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

Abstract

Numerical analyses of the second-order Knudsen layer are carried out on the basis of the linearized Boltzmann equation for hard-sphere molecules under the diffuse reflection boundary condition. The effects of the boundary curvature have been clarified in details, thereby completing the numerical data required up to the second order of the Knudsen number for the asymptotic theory of the Boltzmann equation (the generalized slip-flow theory). A local singularity appears as a result of the expansion at the level of the velocity distribution function, when the curvature exists.

Appendices

Appendix 1: Stress and Heat Flow

The stress tensor and heat-flow vector are also familiar fluid-dynamic quantities that become necessary, most typically, in computing the momentum and energy exchange with the body surface. Denoting the former by $p_{0}(\delta _{ij}+P_{ij})$ and the latter by $p_{0}(2RT_{0})^{1/2}Q_{i}$, their Hilbert part $h_{\mathrm {H}}=h_{\mathrm {H}0}+h_{\mathrm {H}1}\varepsilon +\cdots$ and Knudsen-layer correction $h_{\mathrm {K}}=h_{\mathrm {K}0}+h_{\mathrm {K}1}\varepsilon +\cdots$ ($h=P_{ij},Q_{i}$) up to the second order in $\varepsilon$ are summarized as follows:

$$\begin{aligned}&P_{ij\mathrm {H}m}=P_{\mathrm {H}m}\delta _{ij}-\gamma _{1}\overline{\frac{\partial u_{i\mathrm {H}m-1}}{\partial x_{j}}}+\frac{1}{2}\gamma _{3}\overline{\frac{\partial ^{2}\tau _{\mathrm {H}m-2}}{\partial x_{i}\partial x_{j}}},\ (m=0,1,2),\end{aligned}$$

(21a)

$$\begin{aligned}&Q_{i\mathrm {H}m}=-\frac{5}{4}\gamma _{2}\frac{\partial \tau _{\mathrm {H}m-1}}{\partial x_{i}}+\frac{1}{2}\gamma _{3}\frac{\partial ^{2}u_{i\mathrm {H}m-2}}{\partial x_{j}^{2}},\ (m=0,1,2), \end{aligned}$$

(21b)

and

$$\begin{aligned} P_{ij\mathrm {K}m}=&\frac{3}{2}\frac{\partial \tau _{\mathrm {H}m-1}}{\partial x_{k}}n_{k}(\delta _{ij}-n_{i}n_{j})[\Omega _{1}^{(0)}(\eta )+\Theta _{1}^{(0)}(\eta )],\ (m=0,1),\end{aligned}$$

(22a)

$$\begin{aligned} P_{ij\mathrm {K}2}n_{i}n_{j}&=-3\bar{\kappa }\frac{\partial \tau _{\mathrm {H}0}}{\partial x_{i}}n_{i}\int _{\eta }^{\infty }[\Omega _{1}^{(0)}(z)+\Theta _{1}^{(0)}(z)]dz,\end{aligned}$$

(22b)

$$\begin{aligned} P_{ij\mathrm {K}2}n_{i}t_{j}&=\frac{3}{2}\Big (\frac{\partial ^{2}\tau _{\mathrm {H}0}}{\partial x_{i}\partial x_{j}}n_{i}t_{j}+\kappa _{ij}t_{j}\frac{\partial \tau _{\mathrm {H}0}}{\partial x_{i}}\Big )\int _{\eta }^{\infty }[\Omega _{1}^{(0)}(z)+\Theta _{1}^{(0)}(z)]dz, \end{aligned}$$

(22c)

$$\begin{aligned} Q_{i\mathrm {K}m}t_{i}&= \overline{\frac{\partial u_{i\mathrm {H}m-1}}{\partial x_{j}}}n_{i}t_{j}H_{1}^{(1)}(\eta )+\frac{\partial \tau _{\mathrm {H}m-1}}{\partial x_{i}}t_{i}H_{2}^{(1)}(\eta )\nonumber \\&\quad +\, \frac{\partial ^{2}\tau _{\mathrm {H}m-2}}{\partial x_{i}\partial x_{j}}n_{i}t_{j}H_{3}^{(1)}(\eta )+\frac{\partial }{\partial x_{i}}\overline{\frac{\partial u_{j\mathrm {H}m-2}}{\partial x_{k}}}n_{i}n_{j}t_{k}H_{4}^{(1)}(\eta )\nonumber \\&\quad +\bar{\kappa }\overline{\frac{\partial u_{i\mathrm {H}m-2}}{\partial x_{j}}}n_{i}t_{j}H_{5}^{(1)}(\eta )+\kappa _{ij}\overline{\frac{\partial u_{j\mathrm {H}m-2}}{\partial x_{k}}}n_{k}t_{i}H_{6}^{(1)}(\eta )\nonumber \\&\quad + \, \kappa _{ij}t_{j}\frac{\partial \tau _{\mathrm {H}m-2}}{\partial x_{i}}H_{7}^{(1)}(\eta )+\bar{\kappa }\frac{\partial \tau _{\mathrm {H}m-2}}{\partial x_{i}}t_{i}H_{8}^{(1)}(\eta ),\ (m=0,1,2), \end{aligned}$$

(22d)

$$\begin{aligned} Q_{i\mathrm {K}m}n_{i}&= \biggl [\frac{\partial ^{2}\tau _{\mathrm {H}m-2}}{\partial x_{i}\partial x_{j}}(\delta _{ij}-n_{i}n_{j})-2\bar{\kappa }\frac{\partial \tau _{\mathrm {H}m-2}}{\partial x_{i}}n_{i}\biggr ]\int _{\eta }^{\infty }H_{2}^{(1)}(z)dz\nonumber \\&\quad -\, \frac{1}{2}\frac{\partial }{\partial x_{i}}\overline{\frac{\partial u_{j\mathrm {H}m-2}}{\partial x_{k}}}n_{i}n_{j}n_{k}\int _{\eta }^{\infty }H_{1}^{(1)}(z)dz,\ (m=0,1,2). \end{aligned}$$

(22e)

Here the quantities with the subscript $\mathrm {H}$ in (22) denote their value on the boundary. The functions $H_{1}^{(1)}\sim H_{4}^{(1)}$ have already been obtained in [18, 20, 21]. The present work newly provides the data of $H_{5}^{(1)}\sim H_{8}^{(1)}$, which are included in Table 2 and Fig. 5.

