2.1 The Boltzmann equation

The state of a dilute monatomic gas can be described by the VDF $f(t,\boldsymbol{x},\boldsymbol{v})$ , where $t$ is the time, $\boldsymbol{x}=(x_{1},x_{2},x_{3})$ is the space coordinate and $\boldsymbol{v}=(v_{1},v_{2},v_{3})$ is the molecular velocity. The number of gas molecules in the six-dimensional phase space $\text{d}\boldsymbol{v}\,\text{d}\boldsymbol{x}$ is given by $f(t,\boldsymbol{x},\boldsymbol{v})\,\text{d}\boldsymbol{v}\,\text{d}\boldsymbol{x}$ , and macroscopic quantities can be calculated via the velocity moments of the VDF: the molecular number density is $n=\int f\,\text{d}\boldsymbol{v}$ , the flow velocity is $\boldsymbol{V}=\int \boldsymbol{v}f\,\text{d}\boldsymbol{v}/n$ , the temperature is $T=m\int |\boldsymbol{v}-\boldsymbol{V}|^{2}f\,\text{d}\boldsymbol{v}/3kn$ , the pressure tensor is $\unicode[STIX]{x1D617}_{ij}=m\int (v_{i}-V_{i})(v_{j}-V_{j})f\,\text{d}\boldsymbol{v}$ and the heat flux is $\boldsymbol{q}=m\int |\boldsymbol{v}-\boldsymbol{V}|^{2}(\boldsymbol{v}-\boldsymbol{V})f\,\text{d}\boldsymbol{v}/2$ , where $m$ is the mass of the gas molecules, $k$ is the Boltzmann constant and the subscripts $i,j$ denote the spatial directions. The ideal gas law $p=(P_{11}+P_{22}+P_{33})/3=nkT$ is satisfied.

The dynamics of a dilute monatomic gas in the whole flow regime is governed by the Boltzmann equation,

where $\boldsymbol{v}$ and $\boldsymbol{v}_{\ast }$ are the pre-collision velocities of the two colliding gas molecules, while $\boldsymbol{v}^{\prime }$ and $\boldsymbol{v}_{\ast }^{\prime }$ are their corresponding post-collision velocities. Pre- and post-collision velocities are related through the conservation of momentum and energy,

where $\boldsymbol{u}=\boldsymbol{v}-\boldsymbol{v}_{\ast }$ is the relative pre-collision velocity and $\unicode[STIX]{x1D734}$ is a unit vector along the relative post-collision velocity $\boldsymbol{v}^{\prime }-\boldsymbol{v}_{\ast }^{\prime }$ . The deflection angle $\unicode[STIX]{x1D703}$ between the pre- and post-collision relative velocities satisfies $\cos \unicode[STIX]{x1D703}=\unicode[STIX]{x1D734}\boldsymbol{\cdot }\boldsymbol{u}/|\boldsymbol{u}|$ , $0\leqslant \unicode[STIX]{x1D703}\leqslant \unicode[STIX]{x03C0}$ . Finally, $B(\unicode[STIX]{x1D703},|\boldsymbol{u}|)$ is the non-negative collision kernel, which is determined by the intermolecular potential. For a general spherically symmetrical intermolecular potential $\unicode[STIX]{x1D719}(r)$ , the deflection angle is given as (Chapman & Cowling Reference Chapman and Cowling1970)

and the collision kernel is

Here, $W=b/r$ , with $b$ and $r$ being the aiming and centre-of-mass distances between two colliding molecules, respectively, and $W_{1}$ is the positive root of the term in the brackets in (2.3). For HS molecules of diameter $d$ , the deflection angle is determined through $b=d\cos (\unicode[STIX]{x1D703}/2)$ , hence the collision kernel is $B=d^{2}|\boldsymbol{u}|/4$ . For the (6–12) Lennard–Jones potential,

where $\unicode[STIX]{x1D716}$ is a potential depth, and $d$ is the distance at which the potential is zero, detailed calculations/forms of $B(\unicode[STIX]{x1D703},|\boldsymbol{u}|)$ can be found in Sharipov & Bertoldo (Reference Sharipov and Bertoldo2009a ) and Venkattraman & Alexeenko (Reference Venkattraman and Alexeenko2012).

2.2 Gas kinetic boundary conditions

The scattering kernel $R(\boldsymbol{v}^{\prime }\rightarrow \boldsymbol{v})$ in (1.1) gives the probability that a molecule which hits the wall surface with velocity in $[\boldsymbol{v}^{\prime },\boldsymbol{v}^{\prime }+\text{d}\boldsymbol{v}^{\prime }]$ will return to the gas with velocity in $[\boldsymbol{v},\boldsymbol{v}+\text{d}\boldsymbol{v}]$ . Without considering adsorption/desorption or chemical reactions, the scattering kernel obeys the normalization condition: $\int _{v_{n}>0}R(\boldsymbol{v}^{\prime }\rightarrow \boldsymbol{v})\,\text{d}\boldsymbol{v}=1$ , and the reciprocity relation (which states that if the gas is in equilibrium with the surface, both the incident and reflected molecules must obey the Maxwellian distribution at the surface temperature $T_{w}$ ):

where

is the Maxwellian VDF with zero velocity in the rest frame of the surface.

