Research Article Event-Driven Molecular Dynamics...
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Research Article Event-Driven Molecular Dynamics Simulation of Hard-Sphere Gas Flows in Microchannels Volkan Ramazan Akkaya 1,2 and Ilyas Kandemir 2 1 Department of Energy Systems Engineering, Mu˘ gla Sıtkı Koc ¸man University, 48100 Mu˘ gla, Turkey 2 Department of Mechanical Engineering, Gebze Technical University, Gebze, 41400 Kocaeli, Turkey Correspondence should be addressed to Volkan Ramazan Akkaya; [email protected] Received 20 July 2015; Accepted 1 December 2015 Academic Editor: Manfred Krafczyk Copyright © 2015 V. R. Akkaya and I. Kandemir. is 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. Classical solution of Navier-Stokes equations with nonslip boundary condition leads to inaccurate predictions of flow characteristics of rarefied gases confined in micro/nanochannels. erefore, molecular interaction based simulations are oſten used to properly express velocity and temperature slips at high Knudsen numbers (Kn) seen at dilute gases or narrow channels. In this study, an event- driven molecular dynamics (EDMD) simulation is proposed to estimate properties of hard-sphere gas flows. Considering molecules as hard-spheres, trajectories of the molecules, collision partners, corresponding interaction times, and postcollision velocities are computed deterministically using discrete interaction potentials. On the other hand, boundary interactions are handled stochastically. Added to that, in order to create a pressure gradient along the channel, an implicit treatment for flow boundaries is adapted for EDMD simulations. Shear-Driven (Couette) and Pressure-Driven flows for various channel configurations are simulated to demonstrate the validity of suggested treatment. Results agree well with DSMC method and solution of linearized Boltzmann equation. At low Kn, EDMD produces similar velocity profiles with Navier-Stokes (N-S) equations and slip boundary conditions, but as Kn increases, N-S slip models overestimate slip velocities. 1. Introduction Over the last decades, micro- and nanoelectromechanical systems (MEMS, NEMS) have been rapidly developed. Today, many applications like flow sensors or miniaturized jet engines are taking place in industry. erefore, investigation of gas flows in micro- and nanochannels is essential for design and operation of microdevices such as micropumps, microvalves, and microturbines [1]. As devices get smaller, Knudsen number, Kn (the ratio of the mean free path of gas molecules, , to the characteristic length of the channel, ), increases. Magnitude of Kn determines the flow regime: continuum regime for Kn ≤ 0.01, slip regime for 0.01 < Kn ≤ 0.1, transitional regime for 0.1 < Kn ≤ 10, and free molecular flow for Kn ≥ 10. In many macroscale applications, is negligible compared to the channel size (Kn ≪ 0.01) and the continuum assumption in Navier-Stokes equations is valid. For smaller scales, channel size is in the order of ; flow is either in slip or in transitional regime, and compressibil- ity and rarefaction effects are present. Since Navier-Stokes equations are hardly valid in these regimes, fluid can be treated as an ensemble of particles interacting with each other and the boundaries. Molecular Dynamics (MD) and Direct Simulation Monte Carlo (DSMC) methods are two simulation methods based on the computation of motions and interactions of particles involved in such ensembles. In molecular simulations, computational difficulties arise as the number of molecules gets larger simply because interacting particles and their positions need to be computed for each interaction. DSMC method [2] uses a number of representative molecules to simulate larger number of real molecules. Motions of molecules are exact but collisions are generated in a probabilistic manner. On the other hand, MD simulations are much more realistic and accurate since each particle represents a real molecule and its position and velocity are exactly known. Classically, trajectories of molecules are calculated from integration of Newton’s equations of motion over a time step regarding interaction potential. Hence, this process is time driven. Although the potential is smooth Hindawi Publishing Corporation Mathematical Problems in Engineering Volume 2015, Article ID 842837, 12 pages http://dx.doi.org/10.1155/2015/842837
Transcript of Research Article Event-Driven Molecular Dynamics...
Research ArticleEvent-Driven Molecular Dynamics Simulation ofHard-Sphere Gas Flows in Microchannels
Volkan Ramazan Akkaya12 and Ilyas Kandemir2
1Department of Energy Systems Engineering Mugla Sıtkı Kocman University 48100 Mugla Turkey2Department of Mechanical Engineering Gebze Technical University Gebze 41400 Kocaeli Turkey
Correspondence should be addressed to Volkan Ramazan Akkaya volkanakkayamuedutr
Received 20 July 2015 Accepted 1 December 2015
Academic Editor Manfred Krafczyk
Copyright copy 2015 V R Akkaya and I KandemirThis is an open access article distributed under theCreativeCommonsAttributionLicense which permits unrestricted use distribution and reproduction in anymedium provided the originalwork is properly cited
Classical solution ofNavier-Stokes equationswith nonslip boundary condition leads to inaccurate predictions of flow characteristicsof rarefied gases confined in micronanochannels Therefore molecular interaction based simulations are often used to properlyexpress velocity and temperature slips at highKnudsen numbers (Kn) seen at dilute gases or narrow channels In this study an event-drivenmolecular dynamics (EDMD) simulation is proposed to estimate properties of hard-sphere gas flows Consideringmoleculesas hard-spheres trajectories of the molecules collision partners corresponding interaction times and postcollision velocitiesare computed deterministically using discrete interaction potentials On the other hand boundary interactions are handledstochastically Added to that in order to create a pressure gradient along the channel an implicit treatment for flow boundariesis adapted for EDMD simulations Shear-Driven (Couette) and Pressure-Driven flows for various channel configurations aresimulated to demonstrate the validity of suggested treatment Results agree well with DSMC method and solution of linearizedBoltzmann equation At low Kn EDMD produces similar velocity profiles with Navier-Stokes (N-S) equations and slip boundaryconditions but as Kn increases N-S slip models overestimate slip velocities
1 Introduction
Over the last decades micro- and nanoelectromechanicalsystems (MEMSNEMS) have been rapidly developed Todaymany applications like flow sensors or miniaturized jetengines are taking place in industry Therefore investigationof gas flows in micro- and nanochannels is essential fordesign and operation of microdevices such as micropumpsmicrovalves and microturbines [1] As devices get smallerKnudsen number Kn (the ratio of the mean free path ofgas molecules 120582 to the characteristic length of the channel119871119888) increases Magnitude of Kn determines the flow regime
continuum regime for Kn le 001 slip regime for 001 lt Kn le
01 transitional regime for 01 lt Kn le 10 and freemolecularflow for Kn ge 10 In many macroscale applications 120582 isnegligible compared to the channel size (Kn ≪ 001) and thecontinuum assumption in Navier-Stokes equations is validFor smaller scales channel size is in the order of 120582 flowis either in slip or in transitional regime and compressibil-ity and rarefaction effects are present Since Navier-Stokes
equations are hardly valid in these regimes fluid can betreated as an ensemble of particles interacting with eachother and the boundaries Molecular Dynamics (MD) andDirect Simulation Monte Carlo (DSMC) methods are twosimulation methods based on the computation of motionsand interactions of particles involved in such ensembles Inmolecular simulations computational difficulties arise as thenumber of molecules gets larger simply because interactingparticles and their positions need to be computed for eachinteraction
DSMC method [2] uses a number of representativemolecules to simulate larger number of real moleculesMotions of molecules are exact but collisions are generated ina probabilistic manner On the other hand MD simulationsare much more realistic and accurate since each particlerepresents a real molecule and its position and velocityare exactly known Classically trajectories of molecules arecalculated from integration of Newtonrsquos equations of motionover a time step regarding interaction potential Hence thisprocess is time driven Although the potential is smooth
Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2015 Article ID 842837 12 pageshttpdxdoiorg1011552015842837
2 Mathematical Problems in Engineering
and continuous for real interactions approximate approachesare used especially in dilute gas simulations for the sake ofcomputational efficiency Lennard-Jones potential is themostcommon simplified interaction potential model used in MDsimulations Standard MD simulation based on Lennard-Jones interaction potential was first introduced by Rahman[3] and Verlet [4] This model has been used in numerousstudies up to present time The major drawbacks of thisapproach are limitations of simulation time and size [5]Integration time step is so small that during a simulationof ten microseconds the enormous number of time steps isrequired even for ten thousandmolecules representing a verysmall volume
Modelling the system as an ensemble of hard particles isan alternative to continuous interaction potential approachInteraction potential between hard particles is discrete thatis no force is exerted on the molecules except impactsAssumingno external force field resulting trajectories are lin-ear Intermolecular collisions are instantaneous and binaryWhen a collision occurs postcollisional velocities are analyti-cally determined via conservation of energy andmomentumAnalytically predicted collision times yield the decompo-sition of simulation into a series of asynchronous eventsThis process is event driven since simulation time advancesdirectly from one event to the next Due to its natureevent-driven approach eases simulation of larger systems forlonger periods compared to time-driven one Since the firstintroduction of event-driven molecular dynamics (EDMD)simulations [6] developments of more effective algorithmshave further enhanced performance of EDMD [7ndash12] Sim-ulation of millions of particles for longer periods is possiblewith current computational power of a desktop computer[13]
In order to simulate a confined gas flow using molecularsimulations system boundaries and interactions with the gasmolecules must be modelled appropriately Three types ofboundary are usually sufficient to represent a flow in amicro-or nanochannel These well-known boundaries are periodicwall and flow boundaries
Since the performance of MD simulations is highlyaffected by the number of simulated molecules it is desiredto represent the computational domain with the smallestpossible sample Periodicity is the most common boundarymodel for such representations It also removes surface effectsin the simulations of infinite systems In the presence ofperiodic boundaries computational domain is repeated inthe direction of periodicity to form an infinite latticeWhen amolecule reaches a periodic boundary it continues its courseon the opposite face with exactly the same velocity vector
On the wall boundary a molecule hits the wall if thedistance between the wall and molecular centre is a halfmolecular diameterThemolecule is reflected backwith a newvelocity determined according to the wall model Specularreflection is themost basic wallmodel inwhich surface is per-fectly smooth atmolecular levelWhen amolecule undergoesspecular reflection normal velocity component is invertedwhile tangential components remain unchanged conservingtangential momentum However in reality wall surfaceis rough and the molecules are reflected at random angles
from the wall Diffuse reflection is the most common modelto represent such surfaces In diffuse model postreflectionvelocity of the molecule is substantially independent ofthe incoming velocity and determined stochastically from adistribution based on wall temperature and velocity
Several treatments for flow boundaries have been devel-oped to induce a flow inside the channel These are selectivereflection treatment [14ndash18] acceleration body force treat-ment [19 20] and the implicit treatment for flow boundaries[21 22] In selective reflection and acceleration body forcetreatments total energy and the number of molecules in thesystem are conserved since no new molecule or energy isintroduced In the implicit treatment for flow boundariesmolecules reaching either end (upstream or downstream)of the system leave the computational domain permanentlyMoreover new molecules are introduced to the computa-tional domain according to local domain properties Henceunlike aforementioned methods a number of molecules inthe computational domain and total energy are not preservedAlthough these treatments are convenient to use both intime and event-driven simulations the implicit treatment inDSMCmethod has not been applied toMD simulations untilthis study
Investigation of Shear- and Pressure-Driven flows of ahard-sphere gas in microchannels is beyond the capabilitiesof MD simulation methods using continuous interactionpotentials Additionally implicit treatment of flow bound-aries is not easily applicable since the method is time drivenTherefore simulations were carried out with the use of event-driven molecular dynamics simulations in this study EDMDhas similarities in implementation with both classical MDand DSMC methods In contrast to DSMC method colli-sional dynamics such as movements of molecules determi-nation of collision partners and calculation of postcollisionalvelocities are deterministic in the hard-sphere EDMD Allcollisions are real and computed in MD simulations Inthis study simulations were conducted with real numberof molecules taking advantage of an event scheduling algo-rithm with 119874(1) computational complexity Added to this acell division method increased computational speed greatlywithout missing any event On the other hand as a majorcontribution of this study wall and flow boundaries werehandled stochastically as in DSMCmethod which created anopportunity to adapt implicit treatment for flow boundariesto EDMD simulations for the first time
This study is organized as follows EDMD algorithmand application of boundary treatments are described inSection 2 In Sections 3 and 4 theory and simulation resultsof Shear- (Couette) and Pressure-Driven flows are presentedConcluding remarks are given in Section 5
2 Methodology and Model
21 Computational Model In this study EDMD simulationof monatomic molecules was conducted Computationaldomain was a rectangular parallelepiped region Moleculeswere modelled as hard-spheres thus only translational ener-gies were considered Only binary collisions as a validassumption for dilute gases were employed Postcollisional
Mathematical Problems in Engineering 3
velocities were calculated according to mass diameter andprecollisional velocity information of colliding pair by satis-fying conservation equations of energy and momentum
Argon (Ar) was selected as the monatomic gas of thisstudy In order to reduce numerical errors all magnitudeswere nondimensionalized by considering unit Boltzmannconstant (119896
119887= 10) and scaling with the reference variables
temperature (119879ref = 300K) mass (119898 = 662 times 10minus26 kg) and
diameter (119889 = 341times10minus10m) It is worthmentioning that the
results of the study can be extended to any monatomic gas byselecting proper scaling variables
Due to the nature of MD simulations immense compu-tational resources are required Simulation speed decreasesalmost linearly as the number of molecules in the systemincreases In order to speed up the simulations computa-tional domain was partitioned into cubical cells using themethod described in detail by Kandemir [9] A cell has 26neighbours at maximum fewer if it is at an edge of com-putational domain In this method molecules are assignedto the cells according to their current positions An inter-molecular collision is probable only if the distance betweentwo molecules is less than cell width Otherwise moleculeswould encounter cell crossing events beforehand Thus onlythe molecules of same or neighbour cells are considered ascandidate collisional pairs at a given time Implementationof cell partitioning method reduces the computational effortto 119874(119899
119888) for one molecule where 119899
119888is the average number of
