Rarefaction effects on the flow characteristics in microchannels on asymmetric wall thermal condition |
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Md. Tajul Islam |
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Department of Mathematics, Begum Rokeya University, Rangpur, Bangladesh E-mail:tajul000@yahoo.com |
Copyright © 2015 Md. Tajul Islam. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Rarefaction effects on the flow characteristics in 2D microchannels on asymmetric wall thermal conditions are investigated by control volume technique. In order to examine the influence of Knudsen numbers on the flow characteristics, a series of simulations for both compressible and incompressible flow with different Reynolds and Knudsen numbers are performed. Nitrogen gas is used as working fluid and the slip boundary conditions are used on the walls. The Navier-Stokes and energy equations are solved simultaneously. The results are found in good agreement with those predicted by analytical solutions in 2D continuous flow model employing first order slip boundary conditions. It is shown here that the Knudsen number has effects on velocity and temperature distribution for both the compressible and incompressible flows. It causes the velocity slip on the wall and causes the temperature difference between the wall temperature and the gas temperature on the wall.
Keywords: Compressible; Incompressible; Knudsen Number; Reynolds Number; Slip Flow.
1. Introduction
Micromachining technology is a rapidly emerging technology, where new
potential applications are being continuously developed.
Micro-Electro-Mechanical-Systems (MEMS) applications range in a widening field
of disciplines from consumer products to industrial tools. Sensors, actuators,
micropumps, microvalves, microturbines etc. are the examples of some of the
MEMS devices. Recently noticeable progresses have been made in design and
fabrication of Micro-Electro-Mechanical-Systems (MEMS). In response to the
rapid progress in design and fabrication of MEMS devices, the need for
understanding the momentum and heat transfer in microsystems is essential. The
flows in macro and micro systems are not the same. The characteristic length
scales that govern the energy and momentum transfer in MEMS and their
environments are on the order of microns. Surface effects dominate in small
devices. “The surface-to-volume ratio for a machine with a characteristic
length of 1.0 m is 1.0, while that for a
MEMS device having a size of 1.0 μm is
. The millionfold increase in surface
area relative to the mass of the minute device substantially affects the
transport of mass, momentum, and energy through the surface. The small length
scale of microdevices may invalidate the continuum approximation altogether”
Gad-el-Hak [13].
Transport phenomena at microscale reveal many features that are not observed in macroscale devices. These features are quite different for gas and liquid flows. According to Karniadakis et al. [10], four important effects are encountered in gas microflows. They are compressibility, viscous heating, thermal creep and rarefaction. Liquid flows are encountered with other microscale features such as surface tension and electrokinetic effects.
The deviation of the state of the gas from continuum behavior is measured
by the Knudsen number, Kn. For a microchannel, the Knudsen number is defined as
where
is
the hydraulic diameter of the channel.
In parallel plate channel it is defined as,.
Here
is the mean free path
corresponding to the distance travelled by the molecules between collisions and
is defined by
.
The local value of Knudsen number determines the degree of rarefaction and the degree of validity of the continuum model. Knudsen number is the criterion to indicate whether the flow problem can be solved by continuum approach. Flow regimes are specified according to Knudsen number as shown in Table 1.
Table 1: Different flow regimes based on knudsen numbers
Regime |
Method of Calculation |
Kn Range |
Continuum flow |
Navier-Stokes and energy equations with noslip boundary conditions |
|
Slip flow |
Navier-Stokes and energy equations with slip boundary conditions |
|
Transition flow |
Boltzmann Transport Equations, DSMC |
|
Free molecule flow |
Boltzmann Transport Equations, DSMC |
|
“In the limit of zero Knudsen number, the transport terms in the momentum and energy equations are negligible and the Navier-Stokes equations then reduce to the inviscid Euler equations. Both heat conduction and viscous dissipation are negligible and the flow is then approximately isentropic from the continuum viewpoint, while the corresponding molecular viewpoint is that the velocity distribution function is everywhere of the local equilibrium” Gad-el-Hak [13]. As the Knudsen number increases, rarefaction effects become more important and the continuum approach breaks down for Kn> 0.001.
