Rarefied gas flow through channels of finite length at various pressure ratios

Vacuum 86 (2012) 1952e1959

Contents lists available at SciVerse ScienceDirect

Vacuum journal homepage: www.elsevier.com/locate/vacuum

Rarefied gas flow through channels of finite length at various pressure ratios Stylianos Varoutis a, *, Christian Day a, Felix Sharipov b a b

Karlsruhe Institute of Technology (KIT), Institute for Technical Physics, Hermann-von-Helmholtz-Platz 1, 76344 Karlsruhe, Germany Departamento de Física, Universidade Federal do Paraná, Caixa Postal 19044, Curitiba 81531-990, Brazil

a r t i c l e i n f o

a b s t r a c t

Article history: Received 7 March 2012 Received in revised form 20 April 2012 Accepted 21 April 2012

A rarefied gas flow through channels (i.e. flow through parallel plates) of finite length has been modeled based on the direct simulation Monte Carlo method. The reduced flow rate and the flow field have been calculated as function of the gas rarefaction, the length-to-height ratio and the pressure ratio upstream and downstream of the channel. The whole range of the gas rarefaction including the free-molecular, transitional and hydrodynamic regimes and a wide range of the length-to-height ratio representing both short and long channels have been considered. Several values of the pressure ratio between 0 and 0.5 have been used in the calculations. It is shown that the rarefaction parameter has the most significant effect on the flow field characteristics and patterns, followed by the pressure ratio, while the length-toheight ratio has a rather modest impact. The Mach belt phenomenon is discussed in detail. Ó 2012 Elsevier Ltd. All rights reserved.

Keywords: Vacuum flows Slit flow Microchannels DSMC Knudsen number Transition regime

1. Introduction Rarefied gas flow through channels (i.e. flow through parallel plates) of finite length is considered to be of major practical and theoretical importance in many industrial applications of rarefied gas dynamics, as for instance in measuring devices under arbitrary vacuum conditions [1], in micro propulsion systems [2], in membranes and porous media and in the fabrication and optimization of micro electro mechanical systems [3]. Due to the simplicity of the present flow configuration, the channel flow was proposed as a benchmark problem in rarefied gas dynamics [4]. Several experimental and numerical investigations have been performed in the past, following the pioneering work of Knudsen [5] and Clausing [6]. Recently, the slit flow (i.e. flow through a channel with zero length-to-height ratio) was numerically investigated in Refs. [7e13] for various pressure ratios by applying the stochastic direct simulation Monte Carlo (DSMC) approach [14], while in Refs. [15e21] by solving the corresponding kinetic equation [22,23]. In the case of gas flow through channels of finite length into vacuum, some results based on the DSMC method are reported by Sazhin [24,25]. Few values of the flow rate through a channel into vacuum calculated from the kinetic equation are presented in Ref. [20], where the Unified Flow Solver with an adaptive mesh was used.

* Corresponding author. E-mail address: [email protected] (S. Varoutis). 0042-207X/$ e see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.vacuum.2012.04.032

The opposite limit, i.e. the gas flow through a channel driven by a small pressure drop, is considered in Refs. [26e29] applying the linearized kinetic equation. However, to the best of our knowledge no results on the channel flow in wide rages of the pressure drop, gas rarefaction and aspect ratio are available in the literature. A comparison of computational time needed for DSMC to simulate the slit flow [13] (2D geometry) and the orifice flow [30] (axi-symmetrical geometry) showed that the former requires more effort, because of a larger computational domain in the case of 2D flows. The same increase of the computational time is expected for the simulation of the channel flow in comparison with the short tube flows [31,32]. The aim of the present work is to calculate the gas flow through channels of finite length at various values of the pressure ratio, in the whole range of the Knudsen number by applying the DSMC method [14]. 2. Statement of the problem Consider a steady state flow of monatomic gas through a channel (i.e. through parallel plates) of height H and length L, connecting two semi-infinite reservoirs as is shown in Fig. 1. The channel is assumed to be so large in the z-direction, e.g. the width of the channel is significantly larger than its height, so that the lateral wall influence is neglected and the flow becomes twodimensional. As was shown in Ref. [33], the lateral wall influence on the flow rate through a rectangular channel depends on the gas rarefaction. Since in practice the width is always finite, its influence

S. Varoutis et al. / Vacuum 86 (2012) 1952e1959

1953

y´

L1

L2 P0, T0 P1, T0

weight 1 H1

weight 2

L

H2

weight 3

H/2

x´

(0,0) Fig. 1. Scheme of the flow, coordinates and weighting zones.