molecules in the neighbourhood Since there are119873moleculesin the system total computational effort is about 119874(119873)
assuming that 119874(119899119888) asymp 119874(1) Simulations of the same system
on the basis of single and multicell approaches yield exactlyidentical numerical results using the same seed and randomnumber generator Hence one can conclude that themulticellmethod can guarantee that no collision is missed
Three types of events are possible in event-drivensimulations These are intermolecular collisions molecule-boundary interactions and cell crossings A molecule canbe involved in many future events but only the earliestone is registered as the candidate event for the moleculeThus the number of candidate events is always equal to thenumber of simulated molecules The goals are determinationand execution of the earliest event Following the executioncandidate events of related molecules are invalidated andpossible new candidate events are calculated This processrepeats throughout the simulation
Determination of earliest event by linear search has acomplexity of 119874(119873) and makes EDMD simulations practi-cally impossible for large number of molecules In this studya priority queue method suggested by Paul [11] was imple-mented In his method all events are distributed among timeinterval bins and events belonging to earliest bin are sortedusing a complete binary tree (CBT) structure This approachreduces computational effort approximately to 119874(1) on aver-age
Macroscopic thermodynamic properties of a simulatedsystem can be estimated by ensemble averaging over par-ticlesrsquo behaviour However this would be expensive interms of computational effort because of the requirementof large number of molecules to obtain smooth profiles
The alternative approach is the time averaging for a smallernumber of particles since ensemble average and time averageof a phase variable yield equal values in an ergodic system[23] Since this study deals with large number of moleculesan ensemble averaging scheme was performed by averaging aseries of snapshots of computational domain taken in equalintervals For the sake of profiling computational domainwas partitioned into small subdomains (bins) Eachmoleculeis associated with a certain bin based on its molecular centreAlthough the selection of bin size is arbitrary the averagenumber of molecules inside a bin is an important parameterfor an accurate profiling of system properties Small numberof molecules may demonstrate large fluctuations due tostatistical nature of EDMD simulations On the other handresolution of distribution is reduced for larger bins Inorder to satisfy both criteria computational domain wasdivided into 51 spatial bins along a direction (51 times 51 for 2Ddistributions) in this study Snapshotting was started after atleast 100 collisions per particle (CPP) with a frequency of atleast 1 CPP In order to obtain smoother profiles longer CPPvalues were used for averaging
22 Implicit Treatment for Flow Boundaries From micro-scopic viewpoint in addition to bulk velocity gas moleculesare translated by thermal velocity also In high speed (hyper-sonic) flows the magnitude of the bulk velocity is higherthan that of the thermal velocity For such flows conventionalapproach is the use of vacuum boundary at the exit of thechannel (when a molecule reaches the vacuum boundary itleaves the computational domain permanently and there isno inward flux at that boundary)
In contrast to hypersonic flows themagnitude of thermalmotion is comparable with bulk velocity in low speed flowand inward flux because thermal motion of molecules shouldbe taken into consideration In order to cover a wide rangeof flow regimes Liou and Fang [21] introduced an implicittreatment for flow boundaries in DSMC simulations Thistreatment was adapted for EDMD simulations in this study
In this approach the number of incoming molecules andtheir corresponding velocities are estimated implicitly by thelocal flow properties For either upstream or downstreamboundary molecular flux (119865) into the computational domaincan be determined using the Maxwellian distribution func-tion
119865119895=
119899119895
2radic120587120573119895
exp (minus1199042
119895cos2120601)
+ radic120587119904119895cos120601 [1 + erf (119904
119895cos120601)]
(1)
where
119904119895= 119880119895120573119895
120573119895=
1
119888mp119895=
1
radic2119896119887119879119895119898
(2)
Flux is calculated for each cell face of flow boundary cell 119895Here streamwise velocity and local temperature are denoted
4 Mathematical Problems in Engineering
by119880 and119879 respectively Value of120601 is 0 for upstream and120587 fordownstream boundary 119899
119895is the number density of molecules
in the cell and 119888mp is the most probable speed For a timeinterval 120575119905 the number of molecules (119873
119895) entering domain
from surface area 119860119895of cell 119895 is
119873119895= 119865119895120575119905119860119895 (3)
Incoming molecules are introduced at random positionswithin the corresponding cells touching the outside of thecell surface In case of any overlap (ie centre-to-centredistance of the molecules being less than sum of the radii)molecule is repositioned
Tangential velocity components (V and119908) of an incomingmolecule are independent of streamwise velocity and gener-ated as follows
V = 119881 + 119888mpR119899
radic2
119908 = 119882 + 119888mpR1015840119899
radic2
(4)
Here 119881 and 119882 are mean tangential velocity componentsR119899and R1015840
119899are random numbers generated from normal
distribution with zero mean and unit varianceIf streamwise inlet velocity is zero (119880 = 0) normal
component is generated as follows
119906 = 119888mpradicminus logR119906 (5)
Here R119906is a random number generated from uniform
distribution of interval [0 1) For nonzero streamwise inletvelocities (119880 = 0) Garcia andWagner [24] introduced severalefficient acceptance-rejection methods with the general formof
119906 = 119880 minus 119891(119880
119888mp) 119888mp (6)
Here 119891(119880119888mp) is a random number selected throughan acceptance-rejection method In this study the recom-mended method for low speed flows (minus04119888mp lt 119880 lt 13119888mp)[24] was used
Flow properties of both upstream and downstreamboundaries must be determined in order to be used in (1)ndash(6) For upstream boundary temperature and pressure areknown The number density 119899in is obtained from ideal gasequation
119899in =119875in119896119879in
(7)
Transverse velocities 119881in and 119882in are set to zero Streamwisevelocity (119880in)119895 is extrapolated from neighbour boundary cell119895 where 120588 and 119886 are density and speed of sound respectively[22]
(119880in)119895 = 119880119895+
119875in minus 119875119895
120588119895119886119895
(8)
At downstream boundary only pressure is known Thusnumber density (119899
119890)119895 temperature (119879
119890)119895 and velocity compo-
nents (119880119890)119895 (119881119890)119895 and (119882
119890)119895are extrapolated from neighbour
cell
(119899119890)119895= 119899119895+
119875119890minus 119875119895
1198981198862119895
(119879119890)119895=
119875119890
(119899119890)119895119896
(119880119890)119895= 119880119895+
119875119895minus 119875119890
119898119899119895119886119895
(119881119890)119895= 119881119895
(119882119890)119895= 119882119895
(9)
3 Verifications
Shear-Driven and Pressure-Driven flows in confinedmicrochannels are well-studied problems They are fre-quently used as test cases for the robustness and validations offlow simulations In this section the theoretical backgroundand the simulation setup for such flows are presented Anew implementation of EDMD simulation including cellpartitioning event scheduling and implicit boundary treat-ment was developed in C with serial processing Benefitingfrom object oriented programming (OOP) code can befurther extended to cover different molecular types andboundary treatments On the other hand considering thecomputational performance the parallelization of the code isan essential issue in molecular simulations However thisdeserves a detailed study of its own All simulations werecarried out on a fourth-generation Intel i7 4700CPU com-puter with 16GB of RAM