Appendix 2: Sketch of Derivation of (16)

The non-trivial equality in (16a) is

$$\begin{aligned} \lim _{\eta \rightarrow 0}\int _{0}^{\infty }\int _{0}^{1}(1-\mu ^{2})\frac{\zeta ^{4}}{\nu ^{2}}\frac{1}{\eta }|\frac{\nu \eta }{\mu \zeta }|^{3}e^{-\frac{\nu \eta }{\mu \zeta }}\mathcal {P}(\mu ,\zeta )h_{0}(\mu ,\zeta )d\mu d\zeta =\int _{0}^{\infty }\frac{\zeta ^{3}}{\nu }\mathcal {P}(0,\zeta )h_{0}(0,\zeta )d\zeta . \end{aligned}$$

(23)

Once (23) is proved, the still non-trivial equality in (16c) is

$$\begin{aligned}&2\lim _{\eta \rightarrow 0}\int _{0}^{\infty }\int _{0}^{1}\zeta ^{2}\mathcal {P}(\mu ,\zeta )T[C[h]]d\mu d\zeta \nonumber \\&\qquad \qquad =\lim _{\eta \rightarrow 0}\frac{1}{\eta }\int _{0}^{\infty }\int _{0}^{1}|\frac{\nu \eta }{\mu \zeta }|^{3}e^{-|\frac{\nu \eta }{\mu \zeta }|}\mathcal {P}(\mu ,\zeta )a(\mu ,\zeta )\frac{(1-\mu ^{2})\zeta ^{4}}{\nu ^{3}}d\mu d\zeta . \end{aligned}$$

(24)

We show below the outline of the proof for (23), (24), and (16b).

Proof of  (23) We first split the region of integration with respect to $\mu$ into $(0,\delta )$ and $(\delta ,1)$, where $0<\delta <1$ is a constant. Then for the second part, we see that

$$\begin{aligned}&|\int _{0}^{\infty }\int _{\delta }^{1}\frac{\zeta ^{4}}{\nu ^{2}}\frac{1}{\eta }|\frac{\nu \eta }{\mu \zeta }|^{3}e^{-\frac{\nu \eta }{\mu \zeta }}\{(1-\mu ^{2})\mathcal {P}(\mu ,\zeta )h_{0}(\mu ,\zeta )-\mathcal {P}(0,\zeta )h_{0}(0,\zeta )\}d\mu d\zeta |\\&\quad \le \int _{0}^{\infty }\int _{\delta }^{1}\frac{\zeta ^{4}}{\nu ^{2}}\frac{1}{\eta }|\frac{\nu \eta }{\delta \zeta }|^{3}e^{-\frac{\nu \eta }{\zeta }}\max _{\mu }(|\mathcal {P}(\mu ,\zeta )h_{0}(\mu ,\zeta )|+|\mathcal {P}(0,\zeta )h_{0}(0,\zeta )|)d\mu d\zeta \\&\quad \le \delta ^{-3}\int _{0}^{\infty }\eta \zeta ^{2}|\frac{\nu \eta }{\zeta }|e^{-\frac{\nu \eta }{\zeta }}\max _{\mu }(|\mathcal {P}(\mu ,\zeta )h_{0}(\mu ,\zeta )|+|\mathcal {P}(0,\zeta )h_{0}(0,\zeta )|)d\zeta \\&\quad \le \delta ^{-3}\eta \int _{0}^{\infty }\zeta ^{2}\max _{\mu }(|\mathcal {P}(\mu ,\zeta )h_{0}(\mu ,\zeta )|+|\mathcal {P}(0,\zeta )h_{0}(0,\zeta )|)d\zeta \rightarrow 0,\quad \text{ as } \eta \rightarrow 0. \end{aligned}$$

On the other hand, by applying to $\mathcal {P}$ and $h_{0}$ the mean-value theorem with respect to $\mu$, we have

$$\begin{aligned}&|\int _{0}^{\infty }\int _{0}^{\delta }\frac{\zeta ^{4}}{\nu ^{2}}\frac{1}{\eta }|\frac{\nu \eta }{\mu \zeta }|^{3}e^{-\frac{\nu \eta }{\mu \zeta }}\{(1-\mu ^{2})\mathcal {P}(\mu ,\zeta )h_{0}(\mu ,\zeta )-\mathcal {P}(0,\zeta )h_{0}(0,\zeta )\}d\mu d\zeta |\\&\quad \le \int _{0}^{\infty }\int _{0}^{\delta }\frac{\zeta ^{4}}{\nu ^{2}}\frac{1}{\eta }|\frac{\nu \eta }{\mu \zeta }|^{3}e^{-\frac{\nu \eta }{\mu \zeta }}(|\mathcal {P}^{\prime }h_{0}(0,\zeta )|\mu +|\mathcal {P}(0,\zeta )h_{0}^{\prime }|\mu +|\mathcal {P}h_{0}|\mu ^{2}+|\mathcal {P}^{\prime }h_{0}^{\prime }|\mu ^{2})d\mu d\zeta \\&\quad \le \int _{0}^{\infty }\int _{0}^{\delta }\frac{\zeta ^{3}}{\nu }|\frac{\nu \eta }{\mu \zeta }|^{2}e^{-\frac{\nu \eta }{\mu \zeta }}\max _{\mu }(|\mathcal {P}^{\prime }h_{0}(0,\zeta )|+|\mathcal {P}(0,\zeta )h_{0}^{\prime }|)d\mu d\zeta \\&\quad +\,\eta \int _{0}^{\infty }\int _{0}^{\delta }\zeta ^{2}|\frac{\nu \eta }{\mu \zeta }|e^{-\frac{\nu \eta }{\mu \zeta }}\max _{\mu }(|\mathcal {P}h_{0}|+|\mathcal {P}^{\prime }h_{0}^{\prime }|)d\mu d\zeta \\&\quad \le \delta \int _{0}^{\infty }\frac{\zeta ^{3}}{\nu }\max _{\mu }(|\mathcal {P}^{\prime }h_{0}(0,\zeta )|+|\mathcal {P}(0,\zeta )h_{0}^{\prime }|)d\zeta +\eta \delta \int _{0}^{\infty }\zeta ^{2}\max _{\mu }(|\mathcal {P}h_{0}|+|\mathcal {P}^{\prime }h_{0}^{\prime }|)d\zeta \\&\quad \rightarrow \delta \int _{0}^{\infty }\frac{\zeta ^{3}}{\nu }\max _{\mu }(|\mathcal {P}^{\prime }h_{0}(0,\zeta )|+|\mathcal {P}(0,\zeta )h_{0}^{\prime }|)d\zeta ,\quad \text{ as } \eta \rightarrow 0, \end{aligned}$$

