Non-stationary rarefied gas flow into vacuum from a circular pipe closed at one end

Vacuum 109 (2014) 284e293

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Non-stationary rarefied gas flow into vacuum from a circular pipe closed at one end E.M. Shakhov a, V.A. Titarev b, c, * a

Bauman Moscow State Technical University, Moscow, Russia Dorodnicyn Computing Centre of Russian Academy of Sciences, Moscow, Russia c Moscow Institute of Physics and Technology, Moscow, Russia b

a r t i c l e i n f o

a b s t r a c t

Article history: Received 24 March 2014 Received in revised form 6 June 2014 Accepted 8 June 2014 Available online 16 June 2014

The paper is devoted to the analysis of the time-dependent rarefied gas flow into vacuum from a circular pipe closed at one end. The problem is studied numerically on the basis of the S-model kinetic equation. Limiting solutions corresponding to the one-dimensional free-molecular and hydrodynamic flow regimes are also considered. The results demonstrate the dependence of flow pattern and evacuation time on the Knudsen number. © 2014 Elsevier Ltd. All rights reserved.

Keywords: Rarefied gas flow S-Model kinetic equation Gas evacuation Time-dependent flow

1. Introduction Steady flows of rarefied gas in channels and pipes are covered in reviews [1e3]. Recently, non-stationary time-dependent flows have also drawn attention, e.g. Refs. [4e7]. An important class of such problems concerns the calculation of escape time of gas into vacuum from a reservoir initially filled with this gas. Possibly the simplest problem of this class is that of the gas outflow into vacuum from a circular pipe closed at one end. This is the problem studied in the present work. The analysis is based on the direct numerical solution of the Boltzmann kinetic equation with the S-model collision integral [8,9]. The main goal is to determine the law of gas evacuation from the pipe and general flow pattern as functions of the Knudsen number. 2. Formulation of the problem The problem is specified in the Cartesian coordinate system x,y,z with the Oz axis directed along the pipe. At the initial time moment the rarefied gas at rest characterized by number density n0 and temperature T0 occupies a circular pipe of radius a and length L. The pipe is permanently closed at one end z ¼ L. At the outlet position z ¼ 0 (other end of the pipe) a diaphragm separates the pipe and an infinitely large reservoir, in which there is no gas. At the start of the * Corresponding author. Dorodnicyn Computing Centre of Russian Academy of Sciences, Moscow, Russia. E-mail addresses: [email protected] (E.M. Shakhov), [email protected], titarev@ mail.ru (V.A. Titarev). http://dx.doi.org/10.1016/j.vacuum.2014.06.007 0042-207X/© 2014 Elsevier Ltd. All rights reserved.

process t ¼ 0 the diaphragm is removed and a non-stationary flow of the gas from the pipe into the reservoir starts. The details of the process depend on the length of the pipe and degree of gas rarefaction. For the rest of the paper, the non-dimensional formulation is used, in which time t, the spatial coordinates x ¼ (x,y,z) ¼ (x1,x2,x3), mean u ¼ (ux,uy,uz)¼(u1,u2,u3) and particle x ¼ (x1,x2,x3) velocities, number density n, temperature T, heat flux vector q ¼ (qx,qy,qz) ¼ (q1,q2,q3), viscosity m and distribution function f are scaled using the following quantities:

a=v0 ; a; v0 ; n0 ; T0 ; mn0 v30 ; m0 ¼ mðT0 Þ; n0 v3 0 ;

(1)

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where m is the mass of a molecule, v0 ¼ 2kT0 =m is the most probable molecular speed, k is the Boltzmann constant. Below, the nondimensional variables are denoted by the same letters as the dimensional ones. A time-dependent three-dimensional state of the rarefied gas is determined by the velocity distribution function f(t,x,x), which is assumed to satisfy the S-model kinetic equation [8,9]:

  vf vf nT ap þxa ¼ n f ðSÞ f ; n ¼ d0 ; d0 ¼ 0 ; vt vxa m m0 v0      4 5 n 2 ; fM ¼ ; exp c f ðSÞ ¼ fM 1þ ð1PrÞSa ca c2  5 2 ðpTÞ3=2 v 2qi vi ¼ xi ui ; ci ¼ piffiffiffi; Si ¼ 3=2 ; c2 ¼ cb cb : T nT (2)