The most popular gas–surface BC was proposed in Maxwell (Reference Maxwell1879), and is known as the Maxwell or diffuse–specular BC. The scattering kernel reads

where the constant $\unicode[STIX]{x1D6FC}_{M}$ is the TMAC, with a value in the range of $0\leqslant \unicode[STIX]{x1D6FC}_{M}\leqslant 1$ , and $\unicode[STIX]{x1D6FF}$ is the Dirac delta function. This BC assumes that, after collision with the surface, a molecule is specularly reflected with the probability $1-\unicode[STIX]{x1D6FC}_{M}$ , otherwise it is reflected diffusely (i.e. reflected towards every direction with equal probability, in a Maxwellian velocity distribution). Purely diffuse and specular reflections take place for $\unicode[STIX]{x1D6FC}_{M}=1$ and $\unicode[STIX]{x1D6FC}_{M}=0$ , respectively.

In the Maxwell model, the TMAC is independent of the velocities (or energies) of the impinging molecules, which contradicts both theoretical and experimental investigations. To remove the deficiency in the Maxwell model, Epstein (Reference Epstein1967) proposed the generalized scattering kernel

where the probability of a gas molecules being reflected diffusely is given by $\unicode[STIX]{x1D6E9}(\boldsymbol{v})$ , which is a function of the molecular velocity. For $\unicode[STIX]{x1D6E9}(\boldsymbol{v})=\unicode[STIX]{x1D6FC}_{M}$ , the Epstein model (2.9) reduces to the Maxwell model (2.8).

Various forms of $\unicode[STIX]{x1D6E9}(\boldsymbol{v})$ can be chosen. Following the arguments that (i) for incident molecules with very low energies, most of the molecules are trapped by the attractive part of the gas–surface interaction and hence are almost completely accommodated, (ii) at high energies the degree of accommodation decreases because an increasing fraction of the molecules have sufficient energy to overcome the trapping effect and (iii) at sufficiently high energies it is found that the accommodation coefficient increases again toward a high-energy asymptote, Epstein adopted the following form of $\unicode[STIX]{x1D6E9}(\boldsymbol{v})$ :

where $\unicode[STIX]{x1D6E9}_{0}=1$ and $\unicode[STIX]{x1D6FC}$ , $\unicode[STIX]{x1D6FD}$ and $\unicode[STIX]{x1D6E9}_{1}$ are three constants. If $\unicode[STIX]{x1D6FC}>\unicode[STIX]{x1D6FD}$ , the first term in the right-hand side of (2.10) controls the low-energy behaviour of $\unicode[STIX]{x1D6E9}(\boldsymbol{v})$ , while the second term controls the high-energy behaviour. The accommodation coefficient approaches $\unicode[STIX]{x1D6E9}_{1}$ at the high-energy asymptote. This simple expression gives good agreement for the thermal accommodation coefficient (that relates the temperature jump to the heat passing through the wall) between the theory and experimental data, for various kinds of gases interacting with tungsten over a wide range of temperature (Epstein Reference Epstein1967).

In addition to the Maxwell model, the BC developed by Cercignani & Lampis (Reference Cercignani and Lampis1971) has also been widely used. The Cercignani–Lampis scattering kernel reads:

where $\boldsymbol{v}_{t}$ is the tangential velocity and

The two parameters $\unicode[STIX]{x1D6FC}_{n}$ and $\unicode[STIX]{x1D6FC}_{t}$ in the Cercignani–Lampis model are limited to $[0,1]$ and $[0,2]$ , respectively. When $\unicode[STIX]{x1D6FC}_{n}=\unicode[STIX]{x1D6FC}_{t}=1$ or $\unicode[STIX]{x1D6FC}_{n}=\unicode[STIX]{x1D6FC}_{t}=0$ , the purely diffuse or specular BCs are recovered, respectively, while for $\unicode[STIX]{x1D6FC}_{n}=0$ and $\unicode[STIX]{x1D6FC}_{t}=2$ , the Cercignani–Lampis scattering kernel describes ‘backwards’ scattering.

We believe that the Cercignani–Lampis model is popular because of the following two major factors: first, the Cercignani–Lampis model can recover the plume-like structure around the line of specular reflection in the experiment of thermal beam scattering (Cercignani Reference Cercignani1971). Second, in the free molecular limit of thermal transpiration flow, the TPD exponent (an important indicator of the performance of the Knudsen pump) can be less than 0.5 in the Cercignani–Lampis model, which agrees well with some experimental measurements (Sharipov Reference Sharipov2003b ), while the Maxwell model always predicts a TPD exponent of 0.5 at any value of the TMAC and any shape of the flow cross-section.