L1

y L2

H1 H2

L

H/2

x (0,0) Fig. 2. Three-level computational grid.

on the flow rate should be estimated using the data presented in [33]. Both reservoirs contain the same monatomic gas and constant pressures p0 and p1 are maintained far from the channel entrances in the left and right containers, respectively. For the sake of definiteness we assume p1 < p0. The temperatures of the gas far from channel in both containers are the same and equal to T0, while the temperature of all surfaces is also maintained to T0. The solution of the problem is determined by three parameters, namely, the length-to-height ratio L/H of the channel, the pressure ratio p1/p0 and rarefaction parameter defined as

is the mass flow rate through a slit (L/H ¼ 0) with an expansion into vacuum (p1/p0 ¼ 0) in the free-molecular limit (d ¼ 0).

d ¼ Hp0 =m0 v0 ;

Table 1 Reduced flow rate W vs rarefaction parameter d and aspect ratio L/H at p1/p0 ¼ 0.

(1)

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where m0 is the shear viscosity of the gas at T0 and v0 ¼ 2kT0 =m is the most probable molecular speed with k and m denoting the Boltzmann constant and molecular mass of the gas, respectively. Since the quantity m0v0/p0 represents the equivalent free path, the rarefaction parameter d is inversely proportional to the Knudsen number, i.e. d f 1/Kn. Hence, the limit d / 0 represents the freemolecular regime, while the case of d [ 1 corresponds to the viscous regime. The results will be given in terms of the reduced flow rate W through the channel defined as

_ M _ ; W ¼ M= 0

(2)

_ is the mass flow rate through the channel at any L/H, p1/p0 where M and d, while

pffiffiffi _ ¼ Hp = pv M 0 0 0

(3)

3. The DSMC algorithm The problem under investigation was solved by the DSMC method, which consists of a simulation of a huge number of model particles decoupling their free motion and collisions between them. The method is described in detail in previous papers [12,13,30e32]

d

W L/H ¼ 0.1

0.5

1

2

5

0 0.01 0.05 0.1 0.2 0.5 1 2 5 10 20 50 100 200 500 1000

0.952 0.953 0.962 0.973 0.993 1.03 1.09 1.17 1.31 1.42 1.49 1.53 1.54 1.54 1.54 1.54

0.805 0.806 0.813 0.821 0.835 0.870 0.913 0.980 1.10 1.21 1.30 1.39 1.44 1.48 1.50 1.50

0.684 0.686 0.692 0.698 0.708 0.732 0.767 0.818 0.927 1.04 1.15 1.29 1.36 1.42 1.45 1.46

0.542 0.542 0.545 0.549 0.555 0.570 0.590 0.622 0.706 0.811 0.953 1.14 1.25 1.32 1.39 1.39

0.357 0.356 0.357 0.357 0.356 0.355 0.358 0.368 0.411 0.490 0.626 0.860 1.02 1.14 1.22 1.25

1954

S. Varoutis et al. / Vacuum 86 (2012) 1952e1959

Table 2 Reduced flow rate W vs rarefaction parameter d and aspect ratio L/H at p1/p0 ¼ 0.1.

d

W L/H ¼ 0.1

0.5

1

2

5

0 0.01 0.05 0.1 0.2 0.5 1 2 5 10 20 50 100 200 500 1000

0.857 0.858 0.870 0.878 0.900 0.943 1.01 1.11 1.29 1.40 1.48 1.52 1.54 1.54 1.54 1.54

0.724 0.727 0.734 0.741 0.757 0.793 0.843 0.923 1.09 1.20 1.30 1.40 1.44 1.48 1.50 1.50

0.616 0.616 0.622 0.630 0.641 0.668 0.706 0.768 0.902 1.03 1.15 1.28 1.36 1.41 1.45 1.46

0.498 0.489 0.493 0.496 0.502 0.519 0.541 0.580 0.679 0.800 0.945 1.14 1.25 1.32 1.38 1.39