E.M. Shakhov, V.A. Titarev / Vacuum 109 (2014) 284e293

Here p0 ¼ n0T0 is initial gas pressure in the pipe, d0 is the socalled rarefaction parameter, which is inversely proportional to the Knudsen number. Summation over repeated Greek indices is assumed. For a monatomic gas the Prandtl number Pr ¼ 2/3. The hard-sphere pffiffiffi intermolecular interaction is used in all calculations so that m ¼ T . The non-dimensional macroscopic quantities are defined as the integrals of the velocity distribution function with respect to the molecular velocity:



v2 ¼ va va ;

x2 ¼ xa xa ;

1 g

(3)

p ¼ nT:

z  0;

(4)

z > 0:

On the surface of the pipe the condition of diffuse molecular scattering on the pipe surface with complete thermal 1

-10

0.5

-5

0

z

5

Fig. 2. Axial distributions of number normalized mass flow g (solid lines) and normalized flow rate Q(t, z) (symbols) for d0 ¼ 0. Curves 1 … 4 correspond to t ¼ 2, 10, 50 and 100.

accommodation to the non-dimensional surface temperature T0 ≡ 1 is assumed. The density of reflected molecules is found from the impermeability condition stating that the mass flux through the wall is equal to zero. The same condition is used for the parts of the 1

1

2

4

0

The initial condition for the distribution function f(t,x,x) is given

n

2

0.5

by

8   < 1 exp x2 ; f ð0; x; xÞ ¼ p3=2 : 0;

1

3

   Z   3 1 T þ u2 ; q ¼ n; nu; n 1; x; x2 ; vv2 fdx; 2 2

u2 ¼ ua ua ;

285

1 2 3 4

n

3

0.5

4 0 0

-10

-5

0

z

5 -10

(a) number density n

-5

0

z

5

z

5

(a) number density n 1 2

g

1

1.2 g

2 0.5

0.8

3 0.4

4 0 -10

-5

0

z

5

(b) normalized mass flow g

0 -10

Fig. 1. Axial distributions of number density and normalized mass flow g (Eq. (12)) for d0 ¼ 0. Curves 1 … 4 correspond to t ¼ 2, 10, 50 and 100; solid black line e kinetic solution for n and g, red dashed line e Eq. (11) for t ¼ 2, 10, blue dash-dotted line e hydrodynamic solution (9) for t ¼ 2, 10. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

-5

0

(b) normalized mass flow Fig. 3. Axial distributions of number density n and normalized mass flow g at output time t ¼ 10. The curve numbers correspond to: (1), (2) e numerical solutions for d0 ¼ 1 and d0 ¼ 100, respectively, (3) e Eqs. (11) and (4) e hydrodynamic solution (9).

286

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1.5

reservoir walls directly adjacent to the pipe; these are located at z ¼ 0 (outlet position). At the rest of the reservoir walls the distribution function of the molecules moving into the flow domain is set to zero. An integral computed quantity is the non-dimensional mass flow rate. For a given pipe section at a position z it is defined as integral of the mass flow in the z coordinate direction over the cross section:

g 1

1

0.5

_ Mðt; zÞ ¼

Z nðt; xÞuz ðt; xÞdxdy:

(5)

2

In the presentation of the results it is more convenient to use the normalized flow rate Q(t,z), which is defined as (here it is taken into account that the non-dimensional pipe radius a ¼ 1):

0 -10

-5

0

z

5

Fig. 4. Comparison of axial distributions of normalized mass flow g (solid line) and flow rate Q (circles) at output time t ¼ 10. The curve numbers correspond to: (1) e d0 ¼ 1, (2) e d0 ¼ 100.

n

Q ðt; zÞ ¼

_ Mðt; zÞ _ orifice M 0

1.6 uz

0.6 1 2 3 4

0.4

;

_ orifice ¼ pqorifice ; M 0 0

1 qorifice ¼ pffiffiffi : 0 2 p

(6)

1 2 3 4

1.2 0.8

0.2 0.4

0

0

-10

-5

0

z

5

-10

-5

(a) number density n

0

z

5

0

z

5

0

z

5

(a) velocity

p

6

0.6

M 4

0.4 2

0.2

0

0 -10

-5

0

z

-10

5

(b) Mach number

(b) pressure p 1

0.4

T

g

0.5

0.2

0

0

-10

-5

-5

0

z

5

(c) temperature Fig. 5. Axial distributions of number density, pressure and temperature at output time t ¼ 50. Curves 1e4 correspond to d0 ¼ 0, 1, 10 & 100, respectively.