where $\mathcal {P}^{\prime }$ (or $h_{0}^{\prime }$) is the partial derivative of $\mathcal {P}$ (or $h_{0}$) with respect to $\mu$ at $\mu =\mu _{P}$ (or $\mu _{h}$), and $\mu _{P}$ (or $\mu _{h}$) is a certain value in the interval $[0,\mu ]$ that depends on $\mu$ and $\zeta$. Therefore, we have

$$\begin{aligned}&\lim _{\eta \rightarrow 0}\int _{0}^{\infty }\int _{0}^{1}(1-\mu ^{2})\frac{\zeta ^{4}}{\nu ^{2}}\frac{1}{\eta }|\frac{\nu \eta }{\mu \zeta }|^{3}e^{-\frac{\nu \eta }{\mu \zeta }}\mathcal {P}(\mu ,\zeta )h_{0}(\mu ,\zeta )d\mu d\zeta \nonumber \\&\quad =\lim _{\eta \rightarrow 0}\int _{0}^{\infty }\int _{0}^{1}\frac{\zeta ^{4}}{\nu ^{2}}\frac{1}{\eta }|\frac{\nu \eta }{\mu \zeta }|^{3}e^{-\frac{\nu \eta }{\mu \zeta }}\mathcal {P}(0,\zeta )h_{0}(0,\zeta )d\mu d\zeta +O(\delta )\nonumber \\&\quad =\lim _{\eta \rightarrow 0}\int _{0}^{\infty }\frac{\zeta ^{3}}{\nu }\left( 1+\frac{\nu \eta }{\zeta }\right) e^{-\frac{\nu \eta }{\zeta }}\mathcal {P}(0,\zeta )h_{0}(0,\zeta )d\zeta +O(\delta )\nonumber \\&\quad =\lim _{\eta \rightarrow 0}\int _{0}^{\infty }\frac{\zeta ^{3}}{\nu }e^{-\frac{\nu \eta }{\zeta }}\mathcal {P}(0,\zeta )h_{0}(0,\zeta )d\zeta +O(\delta ). \end{aligned}$$

(25)

Now, in the most right-hand side of (25), we can change the order of the limit and integration, because

$$\begin{aligned} |\int _{0}^{\infty }\frac{\zeta ^{3}}{\nu }(1-e^{-\frac{\nu \eta }{\zeta }})\mathcal {P}(0,\zeta )h_{0}(0,\zeta )d\zeta |\le&\int _{0}^{\infty }\frac{\zeta }{\nu }(1-e^{-\frac{\nu \eta }{\zeta }})\zeta ^{2}|\mathcal {P}(0,\zeta )h_{0}(0,\zeta )|d\zeta \\ \le&\eta \int _{0}^{\infty }\zeta ^{2}|\mathcal {P}(0,\zeta )h_{0}(0,\zeta )|d\zeta \rightarrow 0,\quad \text{ as } \eta \rightarrow 0. \end{aligned}$$

Here, we have used the fact that $0\le x^{-1}(1-e^{-x\eta })\le \eta$ for $x\ge 0$. Therefore, we finally arrive at

$$\begin{aligned}&\lim _{\eta \rightarrow 0}\int _{0}^{\infty }\!\!\!\int _{0}^{1}(1-\mu ^{2})\frac{\zeta ^{4}}{\nu ^{2}}\frac{1}{\eta }|\frac{\nu \eta }{\mu \zeta }|^{3}e^{-\frac{\nu \eta }{\mu \zeta }}\mathcal {P}(\mu ,\zeta )h_{0}(\mu ,\zeta )d\mu d\zeta \\&\quad =\int _{0}^{\infty }\!\!\frac{\zeta ^{3}}{\nu }\mathcal {P}(0,\zeta )h_{0}(0,\zeta )d\zeta +O(\delta ). \end{aligned}$$

In the course of estimates, $\delta$ is a positive constant, arbitrary as far as smaller than unity; thus it can be made small as we wish, which proves (23).

