Bivelocity gas dynamics of micro-channel couette flow

International Journal of Engineering Science 79 (2014) 21–29

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Bivelocity gas dynamics of micro-channel couette flow Peter L.L. Walls ⇑, Behrouz Abedian Department of Mechanical Engineering, Tufts University, Medford, MA 02155, USA

a r t i c l e

i n f o

Article history: Received 8 August 2013 Received in revised form 22 January 2014 Accepted 3 February 2014

Keywords: Microfluidics Bivelocity gas dynamics Burnett equations Micro couette flow Rarefied flows

a b s t r a c t The bivelocity theory is adopted in this paper to predict the characteristics of a monoatomic Maxwellian gas flow in a micro-channel geometry. Our analysis utilizes a full set of mechanically and thermodynamically consistent volume-diffusion hydrodynamic equations. It is argued that the bivelocity theory predicts, contrary to current investigations, a no-slip velocity at the wall in planar Couette flow for rarefied gas flows. The results indicate that the numerical predictions for the density, temperature, and velocity distributions are different from those previously obtained using Burnett equations. Suggestions for experimental verification, motivated by our numerical results, are also presented. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Microelectromechanical systems have had a major impact on our daily lives and many disciplines (Ho & Tai, 1998) (e.g. biology, medicine, optics, aerospace, and mechanical and electrical engineering) over the past three decades. For example hard-disk heads, micro-pumps, and lab-on-a-chip technologies all rely on gas flows around micro-scale geometry, therefore a fundamental understanding of these flows, referred to as microfluidics, is important for the design and characterization of these devices. While it is agreed that this class of gas flow is vital to many industries, there is currently no generally accepted fluid-mechanical theory for this regime of flow (Brenner, 2013; Shen, 2005). It is this gap in knowledge that been the primary motivation of this work. Adaptation of bivelocity theory in the present analysis is motivated by its recent success in explaining traditionally non-continuum phenomenon in a physically consistent manner. The critical parameter used in determining whether a particular flow is considered a continuum or non-continuum gas flow, i.e. rarefied, is the knudsen number. The knudsen number is typically defined as Kn ¼ k=L where k is the mean free path of the gas and L is a characteristic length scale of the geometry of interest. For example, in the case of micro-poiseuille flow L would be the diameter of the tube. The knudsen number is commonly taken as a measure of the flows level of local equilibrium with flows possessing a higher knudsen number being further from local thermal equilibrium. Specifically when Kn  O (1) it is thought that there are not enough particle-to-particle collisions to equilibrate heat and momentum at every point in the flow and the local equilibrium assumption becomes increasingly inaccurate. It should also be stated that there is no objective distinction between a continuum and a non-continuum gas flow (Brenner, 2013). That is, there is no definite value of the knudsen number which determines whether a gas is rarefied or otherwise. However, for the purposes of this paper we will use the oft cited value of 0.01 to mark the transition between a continuum and a rarefied gas (Gad-el Hak, 1999).

⇑ Corresponding author. Tel.: +1 617 353 9645. E-mail addresses: [email protected] (P.L.L. Walls), [email protected] (B. Abedian). http://dx.doi.org/10.1016/j.ijengsci.2014.02.002 0020-7225/Ó 2014 Elsevier Ltd. All rights reserved.