In the slip flow regime where, the fluid
is not in equilibrium state in the region next to the wall which is known as
Knudsen boundary layer. Consequently, there are velocity and temperature
discontinuities on the walls. “The Knudsen boundary layer is significant only
up to a distance of one mean free path away from the wall” Hadjiconstantinou
[5]. Beyond the Knudsen boundary layer, the deviation from the state of
equilibrium is negligible and the Navier-Stokes equations are valid there. The
velocity and the temperature can be compensated by applying slip velocity and
temperature jump boundary conditions on the walls and then the Navier-Stokes equations
and energy equations are applicable in the whole domain.
In the transition regime, where, the
molecular approach should be used or Boltzmann equation can be used directly.
For higher Knudsen numbers, in the transition and free molecular flow regimes,
the Boltzmann equation must be solved by appropriate numerical technique such
as the Direct Simulation Monte Carlo (DSMC) methods or the Lattice Boltzmann
Methods (LBM).
Fundamental work in microflows started in much earlier. In 1846,
Poiseuille [2] studied liquid flows in tubes with diameters ranging from to
.
He focused on the relationship among flow rate, pressure drop and tube
geometry. He did not consider viscosity.
The pioneer investigators in the study of gaseous flow in microchannels are Maxwell [12], Smoluchowski [18], Schaaf and Chambre [17], Pong et al. [15], Ali and George [1], Arkilic et al. [3], Gad-el-Hak, M. [4].
In 1909, Knudsen [11] studied gas flow through glass capillary tubes in
the transition and free molecular flow regimes. He normalized the volumetric
flow rate with the inlet to exit pressure difference and plotted against the
average pressure in the capillary which showed minimum at.
Hongwei Sun and Mohammad Faghri [6] studied the rarefaction and compressibility effects of two dimensional gaseous flows in microchannels using DSMC. They examined the effect of compressibility and rarefaction for the inlet to outlet pressure ratios ranging from 1.38 to 4.5 and Knudsen numbers ranging from 0.03 to 0.11 respectively. They reported that compressibility makes the axial pressure variation nonlinear and enhances the local friction coefficient. On the other hand, rarefaction does not affect pressure distribution but causes the flow to slip at the wall and reduces the local friction coefficient.
Orhan Aydin and Mete Avci [14] studied laminar forced convective heat transfer in a microchannel between two parallel plates analytically. The viscous dissipation, the velocity slip and the temperature jump at the wall are included in the analysis. Both hydrodynamically and thermally fully developed flow cases are examined. Either the hot and cold wall case is considered for the two different thermal boundary conditions, namely the constant heat flux and constant wall temperature.
Most of the studies are considered incompressible because of simplicity and low Mach number flows. There are only a few theoretical studies addressing compressible, laminar flow in uniform conduits. The studies of compressible flow, without the incorporation of rarefied behavior, have been conducted by Prud’homme et al. [16].
Prud’homme et al. [16] and H. van den Berg et al. [7] neglected the transverse velocity and used a perturbation expansion to solve the isothermal, compressible Navier-Stokes equations for laminar flow in a circular tube. For low Reynolds and Mach number flows, they obtained a locally similar velocity profile.
John C. Harley et al. [9] conducted experimental and theoretical
investigations of low Reynolds number, high subsonic Mach number, compressible
gas flow in long conduits. Nitrogen, helium, and argon gases were used. The
channels were microfabricated on silicon wafers. The Knudsen number ranged from
to 0.4. The measured friction factor
was in good agreement with theoretical predictions considering isothermal,
locally fully developed, first order slip flow.
The development of micromachining technology enables fabrication of micro-fluidic devices such as microvalves, micropumps, micronozzles, microsensors, microheat exchangers. Microchannels and chambers are the essential part of such devices. In addition to connecting different devices, microchannles are used for reactant delivery such as biochemical reaction chambers, in physical particle separation, in inkjet print heads, and as heat exchangers for cooling computer chips. It is already established that the flows in micro-devices behave differently from that of macro counterparts. Thus in order to design and fabricate micro-devices properly, insight characteristics of fluid flow and heat transfer in microchannels must be understood. A lot of studies have been performed but there are a lot of discrepancies among the results.
The objective of the present study is to investigate the influence of Reynolds and Knudsen numbers specially Knudsen numbers on flow characteristics of gaseous flow in parallel plate channels. The asymmetric constant wall temperature boundary conditions are applied at the walls. To account for the non-continuum effect, the governing equations are solved in conjunction with the slip velocity and temperature jump boundary conditions.