Proof of  (24) We use the expression (13) of C[h] for small $\eta$ and its consequence (14) for $\mu \zeta >0$, namely

$$\begin{aligned} \frac{\nu ^{3}T[C[h]]}{(1-\mu ^{2})\zeta ^{2}}=\frac{1}{\eta }h_{1}(z)a(\mu ,\zeta )+h_{2}(z)b(\mu ,\zeta )\ln \eta +h_{3}(z)b(\mu ,\zeta )+h_{2}(z)c(\mu ,\zeta ), \end{aligned}$$

where

$$\begin{aligned} h_{1}(z)&=\frac{1}{2}z^{3}e^{-z},\quad h_{2}(z)=1-(\frac{1}{2}z^{2}+z+1)e^{-z},\\ h_{3}(z)&=-(\frac{1}{2}z^{2}+z+1)e^{-z}[\mathrm {Ei}(z)-\ln z-\gamma ]+\frac{1}{2}z+\frac{5}{2}(1-e^{-z})-(\frac{1}{2}z^{2}+2z)e^{-z}; \end{aligned}$$

and $h_{1}\sim h_{3}$ are $O(z^{3})$ for $z\ll 1$ and O(1) for $z\gg 1$ (remember that $z=\frac{\nu \eta }{\mu \zeta }$). Then,

$$\begin{aligned}&2\int _{0}^{\infty }\int _{0}^{1}\zeta ^{2}\mathcal {P}(\mu ,\zeta )T[C[h]]d\mu d\zeta =2\int _{0}^{\infty }\int _{0}^{1}\frac{\zeta ^{4}}{\nu ^{3}}(1-\mu ^{2})\mathcal {P}(\mu ,\zeta )[\frac{1}{\eta }h_{1}(z)a(\mu ,\zeta )\nonumber \\&\qquad \qquad \qquad \qquad +h_{2}(z)b(\mu ,\zeta )\ln \eta +h_{3}(z)b(\mu ,\zeta )+h_{2}(z)c(\mu ,\zeta )]d\mu d\zeta . \end{aligned}$$

(26)

For f(z) that follows the same estimate for both $z\ll 1$ and $z\gg 1$ as $h_{1}\sim h_{3}$ and for $Q(\mu ,\zeta )$ that is regular and decays rapidly in $\zeta$, we have

$$\begin{aligned}&|\int _{0}^{\infty }\int _{0}^{1}\frac{\zeta ^{4}}{\nu ^{3}}(1-\mu ^{2})f(z)Q(\mu ,\zeta )d\mu d\zeta |\le \int _{0}^{\infty }\frac{\zeta ^{4}}{\nu ^{3}}\max _{\mu }|Q|\int _{0}^{1}|f(z)|d\mu d\zeta \\&\qquad \le \eta \int _{0}^{\infty }\frac{\zeta ^{3}}{\nu ^{2}}\max _{\mu }|Q|\int _{\frac{\nu \eta }{\zeta }}^{\infty }z^{-2}|f(z)|dzd\zeta \le \eta \int _{0}^{\infty }\frac{\zeta ^{3}}{\nu ^{2}}\max _{\mu }|Q|d\zeta \int _{0}^{\infty }z^{-2}|f(z)|dz. \end{aligned}$$

Therefore, taking the limit $\eta \rightarrow 0$, all the terms, except the first, on the right-hand side of (26) are seen to vanish. We are left with

$$\begin{aligned}&\lim _{\eta \rightarrow 0}2\int _{0}^{\infty }\int _{0}^{1}\zeta ^{2}\mathcal {P}(\mu ,\zeta )T[C[h]]d\mu d\zeta \\&\quad =2\lim _{\eta \rightarrow 0}\frac{1}{\eta }\int _{0}^{\infty }\int _{0}^{1}\frac{\zeta ^{4}}{\nu ^{3}}(1-\mu ^{2})\mathcal {P}(\mu ,\zeta )h_{1}(z)a(\mu ,\zeta )d\mu d\zeta , \end{aligned}$$

which is none other than (24).

Proof of (16b) We use again the expression (13) of C[h] for small $\eta$ and its consequence (15) for $\mu \zeta <0$. We first rewrite (15) as

$$\begin{aligned} T[C[h]]= & {} [ f_{1}(w,\eta ,\eta _{*})a(\mu ,\zeta )+f_{2}(z,\eta )b(\mu ,\zeta )+f_{3}(w,z,y,\eta _{*})b(\mu ,\zeta )\nonumber \\&+f_{4}(w,z)c(\mu ,\zeta )+f_{5}(\mu ,\zeta ,\eta ,\eta _{*})]\frac{(1-\mu ^{2})\zeta ^{2}}{\nu ^{3}},\end{aligned}$$

(27)

$$\begin{aligned} f_{1}= & {} -\frac{w^{3}e^{-w}}{2|\eta -\eta _{*}|},\end{aligned}$$

(28)

$$\begin{aligned} f_{2}= & {} \ln \eta -(z-1)e^{z}\mathrm {E_{1}}(z)+\frac{z}{2}[ze^{z}\mathrm {E_{1}}(z)-1],\end{aligned}$$

(29)

$$\begin{aligned} f_{3}= & {} -(\ln \eta _{*})e^{-w}\left[ \left( \frac{z^{2}}{2}+\frac{y^{2}}{2}-z+1\right) \left( 1+y\right) -zy^{2}\right] -\frac{y}{2}[ye^{y}\mathrm {E_{1}}(y)-1]e^{-w}\nonumber \\&+e^{-w}\Big [\Big (y-1\Big )\Big (1+w\Big )-\frac{w^{2}}{2}\Big ]e^{y}\mathrm {E_{1}}(y)+\frac{5}{2}\Big (1-e^{-w}\Big )-e^{-w}\Big (\frac{w^{2}}{2}+2w\Big ),\nonumber \\\end{aligned}$$

(30)

$$\begin{aligned} f_{4}= & {} 1-\left( 1+w+\frac{w^{2}}{2}(1+z)+\frac{w^{3}}{2}\right) e^{-w},\end{aligned}$$