Klinc & Kuěčer (Reference Klinc and Kuěčer1972) also proposed an isotropic scattering to describe the gas–surface interaction:

which was recently extended in Struchtrup (Reference Struchtrup2013) by combining with the Epstein BC.

In Poiseuille and thermal transpiration flows, it can be shown that the Klinc & Kuěčer (Reference Klinc and Kuěčer1972) model yields the same mass flow rates as the diffuse BC. Therefore, in the following, only the Maxwell, Epstein and Cercignani–Lampis BCs will be considered, and special attention will be paid to the long-overlooked Epstein BC (2.9).

4.1 The linearized boundary conditions

Due to the reciprocity relation (2.6), the BC for the perturbed VDF $h$ can be obtained by simply replacing $f$ with $h$ in (1.1). Since (4.3) possesses the symmetry

the Epstein BC can be greatly simplified, as the reflected gas molecules from the surface are only resulting from the specular reflection. Also, in this case, the Klinc & Kuěčer (Reference Klinc and Kuěčer1972) BC is exactly the same with the diffuse BC. Suppose a flat wall is located at the $x_{2}$ – $x_{3}$ plane and its normal direction is along the $x_{1}$ direction. The perturbed VDF for the reflected gas molecules $h_{r}$ is related to the perturbed VDF of the impinging molecules $h_{i}$ as

For the Cercignani–Lampis BC, the scattering kernel (2.11) is simplified to (Sharipov Reference Sharipov2002, Reference Sharipov2003a ,Reference Sharipov b ):

where

Table 1.

Dimensionless flow rates in the Poiseuille flow of HS molecules between two infinite parallel plates, obtained from the LBE with the Cercignani–Lampis and Maxwell (last two columns) BCs.

4.2 Numerical results using the Cercignani–Lampis boundary condition

In this section, we present solutions of the LBE for the Poiseuille flow between two infinite parallel plates and through a circular cross-section. Although this classical problem has been investigated extensively, accurate numerical results based on the LBE and the Cercignani–Lampis BC are scarce (Garcia & Siewert Reference Garcia and Siewert2009). In our numerical simulations, the accuracy of the flow rates are controlled to within 0.5 % (Wu et al. Reference Wu, Liu, Zhang and Reese2015a ).

We first consider the Poiseuille flow between two infinite parallel plates at a distance $\ell$ . Table 1 shows the MFR and HFR for HS molecules, when the Cercignani–Lampis and Maxwell BCs are used. When the parameters $\unicode[STIX]{x1D6FC}_{n}$ , $\unicode[STIX]{x1D6FC}_{t}$ , and $\unicode[STIX]{x1D6FC}_{M}$ in the Cercignani–Lampis and Maxwell BCs are fixed, the HFR always increases when the rarefaction parameter $\unicode[STIX]{x1D6FF}$ decreases, while the MFR first decreases and then increases with $\unicode[STIX]{x1D6FF}$ , such that the famous Knudsen minimum is observed at $\unicode[STIX]{x1D6FF}\approx 1$ .

For the Maxwell BC, when $\unicode[STIX]{x1D6FF}$ is fixed, both the MFR and HFR increase significantly when the TMAC $\unicode[STIX]{x1D6FC}_{M}$ is reduced. For the Cercignani–Lampis BC, when the values of $\unicode[STIX]{x1D6FF}$ and $\unicode[STIX]{x1D6FC}_{n}$ are fixed, the MFR also increases rapidly when $\unicode[STIX]{x1D6FC}_{t}$ decreases. From (3.3) we know that $\unicode[STIX]{x1D6FC}_{t}$ is the effective TMAC of the Cercignani–Lampis BC. By choosing $\unicode[STIX]{x1D6FC}_{M}=\unicode[STIX]{x1D6FC}_{t}$ , we see in table 1 that the MFR from the Cercignani–Lampis BC increases slower than that of the Maxwell BC, as $\unicode[STIX]{x1D6FC}_{t}$ and $\unicode[STIX]{x1D6FC}_{M}$ decrease.

Figure 2.

The MFR and HFR in the Poiseuille flow between two parallel plates, when the Cercignani–Lampis and Maxwell BCs are used. Triangles: HS molecules. Dashed lines: helium. Dash-dotted lines: xenon. In the Cercignani–Lampis BC, we use $\unicode[STIX]{x1D6FC}_{n}=1$ , (a) $\unicode[STIX]{x1D6FC}_{t}=1$ and (b) $\unicode[STIX]{x1D6FC}_{t}=0.75$ . In the Maxwell BC, $\unicode[STIX]{x1D6FC}_{M}=0.75$ is used (pentagrams). (c) The TPD exponent.