-10

-5

(c) normalized mass flow Fig. 6. Axial distributions of velocity, Mach number and normalized mass flow g at output time t ¼ 50. Curves 1e4 correspond to d0 ¼ 0, 1, 10 & 100, respectively.

E.M. Shakhov, V.A. Titarev / Vacuum 109 (2014) 284e293

Here M0orifice is the mass flow rate in the steady free-molecular flow through an orifice. Of particular interest is the flow rate through the outlet section of the pipe:

Q outlet ðtÞ ¼ Q ðt; 0Þ:

287

1

nend 1 2 3

(7)

At the initial time moment for any d0 it is equal to Qoutlet(0) ¼ 1. The solution of the problem depends on the rarefaction parameter d0 of the initial gas at rest and the non-dimensional pipe length L.

0.5

3. Limiting cases 0

The analysis of the results first considers the limiting cases with respect to the parameter d0: hydrodynamic expansion and freemolecular flow. In the hydrodynamic limit d0 / ∞ gas expansion process is described by the Euler equations of the compressible gas. In the pipe region L < z < 0 there exists an exact solution [10,11]. Immediately after the diaphragm is removed a centred rarefaction wave travels in the gas. The front of the wave moves with the speed of sound c0. At the initial stage of the expansion the following simple relations hold:

z uz  c ¼ ; t

uz þ 3c ¼ 3c0 ;

¼ n2=3 ;

c0 ¼

L < z < 0;

t<

pffiffiffiffiffiffiffiffi 5=6;

t 0

50

100

150

200

150

200

(a) Number density 1

Tend

0.99

T

L c0

(8) 0.98

From here the explicit expression for the solution is given by 1

Qoutlet

0.97

t 0

50

100

(b) Temperature 1 2 3 4

0.5

Fig. 8. Number density and temperature at the centre of the closed end of the pipe x ¼ y ¼ 0, z ¼ L as functions of time. Curves 1e3 correspond to d0 ¼ 1, 10, 100.

uz ¼ 0

t 0.001

0.01

0.1

1

(a) Normalized flow rate Q

10

100

outlet

c ¼ uz  z=t;

6 T ¼ c2 ; 5

n ¼ T 3=2 :

(9)

At the outlet section z ¼ 0 a flow with the speed of sound forms, thanks to which the one-dimensional flow is separated from the three-dimensional supersonic one at z > 0. Using relations (9), it is possible to obtain the value of the hydrodynamic flow rate through a circular pipe in the non-stationary expansion into vacuum:

1

u ¼

W

3 z c0 þ ; 4 t

3 c ; 4 0

n ¼

27 ; 64

_  ¼ n u pz0:907 M

(10)

When the front of the wave reaches the end of the pipe z ¼ L at time moment t0 ¼ L/c0 it reflects from it. After that, the flow process is described by more complicated relations. At the final stage of gas expansion t / ∞ gas density in the pipe is almost constant along

0.5

Table 1 Normalized flow rate Qoutlet and total mass of gas in the pipe W as functions and rarefaction parameter d0.

0

t 0.001

0.01

0.1

1

10

t

100

(b) Normalized total mass of gas in the pipe Fig. 7. Normalized flow rate and total mass of gas in the pipe defined in Eq. (13) as functions of time. Curves 1e4 correspond to d0 ¼ 0, 1, 10, 100.