(31)

$$\begin{aligned} f_{5}= & {} -|\frac{\nu }{\mu \zeta }|^{2}\int _{\eta _{*}}^{\infty }\nu |\frac{\eta -s}{\mu \zeta }|\left( 1-\frac{\nu }{2}|\frac{\eta -s}{\mu \zeta }|\right) C[h](s,\mu ,\zeta )e^{-\nu |\frac{\eta -s}{\mu \zeta }|}ds, \end{aligned}$$

(32)

and consider the limit (i) in Sect. 4.2 (remember that $y=|\frac{\nu \eta _{*}}{\mu \zeta }|$ and $w=y-z$). Then, in this limit we have

$$\begin{aligned}&T[C[h]]\rightarrow [f_{1}(y,0,\eta _{*})a(\mu ,\zeta )+f_{2}(0,0)b(\mu ,\zeta )+f_{3}(y,0,y,\eta _{*})b(\mu ,\zeta )\\&\quad +f_{4}(y,0)c(\mu ,\zeta )+f_{5}(\mu ,\zeta ,0,\eta _{*})]\frac{(1-\mu ^{2})\zeta ^{2}}{\nu ^{3}}\equiv T[C[h]]_{0}. \end{aligned}$$

Here a special attention should be paid to the limit for $f_{2}$. It yields $f_{2}(0,0)=-\gamma -\ln \nu +\ln |\mu \zeta |$. The limits for the others are straightforward. As a result, $\int _{\mu \zeta <0}\mathcal {P}(\mu ,\zeta )T[C[h]]_{0}d\varvec{\zeta }$ converges, because the integrand diverges only at the rate $\ln |\mu |$. Thanks to the regularity in $\zeta$, the integral is identical to $2\pi \lim _{\epsilon \downarrow 0}\int _{0}^{\infty }\int _{-1}^{-\epsilon }\mathcal {P}(\mu ,\zeta )T[C[h]]_{0}\zeta ^{2}d\mu d\zeta$, so that all we have to do is just to show that

$$\begin{aligned} |\int _{\mu \zeta <0}\mathcal {P}(\mu ,\zeta )(T[C[h]]-T[C[h]]_{0})d\varvec{\zeta }|\rightarrow 0,\quad \text{ as } \eta \rightarrow 0. \end{aligned}$$

(33)

To prove this, we estimate the differences occurring in $T[C[h]]-T[C[h]]_{0}$. Firstly,

$$\begin{aligned} \Delta f_{1}&\equiv |a||f_{1}(w,\eta ,\eta _{*})-f_{1}(y,0,\eta _{*})|=\frac{y^{3}e^{-y}}{2\eta _{*}}|1-(1-\frac{\eta }{\eta _{*}})^{2}e^{\frac{\eta }{\eta _{*}}y}||a|\nonumber \\&\le \frac{y^{3}e^{-y}}{2\eta _{*}}|1-e^{\frac{\eta }{\eta _{*}}y}+\frac{\eta }{\eta _{*}}(2-\frac{\eta }{\eta _{*}})e^{\frac{\eta }{\eta _{*}}y}||a|\le [C_{1}(e^{\frac{\eta }{\eta _{*}}y}-1)+C_{2}\frac{\eta }{\eta _{*}}e^{\frac{\eta }{\eta _{*}}y}]|a|e^{-y/2}, \end{aligned}$$

(34)

$$\begin{aligned} \Delta f_{3}\equiv&|b||f_{3}(w,z,y,\eta _{*})-f_{3}(y,0,y,\eta _{*})|\nonumber \\&= |b||(\ln \eta _{*})e^{-y}\Big \{\Big (\frac{y^{2}}{2}+1\Big )\Big (1+y\Big )\nonumber \\&-e^{\frac{\eta }{\eta _{*}}y}\Big [\Big (\frac{y^{2}}{2}\frac{\eta ^{2}}{\eta _{*}^{2}}+\frac{y^{2}}{2}-\frac{\eta }{\eta _{*}}y+1\Big )\Big (1+y\Big )-\frac{\eta }{\eta _{*}}y^{3}\Big ]\Big \}\nonumber \\&\quad +\,\Big (\frac{5}{2}-\frac{y}{2}\Big )e^{-y}\Big (1-e^{\frac{\eta }{\eta _{*}}y}\Big )+\mathrm {E_{1}}(y)\Big \{1+\Big [\Big (y-1\Big )\Big (1+|1-\frac{\eta }{\eta _{*}}|y\Big )\nonumber \\&\quad -\,\frac{y^{2}}{2}|1-\frac{\eta }{\eta _{*}}|^{2}-\frac{y^{2}}{2}\Big ]e^{\frac{\eta }{\eta _{*}}y}\Big \}-\frac{y}{2}e^{-y}\Big [e^{\frac{\eta }{\eta _{*}}y}|1-\frac{\eta }{\eta _{*}}|\Big (|1-\frac{\eta }{\eta _{*}}|y+4\Big )-\Big (y+4\Big )\Big ]|\nonumber \\ \le&|b|\Big \{|\ln \eta _{*}|\Big (1+y\Big )^{3}\Big (e^{\frac{\eta }{\eta _{*}}y}-1+\frac{2\eta }{\eta _{*}}ye^{\frac{\eta }{\eta _{*}}y}\Big )e^{-y}+\Big (3+y\Big )(e^{\frac{\eta }{\eta _{*}}y}-1)e^{-y}\nonumber \\&\quad +\,\Big [e^{\frac{\eta }{\eta _{*}}y}-1+\frac{\eta }{\eta _{*}}y\Big (1+y\Big )e^{\frac{\eta }{\eta _{*}}y}\Big ]\mathrm {E_{1}}(y)+\frac{y}{2}(y+4)(e^{\frac{\eta }{\eta _{*}}y}-1+\frac{2\eta }{\eta _{*}}e^{\frac{\eta }{\eta _{*}}y})e^{-y}\Big \}\nonumber \\ \le&|b|\Big [C_{1}\Big (e^{\frac{\eta }{\eta _{*}}y}-1\Big )+C_{2}\frac{\eta }{\eta _{*}}e^{\frac{\eta }{\eta _{*}}y}\Big ]e^{-y/2}, \end{aligned}$$