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Prior to Knudsen’s pioneering work in 1909 (Knudsen, 1909; Steckelmacher, 1986) it was believed that the compressible form of the Navier–Stokes–Fourier (NSF) equations (Bird, Stewart, & Lightfoot, 1960; Groot & Mazur, 1962; Landau et al., 1987) were accurate in describing rarefied gas flows. However, Knudsen showed, through a series of experiments involving rarefied gas flows, that the compressible NSF equations, subject to the no-slip boundary condition, were inadequate in describing such flows (Chapman & Cowling, 1991). Since this time, the rarefied or non-continuum domain of flow investigated by Knudsen is typically regarded as being governed by the Boltzmann gas-kinetic equation (Chapman & Cowling, 1991; Gad-el Hak, 1999; Ho & Tai, 1998). However, the mathematical complexity of the Boltzmann equation has prevented an exact solution from being derived and has lead to a variety of approximate methods being developed. These approximate solutions can be classified as either particle methods or moment methods (Agarwal, Yun, & Balakrishnan, 2001). The direct simulation Monte Carlo (DSMC) method (Bird, 1963, 1978) falls into the particle method category while the Burnett equations set (Chapman & Cowling, 1991; Burnett, 1935) is an example of a moment method. However, the Burnett equations are known to violate a number of thermodynamic and mechanical principles (Agarwal et al., 2001; Dadzie, 2013). Despite these problems with the Burnett equations, the complexity and computational cost of DSMC and other simulation methods, such as molecular dynamics, has led researchers to continue focus on the development of several different ‘‘Burnett regime’’ expansions of the Boltzmann equation in an attempt to fix these violations and accurately capture the physics of the problem while reducing computational complexity. ‘‘Burnett regime’’ simply refers to the agreed upon range of 0:01 < Kn0 < 10 also referred to as the transition regime. These expansions are referred to as: Burnett equations (Burnett, 1935), conventional Burnett equations (Chapman & Cowling, 1991), augmented Burnett equations (Zhong, 1991), and the BKG-Burnett equations (Balakrishnan & Agarwal, 1997), each of which has attempted to resolve issues with the original Burnett equations. To this day however, in the field of rarefied gas flow there does not exist an uncontested, generally applicable, macroscopic fluidmechanical theory (Brenner, 2013; Shen, 2005). Recently bivelocity theory (Brenner, 2004, 2005, 2009, 2012a), a new theory proposed by Brenner has attempted to close the gap of understanding in rarefied gas flows. To date this new theory has been successful in describing experimental data in both continuum and non-continuum fluid flows (Brenner & Bielenberg, 2005; Dadzie & Brenner, 2012; GreenShields & Reese, 2007). These experiments range from thermophoresis (Aitken, 1884; Tyndall, 1870) to describing the enhanced mass flow rate of micro-channel pressure driven flows (Dadzie & Brenner, 2012). Bivelocity theory’s hypothesis, that there exist two independent velocities in any compressible fluid, differs substantially from the traditional approach for describing not only rarefied gases but compressible flows in general. While the bivelocity theory has been applied to several non-continuum experiments with great success, the theory was originally derived using the macroscopic non-kinetic framework of Linear Irreversible Thermodynamics (Groot & Mazur, 1962) (LIT) and in the past couple of years has been verified using a kinetic-theory approach (Dadzie, 2013; Dadzie & Reese, 2012; Dadzie, Reese, & McInnes, 2008). Given the theory’s success in describing a variety of both compressible and micro-channel flows in a physically sound manner, we have investigated this theory’s predictions regarding the basic features of a micro-couette flow, a simple shear-driven flow between two infinite parallel plates. 2. Bivelocity equations The bivelocity equations governing the transport of mass, momentum, and energy for the flow of a steady, body force free, Maxwellian gas are, respectively (Brenner, 2012a):

r  ðqv Þ ¼ 0;

ð1Þ

qv  rv ¼ r  P

ð2Þ

qv  r^e ¼ r  je

ð3Þ

and

wherein q is the fluid’s density and

P ¼ Ip  T

v is the fluid’s mass velocity. The pressure tensor is expressed in the standard form ð4Þ

in which p is the equilibrium thermodynamic pressure, T is the viscous stress tensor, and I is the idiom factor. Assuming that bulk viscosity is negligible for a Maxwellian gas; T is given as