2. Model development
2.1. Problem statement
We considered nitrogen gas flow through parallel plate channels in Cartesian co-ordinate systems. Fig.1 shows the geometry and co-ordinates of the microchannel.
Fig. 1: Schematic diagram of the channels
The flow
domain is bounded by and
, where L and H are the
length and height of the channels. We defined aspect ratio as
, for constant length L=3000
and different heights
. All the dimensions are expressed in
SI units. A number of investigations are performed with slip and noslip
boundary conditions on the walls. Various Reynolds and Knudsen numbers are
adjusted by changing pressure ratios and are computed at the outlet. Reynolds
numbers vary from 0.25 to 4.0 and the Knudsen numbers from 0.025 to 0.4.
2.2. Governing equations
The 2D gas
flow and heat transfer is assumed to be steady and laminar. The Three basic
laws of conservation of mass, momentum and energy are solved for both
compressible and incompressible flows for Newtonian fluid. The compressible
forms of the governing equations are expressed in the following form and are
solved with the help of state equation.
The continuity equation:
(1)
The Navier-Stokes equations:
(2)
(3)
The energy equation:
(4)
where i is the specific internal energy. The governing equations for incompressible flow are reduced from the above equations. In case of incompressible flow there is no need to establish linkage between the energy equation and the equations of continuity and momentum.
2.3. Basic concepts
Maxwell [12] in his study, considered the kinetic theory of dilute,
monatomic gases. Gas molecules, modeled as rigid spheres, continuously strike
and reflect from a solid surface, just as they continuously collide with each
other. He assumed the surface as something between perfectly reflecting and
perfectly absorbing surface. He assumed that a fraction of the molecules is absorbed by the
surface (due to the roughness of the wall), and then reemitted with velocities
corresponding to those in still gas at the temperature of the wall. The other
fraction, 1 −
, of the molecules is perfectly reflected by the wall. The
dimensionless coefficient
is called
the tangential momentum accommodation coefficient. When
= 0,
the tangential momentum of the incident molecules equals that of the reflected
molecules and no momentum is transmitted to the wall. This kind of reflection is
called specular reflection. Conversely, when
=1, the gas molecules
transmit all their tangential momentum to the wall and the reflection is a
diffuse reflection. For the momentum balance at the wall, Maxwell demonstrated
the slip velocity as follows:
(5)
where is the mean free path and
is the variation of velocity normal to
wall.
In analogy with the slip phenomenon, Poisson assumed that there might be a temperature jump at the wall and proposed that the equation might be equivalent to:
(6)
where is the temperature jump distance.
Smoluchowski [18] experimentally confirmed the hypothesis and suggested that
is proportional to the mean free path
. Like slip velocity, a fraction
of the molecules come in contact with
the walls and the walls adjust their mean thermal energy. These molecules are
reflected from the walls with the temperature of the gas on the walls. The
other fraction
of the molecules
reflects with their incident thermal energy. The dimensionless coefficient
is called the thermal accommodation
coefficient. For an energy balance on the walls, Smoluchowski demonstrated the
temperature jump as follows:
(7)
where is the Prandtl number.
2.4. Boundary conditions
Asymmetric wall thermal condition is implied on the walls. The free stream temperature and the lower wall temperature are kept at the same constant temperature which is lower than that of the upper wall and the difference between the upper and lower wall temperature varies from 100K to 300K. Pressure condition is assumed on the outlet boundary and velocity inlet and pressure condition are implied on the inlet boundary according to incompressible and compressible flow respectively. We impose slip velocity and temperature jump boundary conditions on the walls. The slip velocity condition was proposed by Maxwell [12] as
(8)
and the temperature jump boundary conditions by Smoluchowski [18] as
(9)
whereand
are
the variation of velocity and temperature normal to the wall. We consider the
tangential momentum accommodation coefficient
=1
and the thermal accommodation coefficient
=1
which describe the gas-wall interactions.
3. Methods
3.1. Numerical method
The control volume method is used to discretize the governing equations.
The pressure based segregated solver is employed in order to achieve steady
state analysis. The momentum and energy equations are solved with second-order
up-wind scheme to interpolate the corresponding cell center variables to the
faces of the cells. The SIMPLE (Semi-Implicit Method for Pressure-Linked
Equations) algorithm is used for introducing pressure into the continuity
equation. A second-order pressure interpolation method is used to calculate the
pressure at cell faces from the neighboring nodes. The computations are
considered to be converged when the residues for continuity, momentum and
energy are less than.