10 25 50 100 200

Qoutlet

W

d0 ¼ 0

d0 ¼ 1

d0 ¼ 100

d0 ¼ 0

d0 ¼ 1

d0 ¼ 100

0.472 0.317 0.216 0.112 0.031

0.484 0.320 0.219 0.110 0.029

0.779 0.560 0.210 0.047 0.007

0.812 0.652 0.468 0.244 0.067

0.809 0.646 0.460 0.238 0.065

0.755 0.461 0.216 0.068 0.013

288

E.M. Shakhov, V.A. Titarev / Vacuum 109 (2014) 284e293

the pipe axis and decreases as 1/t. The amount of total mass of gas decreases with the same law [11]. Gas temperature goes to zero according to the adiabatic law. Another limiting case is that of the free-molecular flow with the specular molecular reflection from the pipe surface. This case corresponds to the one-dimensional free expansion of the planar layer of gas initially occupying the region 2L  z  0. For the spatial distribution of density and velocity we have

    1 1  n ¼ ðerfðs1 Þerfðs2 ÞÞ; nuz ¼ pffiffiffi exp s21 þexp s22 ; 2 2 p s1 ¼

zþ2L z 2 ; s2 ¼ ; erfðsÞ ¼ pffiffiffi t t p

Zs

  exp q2 dq:

0

(11) For more details see e.g. Ref. [12] where the process of plane layer expansion has been studied numerically for arbitrary degrees of rarefaction. Both discussed limiting cases correspond to the specular reflection from the pipe surface. Therefore, they are not representative for studying flows in long pipes for which the strong influence of the surface results in gas deceleration. 4. Results for the general flow In the general case the formulated problem is solved numerically using an explicit version of the time-marching algorithm

[13,14]. Although the flow is axisymmetrical in the physical domain, the kinetic equation is solved in the three-dimensional form, which allows to avoid the difficulties, associated with the use of the axisymmetrical form of the kinetic equation. The mesh convergence studies for the steady flow through a short circular pipe carried out in Ref. [14] demonstrated a very good accuracy of such an approach to axe-symmetrical flows. The numerical method, which combines forward in time Euler time approximation and second-order accurate non-linear spatial discretization can be viewed as an extension of the pioneering scheme of Kolgan [15,16]. Calculations were performed for the pipe length to radius ratio L/a ¼ 10 and up to t ¼ 200 using a spatial mesh of 53,865 hexa cells. Details of the mesh construction and some error estimates can be found below in Section 5. Most of the runs were carried out on the high-performance cluster of the FlowModellium lab of the Moscow Institute of Physics and Technology, Moscow, Russia whereas the rest of the calculations were performed on the ”Chebyshev” computer of Lomonosov Moscow State University, Russia. As a rule, 128 cores were utilised for a single run. Since the slow-down influence of the pipe's surface manifests itself best for the free-molecular regime, it is convenient to start analysis from this case. Fig. 1 shows for various time moments axial distributions of number density n and axial mass flow normalized by the orifice unit mass flow

. g ¼ nuz qorifice : 0

Fig. 9. Contour levels of number density and normalized mass flow for d0 ¼ 1 and t ¼ 10.

(12)

E.M. Shakhov, V.A. Titarev / Vacuum 109 (2014) 284e293

For t ¼ 2, 10 also plotted is the one-dimensional free-molecular solution given by Eq. (11). Fig. 2 depicts comparison of the normalized flow rate Q(t,z) and axial distribution of g. It is clearly seen that the computed kinetic solution for the pipe flow is lagging behind the one-dimensional free-molecular one for the appropriate time moments. The mass flow generally decreases with time from its value at t ¼ 0. It will be shown below, however, that for d0 [1 there may be a small increase at the beginning of the process. Density profiles inside the pipe also decrease with time. Figs. 3 and 4 illustrate the influence of the rarefaction parameter on the gas flow in the pipe. Shown are the results for the threedimensional kinetic solution for d0 ¼ 1, 100 as well as limiting cases. For this output time the plots for d0 ¼ 0 coincide with the results for d0 ¼ 1 and are thus omitted. Figs. 5 and 6 show the axial data, but for the later time t ¼ 50 and d0 ¼ 0, 1, 10 and 100. It is seen that the influence of the pipe surface results in significantly slower propagation of the wave into the pipe even for d0 ¼ 100. As d0 increases, the mass flow grows whereas density and pressure fall faster. This is explained by the greater importance of the pressure effect in the nearly continuum flow regime. There is a slight dip in density profiles near the end of the pipe, which is explained by the temperature drop for d0 [1, see Fig. 8 in the analysis below. Temperature drops significant as gas expands into vacuum. It should be noted that for d0 ¼ 100 the mass flow curve goes above the one-dimensional hydrodynamic solution (Euler equations of the compressible gas). However, the total flow rate does not exceed its limiting value for d0 / ∞ as the mass flow in the kinetic solution almost disappears near the pipe surface.