0.321 0.321 0.321 0.321 0.322 0.322 0.325 0.338 0.389 0.477 0.621 0.859 1.02 1.14 1.22 1.24

this number of model particles is adequate to ensure numerical results within the prescribed error in all flow regimes. In addition, several computational sizes of the upstream and downstream reservoirs have been tested and the minimum ones, which guarantee an invariance in the results of less than 1% have been selected. It has been found that this is achieved by taking H1 ¼ L1 ¼ 30H,

so that only some modifications needed to adapt the scheme to the problem under consideration will be given. For the simulation of the intermolecular collisions the no-time counter scheme [14] was used. The hard-sphere model was implemented for the intermolecular potential. Finally it is noted that the purely diffuse gassurface interaction model was applied, since it is assumed that in general the gas-surface model has no significant impact on the flow rate, as it was shown for the case of short tubes [25,30,31]. Since the flow is symmetric with respect to the x-axis, it is enough to simulate only the upper half (y > 0) of the gas flow domain. The computational mesh, shown in Fig. 2, is unstructured with quadrilateral cells of three different sizes. Such a statically adaptive mesh is required in order to capture the steep macroscopic gradients close to the walls and maintain a reasonable computational resolution. In addition, as qualitatively shown in Fig. 1, three weighting zones have been introduced to ensure more uniform distributions of model particles over the computational domain, see [12,30,31]. The computational parameters have been accordingly chosen in order to ensure that the results on the flow rate would have numerical uncertainty within 1%. In particular, in all cases, the time increment and the largest cell size of the grid are taken equal to Dt ¼ 0.01H/v0 and Dx ¼ 0.05H, respectively. Also, in all cases the number of model molecules is N ¼ 4  107. It has been found that

Table 3 Reduced flow rate W vs rarefaction parameter d and aspect ratio L/H at p1/p0 ¼ 0.5.

d

W L/H ¼ 0.1

0.5

1

2

5

0 0.01 0.05 0.1 0.2 0.5 1 2 5 10 20 50 100 200 500 1000

0.476 0.478 0.486 0.494 0.511 0.548 0.608 0.711 0.968 1.19 1.32 1.39 1.42 1.42 1.42 1.42

0.402 0.403 0.411 0.417 0.428 0.458 0.505 0.587 0.787 1.01 1.18 1.32 1.38 1.41 1.44 1.44

0.342 0.345 0.348 0.354 0.363 0.386 0.419 0.478 0.635 0.832 1.03 1.21 1.30 1.35 1.40 1.41

0.271 0.272 0.275 0.278 0.283 0.295 0.316 0.353 0.457 0.611 0.819 1.06 1.18 1.26 1.32 1.33

0.178 0.178 0.180 0.181 0.181 0.181 0.186 0.199 0.247 0.332 0.489 0.767 0.947 1.07 1.15 1.17

Fig. 3. Reduced flow rate W vs d for various values of p1/p0 and L/H.

S. Varoutis et al. / Vacuum 86 (2012) 1952e1959

L2 ¼ 20H and H2 ¼ 10H, see Fig. 2. Finally, it is noted that the same investigation has been performed for different cell sizes of the computational grid and the final parameters also guarantee the invariance in the results less than 1%. The total number of cells is 416,800 and the number of particles in each cell varies between 10 and 100. The evolution of the system (i.e. the number of times steps or samples) is terminated when the relative statistical scattering of the flow rate satisfies the condition

1.8

p1/p0=0

1.6 1.4 1.2

W

1

pffiffiffiffiffiffiffi Nþ < 0:005; þ N  N

0.8 0.6 δ=0.1 δ=1 δ=10 δ=100 δ=1000

0.4 0.2 0

1955

0

1

2

3

4

5

L/H

(4)

where Nþ and N denote the total number of particles crossing the channel from left to right and in the opposite direction, respectively. The calculation of macroscopic quantities is performed by sampling the corresponding particle information according to the procedure described in Refs. [12,13,32].

1.8 4. Results and discussions

p1/p0=0.1

1.6

The numerical calculations were carried out in the following range of the rarefaction parameter 0  d  1000 for five values of

1.4 1.2

5

W

1

δ=1

y/H

0.8 0.6 δ=0.1 δ=1 δ=10 δ=100 δ=1000

0.4 0.2 0

0

1

0 -5

2

3

4

10

5

x/H 15

δ=10

y/H

1.8 δ=0.1 δ=1 δ=10 δ=100 δ=1000

p1/p0=0.5

1.4 1.2

0

1

W

5

5

L/H 1.6

0

-5

0

5

10

x/H 15

0.8 5

0.6

δ=100

y/H

0.4 0.2 0

0

1

2

3

4

5

0

L/H

-5

0

5

10

x/H 15

Fig. 4. Reduced flow rate W vs L/H for various p1/p0 and d.

