detailed investigation of hydrodynamics and thermal

Scientia Iranica B (2013) 20(4), 1228{1240

Sharif University of TechnologyScientia Iranica

Transactions B: Mechanical Engineeringwww.scientiairanica.com

Detailed investigation of hydrodynamics and thermalbehavior of nano/micro shear driven ow using DSMC

O. Ejtehadia, E. Roohia,* and J. Abolfazli Esfahania,b

a. Department of Mechanical Engineering, Faculty of Engineering, Ferdowsi University of Mashhad, Mashhad, P.O. Box:91775-1111, Iran.

b. Center of Excellence on Modeling and Control Systems, Ferdowsi University of Mashhad, Mashhad, P.O. Box: 91775-111, Iran.

Received 9 April 2012; received in revised form 6 February 2013; accepted 21 April 2013

KEYWORDSDSMC;Mon/diatomic gases;Nano/micro Couette ow;Knudsen layer;Compressibilitye�ects;Rarefaction e�ects.

Abstract. In the present work, we simulate rare�ed gas ow between two movingnano/micro parallel plates maintained at the same uniform temperature using the directsimulation Monte Carlo (DSMC) method. We perform simulations for monatomic argonand diatomic nitrogen gas and compare mon/diatomic gas behavior. For both gases, westudy heat transfer and shear stress and investigate the e�ects of compressibility andrarefaction in the entire Knudsen regime and for a wide range of wall Mach numbers.Slip velocity, temperature jump, wall heat ux and wall shear stress are directly sampledfrom the particles striking the surfaces and reported for monatomic and diatomic casesfor di�erent rarefaction regimes. We also study deviation from the equilibrium using theprobability density function for argon and nitrogen gas molecules.c 2013 Sharif University of Technology. All rights reserved.

1. Introduction

Micro/nanochannels are widely encountered in mi-cro/nanoelectromechanical systems (MEMS/NEMS).To enhance the design and performance of such sys-tems, it is necessary to achieve a deeper understandingof their ow and heat transfer behavior. Whenthe Knudsen number is su�ciently large, the uidrarefaction is the main parameter to evaluate thesesystems [1]. The degree of rarefaction is measured interms of the Knudsen number (Kn). The Knudsennumber is de�ned as the ratio of mean free pathof a molecule to a characteristic length of the ow.Flow regimes are also classi�ed on the basis of theKnudsen number into continuum (Kn�0.001), slip ow(0.001<Kn<0.1), transition ow (0.1<Kn<10) and freemolecular (Kn�10) regimes.

When there is a continuum breakdown, the solu-tions are to be established based on kinetic principles

*. Corresponding author. Tel: +98 511 8763304E-mail address: [email protected] (E. Roohi)

such as those in treating the Boltzmann equation.However, deterministic methods based on the Boltz-mann equation are complex and, hence, have motivatedthe development and application of the direct simula-tion Monte Carlo (DSMC) [2] method, which can alsobe regarded as a statistical method for the solution ofthe Boltzmann equation. The DSMC method possessesmany advantages, such as simplicity of application,unconditional stability, and ease of modeling complexphysical ows. However, the computational cost islarger than traditional computational uid dynamics(CFD) approaches.

Shear-driven ow, such as Couette, are encoun-tered in micro-motors, comb mechanisms and micro-bearings. In this study, we choose the well-knownproblem of planar Couette ow. Despite its simplicity,Couette ow includes many features found in complexrare�ed gas dynamics problems. Micro-Couette ow,i.e. a gas con�ned between two in�nite parallel plateswhich are at the same temperature but moving relativeto each other, has been widely studied. For exampleWillis [3] used Krook's kinetic model to investigate