T ¼ 2lrv v

ð5Þ

T ¼ 2lrv

ð6Þ

and

for bivelocity and NSF fluids, respectively.An overbar indicates the tensor’s symmetric and traceless form, i.e. D ¼ 12 ðD þ DT Þ  13 trðDÞ where the superscriptT is the transpose and trðÞ is the trace. Appearing in (5) is the fluid’s volume velocity v v which is related to the mass velocity v through the diffuse volume flux jv as

vv ¼ v m þ jv

ð7Þ

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23

wherein for a gas, jv is given constitutively by

jv ¼

C mr ln q Pr

ð8Þ

wherein C is a dimensionless, fluid-property-dependent constant, with a value believed to be near unity for all gases (Brenner, 2012b, Table 1). Moving forward we will use C ¼ 1 to simplify the analysis. The Prandtl number is defined as Pr ¼ m=a with the kinematic viscosity m ¼ l=q, in which l is the shear viscosity. The fluid’s thermal diffusivity is defined as a ¼ k=q^cp with k and ^cp the gas’s thermal conductivity and isobaric specific (i.e. per unit mass) heat, respectively. Lastly, the energy Eq. (3) is given in terms of the fluid’s specific energy density consisting of specific internal and kinetic energies

^e ¼ u ^ þ ð1=2Þv  v

ð9Þ

and the energy flux je . According to bivelocity theory the energy flux is given constitutively by

je ¼ ju þ P  v v

ð10Þ

on the other hand NSF predicts

je ¼ ju þ P  v

ð11Þ

wherein ju is the diffuse internal energy flux

ju ¼ q  pjv

ð12Þ

which is defined in terms of the entropic heat flux q given as

q ¼ krT þ arp

ð13Þ

For a gas, (12) can be simplified as 0

ju ¼ k rT

ð14Þ

0

with k ¼ k=c, in which c is the specific heat ratio of the gas. 3. Wall boundary condition Traditionally for micro-channel flow, where the knudsen number can be of O(1), a slip condition proportional to the velocity gradient normal to the wall

v slip /

 @ v  @n wall

ð15Þ

would be imposed at the solid boundaries (Maxwell, 1879). In contrast, the boundary condition proposed in bivelocity theory (Brenner, 2011, 2012a, 2012b) is dictated by the volume velocity as follows:

^n ^ Þ  vv ¼ 0 ðI  n

ð16Þ

The above proposed boundary condition has been successful in matching experimental data in rarefied gas flows (for example thermophoresis and isothermal pressure driven micro-channel flow (Brenner & Bielenberg, 2005; Dadzie & Brenner, 2012)), and will be the boundary condition utilized in this paper. Given the definitions of the volume velocity v v (7) and diffuse volume flux jv (8), it can prove useful to express this boundary condition as a slip on the mass velocity:

 @ ln q @s wall

v slip ¼ a

ð17Þ

where s refers to the axis along the solid boundary where the slip or creep occurs. This expression predicts a slip condition at the wall only when a density gradient exists in the flow direction. In a fully-developed flow, no axial density variation exists in the channel and accordingly the condition (17) yields a no-slip condition which will be utilized in the analysis. 4. Plane micro-channel couette flow Steady flow between infinite parallel plates is a simple geometry for feasible testing of micro-channel models. This model provides insights into analogous systems such as gas bearings and disc-head readers. The geometry consists of a fixed lower plate ðy ¼ 0Þ separated by a distance H from an upper plate moving at a constant velocity V (see Fig. 1 below). The steady bivelocity differential equations governing couette flow reduce to the following:

d dy



l

 du ¼0 dy

ð18Þ

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Fig. 1. Micro-channel couette flow geometry.