31 Shear-Driven (Couette) Flow Shear-Driven flow in a con-fined channel is obtained bymoving the top plate at a constantvelocity (Figure 1(a)) Pressure gradient is zero between theupstream and downstream boundaries Initial temperature ofthe argon inside the channel and at the upstream boundaryis 300K Upper wall moves with a velocity of 100ms whilelower wall is stationary Periodicity is applied along 119911-axis Spacing ratios (the ratio of average distance betweenmolecules 119904 to average diameter of molecules 119889) for thisstudy were selected as 40 and 70 Flow configurations werechosen according to Knudsen numbers of 01 05 and 10 inthree cases (Table 1) The number of simulated particles wasaround 200000 for each case
Simulation results were compared with the continuum-based model in which velocity gradient between two wallsof the channel is affected by the Knudsen number and has alinear nature in Shear-Driven flows For a fully diffusive wallnormalized velocity profile is shown as follows where 119880
119908is
wall velocity and 119867 is channel height [25]
119906
119880119908
=1
2Kn + 1(
119910
119867+ Kn) (10)
Mathematical Problems in Engineering 5
Table 1 Flow configurations in Shear-Driven flow
Case A (Kn = 01) Case B (Kn = 05) Case C (Kn = 10)sd = 40 sd = 70 sd = 40 sd = 70 sd = 40 sd = 70
Channel height 119867 (120583m) 0050 0263 0010 0053 0005 0026Global average pressure 119875 1639 kPa (sd = 40) 306 kPa (sd = 70)Global average density 120588 2607 kgm3 (sd = 40) 487 kgm3 (sd = 70)
Boundary conditions Fully diffusive walls 300K along 119910-directionPeriodicity along 119911-direction
Table 2 Flow configurations in Pressure-Driven flow
Case D Case EInlet pressure 119875in 250 kPa 75 kPaPressure ratio PR 25 5Initial global average density (120588) 400 kgm3 120 kgm3
Initial Kn 0080 0268Channel height 119867 04 120583m (1173 molecular diameter)Aspect ratio (119871
119909119871119910) 5
Boundary conditions Fully diffusive walls 300K along 119910-directionPeriodicity along 119911-direction
32 Pressure-Driven Flows Pressure-Driven flow is gener-ated by setting different pressure levels on upstream anddownstream boundaries (Figure 1(b)) Two Pressure-Drivenflows of different pressure ratios (PR = 119875
119894119875119890) in a
microchannel were investigated Flow properties are tabu-lated in Table 2 The number of simulated molecules wasabout 300000 for both cases
For Pressure-Driven flows Arkilic et al defined thepressure distribution along the fully diffusive channel as afunction of Knudsen number at the exit of the channel (Kne)of length 119871 as [26]
119875 (119909)
119875119890
= minus6Kne
+ radic(6Kne)2
+ (1 + 12Kne)119909
119871+ (PR2 + 12KnePR) (1 minus
119909
119871)
(11)
For fully diffusive walls analytical solution of Navier-Stokesequations with second-order slip boundary condition yieldsa normalized velocity profile [27]
119906
119880119888
=[minus (119910119867)
2
+ 119910ℎ + Kn](14 + Kn)
(12)
where 119880119888is centreline velocity Slip-to-centreline velocity
ratio along the channel then becomes
119880119904
119880119888
=Kn
(14 + Kn) (13)
For second-order slip boundary condition velocity profileand corresponding slip-to-centreline ratio are
119906
119880119888
=[minus (119910119867)
2
+ 119910ℎ + Kn (1 + Kn)](14 + Kn (1 + Kn))
(14)
119880119904
119880119888
=Kn (1 + Kn)
(14 + Kn (1 + Kn)) (15)
If slip velocity is normalized with the local average velocity(12) and (15) become [28]
119880119904
119880=
Kn(16 + Kn)
119880119904
119880=
Kn (1 + Kn)(16 + Kn (1 + Kn))
(16)
4 Results and Discussions
In this section we present simulation results of Shear-Driven and Pressure-Driven flows for aforementioned casesResults were comparedwith analytical solutions derived fromNavier-Stokes equations with slip flow boundary conditionsandDSMCresults Simulation configuration (number of sim-ulated molecules spacing ratio boundary types etc) greatlyaffects computational speed As an example simulation ofone million argon molecules in a cubical region when thespacing ratio is 50 takes 450 seconds to reach one collisionper particle on the aforementioned PC configuration
41 Shear-Driven Flow In Cases AndashC six Couette flowconfigurations at different Knudsen numbers were simulatedComparison of the predicted slip velocities between simu-lation results and the analytical solution is given in Table 3
6 Mathematical Problems in Engineering
y
UwTw
H
Tw x
P T
(a) Shear-Driven flow
y
Tw
Tw
H
x
TePin Tin
(b) Pressure-Driven flow
Figure 1 Flow geometries
1
075
05
025
00 25 50 75 100
u (ms)
N-S 1st ordersd = 40
sd = 70
Kn = 01
yH
(a)
1
075
05
025
0minus8 minus4 0 4 8
sd = 40
sd = 70
Kn = 01
yH
u minus uN-S (ms)
(b)
Figure 2 Shear-Driven flow Case A (Kn = 01) (a) velocity profile (b) deviation from analytical solution
Table 3 Comparison of slip velocities
119880119904(ms)
(theory)
119880119904(ms)
(simulation)(sd = 40)
119880119904(ms)
(simulation)(sd = 70)
Case A (Kn = 01) 833 775 756Case B (Kn = 02) 2500 1859 1929Case C (Kn = 05) 3333 2546 2622
On the upper and lower walls slip velocities were estimatedby linearly extrapolating average velocities of neighbour binsKnudsen number was 01 in Case A and the flow was inthe slip-to-transition regime Consequently deviation fromthe analytical solution is reasonably small less than 10 InCases B and C flows were in the transitional regime (01 lt
Kn le 10) Therefore simulated slip velocities were predicted
noticeably lower than the analytical solution Deviation isbetween 213 and 256 in these cases
Predicted velocity profiles and deviations from the ana-lytical solution are presented in Figures 2ndash4 For low Kn flow(Case A) velocity profile is almost linear and correspondingdeviation is insignificant As rarefaction intensifies (Cases Band C) velocity profile deviates particularly at the near wallregions S shaped velocity profile is more apparent in Case C
Another finding is that nearly identical velocity profileswere obtained for different spacing ratios at the same Knud-sen number As the spacing ratio reduces from 70 to 40computational speed quadruples Utilization of this pieceof information can ease the computational cost of largersystems that is EDMD performs faster in smaller spacingratios [9]
42 Pressure-Driven Flow Pressure distribution along thechannel is given in Figures 5(a) and 6(a) for cases D and E
Mathematical Problems in Engineering 7
1
075
05
025
00 25 50 75 100
u (ms)
N-S 1st ordersd = 40
sd = 70
Kn = 05
yH
(a)
1
075
05
025
0minus8 minus4 0 4 8
sd = 40
sd = 70
Kn = 05
yH
u minus uN-S (ms)
(b)
Figure 3 Shear-Driven flow Case B (Kn = 05) (a) velocity profile (b) deviation from analytical solution
1
075
05
025
00 25 50 75 100
u (ms)
N-S 1st ordersd = 40
sd = 70
Kn = 10
yH
(a)
1
075
05
025
0minus8 minus4 0 4 8
sd = 40
sd = 70
Kn = 10
yH
u minus uN-S (ms)
(b)
Figure 4 Shear-Driven flow Case C (Kn = 10) (a) velocity profile (b) deviation from analytical solution
respectively The results are compared with linear pressuredistribution and slip formulation given by Arkilic et al [26]Figures 5(b) and 6(b) showdeviation betweenMDsimulationresults and analytical solutions Relative pressure differenceis less than 5 for both cases This indicates that adaptedboundary treatment is appropriate for EDMD simulations
Deviation from the linear pressure distribution is given inFigures 5(c) and 6(c) In both cases nonlinearity is slightlylower in EDMD predictions
Local mean free path increases along the channel due topressure gradient This yields an increase in local Knudsennumber (Figure 7) In Case D Kn was between 0080 and
8 Mathematical Problems in Engineering
25
2
15
10 025 05 075 1
N-SLinear MD simulation
PP
e
xL
(a)
0015
0
minus00150 025 05 075 1
N-SMD simulation
xL
(Psim
minusP
N-S)P
N-S
(b)
0 025 05 075 1
N-SMD simulation
xL
(PminusP
lin)P
lin
008
006
004
002
0
(c)
Figure 5 Pressure distribution in Pressure-Driven flow Case D (a) comparison of pressure distribution along 119909-axis (b) deviation betweensimulation and N-S results (c) deviation from linear pressure distribution
5
4
3
2
10 025 05 075 1
N-SLinear MD simulation
PP
e
xL
(a)
0 025 05 075 1
N-SMD simulation
xL
(PminusP
lin)P
lin
01
005
0
(b)
002
0
minus0020 025 05 075 1
N-SMD simulation
xL
(Psim
minusP
N-S)P
N-S
(c)
Figure 6 Pressure distribution in Pressure-Driven flow Case E (a) comparison of pressure distribution along 119909-axis (b) deviation betweensimulation and N-S results (c) deviation from linear pressure distribution