O. Ejtehadi et al./Scientia Iranica, Transactions B: Mechanical Engineering 20 (2013) 1228{1240 1229

Couette ow at low Mach numbers. He showed thatvelocity and shear stress could be calculated from thesolution of an integral equation for the macroscopicvelocity. He derived a slip boundary condition forMaxwell molecules and compared the derived solutionwith those of other moment-based kinetic models.Sirovich [4] developed an approximate method in ki-netic theory for Couette ow and found expressions forslip coe�cients as a function of degree of specular anddi�use re ections. The method was especially accuratefor near-continuum ows. A slip coe�cient within1.5% of the exact value for purely di�use re ection wascalculated by this method. Beskok et al. [5] gave ananalysis of the e�ects of compressibility and rarefactionon pressure-driven and shear-driven micro ow. Theyfound that compressibility and rarefaction are com-peting phenomena and both require consideration inmicro uidic analysis. Marques et al. [6] investigatedgaseous micro-Couette ow with the NS equations andcompared results with the DSMC method in continuumand transition regimes. Their comparison between NSand DSMC solutions for the heat ux vector perpendic-ular to the plates showed that slip and jump boundaryconditions are also important in the transition regimeand do not necessarily depend on the velocities of theplates. Xue et al. [7-9] analyzed the micro-Couette owusing the DSMC method, the NS equations, and theBurnett equations, and found that rarefaction has asigni�cant e�ect on velocity, temperature and pressure,especially for high Knudsen numbers. They stated thatthe non-dimensional parameter Prandtl-Ekcert (PrEc),characterizing convective heat transfer in Couette ow,increases as ow becomes more rare�ed. Their predic-tion of shear stress and heat ux at the wall showed thesuperiority of Burnett equations over the NS equationsin the micro-Couette ow simulations. Meanwhile, theBurnett equations with the slip boundary condition,which is proportional to the Knudsen number, cannotbe extended to the transition ow regime. Frezzottiand Gibelli [10] proposed a kinetic model of uid-wallinteraction by simple extension of the Enskog theoryof dense uids, and applied the model to the caseof thermal Couette ow. They also investigated theinteraction of a dilute monatomic gas with a solid sur-face, compared Molecular Dynamics (MD) simulationswith numerical solutions of the proposed kinetic modeland showed that the results of the two simulations areclose. In an another attempt, Frezzotti et al. [11] usedtheir proposed uid-wall interaction kinetic model tostudy the heat transport between parallel walls andCouette ow, and compared their results with MDsimulations. The distribution functions of reemittedmolecules and the accommodation coe�cients obtainedfrom the two techniques are compared, and it is shownthat the kinetic model predictions are close to MDresults. More recently, Frezzotti et al. [12] proposed

a moment method based on the linearized Bhatnagar-Gross-Krook (BGK) kinetic model equation and ap-plied the method to the cases of Couette, Poiseuilleand cavity ow. Their results showed that excellentapproximations of exact solutions of the kinetic modelequation could be obtained using a small number ofmoments equations. Gu et al. [13] compared the DSMCsolution of the Couette ow with a high order momentapproach, i.e. 26 moment equations for capturingnon-equilibrium phenomena in the transition regime.Their new set of equations overcame many of thelimitations observed in the regularized 13 momentequations. Kumar et al. [14] investigated the accuracyof the statistical BGK and ellipsoidal statistical BGK(ESBGK) equations in comparison with the DSMCsolution in the Couette ow. They showed that thesemethods are more e�cient than the DSMC method fora ow condition that is at the comfort-level limit fordirect simulation Monte Carlo computations.

The objective of the present work is to pro-vide a deeper understanding of shear driven ow inmicro/nano scale channels. Although simulation ofCouette ow has been implemented before, the speci�ccontribution of this work is a detailed comparisonof the behavior of monatomic argon and diatomicnitrogen uids, which is not reported in the literature.For both gases, we investigated important engineeringparameters like heat ux and shear stresses, as wellas compressibility and rarefaction e�ects over a widerange of Knudsen and Mach numbers. Slip velocity,temperature jump, wall heat ux and shear stress aredirectly sampled from particles impinging the walls andare reported for both gases. The approximate solutionsfor planar Couette ow are available within the frame-work of continuum uid mechanics subject to velocityslip and temperature jump boundary conditions andvalid for the low-velocity (subsonic) and low Knudsennumber (Kn�0.1) ows [1,6,15]. Consequently, wecompare our DSMC temperature, velocity, shear stressand heat ux pro�le with the approximate derivationat low Kn/Mach numbers. We also study deviationfrom the Maxwellian equilibrium distribution using thevelocity distribution function of gas particles for bothargon and nitrogen.

2. Present numerical analysis

2.1. Problem statementPlanar Couette ow involves one dimensional owthrough a channel comprising two parallel plates aty = �h=2 moving relative to each other, with aconstant velocity, �Uw, and maintained at the sametemperature. It is depicted in Figure 1 schematically.The ow velocity is assumed to have only an axialcomponent varying only in the y direction, and theexternal forces are assumed to be absent.

1230 O. Ejtehadi et al./Scientia Iranica, Transactions B: Mechanical Engineering 20 (2013) 1228{1240

Figure 1. Schematic of the Couette ow.

2.2. The DSMC methodDSMC is a numerical tool to solve the Boltzmannequation based on direct statistical simulation of themolecular processes described by the kinetic theory [2].It is considered a particle method, in which a particlerepresents a large bulk of real gas molecules. Theprimary principle of DSMC is to decouple the motionand collision of particles during one time step. Theimplementation of DSMC requires that the compu-tational domain be broken down into a collection ofgrid cells. The cells are divided into subcells in eachdirection, and the subcells are then utilized to facilitatethe selection of collision pairs. After ful�lling all molec-ular movements in the domain, the collisions betweenmolecules are simulated in each cell, separately. In thecurrent study, a variable hard sphere (VHS) collisionmodel is used and the collision pair is chosen basedon the no time counter (NTC) method. The channelwalls are treated as di�use re ectors using the fullmomentum/thermal accommodation coe�cient. A fullmomentum accommodation coe�cient means that thetangential momentum of re ected molecules is equal tothe tangential momentum of the wall. Similarly, fullthermal accommodation means that the energy ux ofre ected molecules is equal to the energy ux corre-sponding to the wall temperature. The velocity of there ected molecules (c0) is randomly assigned accordingto the full thermal and momentum accommodation anda half-range Maxwellian distribution determined by thewall temperature [16] as follows:

c0x =p� log(RF (0))Vmpf sin(2� �RF (0)); (1)

c0y = �p� log(RF (0))Vmpf ; (2)

c0z =p� log(RF (0))Vmpf cos(2� �RF (0)): (3)

where Vmpf =p

2RT is the most probable speed ofthe molecules at wall temperature and RF (0) is auniformly distributed random fraction between 0 and1. The positive and negative symbols correspond tothe lower and upper walls, respectively, and R is thegas constant.

An accurate DSMC solution requires some con-straints on the cell size, time step and number ofparticles. The random selection of the particles froma cell for binary collisions requires the cell size to be a

small fraction of the gas mean free path. The decou-pling between the particles movements and collisionsis correct if the time step is a small fraction of themean collision time. The number of particles per cellshould be high enough, around 20, to avoid repeatedbinary collisions between the same particles. Thefollowing procedure is used to solve a stationary prob-lem with DSMC. In the entire computational domain,an arbitrary initial state of gas particles is speci�edand the desired boundary conditions are imposed attime zero. Particle movement and binary collisionsare performed separately. After achieving steady owcondition, sampling of molecular properties within eachcell is ful�lled during a su�cient time period to avoidstatistical scattering. All thermodynamic parameters,such as temperature, velocity, density and pressure, arethen determined from this time-averaged data. Moredetails on the DSMC algorithm are given in [2,17].

3. Results and discussion

To simulate micro/nano-Couette ow, we modi�edthe DSMC code used by Roohi and coworkers forsimulation of di�erent geometries, like micronozzle, mi-crochannel, micro-cavity, and backward facing step [18-22]. In all simulations, monatomic argon and diatomicnitrogen gases are considered at a reference tempera-ture of Tref = 273 K with a reference dynamic viscosityof �Ar = 2:117� 10�5 Ns/m2 and �N2 = 1:656� 10�5

Ns/m2, respectively [2]. We considered argon gas withmolecular mass of m = 6:63 � 10�26 kg, viscositytemperature index of ! = 0:81 and variable hard-sphere diameter of dVHS = 4:17�10�10 m, and nitrogengas with molecular mass of m = 4:65 � 10�26 kg,viscosity temperature index of ! = 0:74 and variablehard-sphere diameter of dVHS = 4:11 � 10�10 m. Thewalls are set to have equal temperature of Tw = 273 K.Our simulations are performed for a wide range ofKnudsen, varying from 0.01 to 10, and Mach numbersranging from 0.2 to 1.2, and both compressibility andrarefaction e�ects, besides comparison of monatomicand diatomic molecule behavior, are investigated.

The macroscopic properties of interest in thepaper are derived using molecular gas dynamics rela-tions [2]. The viscous stress tensor, � , is de�ned as thenegative of the pressure tensor with the static pressuresubtracted from the normal components.

� ij = �(�cicj � �ijp); (4)

where ci and cj are the components of c, the velocity ofthe molecule relative to stream velocity, called thermalvelocity, and p is the scalar pressure de�ned as:

p =1

3%c2: (5)


The heat ux vector is de�ned as:

q = 1=2%c2c+ n"intc; (6)

where "int is the internal energy of a single molecule.In �gures, the ordinate is normalized by the

distance between the parallel plates, h. Flow velocityand temperature are normalized by the relative velocityof the plates, Uw, and the temperature of the wall, Tw,respectively. Also, heat ux and shear stress are nor-malized by the parameters, �1UwCpTw and 1

2�1U2w,

respectively. Here, �1 and Cp are the free streamdensity and the speci�c heat capacity, respectively.The normalized heat ux and shear stress are referredto as the heat ux coe�cient, Ch, and the frictioncoe�cient, Cf , respectively, in the �gures. Also,because of the symmetry in the upper and lower partsof the channel, in �gures which include mon/diatomiccomparison, we only report the solution for the upperhalf of the domain. We also calculated the velocityslip, temperature jump, heat ux and shear stress onthe wall, based on the particles that strike the platesin the DSMC code, and compared the results with themacroscopic amounts near the plates. Slip velocity andtemperature jump are de�ned as follows [23]:

Uslip =P

((m= jcj j) ci)P(m= jcj j) ; (7)

