_i$ ($i=1,2,3$) in $I_1$ and $I_2$ with $\mathop {\mathscr {J}}\limits{^\circ}_i^{BGK}$ (see Appendix B for their explicit forms). Consequently, many integrals with respect to the molecular velocity can be carried out, and we obtain for the BGK model the following:

(4.12a)\begin{gather} I_1 ={-} \int_1^{\infty} r^{2} \tilde{u}_{t,\mathrm{U} b}^{(1)} \tilde{u}_{t,\mathrm{S}}^{(1)} (\tilde{P}_{rr,\mathrm{U} a}^{(1)} - \tilde{P}_{\mathrm{U} a}^{(1)} ) \,\mathrm{d}r, \end{gather}

(4.12b)\begin{align} I_2 & = 2 \int_1^{\infty} r^{2} \tilde{u}_{t,\mathrm{S}}^{(1)} (-\tilde{u}_{r,\mathrm{U} a}^{(1)} \tilde{P}_{rt,\mathrm{U} b}^{(1)} + \tilde{\tau}_{\mathrm{U} a}^{(1)} \tilde{Q}_{t,\mathrm{U} b}^{(1)} ) \,\mathrm{d} r \nonumber\\ &\quad + 2 \int_1^{\infty} r^{2} \tilde{\omega}_{\mathrm{U} a}^{(1)} \left[\tilde{u}_{t,\mathrm{U} b}^{(1)} \tilde{u}_{t,\mathrm{S}}^{(1)} \vphantom{\int_0^{\infty}}\right.\nonumber\\ &\quad \left.- {\rm \pi}\int_0^{\infty} \int_0^{\rm \pi} \zeta^{4} \sin^{3} \theta_\zeta \varphi_{\mathrm{U} b}^{(1)}(r,\theta_\zeta,\zeta) \varphi_{\mathrm{S}}^{(1)}(r,{\rm \pi}-\theta_\zeta,\zeta) E \,\mathrm{d} \theta_\zeta \,\mathrm{d} \zeta \right] \mathrm{d} r. \end{align}

Here, $\tilde {u}_{t,\mathrm {U} b}^{(1)}(r)$, $\tilde {u}_{t,\mathrm {S}}^{(1)}(r)$, $\tilde {P}_{rr,\mathrm {U} a}^{(1)}(r)$, $\tilde {P}_{\mathrm {U} a}^{(1)}(r)$, $\tilde {u}_{r,\mathrm {U} a}^{(1)}(r)$, $\tilde {P}_{rt,\mathrm {U} b}^{(1)}(r)$, $\tilde {\tau }_{\mathrm {U} a}^{(1)}(r)$, $\tilde {Q}_{t,\mathrm {U} b}^{(1)}(r)$ and $\tilde {\omega }_{\mathrm {U} a}^{(1)}(r)$, defined in (3.23), are the macroscopic variables arising in problems U and S. In the last term of $I_2$, the variable $\theta _\zeta = \cos ^{-1}(\zeta _r/\zeta )$ ($0 \le \theta _\zeta \le {\rm \pi}$) is used in place of $\zeta _r$. Accordingly, $\varphi _{\mathrm {U} b}^{(1)}$ and $\varphi _{\mathrm {S}}^{(1)}$ are regarded as functions of $r$, $\theta _\zeta$ and $\zeta$. Note that the integral with respect to $\theta _\zeta$ is a convolution of $\varphi _{\mathrm {U} b}^{(1)}$ and $\varphi _{\mathrm {S}}^{(1)}$.

The numerical solutions of $(\varphi _{\mathrm {U} a}^{(1)},\varphi _{\mathrm {U} b}^{(1)})$ and $\varphi _{\mathrm {S}}^{(1)}$ based on the BGK model have been prepared beforehand in Taguchi & Suzuki (Reference Taguchi and Suzuki2017) for problem U and in Taguchi et al. (Reference Taguchi, Saito and Takata2019) for problem S, using an accurate finite difference method (Takata et al. Reference Takata, Sone and Aoki1993). Therefore, we can make use of these data to carry out the numerical integration in (4.12). We also made additional computations to supplement the data, in particular for $k<0.1$ and $k > 10$ (see also the second paragraph of § 3.7). See Appendix H for further details of the numerical computations.