0211 (ie the flow was in slip-to-transition regime) In CaseE flow was in the transitional regime with corresponding Knup to 12
Ratio of the slip velocity to the local average velocity isgiven in Figure 8 Knudsen number values in the abscissa cor-respond to local values formed in Figure 7 along the channel
Slip velocities were linearly extrapolated from outermost binssimilar to Couette flow calculations Simulation results werecompared with analytical solutions with first- and second-order slip boundary conditions (16) linearized Boltzmannsolution andDSMC results [28] It is seen that EDMD resultsare in good agreement with DSMC and linearized Boltzmann
Mathematical Problems in Engineering 9
12
09
06
03
0
Case DCase E
0 025 05 075 1
Kn
xL
Figure 7 Local Kn distribution in Pressure-Driven flow
12
09
06
03
0001
N-S 1st orderN-S 2nd orderLinearized Boltzmann
DSMC
Case D
Case E
Case ECase D
01 1 10
Kn
UsU
Figure 8 Normalized slip velocity for slip-to-transition region
solution On the other hand breakdown of slip boundarymodels starts around Kn = 01 Both first- and second-order models overestimate the velocity slip Note that forboth Cases D and E upstream and downstream slip velocityratios are smaller than in other regions of the computationaldomain Use of Maxwell distribution to sample number
density and velocities in pressure boundary treatment isthe main reason of this behaviour Such implementation ofboundary treatment works better when the rarefaction effectsare small For highly rarefied (highKn) flows Ahmadzadeganet al [29] advise usingChapman-Enskog velocity distributionto improve pressure boundary treatment
10 Mathematical Problems in Engineering
1
075
05
025
0
yH
0 025 05 075 1
uUc
Kn = 0115
xL = 0559
N-S 2nd orderMD simulation
1
075
05
025
0
yH
0 025 05 075 1
uUc
xL = 0128
N-S 2nd orderMD simulation
0085Kn =
1
075
05
025
0
yH
0 025 05 075 1
uUc
xL = 0833
N-S 2nd orderMD simulation
0159Kn =
Figure 9 Velocity profiles in 119910-direction of Pressure-Driven flow Case D
1
075
05
025
0
YH
0 025 05 075 1
UUC
XL = 0814
N-S 2nd orderMD simulation
0686Kn =
1
075
05
025
0
YH
0 025 05 075 1
UUC
XL = 0206
N-S 2nd orderMD simulation
1
075
05
025
0
YH
0 025 05 075 1
UUC
XL = 0579
N-S 2nd orderMD simulation
0457Kn =0308Kn =
Figure 10 Velocity profiles in 119910-direction of Pressure-Driven flow Case E
Normalized velocity profiles (119906119880119888where 119880
119888is local
centreline velocity) in cross-stream direction at three loca-tions along the channel are shown in Figures 9 and 10 forCase D and Case E respectively Note that flow velocityalong the channel increases as pressure decreases Thusfor a given position along the channel normalization wascarried out by corresponding centreline velocity Here resultsof EDMD simulations were compared with second-orderanalytical solution In CaseD where Kn is small curves agreewell whereas in Case E estimation of higher slip velocity inanalytical solution becomes more apparent as Kn increases
5 Conclusion
This paper presents an event-driven molecular dynamics(EDMD) simulation of monatomic gas flows in microchan-nels In this study molecules are considered as hard-sphereswith discrete interaction potentials and interaction times areanalytically predictable Therefore each collision is real andtakes place in its calculated position Postcollisional velocitiesare also analytically calculated according to the conservationofmomentumand energy By its deterministic nature EDMDdiffers from DSMC method in which simulated system is
Mathematical Problems in Engineering 11
represented by a smaller group of molecules and collisionpairs are selected stochastically Additionally by using cellpartitioning technique and use of priority queues for eventsorting EDMD is now a strong alternative to DSMCmethodin terms of not only accuracy but also computational speed
In this study an implicit treatment for flow boundaries isadapted from DSMC method to EDMD for the first time inorder to induce Shear- and Pressure-Driven flows in confinedchannels The number of introduced molecules throughupstream and downstream boundaries and their correspond-ing velocities and locations are determined implicitly fromlocal number density mean flow velocity and temperature
Velocity slip in Pressure-Driven flows is compared withDSMCresults and solution of linearizedBoltzmann equationThe good agreement confirms the robustness of the treatmentand its applicability to EDMD simulations For both Shear-and Pressure-Driven flows velocity slip is also comparedwith the one derived from Navier-Stokes equations of slipboundary conditions For low Knudsen number (Kn) flowsvelocity profiles and calculated slip velocities agree well Butas the flow rarefies slip boundarymodels estimate higher slipvelocities
Additionally simulations of twoflows having the sameKnbut different spacing ratios produce almost identical velocityprofiles in Shear-Driven cases for the investigated range ofKn Since simulations with lower spacing ratios run fasterthis finding can be further extended to scale rarefied flowsand to reduce the computational time
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] T-R Hsu MEMS amp Microsystems Design Manufacture andNanoscale Engineering JohnWiley amp Sons Hoboken NJ USA2nd edition 2008
[2] G A BirdMolecular GasDynamics and theDirect Simulation ofGas Flows Clarendon Press Oxford UK 1994
[3] A Rahman ldquoCorrelations in the motion of atoms in liquidargonrdquo Physical Review vol 136 no 2 pp A405ndashA411 1964
[4] L Verlet ldquoComputer lsquoexperimentsrsquo on classical fluids I Ther-modynamical properties of Lennard-Jones moleculesrdquo PhysicalReview vol 159 no 1 pp 98ndash103 1967
[5] MMareschal and B L Holian ldquoFrom fluid particles to physicalparticles computing hydrodynamicsrdquo in Microscopic Simula-tions of Complex Hydrodynamic Phenomena M Mareschal andB LHolian Eds vol 292 ofNATOASI Series pp 1ndash12 SpringerScience amp Business Media 1992
[6] B J Alder and T E Wainwright ldquoPhase transition for a hardsphere systemrdquo The Journal of Chemical Physics vol 27 no 5pp 1208ndash1209 1957
[7] D C Rapaport ldquoThe event scheduling problem in moleculardynamic simulationrdquo Journal of Computational Physics vol 34no 2 pp 184ndash201 1980
[8] M Marın and P Cordero ldquoAn empirical assessment of priorityqueues in event-driven molecular dynamics simulationrdquo Com-puter Physics Communications vol 92 no 2-3 pp 214ndash2241995
[9] I Kandemir A multicell molecular dynamics method [PhDthesis] CaseWesternReserveUniversity ClevelandOhioUSA1999
[10] W T Tang R S M Goh and I L-J Thng ldquoLadder queuean O(1) priority queue structure for large-scale discrete eventsimulationrdquoACMTransactions onModeling and Computer Sim-ulation vol 15 no 3 pp 175ndash204 2005
[11] G Paul ldquoA Complexity O(1) priority queue for event drivenmolecular dynamics simulationsrdquo Journal of ComputationalPhysics vol 221 no 2 pp 615ndash625 2007
[12] A Donev A L Garcia and B J Alder ldquoStochastic event-drivenmolecular dynamicsrdquo Journal of Computational Physics vol 227no 4 pp 2644ndash2665 2008
[13] M N Bannerman R Sargant and L Lue ldquoDynamO a freeO(N) general event-driven molecular dynamics simulatorrdquoJournal of Computational Chemistry vol 32 no 15 pp 3329ndash3338 2011
[14] J Li D Liao and S Yip ldquoCoupling continuum to molecular-dynamics simulation reflecting particle method and the fieldestimatorrdquo Physical Review E vol 57 no 6 pp 7259ndash7267 1998
[15] F Bao YHuang Y Zhang and J Lin ldquoInvestigation of pressure-driven gas flows in nanoscale channels usingmolecular dynam-ics simulationrdquo Microfluidics and Nanofluidics vol 18 no 5-6pp 1075ndash1084 2014
[16] F Bao Y Huang L Qiu and J Lin ldquoApplicability of moleculardynamics method to the pressure-driven gas flow in finitelength nano-scale slit poresrdquo Molecular Physics vol 113 no 6pp 561ndash569 2015
[17] J R Whitman G L Aranovich and M D DonohueldquoAnisotropic mean free path in simulations of fluids travelingdown a density gradientrdquo Journal of Chemical Physics vol 132no 22 Article ID 224302 2010
[18] Q D To T T Pham G Lauriat and C Leonard ldquoMoleculardynamics simulations of pressure-driven flows and comparisonwith acceleration-driven flowsrdquo Advances in Mechanical Engi-neering vol 2012 Article ID 580763 10 pages 2012
[19] M Lupkowski and F van Swol ldquoComputer simulation of fluidsinteracting with fluctuating wallsrdquo The Journal of ChemicalPhysics vol 93 no 1 pp 737ndash745 1990