Ttr;jump =1

3R

P�(1= jcj j) kck2

��P (1= jcj j)U2slipP

(1= jcj j)� Ttr;wall; (8)

Trot;jump =1k

P(erot= jcj j)P(1= jcj j) � Trot;wall; (9)

Tjump =�traTtra + �rotTrot

�tra + �rot: (10)

In the preceding equations, Uslip, Ttra;jump, Trot;jumpand Tjump are the slip velocity, and translational,rotational and overall temperature jumps, respectively.Ttra;wall and Trot;wall are the translational and rota-tional temperatures of plates. For our cases, Ttra;wall =Twall and Trot;wall = 0:R are the ideal gas constants,equal to 208 and 297 (JKg�1K�1) for argon and nitro-gen, respectively. �tra and �rot are the translational androtational degrees of freedom and erot is the rotationalenergy. The wall shear stress and heat uxes arede�ned as below [16]:

�w =Fnumts:�A

Xm(cinci � cref

i ); (11)

qw=Fnumts:�A

"X�12mc2

�inc�X�

12mc2

�ref#: (12)

Here, Fnum is the number of real molecules that eachsimulated particle represents, ts is the sampling time,and �A is the interaction area. Superscripts, i and r,denote the incident and re ected particles, respectively.

3.1. Grid study and comparisonTo achieve cell independent solutions, we simulateargon ow in a channel using three di�erent gridresolutions in a y direction. We kept the numberof particles per cell constant, i.e. an average of 300particles per cell, and set Knudsen and Mach numbersto 0.2 and 1.05, respectively. Although the problemis one-dimensional, we employed a two-dimensionalcode and imposed periodic boundary condition on bothsides of the domain to correctly simulate in�nite lengthplates. Figure 2 shows the velocity and temperaturepro�les of argon gas for Grid 1 (150,000 particles in10 � 50 cells), Grid 2 (300,000 particles in 10 � 100cells) and Grid 3 (450,000 particles in 10 � 150 cells),respectively. Since ow properties change only in thevertical direction, the e�ect of cell size in the x directionis not considered. It is observed that Grids 2 and 3

Figure 2. Grid study test: (a) Velocity pro�le, and (b) temperature pro�le.


give almost identical velocity and temperature pro�les.Therefore, we continue our study using Grid 2.

The approximate Navier-Stokes (NS) solutionsfor planar Couette ows are available within theframework of continuum uid mechanics subject tovelocity slip and temperature jump boundary condi-tions, and are valid for the low-velocity (subsonic) ows where variations of density are insigni�cant inthe ow �eld [1]. Marques et al. [6] showed that theapproximate NS solution for the streamwise velocityand the temperature pro�le in the Couette ow couldbe written as:

U =Uw

1 + k0Kn2yh; (13)

T =Tw +2m15k

(1 + 2k1Kn) (Uw

1 + k0Kn)2

� 2m15k

(Uw

1 + k0Kn)2(

2yh

)2: (14)

For hard sphere gases, k0 and k1 are two constantswith values, 1.111 and 2.127, respectively [24,25]. Fora Newtonian, viscous heat-conducting uid rare�edCouette ow in the early slip regime, the shear stress

and the heat ux could be derived using [6]:

�xy = �Uw

1 + k0Kn2h; (15)

qy = �(Uw

1 + k0Kn)2 4yh2 : (16)

It should be noted that the Marques et al. [6] equationfor the velocity pro�le is not self-consistent, since thevelocity slope at the channel center is �tted with theDSMC result. Figure 3(a) compares the DSMC so-lution for velocity, temperature, non-dimensional heat ux and shear stress with approximate NS solutionsgiven by Eqs. (13) - (16). In this comparison, theKnudsen number is set equal to 0.01 and the wall Machnumber is 0.16. At this low Mach number, approximatesolutions are quite close to DSMC's. There are someoscillations in the DSMC solutions, which are dueto statistical scatter inherent in the DSMC method.The DSMC's statistical scatter decreases with theinverse square root of the sample size. Therefore, alarge sample size is required to reduce the scatter.Figure 3(b) compares our DSMC solution with theDSMC solution of Ref. [26], numerical solution of thefull NS equations [13] and approximate solutions givenby Eqs. (13) and (16). This �gure shows that as the

Figure 3. Comparison of current results with di�erent analytical and numerical solutions.


ow rari�es, approximate solutions deviate from bothDSMC solutions. However, numerical NS solutionsare closer to DSMC's, which is due to employingmore accurate �rst order velocity slip and temperaturejump boundary conditions in the numerical NS code.Our DSMC solution is close to the DSMC solution ofRef. [8], except for larger scatter in our simulation, dueto smaller sample size compared with Gu and Emersonsimulations [13].