(35)

$$\begin{aligned} \Delta f_{4}&\equiv |c||f_{4}(w,z)-f_{4}(y,0)|=|c|e^{-y}|y\Big \{1-\frac{1}{2}y\Big [y-2+\frac{\eta }{\eta _{*}}(1-2y)+\frac{\eta ^{2}}{\eta _{*}^{2}}y\Big ]\nonumber \\&\quad +\frac{1}{2}y^{2}\Big (3-3\frac{\eta }{\eta _{*}}+\frac{\eta ^{2}}{\eta _{*}^{2}}\Big )\Big \}\frac{\eta }{\eta _{*}}e^{\frac{\eta }{\eta _{*}}y}+\Big (1+y+\frac{1}{2}y^{2}+\frac{1}{2}y^{3}\Big )\Big (1-e^{\frac{\eta }{\eta _{*}}y}\Big )|\nonumber \\ \le&|c|(1+y)^{2}\Big [6y\frac{\eta }{\eta _{*}}e^{\frac{\eta }{\eta _{*}}y}+(1+y)(e^{\frac{\eta }{\eta _{*}}y}-1)\Big ]e^{-y}\nonumber \\ \le&|c|\Big [C_{1}(e^{\frac{\eta }{\eta _{*}}y}-1)+C_{2}\frac{\eta }{\eta _{*}}e^{\frac{\eta }{\eta _{*}}y}\Big ]e^{-y/2}, \end{aligned}$$

(36)

$$\begin{aligned} \Delta f_{5}\equiv&|f_{5}(\mu ,\zeta ,\eta ,\eta _{*})-f_{5}(\mu ,\zeta ,0,\eta _{*})|\nonumber \\&= |\frac{\nu }{\mu \zeta }|^{3}|\int _{\eta _{*}}^{\infty }\Big \{\Big (s-\eta \Big )\Big (1-\frac{\nu }{2}\frac{s-\eta }{|\mu \zeta |}\Big )e^{\frac{\nu \eta }{|\mu \zeta |}}-s\Big (1-\frac{\nu }{2}\frac{s}{|\mu \zeta |}\Big )\Big \}e^{-\frac{\nu s}{|\mu \zeta |}}C[h](s,\mu ,\zeta )ds|\nonumber \\&\le |\frac{\nu }{\mu \zeta }|^{3}\int _{\eta _{*}}^{\infty }|s(1-\frac{\nu }{2}\frac{s}{|\mu \zeta |})(e^{\frac{\nu \eta }{|\mu \zeta |}}-1)-\eta (1-\frac{\nu }{2}\frac{2s-\eta }{|\mu \zeta |})e^{\frac{\nu \eta }{|\mu \zeta |}}|e^{-\frac{\nu s}{|\mu \zeta |}}ds\max _{s\ge \eta _{*}}|C[h]|\nonumber \\&\le |\frac{\nu }{\mu \zeta }|^{3}\int _{\eta _{*}}^{\infty }\Big \{s\Big (1+\frac{\nu }{2}\frac{s}{|\mu \zeta |}\Big )\Big (e^{\frac{\nu \eta }{|\mu \zeta |}}-1\Big )+\eta (1+\frac{\nu s}{|\mu \zeta |})e^{\frac{\nu \eta }{|\mu \zeta |}}\Big \}e^{-\frac{\nu s}{|\mu \zeta |}}ds\max _{s\ge \eta _{*}}|C[h]|\nonumber \\&= \frac{y}{\eta _{*}}\{\frac{1}{2}(2+y)^{2}(e^{\frac{\eta }{\eta _{*}}y}-1)+\frac{\eta }{\eta _{*}}y(2+y)e^{\frac{\eta }{\eta _{*}}y}\}e^{-y}\max _{s\ge \eta _{*}}|C[h]|\nonumber \\&\le \Big [C_{1}(e^{\frac{\eta }{\eta _{*}}y}-1)+C_{2}\frac{\eta }{\eta _{*}}e^{\frac{\eta }{\eta _{*}}y}\Big ]e^{-y/2}\max _{s\ge \eta _{*}}|C[h]|, \end{aligned}$$

(37)

where $C_{1}$ and $C_{2}$ are positive constants, common to $\Delta f_{1}$ and $\Delta f_{3}\sim \Delta f_{5}$, and the arguments of $a(\mu ,\zeta )$, $b(\mu ,\zeta )$, and $c(\mu ,\zeta )$ are omitted for conciseness. In the above estimates, we have used that $z=(\eta /\eta _{*})y$ and $0<\eta /\eta _{*}<1/3(<1)$. Therefore, for $i=1,\,3\sim 5$, we have