[20] C Huang K Nandakumar P Y K Choi and L W KostiukldquoMolecular dynamics simulation of a pressure-driven liquidtransport process in a cylindrical nanopore using two self-adjusting platesrdquo Journal of Chemical Physics vol 124 no 23Article ID 234701 2006
[21] W W Liou and Y C Fang ldquoImplicit boundary conditions fordirect simulation Monte Carlo method in MEMS flow pre-dictionsrdquo Computer Modeling in Engineering and Sciences vol1 no 4 pp 118ndash128 2000
[22] M Wang and Z Li ldquoSimulations for gas flows in microge-ometries using the direct simulation Monte Carlo methodrdquoInternational Journal of Heat and Fluid Flow vol 25 no 6 pp975ndash985 2004
[23] D J Evans and G P Morriss Statistical Mechanics of Nonequi-librium Liquids Cambridge University Press 2nd edition 2008
[24] A L Garcia and W Wagner ldquoGeneration of the Maxwellianinflow distributionrdquo Journal of Computational Physics vol 217no 2 pp 693ndash708 2006
12 Mathematical Problems in Engineering
[25] G Karniadakis A Beskok and N Aluru Microflows andNanoflows Fundamentals and Simulation Springer New YorkNY USA 2006
[26] E B Arkilic M A Schmidt and K S Breuer ldquoGaseous slipflow in long microchannelsrdquo Journal of MicroelectromechanicalSystems vol 6 no 2 pp 167ndash178 1997
[27] A Beskok G E Karniadakis and W Trimmer ldquoRarefactionand compressibility effects in gas microflowsrdquo Journal of FluidsEngineering vol 118 no 3 pp 448ndash456 1996
[28] A Beskok and G E Karniadakis ldquoReport a model for flows inchannels pipes and ducts at micro and nano scalesrdquoMicroscaleThermophysical Engineering vol 3 no 1 pp 43ndash77 1999
[29] A Ahmadzadegan J Wen and M Renksizbulut ldquoThe role ofthe velocity distribution in the DSMC pressure boundary con-dition for gasmixturesrdquo International Journal ofModern PhysicsC vol 23 no 12 Article ID 1250087 2012
Submit your manuscripts athttpwwwhindawicom
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Stochastic AnalysisInternational Journal of
2 Mathematical Problems in Engineering
and continuous for real interactions approximate approachesare used especially in dilute gas simulations for the sake ofcomputational efficiency Lennard-Jones potential is themostcommon simplified interaction potential model used in MDsimulations Standard MD simulation based on Lennard-Jones interaction potential was first introduced by Rahman[3] and Verlet [4] This model has been used in numerousstudies up to present time The major drawbacks of thisapproach are limitations of simulation time and size [5]Integration time step is so small that during a simulationof ten microseconds the enormous number of time steps isrequired even for ten thousandmolecules representing a verysmall volume
Modelling the system as an ensemble of hard particles isan alternative to continuous interaction potential approachInteraction potential between hard particles is discrete thatis no force is exerted on the molecules except impactsAssumingno external force field resulting trajectories are lin-ear Intermolecular collisions are instantaneous and binaryWhen a collision occurs postcollisional velocities are analyti-cally determined via conservation of energy andmomentumAnalytically predicted collision times yield the decompo-sition of simulation into a series of asynchronous eventsThis process is event driven since simulation time advancesdirectly from one event to the next Due to its natureevent-driven approach eases simulation of larger systems forlonger periods compared to time-driven one Since the firstintroduction of event-driven molecular dynamics (EDMD)simulations [6] developments of more effective algorithmshave further enhanced performance of EDMD [7ndash12] Sim-ulation of millions of particles for longer periods is possiblewith current computational power of a desktop computer[13]
In order to simulate a confined gas flow using molecularsimulations system boundaries and interactions with the gasmolecules must be modelled appropriately Three types ofboundary are usually sufficient to represent a flow in amicro-or nanochannel These well-known boundaries are periodicwall and flow boundaries
Since the performance of MD simulations is highlyaffected by the number of simulated molecules it is desiredto represent the computational domain with the smallestpossible sample Periodicity is the most common boundarymodel for such representations It also removes surface effectsin the simulations of infinite systems In the presence ofperiodic boundaries computational domain is repeated inthe direction of periodicity to form an infinite latticeWhen amolecule reaches a periodic boundary it continues its courseon the opposite face with exactly the same velocity vector
On the wall boundary a molecule hits the wall if thedistance between the wall and molecular centre is a halfmolecular diameterThemolecule is reflected backwith a newvelocity determined according to the wall model Specularreflection is themost basic wallmodel inwhich surface is per-fectly smooth atmolecular levelWhen amolecule undergoesspecular reflection normal velocity component is invertedwhile tangential components remain unchanged conservingtangential momentum However in reality wall surfaceis rough and the molecules are reflected at random angles
from the wall Diffuse reflection is the most common modelto represent such surfaces In diffuse model postreflectionvelocity of the molecule is substantially independent ofthe incoming velocity and determined stochastically from adistribution based on wall temperature and velocity
Several treatments for flow boundaries have been devel-oped to induce a flow inside the channel These are selectivereflection treatment [14ndash18] acceleration body force treat-ment [19 20] and the implicit treatment for flow boundaries[21 22] In selective reflection and acceleration body forcetreatments total energy and the number of molecules in thesystem are conserved since no new molecule or energy isintroduced In the implicit treatment for flow boundariesmolecules reaching either end (upstream or downstream)of the system leave the computational domain permanentlyMoreover new molecules are introduced to the computa-tional domain according to local domain properties Henceunlike aforementioned methods a number of molecules inthe computational domain and total energy are not preservedAlthough these treatments are convenient to use both intime and event-driven simulations the implicit treatment inDSMCmethod has not been applied toMD simulations untilthis study
Investigation of Shear- and Pressure-Driven flows of ahard-sphere gas in microchannels is beyond the capabilitiesof MD simulation methods using continuous interactionpotentials Additionally implicit treatment of flow bound-aries is not easily applicable since the method is time drivenTherefore simulations were carried out with the use of event-driven molecular dynamics simulations in this study EDMDhas similarities in implementation with both classical MDand DSMC methods In contrast to DSMC method colli-sional dynamics such as movements of molecules determi-nation of collision partners and calculation of postcollisionalvelocities are deterministic in the hard-sphere EDMD Allcollisions are real and computed in MD simulations Inthis study simulations were conducted with real numberof molecules taking advantage of an event scheduling algo-rithm with 119874(1) computational complexity Added to this acell division method increased computational speed greatlywithout missing any event On the other hand as a majorcontribution of this study wall and flow boundaries werehandled stochastically as in DSMCmethod which created anopportunity to adapt implicit treatment for flow boundariesto EDMD simulations for the first time
This study is organized as follows EDMD algorithmand application of boundary treatments are described inSection 2 In Sections 3 and 4 theory and simulation resultsof Shear- (Couette) and Pressure-Driven flows are presentedConcluding remarks are given in Section 5
2 Methodology and Model
21 Computational Model In this study EDMD simulationof monatomic molecules was conducted Computationaldomain was a rectangular parallelepiped region Moleculeswere modelled as hard-spheres thus only translational ener-gies were considered Only binary collisions as a validassumption for dilute gases were employed Postcollisional