It should be noted that there are alternativeanalytical formula for the velocity pro�le of the Couette ow. For example, Lilley and Sader [27] showed thatthe velocity gradient is singular at the surface andsuggested a power law velocity pro�le in the Knudsenlayer for Couette ow. They used a DSMC solution to�nd a suitable �t parameter for their power formula fora case with Kn=0.006 and Uwall = 30 m/s.

3.2. Compressibility e�ectsThe reduction in dimensions of the devices results inthe importance of surface e�ects, due to the increasein surface area. This leads to a dominance of viscousforces over inertial forces. Therefore, variations ofpressure and density could be signi�cant, even thoughthe Mach number is well below the conventional limitof 0.3 [28]. The wall Mach number, Mwall, is de�nedas the ratio of Uw to parameter

p kTw=m and is

speci�ed by varying the driving velocity of both plates,Uw. The pressure in linear Couette ow is constant,consequently, compressibility e�ects are due to thetemperature changes only. At �rst, we performedsimulations in such a way that by keeping the Knudsennumber constant and varying the Mach number, thecompressibility e�ects are the e�ective issue. Weset the Mach number in such a way that di�erentMach regimes from subsonic to supersonic are covered.In Figures 4 through 7, velocity, temperature, heat ux and shear stress in domain are plotted for bothmonatomic argon and diatomic nitrogen gases, forwall Mach numbers changing from 0.2 to 1.2. InFigure 4, the velocity pro�les are almost identical toeach other. In fact, there are small variations invelocity pro�les from low subsonic wall Mach numberto early supersonic wall speed, for both gases. Howeveras the normalized velocity, i.e. U=Uw, is plotted inFigure 4, one can conclude that the increments ofboth the velocity in the domain and the gas velocityslip on the wall are proportional to the increase inwall velocity. Therefore, the ratio of U=Uw is nearlyconstant for all cases. It is observed that the velocitypro�le deviates from the linear incompressible velocitypro�le for the compressible, rare�ed gas near thewall. For the range of simulations performed here,the deviation from the linear pro�le is small, and themaximum deviation occurs for the case of maximumplate velocity. Here, the maximum deviation in the

Figure 4. Study of compressibility e�ects: Velocity pro�lefor di�erent mach numbers; Kn = 0:1, Tw = 273 (K).

Figure 5. Study of compressibility e�ects: Temperaturepro�le for di�erent mach numbers; Kn = 0:1,Tw = 273 (K).

non-dimensional velocities is about 0.008, which isobserved near the plates at Mwall = 1:2. This isbecause the macroscopic velocity only depends onthe microscopic velocity of the molecules, which isindependent of the degree of freedom of molecules.The microscopic velocity is determined according tothe amount of kinetic energy transferred from the wallto gas molecules, which increases for higher wall speedvalues.


Figure 6. Study of compressibility e�ects: Heat uxpro�le for di�erent Mach numbers; Kn = 0:1,Tw = 273 (K).

Figure 7. Study of compressibility e�ects: shear stresspro�le for di�erent Mach numbers; Kn = 0:1,Tw = 273 (K).

Figure 5 shows the variation of temperature in thedomain with wall Mach variations. The temperaturepro�les show that, as Mwall increases, the temperaturedi�erence between the plates and the center of thechannel becomes larger due to higher viscous dissipa-tion. Thus, compressibility e�ects become signi�cant.The increase of temperature jump at the walls byincreasing the Mach number is quite obvious, especially

for the monatomic argon gas in Figure 5(a). Themagnitude of the temperature jump for argon gas atMwall = 1:2 is almost 47% of the wall temperature.This amount for nitrogen gas at the same Mach numberis about 27% of the wall temperature. This decrease oftemperature in the domain is explained by an increasein heat capacity, due to using a gas with �ve degrees offreedom instead of three. The relation between Cp and� is given by Cp = (�=2 + 1)R. Therefore, as the heatcapacity of nitrogen is higher, it has lower temperaturecompared to the argon for the same Knudsen and Machnumbers. Figure 6 shows the non-dimensional heat uxpro�le. It shows that the slope of the heat ux pro�lesincreases by increasing the wall Mach number. This isexplained by the temperature increase in the domain asa result of viscous heating. By increase of temperature,larger thermal velocities would accelerate the frequencyof the momentum exchange between the gas and plates.As a result, higher heat uxes for higher wall Machnumbers are observed in Figure 6. Comparison ofmon/diatomic gases in Figure 6(a) and (b) shows thatthe magnitude of Ch decreases when dealing withdiatomic nitrogen gas. This is also explained by thee�ect of degrees of freedom on speci�c heat capacities,as discussed before.