289

Fig. 7 illustrates the behaviour of normalized flow rate Qoutlet and normalized total mass of gas in the pipe W as functions of time and the rarefaction parameter. Some numerical values are given in Table 1. Here the quantity Qoutlet is defined in Eq. (6) whereas W is the total mass of gas in the pipe divided by its initial value:

WðtÞ ¼

Mtot ðtÞ ; Mtot ð0Þ

Z Mtot ðtÞ ¼

nðt; xÞdx:

(13)

Lz0

It is seen that at the very early stage of the process, the flow rate is practically constant and does not depend on the rarefaction parameter. However, for later times and d0  10 the value of Qoutlet starts to fall due to the influence of the pipe surface. For large times Qoutlet is linearly proportional to the logarithm of time. The behaviour of the flow rate is distinctly different for the nearly hydrodynamic flow regime d0 ¼ 100. Unlike the rarefied and transitional flow regime, the value of Qoutlet initially increases with time, but then falls as well. There is a visible kink (jump in the derivative) in the flow rate curve at around t z 20, which approximately corresponds to the moment of time when the wave reflected from the closed end z ¼ L reaches the outlet section of the pipe. This effect perhaps requires further investigation which is beyond the scope of the present work. Fig. 8 shows temporal behaviour of number density and temperature at the centre of the closed end of the pipe

nend ðtÞ ¼ nðt; 0; 0; LÞ;

Tend ðtÞ ¼ Tðt; 0; 0; LÞ

Fig. 10. Contour levels of number density and normalized mass flow for d0 ¼ 100 and t ¼ 10.

290

E.M. Shakhov, V.A. Titarev / Vacuum 109 (2014) 284e293

for various values of the rarefaction parameter d0 ¼ 1, 10, 100; the curve for d0 ¼ 0 is omitted as it practically coincides with d0 ¼ 1. It is seen that the macroscopic quantities stay constant until t z L/c0, then start to fall. For later times the temperature starts to regain its initial value T ¼ 1. It should be noted that in both limiting solutions described by Eqs. (9) and (11) the temperature falls significantly with time at z ¼ L. However, in the case of the multidimensional kinetic solution the fixed surface temperature heats up the gas so that the flow inside the pipe is close to isothermal for all values of the rarefaction parameter. Figs. 9 and 10 show contour levels of number density and normalized mass flow the output time t ¼ 10 and d0 ¼ 1, 100. Note, that the selected output time is approximately sufficient for the wave resulting from the break-up of the initial discontinuity to reach the closed end of the pipe z ¼ L. Figs. 11 and 12 show the same contour levels, but for larger output time t ¼ 100, for which the wave has already reflected. The complete x  z plane is shown to demonstrate that the numerical solution is symmetrical. The general flow pattern is similar to the steady flow in the circular pipe [17] for all presented output times. For the rarefied flow regime d0 ¼ 1 the density distribution inside the pipe is almost onedimensional (function of z only) whereas for the nearly hydrodynamic regime the distribution becomes highly non-linear. For both values of d0 there is significant deceleration of mass flow due to the influence of the pipe surface. Such a distribution of mass flow explains the behaviour of the flow rate curves in Fig. 5, namely the fall of the flow rate for t > 0.

5. Accuracy of calculations As a basic verification test, the three-dimensional numerical method and the corresponding code were used to solve some basic one-dimensional shock tube problems. The relevant discussion can be found in Ref. [13]. A rectangular domain with reflective boundary conditions was used. Comparison with the analytical solution of the compressible Euler equations showed that the numerical method correctly reproduces the velocities of all waves in the solutions with correct post-shock values of macroscopic variables. It should be stated that it is very difficult to carry out a proper convergence study for the problem considered in the present work, as it is a time-dependent 6-dimensional problem. Therefore, we have run only a limited number of test calculations and rely on our experience in computing steady flows through various circular and non-circular pipes into vacuum. Firstly, it has been established that the reservoir size can be decreased to about five pipe radii. This is in line with the steadystate solutions reported by several authors. Next, two spatial meshes have been considered. Their topology in the cross sectional plane of the pipe and reservoir is similar to our previous studies of stationary flows in pipes [17,18,14]. It is of the O-grid type, with a square patch in the centre of the cross section and clustering applied towards the pipe surface and its ends. The meshes contain approximately 17 and 54 thousands hexahedrons and differ in the radial resolution (clustering towards the pipe's surface) and longitudinal resolution inside the pipe. No refinement in the angular

Fig. 11. Contour levels of number density and normalized mass flow for d0 ¼ 1 and t ¼ 100.