Mathematical Problems in Engineering 3
velocities were calculated according to mass diameter andprecollisional velocity information of colliding pair by satis-fying conservation equations of energy and momentum
Argon (Ar) was selected as the monatomic gas of thisstudy In order to reduce numerical errors all magnitudeswere nondimensionalized by considering unit Boltzmannconstant (119896
119887= 10) and scaling with the reference variables
temperature (119879ref = 300K) mass (119898 = 662 times 10minus26 kg) and
diameter (119889 = 341times10minus10m) It is worthmentioning that the
results of the study can be extended to any monatomic gas byselecting proper scaling variables
Due to the nature of MD simulations immense compu-tational resources are required Simulation speed decreasesalmost linearly as the number of molecules in the systemincreases In order to speed up the simulations computa-tional domain was partitioned into cubical cells using themethod described in detail by Kandemir [9] A cell has 26neighbours at maximum fewer if it is at an edge of com-putational domain In this method molecules are assignedto the cells according to their current positions An inter-molecular collision is probable only if the distance betweentwo molecules is less than cell width Otherwise moleculeswould encounter cell crossing events beforehand Thus onlythe molecules of same or neighbour cells are considered ascandidate collisional pairs at a given time Implementationof cell partitioning method reduces the computational effortto 119874(119899
119888) for one molecule where 119899
119888is the average number of
molecules in the neighbourhood Since there are119873moleculesin the system total computational effort is about 119874(119873)
assuming that 119874(119899119888) asymp 119874(1) Simulations of the same system
on the basis of single and multicell approaches yield exactlyidentical numerical results using the same seed and randomnumber generator Hence one can conclude that themulticellmethod can guarantee that no collision is missed
Three types of events are possible in event-drivensimulations These are intermolecular collisions molecule-boundary interactions and cell crossings A molecule canbe involved in many future events but only the earliestone is registered as the candidate event for the moleculeThus the number of candidate events is always equal to thenumber of simulated molecules The goals are determinationand execution of the earliest event Following the executioncandidate events of related molecules are invalidated andpossible new candidate events are calculated This processrepeats throughout the simulation
Determination of earliest event by linear search has acomplexity of 119874(119873) and makes EDMD simulations practi-cally impossible for large number of molecules In this studya priority queue method suggested by Paul [11] was imple-mented In his method all events are distributed among timeinterval bins and events belonging to earliest bin are sortedusing a complete binary tree (CBT) structure This approachreduces computational effort approximately to 119874(1) on aver-age
Macroscopic thermodynamic properties of a simulatedsystem can be estimated by ensemble averaging over par-ticlesrsquo behaviour However this would be expensive interms of computational effort because of the requirementof large number of molecules to obtain smooth profiles
The alternative approach is the time averaging for a smallernumber of particles since ensemble average and time averageof a phase variable yield equal values in an ergodic system[23] Since this study deals with large number of moleculesan ensemble averaging scheme was performed by averaging aseries of snapshots of computational domain taken in equalintervals For the sake of profiling computational domainwas partitioned into small subdomains (bins) Eachmoleculeis associated with a certain bin based on its molecular centreAlthough the selection of bin size is arbitrary the averagenumber of molecules inside a bin is an important parameterfor an accurate profiling of system properties Small numberof molecules may demonstrate large fluctuations due tostatistical nature of EDMD simulations On the other handresolution of distribution is reduced for larger bins Inorder to satisfy both criteria computational domain wasdivided into 51 spatial bins along a direction (51 times 51 for 2Ddistributions) in this study Snapshotting was started after atleast 100 collisions per particle (CPP) with a frequency of atleast 1 CPP In order to obtain smoother profiles longer CPPvalues were used for averaging
22 Implicit Treatment for Flow Boundaries From micro-scopic viewpoint in addition to bulk velocity gas moleculesare translated by thermal velocity also In high speed (hyper-sonic) flows the magnitude of the bulk velocity is higherthan that of the thermal velocity For such flows conventionalapproach is the use of vacuum boundary at the exit of thechannel (when a molecule reaches the vacuum boundary itleaves the computational domain permanently and there isno inward flux at that boundary)
In contrast to hypersonic flows themagnitude of thermalmotion is comparable with bulk velocity in low speed flowand inward flux because thermal motion of molecules shouldbe taken into consideration In order to cover a wide rangeof flow regimes Liou and Fang [21] introduced an implicittreatment for flow boundaries in DSMC simulations Thistreatment was adapted for EDMD simulations in this study
In this approach the number of incoming molecules andtheir corresponding velocities are estimated implicitly by thelocal flow properties For either upstream or downstreamboundary molecular flux (119865) into the computational domaincan be determined using the Maxwellian distribution func-tion
119865119895=
119899119895
2radic120587120573119895
exp (minus1199042
119895cos2120601)
+ radic120587119904119895cos120601 [1 + erf (119904
119895cos120601)]
(1)
where
119904119895= 119880119895120573119895
120573119895=
1
119888mp119895=
1
radic2119896119887119879119895119898
(2)
Flux is calculated for each cell face of flow boundary cell 119895Here streamwise velocity and local temperature are denoted
4 Mathematical Problems in Engineering
by119880 and119879 respectively Value of120601 is 0 for upstream and120587 fordownstream boundary 119899
119895is the number density of molecules
in the cell and 119888mp is the most probable speed For a timeinterval 120575119905 the number of molecules (119873
119895) entering domain
from surface area 119860119895of cell 119895 is
119873119895= 119865119895120575119905119860119895 (3)
Incoming molecules are introduced at random positionswithin the corresponding cells touching the outside of thecell surface In case of any overlap (ie centre-to-centredistance of the molecules being less than sum of the radii)molecule is repositioned
Tangential velocity components (V and119908) of an incomingmolecule are independent of streamwise velocity and gener-ated as follows
V = 119881 + 119888mpR119899
radic2
119908 = 119882 + 119888mpR1015840119899
radic2
(4)
Here 119881 and 119882 are mean tangential velocity componentsR119899and R1015840
119899are random numbers generated from normal
distribution with zero mean and unit varianceIf streamwise inlet velocity is zero (119880 = 0) normal
component is generated as follows
119906 = 119888mpradicminus logR119906 (5)
Here R119906is a random number generated from uniform
distribution of interval [0 1) For nonzero streamwise inletvelocities (119880 = 0) Garcia andWagner [24] introduced severalefficient acceptance-rejection methods with the general formof
119906 = 119880 minus 119891(119880
119888mp) 119888mp (6)
Here 119891(119880119888mp) is a random number selected throughan acceptance-rejection method In this study the recom-mended method for low speed flows (minus04119888mp lt 119880 lt 13119888mp)[24] was used
Flow properties of both upstream and downstreamboundaries must be determined in order to be used in (1)ndash(6) For upstream boundary temperature and pressure areknown The number density 119899in is obtained from ideal gasequation
119899in =119875in119896119879in
(7)
Transverse velocities 119881in and 119882in are set to zero Streamwisevelocity (119880in)119895 is extrapolated from neighbour boundary cell119895 where 120588 and 119886 are density and speed of sound respectively[22]