  d 4 dj p l v ¼0 dy 3 dy

ð19Þ

  d dT du dp 4 dj þ lu  a þ ljv v ¼ 0 k dy dy dy dy 3 dy

ð20Þ

wherein the x-direction mass velocity is represented by u and y is the coordinate normal to the stationary wall. Eqs. (18)– (20) constitute a complete set of ordinary differential equations and now require an appropriate equation of state. In this paper the fluid being analyzed is assumed to be a Maxwellian monatomic gas, where Maxwellian is referring to the intermolecular potential. This assumption will simplify the analysis and allows for straightforward incorporation of temperature effects on the gas’s transport properties. The state of a monatomic gas flow can be fully characterized by its density q, temperature T, and mass velocity u. We assume that the gas is in a state of local equilibrium to enable use of the ideal gas law

p ¼ qRT

ð21Þ

wherein R is the specific gas constant. While we recognize that for flows approaching a knudsen number on the order of unity the local equilibrium assumption breaks down, we will still use the ideal gas law as an approximation, as has been previously done (Lockerby & Reese, 2003; Xue, Ji, & Shu, 2001). For a monatomic gas the isochoric and isobaric specific heats ^cv and ^cp are related to the specific gas constant through the relations

^cv ¼

3 5 R and ^cp ¼ R 2 2

such that the specific heat ratio c is defined as ^cp =^cv ¼ 5=3. The speed of sound in the gas is given by the simple relation as

c2 ¼ c

p

ð22Þ

q

Lastly, for a Maxwellian monatomic gas modeled by point centers of force, the thermal conductivity and dynamic viscosity are directly proportional to temperature (Garz & Santos, 2003; Lockerby & Reese, 2003; Xue et al., 2001), i.e. ðl; kÞ / T. For convenience in calculations we will non-dimensionalize the flow variables and properties as follows, in a manner similar to (Xue et al., 2001):

y u T ~ ¼ pffiffiffiffiffiffiffiffi ; T~ ¼ ; ; u H T0 RT 0 p l ~ k ~¼ ~¼ ; l ; k¼ p p0 l0 l0 ^cp ~¼ y

q~ ¼

q q0

where the subscript ‘‘0’’ indicates the value of the property or variable at the stationary wall ðy ¼ 0Þ. The constant Prandtl number for a monatomic gas Pr ¼ l^cp =k ¼ 2=3 is used in the following relations for the dimensionless viscosity and thermal conductivity.

3 l~ ¼ T~ and k~ ¼ T~ 2

The knudsen and Mach numbers are defined as

Kn0 ¼

l0 pffiffiffiffiffiffiffiffi q0 RT 0 H

Ma ¼

V c0

and

Using the above defined dimensionless parameters (18)–(20) can be non-dimensionalized as

ð23Þ

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  ~ d ~ du T ¼0 ~ dy ~ dy " # d 4 2 ~ d~jv ~ p  Kn0 T ¼0 ~ ~ dy 3 dy " # ~ 3 T~ dp ~ 2 2 ~ d~j2v d 15 ~ dT~ ~ du ~ T þ Tu  þ Kn T ¼0 ~ 4 dy ~ ~ 2q ~ 3 0 dy ~ ~ dy dy dy

25

ð24Þ

ð25Þ

ð26Þ

wherein we have introduced the dimensionless diffuse volume flux

~ ~ ~jv ¼ 3 T dq ~ ~ 2 dy 2q and will be subject to the following boundary conditions:

~ ¼ 0Þ : Fixed Bottom Plate ðy q~ ¼ 1; T~ ¼ 1; u~ ¼ 0 ~ ¼ 1Þ : Moving Top Plate ðy q~ ¼ 1; T~ ¼ 1; u~ ¼ c0:5 Ma where, as discussed in Section 3, there is a no-slip condition imposed on both solid boundaries. Upon inspection of (24), (25), and (26) we can see that these equations are of third order in terms of the density and second order in temperature and velocity, thus the order must be reduced to facilitate application of the above boundary conditions. We will use the method of Lee (1994), which was adopted and implemented by both Xue et al. (2001) and Lockerby and Reese (2003) in their studies of the Burnett equations. The method relies on the assumption that after integrating the y-direction momentum Eq. (25) the resulting constant is independent of the knudsen number. Integrating once results in:

4 d~j ~1 ~  Kn20 T~ v ¼ K p ~ 3 dy

ð27Þ

In the limit Kn0 ! 0 the dimensionless pressure gradient disappears identically in (25), which corresponds to the NSF case of ~ 1 ¼ 1 and continue our analysis by constant pressure across the channel. This assumption allows us to assign the value of K reducing the total order of our system of equations, which can now be solved numerically. 4.1. Numerical solutions To solve the set of nonlinear ODE’s the bvp4c solver in MATLAB was utilized. For this study we ensured the accuracy of the solution in two ways: the values of the solution residuals were kept below 105 , and the solutions mesh independence was ensured. Below we have presented the dimensionless densities (Figs. 2 and 3), along with the temperature and velocity profiles predicted by Burnett and bivelocity hydrodynamics (Fig. 4). A separate plot for the temperature, which is independent of the knudsen number, and velocity profiles predicted by the Burnett equations is not shown as they align exactly with the Kn0 ¼ 0:01 instance. In Fig. 4, while the velocity profile will vary based on the knudsen number, more than a single case was not plotted since there was a negligible variation between cases. A cross channel pressure increase is predicted by both theories (see Fig. 5 for bivelocity results), which differs from the constant pressure solution of the Navier–Stokes–Fourier equations. This result is not surprising in that a cross-channel density and temperature increase is predicted. 5. Discussion 5.1. Heat transfer The temperature distributions predicted by the Burnett and bivelocity theories indicate, at first glance, that their respective entropic heat transfers q will be fundamentally different. However, as is seen in Fig. 6 both theories (Burnett and bivelocity) predict identical entropic heat flux values, owing to the additional contribution from pressure gradients in the bivelocity instance. Note that while both predict equal entropic heat fluxes, bivelocity theory provides a more physically consistent prediction for the density across the channel, i.e. there is no longer a transition period near Kn0 ¼ 0:16 (Fig. 3). Without a slip-velocity boundary condition, the heat flux will continue to increase with the knudsen number since there is no mechanism to reduce the internal shearing of the gas and will not match the ‘‘exact’’ numerical solution of the Boltzmann equations given by Nanbu (1983) above values of the Knudsen number where a slip-velocity is typically imposed.

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Fig. 2. Bivelocity density distributions across the micro-channel for Ma ¼ 3; Kn0 ¼ 0:01; 0:16; 0:39; 0:78.

1

K n0 K n0 K n0 K n0

0.9 0.8

= 0.01 = 0.16 = 0.39 = 0.78

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0.8

1

1.2

ρ˜

1.4

1.6

1.8

2

Fig. 3. Burnett density distributions across the micro-channel for Ma ¼ 3; Kn0 ¼ 0:01; 0:16; 0:39; 0:78.

Fig. 4. Velocity and temperature distributions across the micro-channel for Ma ¼ 3; Kn0 ¼ 0:01; 0:16; 0:39; 0:78.

The internal energy flux ju , defined by its presence in (10), is predicted to be lower in bivelocity gas dynamics, owing to the presence of the diffuse volume flux in (12). 5.2. Experimental validation While the complete profiles for the temperature, density, and pressure have been presented; a conclusion with regards to which theory, Burnett or bivelocity, along with the appropriate boundary condition is more accurate in this transition regime cannot be made without conclusive experimental data. However, in other geometries (e.g. isothermal pressure driven flow), biovelocity theory, along with imposing a no-slip on the volume velocity, has been successful in describing the experimental

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Fig. 5. Bivelocity dimensionless pressure distribution across the micro-channel at Ma ¼ 3; Kn0 ¼ 0:01; 0:16; 0:39; 0:78.

~ ¼ 0 with Ma ¼ 3; Kn0 ¼ 0:01 ! 1. Fig. 6. Dimensionless heat and internal energy fluxes at y

data in a physically sound manner. Unfortunately, experimental data is lacking and experiments will need to be performed in order to settle the debate. When an experiment is inevitably performed, it will become necessary to determine the quantities ~ q ~ , or p ~ would allow one to of importance. For example, while directly measuring the cross channel distributions of T; determine the accuracy of the proposed theories, the scale of this experiment would make this difficult to achieve. However, measuring the mass flow rate would be a more straightforward method which has been defined as

~ dot ¼ m

Z

~¼1 y

~q ~ dy ~ u

ð28Þ

~¼0 y

Fig. 7. Dimensionless mass flow rate across the micro-channel at Ma ¼ 3; Kn0 ¼ 0:01 ! 1:0.