The approximate analytical expression (3.3) predicts no influence of $\unicode[STIX]{x1D6FC}_{n}$ on the MFR. From the numerical simulation we see that the influence is indeed very limited. When $\unicode[STIX]{x1D6FC}_{t}$ and $\unicode[STIX]{x1D6FF}$ are fixed, the MFR decreases slightly when $\unicode[STIX]{x1D6FC}_{n}$ increases. For instance, for $\unicode[STIX]{x1D6FF}=0.01$ , the MFR is decreased by 10 % when $\unicode[STIX]{x1D6FC}_{n}$ is increased from 0.25 to 1; as $\unicode[STIX]{x1D6FF}$ increases, this influence becomes weaker and weaker.

The variation of the HFR with respect to $\unicode[STIX]{x1D6FC}_{n}$ and $\unicode[STIX]{x1D6FC}_{t}$ is more complicated than that of the MFR. First, when $\unicode[STIX]{x1D6FC}_{n}$ and $\unicode[STIX]{x1D6FF}$ are fixed, the HFR increases slightly with $\unicode[STIX]{x1D6FC}_{t}$ at large values of $\unicode[STIX]{x1D6FF}$ , while it increases with decreasing $\unicode[STIX]{x1D6FC}_{t}$ at small values of $\unicode[STIX]{x1D6FF}$ . Second, when $\unicode[STIX]{x1D6FC}_{t}=1$ and $\unicode[STIX]{x1D6FF}$ is fixed, the HFR does not change with $\unicode[STIX]{x1D6FC}_{n}$ : in fact, in this case it can be proven that the Cercignani–Lampis BC is reduced to the diffuse BC (Sharipov Reference Sharipov2002). Third, when $\unicode[STIX]{x1D6FC}_{t}(\neq 1)$ and $\unicode[STIX]{x1D6FF}$ are fixed, the HFR increases slightly with $\unicode[STIX]{x1D6FC}_{n}$ at large values of $\unicode[STIX]{x1D6FF}$ , but it increases with decreasing $\unicode[STIX]{x1D6FC}_{n}$ at smaller values of $\unicode[STIX]{x1D6FF}$ . Similar behaviours have been observed by Sharipov (Reference Sharipov2002) when the linearized Shakhov kinetic model is used instead of the LBE.

The influence of the intermolecular potential between gas molecules is also investigated. For this we choose helium and xenon, since from Sharipov & Bertoldo (Reference Sharipov and Bertoldo2009b ) it is known that the results of other noble gases such as neon, argon and krypton will lie between helium and xenon. Some typical MFR and HFR profiles are shown in figure 2, from which we see that the influence of the intermolecular potential is obvious at small values of $\unicode[STIX]{x1D6FF}$ , irrespective of the gas–surface BCs. For small values of $\unicode[STIX]{x1D6FF}$ (i.e. $\unicode[STIX]{x1D6FF}<1$ ), among the HS gas, helium and xenon, the HS gas has the largest MFR and HFR, while xenon has the smallest: in the diffuse BC, the relative differences in the MFR and HFR between HS gas and xenon are approximately 15 % when $\unicode[STIX]{x1D6FF}=0.01$ . In the Cercignani–Lampis BC with $\unicode[STIX]{x1D6FC}_{n}=1$ and $\unicode[STIX]{x1D6FC}_{t}=0.75$ , the relative differences in the MFR and HFR between HS gas and xenon are approximately 12 % and 23 %, respectively, when $\unicode[STIX]{x1D6FF}=0.01$ .

Figure 2(c) also shows the TPD exponent, which is an important parameter determining the performance of a Knudsen pump. The TPD exponent is defined as follows: consider a closed system consisting of two reservoirs connected by a long channel. If the temperature ratio $T_{1}/T_{2}$ is maintained between the reservoirs, a pressure ratio $p_{1}/p_{2}$ is established between them. When the steady state is reached, the two ratios are related to each other as

where $\unicode[STIX]{x1D6FE}$ is the TPD exponent. If the temperature ratio between the two reservoirs is small, it can be expressed as

It can be found from figure 2(c) that, in the range of $\unicode[STIX]{x1D6FF}$ considered, the HS gas has the largest TPD, while xenon has the smallest. This difference increases when $\unicode[STIX]{x1D6FF}$ decreases. For instance, in the diffuse BC, the relative difference in TPD exponent between the HS gas and xenon is approximately 8 % when $\unicode[STIX]{x1D6FF}=0.01$ . In the Cercignani–Lampis BC, when $\unicode[STIX]{x1D6FF}$ is fixed, the TPD exponent decreases with $\unicode[STIX]{x1D6FC}_{t}$ . In the Maxwell BC, when $\unicode[STIX]{x1D6FC}_{M}$ decreases, the TPD exponent decreases at large values of $\unicode[STIX]{x1D6FF}$ , but at small values of $\unicode[STIX]{x1D6FF}$ (free molecular flow regime), the TMAC $\unicode[STIX]{x1D6FC}_{M}$ does not have any influence on the TPD exponent.

Figure 3.