The pro�les of normalized shear stress in Figure 7show the uniform distribution of normalized shearstress in the domain. The �gure indicates a decreasein the absolute amount of shear stress by increasingthe wall Mach number. This is explained by a decreasein the probability of intermolecular collisions due toa decrease in the total collision cross section (�T ),according to the relation below [16]:

�T = �ref(c2rc2r;ref

)��: (17)

Here, cr is the relative velocity of the collidingmolecules, � = 2=(� � 1), where � is the parameterindicating the hardness of particles and the subscript`ref' denotes the reference values at the referencetemperature, Tref . The values of � for argon andnitrogen for VHS molecules at 273 K are 0.31 and0.24, respectively [2]. As Mach number increases,the relative velocity of the molecules in the domainwould increase. Therefore, the total collision crosssection, and the probability of intermolecular collisionsdecreases. Consequently, higher wall Mach numberswould lead to smaller magnitudes of shear stress. Fromthe macroscopic point of view, the classical relationshipbetween three basic non-dimensional numbers, Mach,Reynolds and Knudsen, is given by:

Kn =MaRe

r �2: (18)

In a constant Knudsen number, as Mach numberincreases, the Reynolds number would increase. The


increase of Reynolds number means the dominance ofthe inertial forces over the viscous forces. Therefore,due to the convective nature of the ow, the e�ect ofmolecular di�usion gets smaller and less magnitudesof Cf are observed. Comparing Figure 7(a) and (b),it is observed that the magnitude of Cf is almost thesame for monatomic and diatomic gases for Kn=0.1.However, as will be discussed later, the magnitudeof non-dimensional shear stress increases for diatomicgas in comparison with the monatomic gas for lagerKnudsen numbers.

3.3. Rarefaction e�ectsRarefaction e�ects gain importance with the reductionin the geometry size, since the sizes of the geometriesbecome comparable to the mean free path of the gasmolecules. These e�ects are speci�ed through theKnudsen number. Now, the plates' velocity is keptconstant (Uw = 250 m/s) and the corresponding wallMach numbers are 0.81 and 0.74 for argon and nitrogen,respectively. This di�erence in the wall Mach number isdue to the di�erent speci�c heat ratio ( ) of two gases.Figure 8 shows normalized velocity pro�les. This �gureconsiders rarefaction e�ects for six di�erent Knudsennumbers in the entire Knudsen regimes, i.e. fromcontinuum to free molecular. We simulate Knudsennumbers corresponding to the early and end parts ofthe slip regime (Kn = 0.01 and 0.1), and early, mid,and end parts of the transition regime (Kn = 0.5, 1,5 and 10). It is observed in Figure 8, that the slopeof the velocity pro�le decreases as Knudsen numberincreases. This means a decrease of velocity in thedomain. In a constant wall Mach number ow, as

Figure 8. Study of rarefaction e�ects: velocity pro�le fordi�erent Kn number ows; Uw = 250 (m/s), Tw = 273 (K).

Figure 9. Schematic diagram showing Knudsen layer andslip velocity.

the ow becomes rare�ed, the number of moleculescolliding into the moving walls and transferring thekinetic momentum from the moving wall to the domaindecreases. Therefore, the velocity in the domaindecreases as the Knudsen number increases. Analyticalsolution of the linearized Boltzmann equation indicatesthat the bulk ow velocity pro�le is essentially linear forall Knudsen numbers [1]. However, a kinetic boundarylayer, named Knudsen layer, on the order of one to afew mean free paths, starts to become dominant be-tween the bulk ow and solid surfaces in the transition ow regime [29,30]. Figure 9 is a schematic diagramshowing the Knudsen layer. In the �gure, ugas is thereal slip velocity of the gas inside the Knudsen layer,and uslip is the approximate value obtained from the NSequations to calculate the velocity pro�le in the entire ow �eld. As observed in Figure 8, when Kn=0.01,the velocity pro�le is linear, but when Kn>0.1, thelinear velocity pro�les no longer exist. This is due tothe increase of the thickness of the Knudsen layer inthe transition ow regime. Gallis et al. [31] reportedthat within the Knudsen layer, molecules collide withthe surface more frequently than they collide with eachother. As a result, when Knudsen number is small(Kn<0.1), the change of velocity in the Knudsen layeris negligible in the entire velocity pro�le; however, inthe slip and transition ow regimes, Knudsen layeroccupies a signi�cant proportion of the channel inthe Couette problem, and a non-Newtonian behaviorbecomes pronounced. In Figure 8, it is observed thatbending in the velocity pro�le occurs most speci�callyat Knudsen number equal to 1. However, with furtherincrease in Knudsen number, the velocity pro�les aretending to the linear pro�les again. This is becausethe ow will enter the free molecular regime, where theKnudsen layer occupies the domain completely. Theseobservations quite conform with the results reported by


Figure 10. Study of rarefaction e�ects: Temperaturepro�le for di�erent Kn number ows; Uw = 250 (m/s),Tw = 273 (K).