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~ dot is the dimensionless mass flow rate per unit depth, which is a typical method for these micro scale geometries where m (Dadzie & Brenner, 2012) Another benefit of choosing to compare the mass flow rates is the degree in which the Burnett and bivelocity solutions differ. As is seen in Fig. 7, when Kn0 ¼ 1 the mass flow rate ratio is approximately

~ dot ÞBurnett ðm  1:63 ~ dot ÞBiv elocity ðm

6. Conclusions Bivelocity gas dynamics, along with the recently introduced no-slip on the volume velocity, has been successful in describing experimental phenomena previously explained only through the use of ad-hoc boundary conditions, i.e. those requiring fitting parameters (Brenner & Bielenberg, 2005; Dadzie & Brenner, 2012). In this analysis we have presented a full set of mechanically and thermodynamically consistent equations for steady micro-channel couette flow. The results of which differ significantly from those given by the traditional Burnett equations. It has been shown that the magnitude of the knudsen number not only affects the density distribution, but the temperature as well. This paper has argued that according to bivelocity theory a slip-velocity boundary condition, in this geometry, is inappropriate and should be replaced with a no-slip on the volume velocity, which also corresponds to a no-slip mass velocity. Therefore in order to determine which of the competing models accurately describes this geometry, experimental data will be needed to make the more informed decision. It is concluded that an investigation of the mass flow rate in this micro-channel would be the best means to determine the accuracy of this theory’s predictions. References Agarwal, R. K., Yun, K.-Y., & Balakrishnan, R. (2001). Beyond Navier Stokes: Burnett equations for flows in the continuum transition regime. Physics of Fluids, 13(10), 3061–3085 . Aitken XV, J. (1884). On the formation of small clear spaces in dusty air. Transactions of the Royal Society of Edinburgh, 32(02), 239–272 . Balakrishnan, R., & Agarwal, R. (1997). Entropy consistent formulation and numerical simulation of the BGK-Burnett equations using a kinetic wave/particle flux splitting algorithm. In: P. Kutler, J. Flores, J. -J. Chattot (Eds.), Fifteenth International Conference on Numerical Methods in Fluid Dynamics (pp. 480– 485), Vol. 490 of Lecture Notes in Physics, Springer Berlin/Heidelberg, http://dx.doi.org/10.1007/BFb0107148. Bird, G. (1963). Approach to translational equilibrium in a rigid sphere gas. Physics of Fluids, 6, 1518. Bird, G. (1978). Monte carlo simulation of gas flows. Annual Review of Fluid Mechanics, 10(1), 11–31. Bird, R. B., Stewart,W. E., & Lightfoot, E. N. (1960). Transport phenomena. John Wiley & Sons, Inc. Brenner, H. (2004). Is the tracer velocity of a fluid continuum equal to its mass velocity? Physics Review E, 70(6), 061201 . Brenner, H. (2005). Navier–Stokes revisited. Physica A: Statistical Mechanics and its Applications, 349(1-2), 60–132 . Brenner, H. (2009). Bi-velocity hydrodynamics. Physica A: Statistical Mechanics and its Applications, 388(17), 3391–3398. Brenner, H. (2011). Beyond the no-slip boundary condition. Physics Review E, 84(4), 046309. Brenner, H. (2012a). Beyond Navier Stokes. International Journal of Engineering Science, 54(0), 67–98 . Brenner, H. (2012b). Fluid mechanics in fluids at rest. Physics Review E, 86(1), 016307. Brenner, H. (2013). Proposal of a critical test of the Navier–Stokes–Fourier paradigm for compressible fluid