$$\begin{aligned}&|\int _{\mu \zeta <0}\mathcal {P}(\mu ,\zeta )\Delta f_{i}\frac{(1-\mu ^{2})\zeta ^{2}}{\nu ^{3}}d\varvec{\zeta }|\le 2\pi \int _{\mu \zeta <0}|\mathcal {P}(\mu ,\zeta )|\Delta f_{i}\frac{\zeta ^{4}}{\nu ^{3}}d\mu d\zeta \\&\le 2\pi \int _{\mu \zeta <0}\tilde{P}(\mu ,\zeta )(C_{1}|e^{\frac{\eta }{\eta _{*}}y}-1|+C_{2}\frac{\eta }{\eta _{*}}e^{\frac{\eta }{\eta _{*}}y})e^{-y/2}\frac{\zeta ^{4}}{\nu ^{3}}d\mu d\zeta \\&= 2\pi \eta _{*}\int _{0}^{\infty }\max _{\mu }|\tilde{P}|\frac{\zeta ^{3}}{\nu ^{2}}\int _{\frac{\eta _{*}\nu }{\zeta }}^{\infty }(C_{1}|e^{\frac{\eta }{\eta _{*}}y}-1|+C_{2}\frac{\eta }{\eta _{*}}e^{\frac{\eta }{\eta _{*}}y})y^{-2}e^{-y/2}dyd\zeta \\&\le 2\pi \eta _{*}\int _{0}^{\infty }\max _{\mu }|\tilde{P}|\frac{\zeta ^{5}}{\nu ^{4}}\frac{1}{\eta _{*}^{2}}\frac{2C_{1}(e^{\frac{\eta \nu }{\zeta }}-1)+2\frac{\eta }{\eta _{*}}(C_{2}e^{\frac{\eta \nu }{\zeta }}+2C_{1})}{1-2\frac{\eta }{\eta _{*}}}e^{-\frac{\eta _{*}\nu }{2\zeta }}d\zeta \\&\le \frac{12\pi }{\eta _{*}}\int _{0}^{\infty }\max _{\mu }|\tilde{P}|\frac{\zeta ^{5}}{\nu ^{4}}[C_{1}(e^{\frac{\eta \nu }{\zeta }}-1)e^{-\frac{\eta _{*}\nu }{2\zeta }}+\frac{\eta }{\eta _{*}}(C_{2}+2C_{1})]d\zeta \\&= \frac{12\pi }{\eta _{*}}\Big \{\int _{0}^{\sqrt{\eta }}+\int _{\sqrt{\eta }}^{\infty }\Big \}\max _{\mu }|\tilde{P}|\frac{\zeta ^{5}}{\nu ^{4}}C_{1}(e^{\frac{\eta \nu }{\zeta }}-1)e^{-\frac{\eta _{*}\nu }{2\zeta }}d\zeta +O(\eta )\\&\le \frac{12\pi }{\eta _{*}}C_{1}\int _{0}^{\sqrt{\eta }}\max _{\mu }|\tilde{P}|\frac{\zeta ^{5}}{\nu ^{4}}(e^{\frac{\eta \nu }{\zeta }}-1)e^{-\frac{\eta _{*}\nu }{2\zeta }}d\zeta \\&\quad +\,\frac{12\pi }{\eta _{*}}C_{1}\int _{\sqrt{\eta }}^{\infty }\max _{\mu }|\tilde{P}|\frac{\zeta ^{5}}{\nu ^{4}}(e^{\eta c_{1}(1+\frac{1}{\sqrt{\eta }})}-1)d\zeta +O(\eta )=O(\sqrt{\eta })\rightarrow 0,\quad \mathrm{as}\ {\eta \rightarrow }\,0, \end{aligned}$$

where $\tilde{P}(\mu ,\zeta )=\max (|a|,|b|,|c|,\max _{s\ge \eta _{*}}|C[h]|)|\mathcal {P}|$. In the last line we have used that there exist positive constants $c_{0}$ and $c_{1}$ s.t. $c_{0}(1+\zeta )<\nu (\zeta )<c_{1}(1+\zeta )$. As to the remaining

$$\begin{aligned} \Delta f_{2}\equiv |f_{2}(z,\eta )-f_{2}(0,0)|=|\ln z-(z-1)e^{z}\mathrm {E_{1}}(z)+\frac{1}{2}z[ze^{z}\mathrm {E_{1}}(z)-1]+\gamma |, \end{aligned}$$

we have, as in the case of (24),

$$\begin{aligned}&\int _{\mu \zeta <0}|\mathcal {P}(\mu ,\zeta )b(\mu ,\zeta )\Delta f_{2}|\frac{(1-\mu ^{2})\zeta ^{2}}{\nu ^{3}}d\varvec{\zeta }\le 2\pi \int _{0}^{\infty }\frac{\zeta ^{4}}{\nu ^{3}}\max _{\mu }|\tilde{P}|\int _{0}^{1}\Delta f_{2}d\mu d\zeta \\&\quad =2\pi \eta \int _{0}^{\infty }\frac{\zeta ^{3}}{\nu ^{2}}\max _{\mu }|\tilde{P}|(\int _{\frac{\nu \eta }{\zeta }}^{\infty }z^{-2}\Delta f_{2}dz)d\zeta \le 2\pi \eta \int _{0}^{\infty }\frac{\zeta ^{3}}{\nu ^{2}}\max _{\mu }|\tilde{P}|(\int _{c_{0}\eta }^{\infty }z^{-2}\Delta f_{2}dz)d\zeta . \end{aligned}$$

Then, using the fact that $\Delta f_{2}$ is O(z) for $z\ll 1$ and $O(\ln z)$ for $z\gg 1$, we see that the first and the second term of the following splitting

$$\begin{aligned} \int _{c_{0}\eta }^{\infty }z^{-2}\Delta f_{2}dz=\int _{c_{0}\eta }^{c_{0}\eta _{*}}z^{-2}\Delta f_{2}dz+\int _{c_{0}\eta _{*}}^{\infty }z^{-2}\Delta f_{2}dz, \end{aligned}$$

are $O(\ln \eta )$ and O(1), respectively. And finally we obtain

$$\begin{aligned} \int _{\mu \zeta <0}|\mathcal {P}(\mu ,\zeta )b(\mu ,\zeta )\Delta f_{2}|\frac{(1-\mu ^{2})\zeta ^{2}}{\nu ^{3}}d\varvec{\zeta }\le O(\eta \ln \eta )\rightarrow 0,\quad \text{ as } \eta \rightarrow 0. \end{aligned}$$

This completes the proof of (33), thus that of (16b).

Appendix 3: Data of Computations and Measure of Accuracy

For the check of numerical accuracy, besides the grids S1–S3 and M1–M7 in [18], we have introduced a new grid S4 for $\eta$ space, which is defined by setting $(N,N_{\eta })=(200,250)$ in (B.1) of [18] and is twice as fine as the standard spatial grid S1.