Xue et al. [8]. It is also observed that by increasing theKnudsen number, the slip velocity increases. It shouldbe noted that the slip velocity at the plates would reachits maximum value as the Knudsen number approaches10. It is also observed, in Figure 8(a) and (b), thatusing diatomic gas instead of monatomic gas has almostno e�ect on the bulk velocity of the domain. Here, themaximum deviation in the velocity pro�les is about0.005 at Kn=1.

Figure 10 illustrates the temperature pro�les atdi�erent Knudsen numbers. As the Knudsen numberincreases, temperature in the domain and also the tem-perature jump at the plates becomes more signi�cant,but the curvature of the temperature pro�les reduces.In a constant wall Mach number ow, the amount ofwall kinetic energy is constant. As the ow becomesmore rare�ed, this constant kinetic energy will be savedin a smaller number of molecules, and shows itselfin terms of temperature rise. However, in a densegas, this amount of constant energy is distributed ina larger number of molecules, and the temperaturerise is smaller than that observed in rare�ed gases.Also for small Knudsen numbers and close to thecontinuum regime, only near surface molecules collidewith the walls, and only in the middle of the channel dointermolecular collisions happen. For larger Knudsennumbers, as the gas becomes more rare�ed, moleculesin the middle of the channel have the same chance formolecular-surface collision and transferring heat withthe walls. Consequently, the maximum curvature inFigure 10 is observed in the Kn=0.01, and, by increasein Knudsen number, the temperature in the domain

Figure 11. Study of rarefaction e�ects: Heat ux pro�lefor di�erent Kn number ows; Uw = 250 (m/s),Tw = 273 (K).

becomes uniform. The maximum temperature in thecenter of the channel for nitrogen is about 95% of themaximum temperature for argon, when Kn = 0.01.This fraction is about 85%, when Kn = 10. This meansthat the e�ect of mon/diatomic structures will be moresigni�cant in rare�ed conditions.

Figure 11 shows the non-dimensional heat uxvariation. It is observed that, by increasing theKnudsen number, the magnitude of heat ux in thedomain increases. This is because of an increase inthe surface-molecular collisions, due to an increase inthe mean free path of the molecules. As a result, notonly can the molecules near the surface, but also themolecules at the center of the channel, collide with thesurfaces. Therefore, more absorbed kinetic energy fromthe wall is transferred into heat. The increase in theslope of the non-dimensional heat ux for Kn>5 slowsdown, which is an absolute result of rarefaction e�ects.It is also observed in Figure 11(a) and (b) that thediatomic gases experience a lower amount of heat uxthan monatomic ones. The reasoning is similar to thatalready discussed for Figure 6.

Our simulations show that, as the gas becomesmore rare�ed, it will experience less shear stress due toa smaller amount of intermolecular collisions. However,the magnitude of non-dimensional shear stress wouldstill increase, as observed in Figure 12. In other words,as shear stress is normalized by the term, including freestream density, �1, it is correct to say that by increas-ing the Knudsen number, the magnitude of normalizedshear stress increases, even though the magnitude ofshear stress decreases. It is observed that, in Eq. (18),


Figure 12. Study of rarefaction e�ects: Shear stresspro�le for di�erent Kn number ows; Uw = 250 (m/s),Tw = 273 (K).

by increasing the Knudsen number, Reynolds numberwould decrease. This means the dominance of viscousforces over inertial forces at higher Knudsen number ows. As viscosity increases, temperature, heat uxand shear stress would increase, which totally supportsmolecular or microscopic theories. Comparison ofFigure 12(a) and (b) shows that the shear coe�cientfor monatomic and diatomic gases is almost the samewhen Kn=0.01. But the di�erence in the magnitudeof shear stress coe�cients becomes larger in rare�edgases.

Figures 13 and 14 show the variations of di�erentproperties, such as velocity slip, temperature jump,wall heat and shear stress, on the surface. Here, our

purpose is to compare mon/diatomic gas behaviors,study Kn number e�ects and compare the valuessampled directly from Eqs. (7) through (12) with themacroscopic (volume based) values of these propertiescalculated in the adjacent cells near the wall. Figure 13shows that the velocity slip and temperature jumpincrease as Knudsen number increases, however, therate of this increase slows down as the ow approachesthe free molecular regime. Both the direct samplingmethod and volume based method are in agreementwith each other for Couette ow. In fact, if owdoes not experience large gradients on the surface cells,volume-based sampling would provide solutions closeto direct sampling solutions. Figure 14 shows thatthe wall heat ux and absolute amount of wall sheardecreases by an increase in Knudsen number. Bothvolume based and microscopic methods show the sametrend. There is small di�erence between the valuespredicted by both methods, while it is expected thatthe direct sampling method will provide the accuratesolution.