5

δ=1000

y/H Table 4 Comparison with Ref. [25] at L/H ¼ 1 and p1/p0 ¼ 0.

d

W Present

Ref. [25]

jDWj=W (%)

0.1 1 10 100

0.698 0.767 1.04 1.36

0.697 0.764 1.024 1.351

0.1 0.4 1.5 0.7

0 -5

0

5

10

Fig. 5. Streamlines for various d, for L/H ¼ 5 and p1/p0 ¼ 0.1.

x/H 15

1956

S. Varoutis et al. / Vacuum 86 (2012) 1952e1959

the aspect ratio L/H ¼ 0.1, 0.5, 1, 2, 5 and for three values of the pressure ratio p1/p0 ¼ 0, 0.1, 0.5. Note that the obtained results cover actually the whole range of the rarefaction parameter d and a reasonably wide range of the aspect ratio L/H and of the pressure ratio p1/p0. As was pointed out in Ref. [4], for the pressure ratio p1/p0 close to unity the statistical scattering of the numerical results becomes so large that the computational efforts to apply the DSMC are not justified. Thus, a numerical solution of the kinetic equation, see Refs. [18e20], is more appropriate in the rage 0.5 < p1/p0  1.

parallel plates). An analytical expression of Wch is provided in Refs. [23,34]. As d is increased from 0 to unity, W increases very slowly, then in the range 1  d  100 there is a significant increase of W and finally, at the large values of d, W keeps growing very weakly reaching asymptotically the continuum results at the hydrodynamic limit (d / N). It is noted that the results for p1/ p0 ¼ 0 and 0.1 are almost identical when d > 10. This is a typical choked flow, when the mass flow rate does not depend of the downflow pressure.

4.1. Flow rate

1.2

Inlet

δ=1 δ=10 δ=100 δ=1000

1 0.8

p/p 0

The results for the reduced flow rate W in terms of d and L/H are presented in Tables 1e3 for p1/p0 ¼ 0, 0.1 and 0.5, respectively. In order to analyze the behavior of the flow rate, its numerical results are plotted in Fig. 3, for three values of the aspect ratio L/H ¼ 0.1, 1 and 5. In all cases, the qualitative dependency of W on d is similar. More specifically, in the free-molecular limit (d / 0) the reduced flow rate tends to its theoretical value of Wch(1  p1/p0), where Wch is the transmission probability of molecules through a channel (i.e.

Outlet

0.6 0.4

5

0.2

δ=1

y/H

0 -10

-5

0

5

x

10

15

20

25

1.2

Inlet

0 -5

0

5

10

5

x/H 15

0.8

δ=10

T/T0

y/H

δ=1 δ=10 δ=100 δ=1000

1

Outlet

0.6 0.4

0 -5

0

5

10

5

0.2

x/H 15

0 -10

δ=100

y/H

-5

0

5

x

10

15

20

25

15

20

25

8 δ=1 δ=10 δ=100 δ=1000

6 -5

0

5

10

5

x/H 15

Ma

0

4

δ=1000

y/H

Inlet 2

Outlet 0 -10

0 -5

0

5

10

Fig. 6. Streamlines for various d, for L/H ¼ 5 and p1/p0 ¼ 0.5.

x/H

15

-5

0

5

x

10

Fig. 7. Distributions of pressure p/p0, temperature T/T0 and local Ma along the axis y ¼ 0 for p1/p0 ¼ 0 and L/H ¼ 5.

S. Varoutis et al. / Vacuum 86 (2012) 1952e1959

In Fig. 4, results of W in terms of L/H are shown for p1/ p0 ¼ 0,0.1,0.5 and d ¼ 101,1,10,102,103. As expected in most cases W has its maximum value at L/H ¼ 0 and then it decreases as the length-to-height ratio is increased. It is observed however, that for p1/p0 ¼ 0.5 and d ¼ 1000, W has a slight maximum at L/H ¼ 0.5. This phenomenon, which is present in the hydrodynamic regime and for high pressure ratios, although it has been reported in the numerical work in [12] and in the experimental work in [35,36], has not been yet physically explained.