E.M. Shakhov, V.A. Titarev / Vacuum 109 (2014) 284e293

Fig. 12. Contour levels of number density and normalized mass flow for d0 ¼ 100 and t ¼ 100.

Fig. 13. Spatial meshes used in calculations.

291

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E.M. Shakhov, V.A. Titarev / Vacuum 109 (2014) 284e293

around 1e2%. Additionally, there are indirect estimates of accuracy, e.g. conservation of mass: loss of mass in the pipe should be equal to the integrated mass flow through the outlet. This is satisfied at the level of 0.1% for up to t ¼ 100. Then, as the flow rate becomes small, relative error increases to around 2e3%. The freemolecular value of flow rate is recovered at t ¼ 0 with very high accuracy. The difference in the flow rate curves for finite values of the rarefaction parameter is illustrated for d0 ¼ 1 and 100 in Fig. 15. It should also be noted that the numerical method correctly captures the effect of significant temperature drop as gas expands into vacuum, see Fig. 5. This demonstrates high accuracy of calculations. 6. Conclusions The time-dependent flow of rarefied gas into vacuum from a circular pipe closed at one end has been studied. The analysis is based on solving numerically the S-model kinetic equation and includes limiting one-dimensional free-molecular and hydrodynamic solutions as well general kinetic solution for the final values of the rarefaction parameter (or Knudsen number). It is shown that the flow rate and evacuation time depend significantly on the rarefaction parameter in the transitional and nearly hydrodynamic flow regimes whereas for nearly free-molecular and rarefied flow regimes this dependence is weak. The flow pattern inside the pipe

Fig. 14. Velocity mesh consisting of 21  16  32 cells.

1

1

Q

Q 1 2

0.5

0.5

0

0 0

50

t

100

0

50

t

100

(b)

(a)

Fig. 15. Comparison of flow rate curves for different meshes. (1) e 17,235 cells, (2) e 53,865 cells.

direction is made as the flow is axisymmetrical in the physical domain. The velocity domain is a cylinder and the velocity mesh is constructed in the cylindrical coordinate system. The velocity mesh resolution can be described by a group of three numbers, corresponding to the numbers of nodes in the radial, angular and xz directions. For d0  10 the mesh consists of 17  16  24 cells whereas 21  16  32 cells are used for d0 ¼ 100. The finer velocity mesh and slightly larger velocity domain size for d0 [1 are needed to properly resolve large temperature drops in the low-density region. Fig. 13 illustrates spatial meshes whereas velocity mesh of 21  16  32 cells is shown on Fig. 14. We have carried out comparative studies of all flow quantities using both spatial meshes. The calculations show that there is small difference in all quantities (both macroscopic variables and integral quantities) between the runs. Density profiles are also very close for all d0 values. However, velocity and temperature and better resolved on new meshes due to finer velocity discretization. We therefore estimate the accuracy of the results to be

for the sufficiently large time is similar to the one occurring in the steady-state flow through the circular pipe of the same length. Acknowledgements This work was supported by the Russian Foundation for Basic Research, projects no. 13-01-00522-A and 14-08-00604-A. The second author also acknowledges the support by the Russian government under grant ”Measures to Attract Leading Scientists to Russian Educational Institutions” (contract No. 11.G34.31.0072). References [1] Sharipov F, Seleznev V. Data on internal rarefied gas flows. J Phys Chem Ref Data 1998;27(3):657e706. [2] Sharipov F, Seleznev V. Flows of rarefied gases in channels and microchannels. Russian Academy of Science, Ural Branch, Institute Thermal Physics; 2008 [in Russian]. [3] Sharipov F. Benchmark problems in rarefied gas dynamics. Vacuum 2012;86(11):1697e700.

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