We have plotted the probability density function(PDF) for a cell in the middle of the geometry inFigure 15(a) and (b). This is a particular location,because the velocity of molecules above this locationwould have a net velocity in the direction of theupper plate, and the molecules below there wouldhave a velocity in the reverse direction. We alsoplotted the Maxwellian equilibrium distribution in that�gure to investigate the deviation from equilibrium [2].Maxwellian distribution describes the probability of�nding a randomly selected particle in a unit volumeelement in the velocity space for equilibrium gases, andis given by [16]:

f0 = (m

2�kT)3=2 exp(�mc02

2kT): (19)

PDF shown in Figure 15(a) illustrates that by increas-

Figure 13. Comparison of volume-based-results with microscopic results.


Figure 14. Comparison of volume-based-results with microscopic results.

Figure 15. Study of rarefaction e�ects: Probability density function for di�erent Kn number ows; Uw = 250 (m/s),Tw = 273 (K).

ing the Knudsen number, the ow deviates from theequilibrium more. It is observed in Figure 15(a) thatall curves have a global maximum. By increasing theKnudsen number, this global maximum deviates fromits identical point in the Maxwellian distribution. Also,the curves shift to the right when the Knudsen numberincreases. It is observed that when Kn=0.01, thecurve is almost identical to Maxwellian distribution,and the maximum deviation occurs when Kn=10.Figure 15(b) illustrates a similar distribution withalmost 15% decrease in the magnitude of the globalmaximum for diatomic nitrogen gas. This is due to thelower mass of nitrogen compared to argon. However,nitrogen molecules have greater PDF for higher velocitymagnitudes.

4. Conclusion

We studied nano/micro Couette ow through a channelfor a wide range of Mach and Knudsen numbers, usingthe DSMC method. We investigated compressibilityand rarefaction e�ects on the behaviors of velocity,temperature, heat ux and shear stress pro�les for

monatomic argon and diatomic nitrogen gases. Com-pressibility tests showed that an increase in wall veloc-ity results in an increase in non-dimensional velocity,temperature and heat ux, but a decrease in the magni-tude of shear stress in the domain. The most importantparameter here was the viscous dissipation, whichincreased as the wall velocity increased and caused atemperature increase in the domain. Rarefaction testsshowed that an increase in Knudsen number results ina decrease in non-dimensional velocity, but an increasein temperature, an absolute amount of non-dimensionalheat ux and shear stress in the domain. It is observedthat the gradient of the velocity pro�les decreases asthe ow becomes more rare�ed. It is also observedthat when Kn=0.01, the velocity pro�le is linear, but,by increasing the Knudsen number, the velocity pro�ledeviated from the linear pro�les and a non-Newtonianbehavior was observed. This is due to the increase inthe thickness of the Knudsen layer in the transition owregime. Comparison of surface parameters calculatedby both microscopic and volume based macroscopicmethods showed good agreement between these twomethods in prediction of slip velocity and temperature


jump. However, slight deviations were observed in theprediction of wall heat ux and shear stress at smallKnudsen numbers. All surface properties approacheda nearly constant value as the ow approached thefree molecular regime. While the velocity slips of bothmonatomic and diatomic gases were the same, the lat-ter had a much smaller temperature jump. This is dueto the higher heat capacity of the diatomic nitrogen.We also illustrated deviation from the equilibrium byplotting the probability density function in a particularcell. Nitrogen gas had a lower global maximum in PDFdue to lower mass, while it has more molecules withhigher velocity magnitudes.

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Biographies

Omid Ejtehadi obtained an MS degree from FerdowsiUniversity of Mashhad, Iran, in 2012. His researchinterests include: rare�ed gas dynamics, DSMC simula-tion of micro/nano ows, and MEMS/NEMS. To date,he has published 3 journal papers and 4 conferencepapers.

Ehsan Roohi obtained a PhD degree in Aerospace En-gineering from Sharif University of Technology, Tehran,Iran, in 2010, and is currently Assistant Professorof Mechanical Engineering at Ferdowsi University ofMashhad, Iran. To date, he has published 28 journalarticles and 38 conference papers. His research inter-ests include: rare�ed gas dynamics, Direct SimulationMonte Carlo (DSMC) and two phase ows.

Javad Abolfazli Esfahani obtained a PhD degreein Mechanical Engineering from the University of NewBrunswick (UNB), Canada, in 1997, and is currentlyProfessor of Mechanical Engineering at Ferdowsi Uni-versity of Mashhad, Iran. To date, he has published5 books/chapters, 52 journal articles and more than80 conference papers. Dr. Esfahani is a member ofCenters of Excellence in Energy Conversion (CEEC),on Modelling and Control Systems, (CEMCS) and inEnergy Management (CEEM). He is also a memberof the Iranian Combustion Institute and the IranianSociety of Mechanical Engineering, and is also on theEditorial Boards of the Journal of Fuel and Combustion(Iranian) and IRECHE. His research interests include:theoretical and numerical heat and mass transfer.

detailed investigation of hydrodynamics and thermal

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