The relative error $(G_{Lennard{-}Jones}/G_{HS}-1)\times 100$ in the MFR between the Lennard–Jones and HS potentials, in the Poiseuille (dashed lines) and thermal transpiration (solid lines) flows through a tube, when the diffuse BC is used.

Table 2.

Dimensionless flow rates in the Poiseuille flow of HS molecules through a circular tube, using the Cercignani–Lampis BC.

Next we consider the Poiseuille flow through a long tube, where the characteristic length $\ell$ is chosen as the radius of the circular cross-section. The flow rates are shown in table 2. Unlike to Poiseuille flow between two parallel plates, where the MFR and HFR increase logarithmically as $-\ln \unicode[STIX]{x1D6FF}$ when $\unicode[STIX]{x1D6FF}\rightarrow 0$ (Takata & Funagane Reference Takata and Funagane2011), both approach constant values when $\unicode[STIX]{x1D6FF}\rightarrow 0$ . The influence of the BC on the dimensionless flow rates is similar to that between two parallel plates.

Figure 3 shows the influence of the intermolecular potential on the MFR in the thermal transpiration flow. For $\unicode[STIX]{x1D6FF}>0.5$ , the HS model underpredicts the MFR of the Lennard–Jones potentials, for example, when $\unicode[STIX]{x1D6FF}=10$ , by approximately 8 % and 4 % for argon and helium, respectively. When $\unicode[STIX]{x1D6FF}<0.5$ , however, the HS model overpredicts the MFR. When $\unicode[STIX]{x1D6FF}\rightarrow 0$ , the intermolecular potential has no influence on the dimensionless mass flow rate. On the other hand, the influence of the intermolecular potential in the MFR of the Poiseuille flow is within 2 % for all the rarefaction parameters considered.

5.1 Thermal transpiration through a rectangular cross-section

Consider the thermal transpiration of helium through a long rectangular channel made of polyether ether ketone (Yamaguchi et al. Reference Yamaguchi, Rojas-Cárdenas, Perrier, Graur and Niimi2014, Reference Yamaguchi, Perrier, Ho, Méolans, Niimi and Graur2016), where the aspect ratio of the rectangular cross-section is 27.27. For such a large aspect ratio, the numerical simulation based on the Shakhov kinetic model revealed that, when $\unicode[STIX]{x1D6FF}\geqslant 0.5$ , there is no difference in the dimensionless flow rates for flows through two infinite parallel plates and rectangular cross-sections (Graur & Ho Reference Graur and Ho2014). Thus, we use the numerical data for flows between two parallel plates to reduce the computational cost, since the experimental measurements are limited to this region of the rarefaction parameter.

Figure 4.

The MFR in the thermal transpiration of helium through a rectangular cross-section of aspect ratio 27.27. Solid dots: experimental data collected from table 1 in Yamaguchi et al. (Reference Yamaguchi, Perrier, Ho, Méolans, Niimi and Graur2016). Pentagrams: the diffuse BC. Squares: the Cercignani–Lampis BC with $\unicode[STIX]{x1D6FC}_{n}=0.25$ and $\unicode[STIX]{x1D6FC}_{t}=0.5$ . Triangles: the Cercignani–Lampis BC with $\unicode[STIX]{x1D6FC}_{n}=0.25$ and $\unicode[STIX]{x1D6FC}_{t}=1.75$ . Lines without symbols: the Epstein BC (2.9) with the thermal slip coefficient $\unicode[STIX]{x1D714}=0.9$ : dash-dotted line for $\unicode[STIX]{x1D6E9}_{0}=0.9$ and $\unicode[STIX]{x1D6FC}=0.0192$ , while solid line for $\unicode[STIX]{x1D6E9}_{0}=1$ and $\unicode[STIX]{x1D6FC}=0.019$ . Other parameters are $\unicode[STIX]{x1D6E9}_{1}=0.1$ and $\unicode[STIX]{x1D6FD}=\unicode[STIX]{x1D6FC}$ , so that according to (3.4), the effective TMAC $\unicode[STIX]{x1D712}$ are 0.85 and 0.95 when $\unicode[STIX]{x1D6E9}_{0}=0.9$ and 1, respectively. Inset: the MFR in the Poiseuille flow between two parallel plates.