continua. Physical Review E, 87(1), 013014. Brenner, H., & Bielenberg, J. R. (2005). A continuum approach to phoretic motions: Thermophoresis. Physica A: Statistical Mechanics and its Applications, 355(24), 251–273. Burnett, D. (1935). The distribution of velocities in a slightly non-uniform gas. Proceedings of the London Mathematical Society, 2(1), 385–430. Chapman, S., & Cowling, T. G. (1991). The mathematical theory of non-uniform gases: An account of the kinetic theory of viscosity, thermal conduction and diffusion in gases. Cambridge university press. Dadzie, S. K. (2013). A thermo-mechanically consistent Burnett regime continuum flow equation without Chapman–Enskog expansion. Journal of Fluid Mechanics, 716, R6. . Dadzie, S. K., & Brenner, H. (2012). Predicting enhanced mass flow rates in gas microchannels using nonkinetic models. Physics Review E, 86(3), 036318 . Dadzie, S. K., & Reese, J. M. (2012). Analysis of the thermomechanical inconsistency of some extended hydrodynamic models at high knudsen number. Physics Review E, 85(4), 041202 . Dadzie, S. K., Reese, J. M., & McInnes, C. R. (2008). A continuum model of gas flows with localized density variations. Physica A: Statistical Mechanics and its Applications, 387(24), 6079–6094 . Gad-el Hak, M. (1999). The fluid mechanics of microdevices-the freeman scholar lecture. Transactions-American Society of Mechanical Engineers Journal of FLUIDS Engineering, 121, 5–33. Garz, V., & Santos, A. (2003). Kinetic theory of gases in shear flows: Nonlinear transport (131). Springer. GreenShields, C. J., & Reese, J. M. (2007). The structure of shock waves as a test of brenner’s modifications to the Navier Stokes equations. Journal of Fluid Mechanics, 580, 407–429 . Groot, S. d., & Mazur, P. (1962). Non-equilibrium Thermodynamics. Ho, C., & Tai, Y. (1998). Micro-electro-mechanical-systems (MEMS) and fluid flows. Annual Review of Fluid Mechanics, 30, 579–612. Knudsen, M. (1909). The law of the molecular flow and viscosity of gases moving through tubes. Annals of Physics, 28, 75–130. Landau, L., & Lifshitz, E. (1987). Fluid mechanics. In L. D. Landau & E. M. Lifshitz (Eds.). Course of theoretical physics (2nd ed.) (6). Bu, ButterworthHeinemann. Lee, C. J. (1994). Unique determination of solutions to the burnett equations. AIAA Journal, 32(5), 985–990. Lockerby, D. A., & Reese, J. M. (2003). High-resolution burnett simulations of micro couette flow and heat transfer. Journal of Computational Physics, 188(2), 333–347.

P.L.L. Walls, B. Abedian / International Journal of Engineering Science 79 (2014) 21–29

29

Maxwell, J. C. (1879). On stresses in rarified gases arising from inequalities of temperature. Philosophical Transactions of the Royal Society of London, 170, 231–256. Nanbu, K. (1983). Analysis of the couette flow by means of the new direct-simulation method. Journal of the Physical Society of Japan, 52(5), 1602–1608. Shen, C. (2005). Rarefied gas dynamics: Fundamentals simulations and micro flows. Springer. Steckelmacher, W. (1986). Knudsen flow 75 years on: The current state of the art for flow of rarefied gases in tubes and systems. Reports on Progress in Physics, 49(10), 1083. Tyndall, J. (1870). On dust and disease. W. Clowes and Sons. Xue, H., Ji, H. M., & Shu, C. (2001). Analysis of micro-couette flow using the burnett equations. International Journal of Heat and Mass Transfer, 44(21), 4139–4146. Zhong, X. (1991). Development and computation of continuum higher order constitutive relations for high-altitude hypersonic flow. Stanford University.