3.2. Grid independency test
To evaluate the grid size effect, grid independency tests are carried out. Three different sizes of grid 30×1200, 40×1500 and 50×1800 are tested for a typical channel of AR=600 for incompressible flow with slip boundary conditions and the results are listed in Table 2. The table shows that the relative difference of average velocity at outlet for grid sizes 30×1200 and 40×1500 is 0.27% and for grid sizes 40×1500 and 50×1800 is 0.04% respectively. For convenience, we used grid size 40×1500 or its multiple according to the length of the dimensions of the domain.
Table 2: Grid independence test of the average velocity under three sizes of grids
Grid size |
Average velocity |
30×1200 |
19.49037 |
40×1500 |
19.54263 |
50×1800 |
19.55102 |
4. Results and discussion
In order to
validate our model we simulated 2D isothermal flow with first order slip
boundary conditions. Nitrogen gas was used with channel AR=2500 and PR=2.36.
The outlet Reynolds and Knudsen numbers were 1.99 and 0.044 respectively. The
Centerline velocity distribution normalized with the speed of sound at the inlet is compared with I. A.
Graur [8] those predicted by analytical solutions derived from the quasi gas
dynamic equations with the same conditions. The results are displayed in Fig.2.
The square symbols represent the analytical results and the solid line
represents the results from current work. The results show good agreement which
validate our model.
In our study the effects of Knudsen number is the main focus of our investigation. We study the effects of this parameter on flow characteristics for the fully developed flow. We examine the case of asymmetric wall thermal boundary condition with constant wall temperature at the walls. The slip velocity and temperature jump boundary conditions are applied on the walls. Both the free stream and the lower wall temperature are kept at 200K and the upper wall temperature at 300K. Our investigation covers both the compressible and incompressible flow.
The cross sectional velocity distributions with slip boundary conditions of incompressible flow are shown in Fig.3. In case of Tup300 condition, the upper and the lower wall temperature are constant at 300K and 200K respectively. In case of Tup200 condition, the upper and the lower wall temperature are constant at 200K and 300K respectively.
For Tup300 condition Knudsen number on the upper wall is 0.024 and that on the lower wall is 0.017. Consequently the fluid velocity on the upper wall is higher than that of the lower wall which is 0.96 and 0.82 on the upper and lower wall respectively. Since the flow is incompressible, the cross sectional average velocity will remain the same. As a result the velocity of the fluid near the upper wall will be lower than the velocity of the corresponding fluid near the lower wall. For Tup200 condition we see the opposite scenario which is displayed by the Fig.3.
Then we consider two cases of symmetric thermal boundary conditions at the walls. In the first case the upper and lower wall temperature are kept at the same constant temperature 300K with slip and noslip boundary conditions. In the second case the wall temperature 300K is increased to 600K remaining all other conditions the same. The cross sectional velocity distributions for slip and noslip boundary conditions for two cases of wall temperature are displayed in Fig.4. At 300K wall temperature the outlet Reynolds number is 0.88 for both slip and noslip conditions and the outlet Knudsen number is 0.029 for slip condition.
At temperature 600K the corresponding Reynolds and Knudsen numbers are 0.53 and 0.059 respectively. We see that for slip condition there is slip on the wall and the fluid velocity near the wall is higher than that of noslip condition. But near the core region the fluid velocity with noslip boundary condition is higher than that of slip condition. As a result the average outlet velocities for both the flows with slip and noslip boundary conditions are equal and fixed. At temperature 300K the average velocity for slip and noslip condition at the outlet is 5.99 which increase to 11.99 for the increase of temperature from 300K to 600K.
To examine the simultaneous effect of Reynolds and Knudsen numbers on flow properties we investigated four cases with different sets of combinations of Reynolds and Knudsen numbers. Among the four cases, the first one is with noslip boundary condition and the rest are with slip boundary condition. The results are depicted by Fig.5. From the figure we see that for noslip condition fluid velocity on the wall is zero. As the rarefaction increases, fluid velocity on the wall increases while the maximum velocity decreases. Consequently the outlet average velocities for all the slip and noslip wall conditions are equal which 19.58 are (m/s).
Table 3: Average outlet velocity and mass flow rate for incompressible flow.