Figure 4 and table 3 compare the experimental data with the numerical results, when the diffuse, Cercignani–Lampis, and Epstein BCs are used. From table 1 we know that, for the Maxwell BC, when $\unicode[STIX]{x1D6FF}<3.5$ is fixed, the lowest MFR is reached for $\unicode[STIX]{x1D6FC}_{M}=1$ . The comparison with the experiment shows that the diffuse BC overpredicts the MFR by more than 10 %, and choosing other values of the TMAC will increase the prediction error drastically. We then turn to the Cercignani–Lampis BC. From table 1 we know that, for large values of $\unicode[STIX]{x1D6FF}$ , better agreement between the simulation and experiment can be achieved when we choose small values of $\unicode[STIX]{x1D6FC}_{n}$ and $\unicode[STIX]{x1D6FC}_{t}$ , see the squares in figure 4 and data denoted by ‘Cercignani–Lampis1’ in table 3, where $\unicode[STIX]{x1D6FC}_{n}=0.25$ and $\unicode[STIX]{x1D6FC}_{t}=0.5$ . However, when $\unicode[STIX]{x1D6FF}<2$ , the agreement becomes even worse than that of the diffuse BC, for example, for $\unicode[STIX]{x1D6FF}=2.22$ and $0.998$ , this Cercignani–Lampis BC overpredicts the MFR by approximately 16 % and 42 %, respectively. One can significantly reduce this difference by choosing large values of $\unicode[STIX]{x1D6FC}_{t}$ (see triangles in figure 4 and data corresponding to ‘Cercignani–Lampis2’ in table 3, where $\unicode[STIX]{x1D6FC}_{n}=0.25$ and $\unicode[STIX]{x1D6FC}_{t}=1.75$ ), but this increases the error when $\unicode[STIX]{x1D6FF}$ is large. For example, when $\unicode[STIX]{x1D6FF}>5$ , numerical results overpredicts the MFR by approximately 20 %. This large difference in the slip regime is caused by the fact that the thermal slip coefficient of the Cercignani–Lampis BC is larger than one (Sharipov Reference Sharipov2003a ), which is much larger than the experimental measured value of approximately $0.9$ .

Table 3.

The MFR in the thermal transpiration of helium between two infinite parallel plates, using the diffuse, Cercignani–Lampis, and Epstein BCs. The experimental data are collected from table 1 in Yamaguchi et al. (Reference Yamaguchi, Perrier, Ho, Méolans, Niimi and Graur2016). The relative error between the experimental and numerical results is defined as $100\times (G_{T}^{numerical}/G_{T}^{exp}-1)$ . The parameters for various BCs are given in figure 4.

The Epstein BC allows more flexibility to choose the velocity slip coefficients and should be able to give better agreement with the experiment. To demonstrate this, we choose the effective thermal slip coefficient $\unicode[STIX]{x1D714}$ to be the experimental measured value (Yamaguchi et al. Reference Yamaguchi, Perrier, Ho, Méolans, Niimi and Graur2016). To achieve this, we choose $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FD}$ in (2.10) for simplicity. We first fix the values of $\unicode[STIX]{x1D6E9}_{0}$ and $\unicode[STIX]{x1D6E9}_{1}$ , and obtain $\unicode[STIX]{x1D6FC}$ and $\unicode[STIX]{x1D6FD}$ from (3.5) by setting $\unicode[STIX]{x1D714}=0.9$ . We then vary the values of $\unicode[STIX]{x1D6E9}_{0}$ and $\unicode[STIX]{x1D6E9}_{1}$ , to see the possible influence of different parameters. Our numerical results show that $\unicode[STIX]{x1D6E9}_{1}$ has very small influence on the MFR, so in figure 4 and table 3 only the results for $\unicode[STIX]{x1D6E9}_{1}=0.1$ are shown. It is clear that for $\unicode[STIX]{x1D714}=0.9$ , the simulation results agree well with the experimental data when $\unicode[STIX]{x1D6FF}\gtrsim 2$ , for a wide range of $\unicode[STIX]{x1D6E9}_{0}$ , see $\unicode[STIX]{x1D6E9}_{0}=0.9$ and 1 in figure 4 and table 3. For $\unicode[STIX]{x1D6FF}\lesssim 2$ , the value of $\unicode[STIX]{x1D6E9}_{0}$ begins to have a strong influence on the MFR: the larger the value of $\unicode[STIX]{x1D6E9}_{0}$ , the smaller the MFR in thermal transpiration flow. Taking into account that the experimental data have large errors when $\unicode[STIX]{x1D6FF}<1$ (the accuracy of the measurement decreases when the pressure decreases because the physical variation of the pressure is no longer very great with respect to the resolution of the pressure sensor), it seems that the case of $\unicode[STIX]{x1D6E9}_{0}=1$ provides good agreement with the experimental measured MFR.

Figure 5.

The TPD exponent and TPR in the thermal transpiration of helium through a rectangular cross-section of aspect ratio 27.27. Solid dots: experimental data collected from figure 8 in Yamaguchi et al. (Reference Yamaguchi, Rojas-Cárdenas, Perrier, Graur and Niimi2014). Pentagrams, squares, triangles and lines without symbols: see the parameters in figure 4.

So far, based solely on the comparison in figure 4 and table 3, it is too early to say that the Epstein BC is better than the Cercignani–Lampis BC. In the inset of figure 4 we find that the MFR in Poiseuille flow varies a lot among different BCs. Therefore, the TPD exponent and thermal molecular pressure ratio (TPR) should vary significantly between the two BCs. Fortunately, these two parameters have been measured experimentally using the same gas and solid surface (Yamaguchi et al. Reference Yamaguchi, Rojas-Cárdenas, Perrier, Graur and Niimi2014), which provides an ideal case to assess the accuracy of the Epstein and Cercignani–Lampis BCs.