The truncation of the $\zeta$ and $\eta$ spaces is justified by confirming the sufficient decay of the velocity distribution function at $\eta =d$ and $\zeta =Z$. Table 3 shows the results in the case of the standard grid (S1,M1), for which $d=44.46$ and $Z=5.0$. The sufficient decay is actually observed. One may think that the decay at $\zeta =Z$ would not be enough for $\phi E$, $\psi _{A}E$, and $\psi _{B}E$, when compared with the others. Actually, however, a small extension of Z improves a lot. For instance, $\max _{i\ge 1}|\phi _{4}E(\eta ^{(i)},\cdot ,Z)|$ improves from $2.6\times 10^{-4}$ with (S1,M1) to $3.9\times 10^{-8}$ with (S1,M6) (remind that $\phi _{4}=\bar{\phi }_{4}+\phi$), where the arrangement of grid points is common between M1 and M6 for $0\le \zeta \le 5.0$ and M6 covers a wider region, i.e., $0\le \zeta \le 5.8$. As will be shown below, the difference of the results between (S1,M1) and (S1,M6) is negligible, at least at the level of the macroscopic quantities like the slip/jump coefficients.

Table 3 Data for the decay assessment of the velocity distribution function at the edge of the computational region

Although the results in Sect. 5 are obtained by the splitting of solution explained at the beginning of Sect. 4, we have also solved $\phi _{4}$ and $\psi _{5}\sim \psi _{8}$ directly without the splitting. The results are hardly different from each other. Indeed, the difference of the results between the two manners in the Knudsen-layer functions is less than $7.9\times 10^{-10}$ in the case of grid (S1,M1). Therefore, we have examined the grid dependence of the results by the computation without splitting only. The grid dependence of the computed slip/jump coefficients is shown in Table 4. The grid in $\zeta$-space most affects the results, especially for $c_{4}^{(0)}$, $b_{7}^{(1)}$, and $b_{8}^{(1)}$ [compare the results by (S1,M1), (S1,M3), and (S1,M7), where M3 (or M1) is the grid about twice (or 3 / 2) as many points as M7]. The accuracy down to the fourth or fifth decimal place is expected from the table. M2 is the grid that refines M1 only in the range that M is small. The comparison between (S1,M1) and (S1,M2) in the table shows that M1 is fine enough for small M. The comparisons among (S1,M1), (S3,M1), and (S1,M6) in the table show that the error due to the truncation of the $\zeta$ and $\eta$ spaces is almost negligible.

Table 4 Slip/jump coefficients $c_{4}^{(0)}$ and $b_{5}^{(1)}\sim b_{8}^{(1)}$ obtained by different grids

The accuracy of the collision integral computation has already been assessed in [18] by checking the identities $C[(1,\mu \zeta ,\zeta ^{2})E]=(1,\mu \zeta ,\zeta ^{2})\nu E$ for $C=E\mathcal {C}E^{-1}$ and $C[E]=\nu E$ for $C=E\mathcal {C}^{S}E^{-1}$. With the standard grid M1, these identities are confirmed to hold within the error of $9.1\times 10^{-8}$, $1.7\times 10^{-8}$, $6.6\times 10^{-8}$, and $8.9\times 10^{-9}$ respectively, while the maximum values of $(1,\mu \zeta ,\zeta ^{2})\nu E$ are 0.13, 0.064, and 0.062 respectively. [18]

The mass, momentum, and energy balances offer another measure of accuracy. They are the following identities that are obtained from (4a) and (5a) by the integration in molecular velocity space after multiplying the collision invariants:

$$\begin{aligned}&\langle \mu \zeta \phi _{4}\rangle =-2\int _{\eta }^{\infty }Y_{2}^{(1)}(z)dz,\ \langle \mu ^{2}\zeta ^{2}\phi _{4}\rangle =-\frac{3}{2}\int _{\eta }^{\infty }[\Omega _{1}^{(0)}(z)+\Theta _{1}^{(0)}(z)]dz,\nonumber \\&\langle \mu \zeta (\zeta ^{2}-\frac{5}{2})\phi _{4}\rangle =-2\int _{\eta }^{\infty }H_{2}^{(1)}(z)dz, \end{aligned}$$

(38a)

$$\begin{aligned}&\langle \mu (1-\mu ^{2})\zeta ^{3}\psi _{7}\rangle =\frac{3}{2}\int _{\eta }^{\infty }[\Omega _{1}^{(0)}(z)+\Theta _{1}^{(0)}(z)]dz,\end{aligned}$$

(38b)

$$\begin{aligned}&\langle \mu (1-\mu ^{2})\zeta ^{3}\psi _{i}\rangle _{+}=-\langle \mu (1-\mu ^{2})\zeta ^{3}\psi _{i}\rangle _{-},\quad (i=5,6,8). \end{aligned}$$

(38c)

Here $\langle \cdot \rangle _{\pm }$ is the half-range integral with respect to the molecular velocity defined by $\langle f(\varvec{\zeta })\rangle _{\pm }=\langle f(\varvec{\zeta })\chi [0,1](\pm \mu )\rangle$. With the standard grid (S1,M1), the identities in (38a) hold within the error of $6.0\times 10^{-7}$, $3.2\times 10^{-6}$, and $5.0\times 10^{-6}$, while the maxima of their left-hand side are 0.96, 0.29, and 2.6. In the case of (38b), the error is within $5.1\times 10^{-6}$, while the maximum of the l.h.s. is 0.29. In the case of (38c), the error is within $7.6\times 10^{-7}$, $3.2\times 10^{-6}$, or $7.2\times 10^{-7}$ ($i=5,6$, or 8), while the maximum of the l.h.s. are 0.27, 0.12, or 0.13 ($i=5,6$, or 8).