Re |
Kn |
Re.Kn |
Outlet average velocity |
Mass flow rate |
4.0 |
0.025 |
0.1 |
19.56 |
3.12e-05 |
1.0 |
0.1 |
0.1 |
19.56 |
7.81e-06 |
0.25 |
0.4 |
0.1 |
19.56 |
1.95e-06 |
Fig. 6: Velocity profiles with different Knudsen and Reynolds numbers for compressible flow.
All the velocity distributions shown in Fig.5 intersect at fixed point. Here the product of Reynolds number and Knudsen number for the slip wall conditions is 0.1.
The Table 3 shows the outlet average velocity and mass flow rate for different Reynolds and Knudsen numbers.
Velocity distributions for different Reynolds and Knudsen numbers for compressible flow are displayed in Fig.6. Here the product of the Reynolds number and the Knudsen number are fixed to 0.1. A similar velocity distribution with the same Reynolds and Knudsen numbers is displayed for incompressible flow in Fig.5. From Fig.6, we see that the velocity distributions intersect at fixed points but the outlet average velocity are not equal as it does for incompressible flow.
The Table 4 shows the outlet average velocity and mass flow rate for different Reynolds and Knudsen numbers.
Table 4: Average outlet velocity and mass flow rate for compressible flow.
Re |
Kn |
Re.Kn |
Outlet average velocity |
Mass flow rate |
4.0 |
0.025 |
0.1 |
19.48 |
3.1158e-05 |
1.0 |
0.1 |
0.1 |
20.12 |
7.8964e-06 |
0.25 |
0.4 |
0.1 |
19.96 |
1.9614e-06 |
Cross
sectional velocity distribution at for
incompressible and compressible flow are depicted in Fig.7 and Fig.8
respectively. Slip boundary conditions are imposed on the walls. In both the
cases the Reynolds number are kept fixed and the Knudsen numbers are varied. In
the figure we see that the higher the rarefaction, the higher the velocity. As
the Knudsen number increases the slip velocity on the wall increases and
consequently the velocity increases.
Normalized cross sectional temperature distributions as a function of with different Knudsen numbers for
incompressible flow are displayed in Fig.9. Here for the flows with slip
conditions we decrease the Reynolds numbers and increase the corresponding
Knudsen numbers gradually so that their products remain fixed. From the figure
we see that with noslip boundary condition the temperature of the gas particles
on the wall is equal to the temperature of the wall. But with slip boundary
conditions the gas particles on the wall cannot reach the temperature of the
wall. As the Knudsen number increases, the difference of temperature of the gas
particles on the wall and the wall temperature increases.
Centerline normalized temperature distribution as a function of x/L for incompressible flow is depicted in Fig.10. The temperature is normalized with the upper wall temperature. The solid line and the dashed lines show the temperature distributions subject to noslip and slip boundary conditions respectively. From the figure we see that the difference of temperature between the wall temperature and the centerline gas temperature is the minimum when noslip boundary condition is used. The temperature difference increases with the increase of Knudsen numbers.
Fig. 13: Friction coefficient distributions with fixed Reynolds but different Knudsen numbers.
The centerline normalized temperature distribution for compressible slip flow along streamwise direction is depicted in Fig.11. The figure shows that the centerline temperature distribution is affected by the Knudsen number as it was in the case of incompressible flow. The difference of wall temperature and the centerline temperature depends on the Knudsen number. As the Knudsen number increases the difference between the wall temperature and the centerline temperature also increases. The maximum difference occurs near the outlet where the Knudsen number is the highest. Comparing with Fig.10, we see that against the same Knudsen number the temperature difference between the centerline temperature and the wall temperature for the compressible flow is higher than that of incompressible flow.
Distributions of normalized friction factor along the streamwise direction for compressible flow are displayed in Fig.12 for different Knudsen numbers. Friction factors are normalized with the maximum friction factor. Since the outlet Knudsen number is changed from 0.025 to 0.1, the friction factor is rapidly decreased near outlet. It is due to the fact that the Knudsen number causes slip on the wall and consequently friction factor decreases.
Distributions of normalized friction coefficient for compressible flow measured on the top wall along the streamwise direction are depicted in Fig.13 for different Knudsen numbers. Like friction factor, friction coefficients also decrease for the increase of Knudsen numbers. Friction coefficients for Knudsen number 0.1 is much lower than that of 0.025.