In the two experiments (Yamaguchi et al. Reference Yamaguchi, Rojas-Cárdenas, Perrier, Graur and Niimi2014, Reference Yamaguchi, Perrier, Ho, Méolans, Niimi and Graur2016), the temperature difference is small compared to the average gas temperature. Therefore, the TPD exponent can be accurately approximated by (4.15), while the TPR is calculated as follows: in steady state, according to (3.2) in Yamaguchi et al. (Reference Yamaguchi, Perrier, Ho, Méolans, Niimi and Graur2016), the gas pressure $p$ along the flow direction satisfies

where $p$ and $T$ have been normalized by the initial pressure and the average pressure of the hot and cold reservoirs, respectively. Assuming a linear temperature variation along the rectangular channel, the pressure distribution can be obtained easily; we find that the result does not change even when using the exponential shape of the temperature distribution (Rojas-Cárdenas et al. Reference Rojas-Cárdenas, Graur, Perrier and Méolans2013), due to the small temperature difference between two reservoirs in the experiments. The TPR is then calculated as the ratio of the pressures of the cold and hot reservoirs.

Figure 5 compares the TPD exponent and TPR between the experimental and numerical results using different gas–surface BCs. It becomes clear that the Epstein BC with $\unicode[STIX]{x1D6E9}_{0}=0.9$ (other parameters shown in figure 4) gives the best agreement, while the Cercignani–Lampis BC either overestimates ( $\unicode[STIX]{x1D6FC}_{t}=1.75$ ) or underestimates ( $\unicode[STIX]{x1D6FC}_{t}=0.5$ ) the TPD exponent significantly. For the TPR, the trend is opposite to that of the TPD exponent. We believe the discrepancies between the very Epstein BC and the experimental data when $\unicode[STIX]{x1D6FF}<2$ are due to the experimental errors, which become larger and larger as the gas pressure reduces.

5.2 Poiseuille flow through a rectangular cross-section

Consider Poiseuille flow of helium through a silicon microchannel, with a rectangular cross-section of aspect ratio 52.45, subject to a pressure ratio of 5 between the inlet and outlet of the long channel. With such a large pressure ratio, the experimental measurement of the MFR can be made very accurate even in the free molecular flow regimes. This provides a more strict test of the gas–surface BC, as the profile of the perturbed VDF varies significantly at small values of $\unicode[STIX]{x1D6FF}$ . For instance, both the theoretical and numerical analysis (Takata & Funagane Reference Takata and Funagane2011; Wu et al. Reference Wu, Reese and Zhang2014) show that the width of the VDF in the normal direction to the wall surface is proportional to the rarefaction parameter $\unicode[STIX]{x1D6FF}$ , when $\unicode[STIX]{x1D6FF}\rightarrow 0$ . Thus, the effective TMAC in § 3 obtained using the VDF from the Chapman–Enskog expansion becomes inaccurate.

Since the pressure ratio is not small, the dimensionless MFR $G_{P}(\unicode[STIX]{x1D6FF})$ , obtained from the LBE for the Lennard–Jones potential, is transformed to the measured MFR $G(\unicode[STIX]{x1D6FF}_{m})$ by (Sharipov & Seleznev Reference Sharipov and Seleznev1994)

where $\unicode[STIX]{x1D6FF}_{m}$ is the gas rarefaction parameter at the average value of the inlet and outlet pressures, with the characteristic flow length $\ell$ being the shorter side of the rectangular cross-section.

Figure 6.

The comparison of the MFR in the Poiseuille flow through a rectangular cross-section of aspect ratio 52.45 with the experimental data (Ewart et al. Reference Ewart, Perrier, Graur and Méolans2007), where the LBE with the Lennard–Jones potential of helium is solved, with the Maxwell, Cercignani–Lampis and Epstein BCs.

At such a large aspect ratio, when $\unicode[STIX]{x1D6FF}>1$ , the numerical results for the Poiseuille flow between two parallel plates can be safely used (Graur & Ho Reference Graur and Ho2014). However, when $\unicode[STIX]{x1D6FF}<1$ , the numerical simulation is performed in the two-dimensional cross-section: the symmetry is considered and only one quarter of the rectangular cross-section is simulated, which is approximated by $31\times 101$ non-uniform cells, with most of the cells adjacent to the surface (Wu et al. Reference Wu, Reese and Zhang2014).