A comparison of the present results with those published previously [25] is performed in Table 4. It can be seen that in most cases the flow rates obtained here and in Ref. [25] are in agreement with the numerical accuracy 1%, except for the case of d ¼ 10, where higher deviation was found. 4.2. Flow field In Figs. 5 and 6 a more detailed look throughout the flow field is performed by providing the streamlines for the case of p1/

1.2

Inlet

0.8

Inlet

0.8

Outlet

0.6 0.4

0.2

-5

0

5

x

10

15

20

0 -10

25

1.2

Outlet

0

5

Inlet

1

x

10

15

T/T0

0.4

Outlet

0.6 0.4

δ=1 δ=10 δ=100 δ=1000

0.2

-5

0

5

x

10

15

20

0 -10

25

-5

0

5

x

10

15

20

25

1.2

δ=1 δ=10 δ=100 δ=1000

3

δ=1 δ=10 δ=100 δ=1000

0.2

4

δ=1 δ=10 δ=100 δ=1000

Inlet 1

Ma

0.8

Ma

25

0.8

0.6

2

0.6 0.4

1

20

1

0.8

T/T0

-5

1.2

Inlet

0 -10

Outlet

0.6 0.4

0.2 0 -10

δ=1 δ=10 δ=100 δ=1000

1

p/p0

p/p0

1.2

δ=1 δ=10 δ=100 δ=1000

1

1957

Outlet

Outlet

Inlet

0.2

0 -10

-5

0

5

x

10

15

20

25

Fig. 8. Distributions of pressure p/p0, temperature T/T0 and local Ma along the axis y ¼ 0 for p1/p0 ¼ 0.1 and L/H ¼ 5.

0 -10

-5

0

5

x

10

15

20

25

Fig. 9. Distributions of pressure p/p0, temperature T/T0 and local Ma along the axis y ¼ 0 for p1/p0 ¼ 0.5 and L/H ¼ 5.

1958

S. Varoutis et al. / Vacuum 86 (2012) 1952e1959

p0 ¼ 0.1,0.5, L/H ¼ 5 and d ¼ 1,10,102,103. Since the behavior of streamlines for the case of L/H ¼ 0.1 is quite similar to the one for the case of slit flow [13], their presentation is omitted. More specifically, for the case of flows in the transition regime and slip regime (i.e. d ¼ 1 and 10) the streamlines are almost symmetric upstream and downstream of the channel, while for d [ 10 the streamlines bend and vortices start to be observed just after the exit of the channel. At d ¼ 100 the vortices have been already created and as d increases these vortices are enlarged and in the hydrodynamic limit i.e. d ¼ 1000 strong jets of gas along the y ¼ 0 axis are observed. The axial distributions of the pressure p1/p0, temperature T/T0 and local Mach number (Ma) along the symmetry axis for L/H ¼ 5 are presented in Figs. 7e9 for the aspect ratio L/H ¼ 5 and for the pressure ratio p1/p0 ¼ 0, 0.1 and 0.5, respectively. The local Ma is calculated based on the pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi axial bulk velocity ux according to Ma ¼ ðux =v0 Þ 2T0 =ðT gÞ, where T is the local temperature and g ¼ 5/3 for a monatomic gas. It can be seen that for p1/p0 ¼ 0, see Fig. 7, the pressure p monotonically decreases starting from its upstream equilibrium values p0 and reaching the vacuum conditions (p1 / 0) in the downstream container. The temperature T also monotonically decreases starting from its equilibrium value T0. Naturally, the local Ma monotonically increases beginning from zero. As was mentioned in Ref. [12], if the downstream reservoir would contain a background gas, then the temperature and local Ma far from the channel outlet could tend to the equilibrium values T0 and to zero, respectively.