5. Conclusions
Temperature increases rarefaction which increases the gas velocity with slip conditions. If the velocity distribution is expressed as the function of Reynolds numbers and Knudsen numbers and if their product is fixed then the corresponding average velocity for incompressible flow is fixed but for compressible flow it is different. Knudsen number causes the increase of velocity for both the compressible and incompressible flow. Knudsen number also increases the temperature difference between the wall temperature and the gas temperature on the walls. The temperature distributions in the core region are also affected by the Knudsen number. Increase of Knudsen number causes the decrease of both the friction factor and friction coefficients.
References
[1] A. Beskok, G. E. Karniadakis, Simulation of Heat and Momentum Transfer in Complex Microgeometries, Journal of Thermophysics and Heat Transfer 8 (1994) 647-655.http://dx.doi.org/10.2514/3.594.
[2] J. L. M. Poiseuille, Experimental investigations upon the flow of liquids in tubes of very small diameter, Lancaster Press Inc., Lancaster, 1846.
[3] E. B. Arkilic, K. S. Bueuer, M. A. Schmidt, Gaseous Slip Flow in Long Microchannles, Application of Microfabrication to Fluid Mechanics, ASME 197 (1994) 57-65.
[4] Gad-el-Hak M., The fluid mechanics of microdevices – The Freeman Scholar Lecture, J. Fluids Eng. 121 (1999) 5–33.http://dx.doi.org/10.1115/1.2822013.
[5] Hadjiconstantinou N.G., The limits of Navier Stokes theory and kinetic extensions for describing small scale gaseous hydrodynamics, Phys. Fluids 18 (2006) 1070-6631.http://dx.doi.org/10.1063/1.2393436.
[6] H. Sun, M. Faghri, Effects of Rarefaction and Compressibility of Gaseous Flow in Microchannel using DSMC, Numerical Heat Transfer, Part a 38 (2000) 153-168.http://dx.doi.org/10.1002/(SICI)1099-0518(20000515)38:10<1852::AID-POLA720>3.0.CO;2-J.
[7] H. van den Berg, C. Seldam, P. Gulik, Compressible Laminar flow in a capillary, J. Fluid mech. 246 (1993) 1-20.http://dx.doi.org/10.1017/S0022112093000011.
[8] I. A. Graur, J. G. Meolans, D. E. Zeitoun, Analytical and numerical descriotion for isothermal gas flows in microchannels 2 (2006) 64-77.
[9] J. C. Harley, Y. Huang, H. H. Bau, J. N. Zemel, Gas flow in micro-channels, J. Fluid Mech. 284 (1995) 251-274.http://dx.doi.org/10.1017/S0022112095000358.
[10] Karniadakis G., Beskok A., Aluru N., Microflows and Nanoflows, Fundamentals and Simulation, Springer, New York, 2005.
[11] Knudsen M., Die gesetze der molekularstrmung und der innerenreibungsstrmung der gasedurchrhren, Annalen der Physik 28 (1909) 75-130.http://dx.doi.org/10.1002/andp.19093330106.
[12] Maxwell, J.C., On Stresses in Rarified Gases Arising From Inequalities of Temperature, Philos. Trans. R. Soc. London 170 (1879) 231-256.http://dx.doi.org/10.1098/rstl.1879.0067.
[13] M. Gad-el-Hak, the MEMS Handbook (2nd edition), CRC press, New York, 2006.
[14] Orhan A., Mete A., Analysis of laminar heat transfer in micro-Poiseille flow, International Journal of Thermal Sciences 46 (2007) 30-37.http://dx.doi.org/10.1016/j.ijthermalsci.2006.04.003.
[15] Pong K. C., Ho C. M., Liu J., Tai Y. C., Non-linear Pressure Distribution in Uniform Microchannels, Application of Microfabrication to Fluid Mechanics, ASME 197 (1994) 51-56.
[16] R. Prud'homme, T. Chapman, J. Bowen, Laminar Compressible Flow in a Tube, Appl. Sci. Res. 43 (1986) 67-74.http://dx.doi.org/10.1007/BF00385729.
[17] Schaaf, S., Chambre, P., Flow of Rarefied Gases, Princeton University Press, Princeton, 1961.
[18] Smoluchowski, M., Über den TemperatursprungbeiWärmeleitung in Gasen.,Akad. Wiss. Wien. CVII (1898) 304-329.