Figure 6 compares the MFR between the experimental and the numerical results, when the Maxwell, Cercignani–Lampis and Epstein BCs are used. Since in the slip flow regime the measured TMAC is approximately 0.92, we choose $\unicode[STIX]{x1D6FC}_{M}=0.92$ in the Maxwell BC, $\unicode[STIX]{x1D6FC}_{t}=0.92$ in the Cercignani–Lampis BC with $\unicode[STIX]{x1D6FC}_{n}=1$ and $\unicode[STIX]{x1D712}=0.92$ in the Epstein BC, together with $\unicode[STIX]{x1D6E9}_{0}=1$ , $\unicode[STIX]{x1D6E9}_{1}=0.1$ and $\unicode[STIX]{x1D6FC}=\unicode[STIX]{x1D6FD}=0.0293$ . At $\unicode[STIX]{x1D6FF}_{m}>2$ , the three BCs yield almost identical results. However, as $\unicode[STIX]{x1D6FF}_{m}$ decreases, the Cercignani–Lampis and Epstein BCs generate smaller MFRs than the Maxwell BC. For the Epstein BC, this behaviour is easy to understand: as $\unicode[STIX]{x1D6FF}$ decreases, the VDF in the normal direction to the wall shrinks (Takata & Funagane Reference Takata and Funagane2011; Wu et al. Reference Wu, Reese and Zhang2014), so that according to (2.10), fewer gas molecules are specularly reflected than that of the Maxwell BC and the MFR is therefore smaller.

In general, it is seen that the Cercignani–Lampis and Epstein BCs perform better than the Maxwell BC in this special case. But since the MFR in the thermal transpiration flow is not available, it is hard to say which one is best. If only MFR in the Poiseuille flow is concerned, we prefer to use the Epstein BC, since in the discrete velocity method the computational cost for the gas–surface interaction is of the order of $N_{v}^{3}$ , while that of the Cercignani–Lampis BC is $N_{v}^{6}$ , where $N_{v}$ is the number of discretized points in each velocity direction.

5.3 Thermal transpiration through a long tube

Rojas-Cárdenas et al. measured the thermal transpiration flow through a glass tube of circular cross-section (Rojas-Cárdenas et al. Reference Rojas-Cárdenas, Graur, Perrier and Méolans2013), where, like the thermal transpiration along the channel of rectangular cross-section (Yamaguchi et al. Reference Yamaguchi, Rojas-Cárdenas, Perrier, Graur and Niimi2014, Reference Yamaguchi, Perrier, Ho, Méolans, Niimi and Graur2016), data of the MFR, TPD exponent and TPR are available. This means that both the viscous and thermal slip coefficients $\unicode[STIX]{x1D712}$ and $\unicode[STIX]{x1D714}$ in the various BCs need to be adjusted simultaneously, and again, provides a tough assessment of the various BCs.

Before the comparison, it is noted that, according the definitions in Rojas-Cárdenas et al. (Reference Rojas-Cárdenas, Graur, Perrier and Méolans2013) and Yamaguchi et al. (Reference Yamaguchi, Perrier, Ho, Méolans, Niimi and Graur2016), the experimental measured MFR $G(\unicode[STIX]{x1D6FF}_{m})$ is related to the simulated MFR $G_{T}=-Q_{P}$ as

where $\unicode[STIX]{x1D6FF}_{m}$ is the gas rarefaction parameter at the initial gas pressure and mean temperature of the two reservoirs, with the characteristic flow length $\ell$ being the radius of the tube.

The experiments are conducted for both argon and helium. In the following we consider them separately since different gases have different interactions with the same glass tube. Also, it is noted that when the temperature difference is small, the TPD exponent and the TPR are closely related: if the BC can accurately describe the TPR, it can definitely describe the TPD exponent with good accuracy, see the example in figure 5. For this reason, in the following only the MFR and TPR are considered. Finally, the experimental error is large when $\unicode[STIX]{x1D6FF}$ is small, i.e. when the gas pressure is low such that the pressure sensor is no longer very sensitive to the pressure variations and the accuracy of the MFR measurement is reduced. Therefore, in the following, we only analyse the region where $\unicode[STIX]{x1D6FF}>2$ .

Figure 7.

The MFR and TPR in the thermal transpiration of argon through a circular glass tube. Solid dots: the experimental MFR is adopted from figure 8 in Rojas-Cárdenas et al. (Reference Rojas-Cárdenas, Graur, Perrier and Méolans2013) and has been normalized by (5.3), while the TPR is adopted from figure 9 in Rojas-Cárdenas et al. (Reference Rojas-Cárdenas, Graur, Perrier and Méolans2013) when the temperature difference is 71 K. Solid lines: diffuse BC. Dashed lines: the Epstein BC with $\unicode[STIX]{x1D6E9}_{0}=\unicode[STIX]{x1D6E9}_{1}=1$ and $\unicode[STIX]{x1D6FC}=20\unicode[STIX]{x1D6FD}=50.6$ , so that the thermal slip coefficient is $\unicode[STIX]{x1D714}=1.1$ and the effective TMAC is $\unicode[STIX]{x1D712}=0.98$ . Dotted lines: the Cercignani–Lampis BC with $\unicode[STIX]{x1D6FC}_{t}=1.2$ and $\unicode[STIX]{x1D6FC}_{n}=0.25$ .