Figure 3 shows the obtained $h_L$ as a function of $k$. In the figure, the symbols represent the numerical results, the values of which are also tabulated in table 5. As seen from the figure and table, $h_L$ decreases monotonically with $k$. The dashed curve shows the asymptotic formula (3.89b) for small $k$, and the solid (horizontal) line represents its leading-order term ($h_L=2{\rm \pi}$), corresponding to the Navier–Stokes result (Rubinow & Keller Reference Rubinow and Keller1961). The dash-dotted line shows the result for the free molecular gas (see (4.10)). The numerical results tend to approach both $2{\rm \pi}$ and $-\tfrac {4}{3}{\rm \pi}$ as $k \to 0$ and $k \to \infty$, respectively. The values of $h_L$ in these limits have different signs, and the curve $h_L(k)$ intersects $h_L=0$ at an intermediate value of $k$. Writing this value $k_{th}$ (i.e. the threshold of $k$ above which the negative lift occurs), we find $k_{th} = 0.710$ for the present computation based on the BGK model and the diffuse reflection boundary condition. In this way, the present asymptotic theory supplemented by the numerical computations for $h_L$ can describe the change of the transverse force in terms of the Knudsen number and, in particular, the transition to negative lift.

Figure 3.

Plot of $h_L$ versus $k$ for the BGK model under the diffuse reflection boundary condition. The circle symbols represent the numerical results. The asymptotic formula (3.89b) with $k_0=-1.01619$ (see (3.90)) is shown by the dashed curve. The asymptotic values at $k \to 0$ (the continuum limit) and $k \to \infty$ (the free molecular limit) are shown by the solid and dash-dotted lines, respectively. The value of $h_L$ decreases with $k$ and intersects $h_L=0$ at $k = k_{th} \approx 0.710$.

Table 5.

Values of $h_L$ for various $k$ for the BGK model under the diffuse refection boundary condition. Note that $h_L\to -\tfrac {4}{3}{\rm \pi} \approx -4.1888$ as $k\to \infty$. $^{a}$The results obtained by using the asymptotic formula (3.89b).

In order to obtain further insight into the present results, we compare the formulas (3.87) and (3.88) with existing numerical results, in particular with those obtained by the DSMC method (Volkov Reference Volkov2011). To this end, we consider the case in which the angular velocity of the sphere $\boldsymbol {\varOmega }_0$ is perpendicular to the flow velocity at infinity $\boldsymbol {v}_\infty$, i.e. $\boldsymbol {\varOmega }_0 = (\varOmega _0,0,0)$ and $\boldsymbol {v}_\infty = (0,v_{\infty },0)$ with $\varOmega _0 > 0$ and $v_\infty > 0$ (or $\alpha _0 = {\rm \pi}/2$) in figure 1. Following Volkov (Reference Volkov2011), we introduce the drag, lift and torque coefficients by

(4.13a–c)\begin{equation} C_D = \frac{F_D}{\frac{1}{2} \rho_0 {\rm \pi}L^{2} {v}_\infty^{2}}, \quad C_L = \frac{-F_L}{\frac{1}{2} \rho_0 {\rm \pi}L^{3} v_\infty \varOmega_0}, \quad C_T = \frac{-M}{\frac{1}{2} \rho_0 {\rm \pi}L^{5} \varOmega_0^{2}}, \end{equation}

where $(0,F_D,0)$, $(0,0,F_L)$ and $(M,0,0)$ are the drag, lift and moment of force (torque) acting on the sphere. Note that these coefficients depend on physical parameters $\textit {Kn}=\ell _0/L$, $\textit {Ma}=v_\infty /(5RT_0/3)^{1/2}$ and $\hat {\varOmega }_0 = L \varOmega _0/(2RT_0)^{1/2}$. If we use the expressions of the forces and torque for small $\textit {Ma} \ll 1$ obtained in the present study, $C_D$, $C_L$ and $C_T$ are given, at leading order in $\textit {Ma}$, as