The behavior of the flow field at p1/p0 ¼ 0.1 (see Fig. 8) for large values of the gas rarefaction d is qualitatively different from that at p1/p0 ¼ 0. Namely, for d > 10 just after the outlet of the channel, an oscillatory behavior of all quantities with reducing amplitude around its equilibrium value is observed. Such a phenomenon is called Mach belts and is due to the compression waves created in the exit from the channel. This behavior was also reported in the case of slit flow (L/H ¼ 0) in Ref. [13]. When the pressure ratio is closer to unity, i.e. p1/p0 ¼ 0.5, the Mach belts disappear as is shown in Fig. 9. In this case, the pressure p reaches its downstream value p1 just after the channel outlet. The temperature T reaches its minimum value in the channel outlet. Then it sharply begins to tend to its equilibrium value T0. The local Ma has the opposite behavior, i.e., it reaches its maximum value in the channel outlet and then sharply tends to zero. In order to have a more detailed picture of the Mach belt phenomenon, the color contours of the flow field just after the exit of the channel for d ¼ 1000, p1/p0 ¼ 0.1 and L/H ¼ 5 are depicted in Fig. 10. It can be seen that the gas forms a series of barrels in the outlet of the channel, which are separated by the Mach belts. The barrels basically are stretched in the region from 2H to 2H in the y direction. This flow behavior has been also reported in Ref. [20] where rarefied gas flows through 2D channels and 3D tubes were simulated by an adaptive kinetic-fluid solver. As it is also stated in Ref. [13], the number of barrels is restricted by the computational domain, but their further increase will not affect the mass flow rate W. The flow fields for other values of the aspect ratio L/H and the same values of d and p1/p0 are quite similar to those for L/H ¼ 5 and not presented here. 5. Concluding remarks The rarefied gas flow through a channel (i.e. parallel plates) of finite length has been investigated computationally in terms of the three parameters determining the flow problem, namely the rarefaction parameter d, the length-to-height ratio L/H and the pressure ratio p1/p0. The numerical scheme is based on the DSMC method. The results are presented in tabulated and graphical forms for the flow rate, pressure, temperature and the Mach number for the following range of the main parameters: 0  d  1000, 0  L/H  5 and 0  p1/p0  0.5. Since the presented results correspond to wide ranges of the all three involved parameters, their effect on the flow patterns and characteristics has been examined in a detailed and systematic manner. It has been shown that the flow quantities and characteristics strongly depend on the rarefaction parameter d and pressure ratio p1/p0, while their dependency on the aspect ratio L/H is weaker. Near the hydrodynamic regime, the behavior of the flow field changes qualitatively by varying the pressure ratio. Namely, at p1/ p0 ¼ 0.1, the Mach belts are formed just past the channel outlet, while for the smaller and lager pressure ratio such belts are not observed. The present results can be used to design and/or optimize vacuum systems and microfluidics devices. Acknowledgments

Fig. 10. Fields of pressure p/p0, temperature T/T0 and local Ma at d ¼ 1000, p1/p0 ¼ 0.1 and L/H ¼ 5.

The work of SV has been supported by the European Fusion Development Agreement (EFDA), within the framework of a Fusion Researcher Fellowship. Also, FS gratefully acknowledges the support by the Conselho Nacional de Desenvolvimento Cientifico e Tecnologico (CNPq, Brazil). The views and opinions expressed herein do not necessarily reflect those of the European Commission.

S. Varoutis et al. / Vacuum 86 (2012) 1952e1959

References [1] Jousten K, editor. Handbook of vacuum technology. Weinheim: WILEY-VCH Verlag GmbH & Co KGaA; 2008. [2] Ketsdever AD, Clabough MT, Gimelshein SF, Alexeenko A. Experimental and numerical determination of micropropulsion device efficiencies at low Reynolds number. AIAA J 2005;43(3):633e41. [3] Gad-el-Hak M, editor. The MEMS handbook. Florida, USA: CRC Press; 2002. [4] Sharipov F. Benchmark problems in rarefied gas dynamics. Vacuum 2012; 86(11):1697e700. [5] Knudsen M. Die molekularstromung der gase durch offnungen und die effusion. Ann Phys 1909;28:999. [6] Clausing P. The flow of highly rarefied gases through tubes of arbitrary length. J Vac Sci Technol 1971;8(5):636e46. [7] Wadsworth DC, Erwin DA. Numerical simulation of rarefied flow through a slit. part I: direct simulation Monte Carlo results. Phys Fluids A 1993;5(1): 235e42. [8] Chung CH, Jeng DR, De Witt KJ, Keith Jr TG. Numerical simulation of rarefied gas flow through a slit. J Thermophys Heat Transfer 1992;6(1): 27e34. [9] Sazhin O. Pressure-driven flow of rarefied gas through a slit at a various pressure ratios. J Vac Sci Tech A 2012;30(2):021603. [10] Sazhin O. Gas flow through a slit into a vacuum in a wide range of rarefaction. J Exp Theor Phys 2008;107(1):162169. [11] Sazhin O. Gas molecule-molecule interaction and the gas-surface scattering effect on the rarefied gas flow through a slit into a vacuum. J Exp Theor Phys 2009;108(5):874e9. [12] Sharipov F, Kozak DV. Rarefied gas flow through a thin slit into vacuum simulated by the Monte Carlo method over the whole range of the Knudsen number. J Vac Sci Technol A 2009;27(3):479e84. [13] Sharipov F, Kozak DV. Rarefied gas flow through a thin slit at an arbitrary pressure ratio. Eur J Mech B/Fluids 2011;30(5):543e9. [14] Bird GA. Molecular gas dynamics and the direct simulation of gas flows. Oxford: Oxford University Press; 1994. [15] Hasegawa M, Sone Y. Rarefied gas flow through a slit. Phys Fluids A 1991;3(3): 466e77. [16] Cercignani C, Sharipov F. Gaseous mixture slit flow at intermediate Knudsen numbers. Phys Fluids A 1992;4(6):1283e9. [17] Sharipov F. Rarefied gas flow through a slit: influence of the gas-surface interaction. Phys Fluids 1996;8(1):262e8. [18] Graur I, Polikarpov AP, Sharipov F. Numerical modelling of rarefied gas flow through a slit into vacuum based on the kinetic equation. Comput Fluids 2011;49:87e92.