(4.14a)$$\begin{gather} \sqrt{\frac{5}{6}} \textit{Ma} C_D = \frac{h_D}{\rm \pi}, \end{gather}$$

(4.14b)$$\begin{gather}C_L = \frac{h_L}{\rm \pi}, \end{gather}$$

(4.14c)$$\begin{gather}\hat{\varOmega}_0 C_T ={-} \frac{h_M}{\rm \pi}. \end{gather}$$

In figure 4, we compare $C_D$, $C_L$ and $C_T$ based on (4.14) with those obtained by Volkov (Reference Volkov2011) for various $\textit {Kn}$ and $\textit {Ma}$ in the case of $\hat {\varOmega }_0 = 0.1$. Here, to evaluate $C_D$, $C_L$ and $C_T$ from (4.14), we have used the numerical data of $h_D$, $h_L$ and $h_M$ for the BGK model shown in tables 3, 5 and 4. For $k < 0.01$, we have used the asymptotic formulas (3.89a)–(3.89c). Note that the DSMC results are for a hard-sphere gas, but the data $h_D$, $h_L$ and $h_M$ are for the BGK model. For the drag, it is known that the agreement between the BGK model and the hard-sphere model is enhanced if we make a conversion of the Knudsen number (see the third paragraph of § 3.7). This conversion has been used in figure 4(a), that is, $C_D$ for the BGK model is shown against $\textit {Kn}/1.270\cdots$. We also mention that the original values of the Knudsen number presented in Volkov (Reference Volkov2011) (e.g. figure 8 there) have been multiplied by $\sqrt {{\rm \pi} }$ in figure 4(a–c); otherwise, the values of $C_D$ by Volkov (Reference Volkov2011) deviate systematically from the known results based on the linearized Boltzmann equation for a hard-sphere gas (Takata et al. Reference Takata, Sone and Aoki1993), resulting in an inconsistency. As seen from figure 4(b), the agreement of $C_L$ between the present results and the DSMC results is quite encouraging. When $\textit {Ma} = 0.03$, $C_L$ obtained by the present formula (the solid curve) and $C_L$ by the DSMC data (the red circle) are close each other for $\textit {Kn} \gtrsim 0.4$. In particular, the Knudsen number at which $C_L$ vanishes in the DSMC simulations seems to be close to the point $\textit {Kn} = \textit {Kn}_{th} \approx 0.801$ obtained by the present theory, despite the difference in the molecular model. As the Mach number increases, the discrepancy between the DSMC and present results becomes larger. We also notice that the discrepancy is more pronounced for smaller $\textit {Kn}$. We recall that the Mach number should be smaller than $\textit {Kn}$ for the present theory to be applicable (the Reynolds number should be small). Therefore, the limit $\textit {Kn} \to 0$ with $\textit {Ma}$ fixed does not correspond to our analysis, which explains the observed discrepancy for small $\textit {Kn}$. The torque coefficient obtained through $h_M$ also compares favourably with the DSMC results when $\textit {Ma} \le 0.2$ (see figure 4c).

Figure 4.

Coefficients (a) $C_D$, (b) $C_L$ and (c) $C_T$ as a function of $\textit {Kn}$ in the case of $L \varOmega _0/(2RT_0)^{1/2} = 0.1$. The solid lines represent (4.14) using the data of $h_D$, $h_L$ and $h_M$ for the BGK model. The symbols show the results of the DSMC computations for a hard-sphere gas (Volkov Reference Volkov2011). The results for $C_D$ based on the linearized Boltzmann equation for a hard-sphere gas (Takata et al. Reference Takata, Sone and Aoki1993) (i.e. (4.14a) using the data of $h_D$ for hard-sphere gas in table 3) are also shown in (a).