1959

[19] Graur I, Polikarpov AP, Sharipov F. Numerical modelling of rarefied gas flow through a slit at arbitrary gas pressure ratio based on the kinetic equation. Z Angew Math Phys (ZAMP), in press. [20] Aristov VV, Frolova AA, Zabelok SA, Arslanbekov RR, Kolobov VI. Simulations of pressure-driven flows through channels and pipes with unified flow solver. Vacuum 2012;86(11):1717e24. [21] Misdanitis S, Pantazis S, Valougeorgis D. Pressure driven rarefied gas flow through a slit and an orifice. Vacuum 2012;86(11):1701e8. [22] Cercignani C. The Boltzmann equation and its application. New York: Springer-Verlag; 1988. [23] Sharipov F, Seleznev V. Data on internal rarefied gas flows. J Phys Chem Ref Data 1998;27(3):657e706. [24] Sazhin O. Rarefied gas flow through a channel of finite length into a vacuum. J Exp Theor Phys 2009;109(4):700706. [25] Sazhin O. Impact of the gas-surface scattering and gas moleculeemolecule interaction on the mass flow rate of the rarefied gas through a short channel into a vacuum. J Vac Sci Tech A 2010;28(6):1393e8. [26] Akinshin VD, Makarov AM, Seleznev VD, Sharipov FM. Flow of a rarefied gas in a plane channel of finite length for a wide range of Knudsen numbers. J Appl Mech Tech Phys 1988;29(1):97e103. [27] Akinshin VD, Makarov AM, Seleznev VD, Sharipov FM. Rarefied gas motion in a short planar channel over the entire Knudsen number range. J Appl Mech Tech Phys 1989;30(5):713e7. [28] Sharipov FM, Seleznev VD, Makarov AM. Nonisothermal motion of a rarefied gas in a short planar channel over a wide range of Knudsen numbers. J Eng Phys Termophys 1990;59(1):869e75. [29] Titarev VA. Implicit high-order method for calculating rarefied gas flow in a planar microchannel. J Comp Phys 2012;231:109e34. [30] Sharipov F. Numerical simulation of rarefied gas flow through a thin orifice. J Fluid Mech 2004;518:35e60. [31] Varoutis S, Valougeorgis D, Sazhin O, Sharipov F. Rarefied gas flow through short tubes into vacuum. J Vac Sci Technol A 2008;26(2):228e38. [32] Varoutis S, Valougeorgis D, Sharipov F. Simulation of gas flow through tubes of finite length over the whole range of rarefaction for various pressure drop ratios. J Vac Sci Technol A 2009;22(6):1377e91. [33] Sharipov F. Rarefied gas flow through a long rectangular channel. J Vac Sci Technol A 1999;17(5):3062e6. [34] Berman AS. Free molecule transmission probabilities. J Appl Phys 1965; 36(10):3356. [35] Fujimoto T, Usami M. Rarefied gas flow through a circular orifice and short tubes. J Fluids Eng-Tran ASME 1984;106(4):367e73. [36] Jitschin W, Weber U, Hartmann HK. Convenient primery gas flow meter. Vacuum 1995;46:821e4.