(119880in)119895 = 119880119895+
119875in minus 119875119895
120588119895119886119895
(8)
At downstream boundary only pressure is known Thusnumber density (119899
119890)119895 temperature (119879
119890)119895 and velocity compo-
nents (119880119890)119895 (119881119890)119895 and (119882
119890)119895are extrapolated from neighbour
cell
(119899119890)119895= 119899119895+
119875119890minus 119875119895
1198981198862119895
(119879119890)119895=
119875119890
(119899119890)119895119896
(119880119890)119895= 119880119895+
119875119895minus 119875119890
119898119899119895119886119895
(119881119890)119895= 119881119895
(119882119890)119895= 119882119895
(9)
3 Verifications
Shear-Driven and Pressure-Driven flows in confinedmicrochannels are well-studied problems They are fre-quently used as test cases for the robustness and validations offlow simulations In this section the theoretical backgroundand the simulation setup for such flows are presented Anew implementation of EDMD simulation including cellpartitioning event scheduling and implicit boundary treat-ment was developed in C with serial processing Benefitingfrom object oriented programming (OOP) code can befurther extended to cover different molecular types andboundary treatments On the other hand considering thecomputational performance the parallelization of the code isan essential issue in molecular simulations However thisdeserves a detailed study of its own All simulations werecarried out on a fourth-generation Intel i7 4700CPU com-puter with 16GB of RAM
31 Shear-Driven (Couette) Flow Shear-Driven flow in a con-fined channel is obtained bymoving the top plate at a constantvelocity (Figure 1(a)) Pressure gradient is zero between theupstream and downstream boundaries Initial temperature ofthe argon inside the channel and at the upstream boundaryis 300K Upper wall moves with a velocity of 100ms whilelower wall is stationary Periodicity is applied along 119911-axis Spacing ratios (the ratio of average distance betweenmolecules 119904 to average diameter of molecules 119889) for thisstudy were selected as 40 and 70 Flow configurations werechosen according to Knudsen numbers of 01 05 and 10 inthree cases (Table 1) The number of simulated particles wasaround 200000 for each case
Simulation results were compared with the continuum-based model in which velocity gradient between two wallsof the channel is affected by the Knudsen number and has alinear nature in Shear-Driven flows For a fully diffusive wallnormalized velocity profile is shown as follows where 119880
119908is
wall velocity and 119867 is channel height [25]
119906
119880119908
=1
2Kn + 1(
119910
119867+ Kn) (10)
Mathematical Problems in Engineering 5
Table 1 Flow configurations in Shear-Driven flow
Case A (Kn = 01) Case B (Kn = 05) Case C (Kn = 10)sd = 40 sd = 70 sd = 40 sd = 70 sd = 40 sd = 70
Channel height 119867 (120583m) 0050 0263 0010 0053 0005 0026Global average pressure 119875 1639 kPa (sd = 40) 306 kPa (sd = 70)Global average density 120588 2607 kgm3 (sd = 40) 487 kgm3 (sd = 70)
Boundary conditions Fully diffusive walls 300K along 119910-directionPeriodicity along 119911-direction
Table 2 Flow configurations in Pressure-Driven flow
Case D Case EInlet pressure 119875in 250 kPa 75 kPaPressure ratio PR 25 5Initial global average density (120588) 400 kgm3 120 kgm3
Initial Kn 0080 0268Channel height 119867 04 120583m (1173 molecular diameter)Aspect ratio (119871
119909119871119910) 5
Boundary conditions Fully diffusive walls 300K along 119910-directionPeriodicity along 119911-direction
32 Pressure-Driven Flows Pressure-Driven flow is gener-ated by setting different pressure levels on upstream anddownstream boundaries (Figure 1(b)) Two Pressure-Drivenflows of different pressure ratios (PR = 119875
119894119875119890) in a
microchannel were investigated Flow properties are tabu-lated in Table 2 The number of simulated molecules wasabout 300000 for both cases
For Pressure-Driven flows Arkilic et al defined thepressure distribution along the fully diffusive channel as afunction of Knudsen number at the exit of the channel (Kne)of length 119871 as [26]
119875 (119909)
119875119890
= minus6Kne
+ radic(6Kne)2
+ (1 + 12Kne)119909
119871+ (PR2 + 12KnePR) (1 minus
119909
119871)
(11)
For fully diffusive walls analytical solution of Navier-Stokesequations with second-order slip boundary condition yieldsa normalized velocity profile [27]
119906
119880119888
=[minus (119910119867)
2
+ 119910ℎ + Kn](14 + Kn)
(12)
where 119880119888is centreline velocity Slip-to-centreline velocity
ratio along the channel then becomes
119880119904
119880119888
=Kn
(14 + Kn) (13)
For second-order slip boundary condition velocity profileand corresponding slip-to-centreline ratio are
119906
119880119888
=[minus (119910119867)
2
+ 119910ℎ + Kn (1 + Kn)](14 + Kn (1 + Kn))
(14)
119880119904
119880119888
=Kn (1 + Kn)
(14 + Kn (1 + Kn)) (15)
If slip velocity is normalized with the local average velocity(12) and (15) become [28]
119880119904
119880=
Kn(16 + Kn)
119880119904
119880=
Kn (1 + Kn)(16 + Kn (1 + Kn))
(16)
4 Results and Discussions
In this section we present simulation results of Shear-Driven and Pressure-Driven flows for aforementioned casesResults were comparedwith analytical solutions derived fromNavier-Stokes equations with slip flow boundary conditionsandDSMCresults Simulation configuration (number of sim-ulated molecules spacing ratio boundary types etc) greatlyaffects computational speed As an example simulation ofone million argon molecules in a cubical region when thespacing ratio is 50 takes 450 seconds to reach one collisionper particle on the aforementioned PC configuration
41 Shear-Driven Flow In Cases AndashC six Couette flowconfigurations at different Knudsen numbers were simulatedComparison of the predicted slip velocities between simu-lation results and the analytical solution is given in Table 3
6 Mathematical Problems in Engineering
y
UwTw
H
Tw x
P T
(a) Shear-Driven flow
y
Tw
Tw
H
x
TePin Tin
(b) Pressure-Driven flow
Figure 1 Flow geometries
1
075
05
025
00 25 50 75 100
u (ms)
N-S 1st ordersd = 40
sd = 70
Kn = 01
yH
(a)
1
075
05
025
0minus8 minus4 0 4 8
sd = 40
sd = 70
Kn = 01
yH
u minus uN-S (ms)
(b)
Figure 2 Shear-Driven flow Case A (Kn = 01) (a) velocity profile (b) deviation from analytical solution
Table 3 Comparison of slip velocities
119880119904(ms)
(theory)
119880119904(ms)
(simulation)(sd = 40)
119880119904(ms)
(simulation)(sd = 70)
Case A (Kn = 01) 833 775 756Case B (Kn = 02) 2500 1859 1929Case C (Kn = 05) 3333 2546 2622
On the upper and lower walls slip velocities were estimatedby linearly extrapolating average velocities of neighbour binsKnudsen number was 01 in Case A and the flow was inthe slip-to-transition regime Consequently deviation fromthe analytical solution is reasonably small less than 10 InCases B and C flows were in the transitional regime (01 lt
Kn le 10) Therefore simulated slip velocities were predicted
noticeably lower than the analytical solution Deviation isbetween 213 and 256 in these cases
Predicted velocity profiles and deviations from the ana-lytical solution are presented in Figures 2ndash4 For low Kn flow(Case A) velocity profile is almost linear and correspondingdeviation is insignificant As rarefaction intensifies (Cases Band C) velocity profile deviates particularly at the near wallregions S shaped velocity profile is more apparent in Case C
Another finding is that nearly identical velocity profileswere obtained for different spacing ratios at the same Knud-sen number As the spacing ratio reduces from 70 to 40computational speed quadruples Utilization of this pieceof information can ease the computational cost of largersystems that is EDMD performs faster in smaller spacingratios [9]
42 Pressure-Driven Flow Pressure distribution along thechannel is given in Figures 5(a) and 6(a) for cases D and E
Mathematical Problems in Engineering 7
1
075
05
025
00 25 50 75 100
u (ms)
N-S 1st ordersd = 40
sd = 70
Kn = 05
yH
(a)
1
075
05
025
0minus8 minus4 0 4 8
sd = 40
sd = 70
Kn = 05
yH
u minus uN-S (ms)
(b)
Figure 3 Shear-Driven flow Case B (Kn = 05) (a) velocity profile (b) deviation from analytical solution
1
075
05
025
00 25 50 75 100
u (ms)
N-S 1st ordersd = 40
sd = 70
Kn = 10
yH
(a)
1
075
05
025
0minus8 minus4 0 4 8
sd = 40
sd = 70
Kn = 10
yH
u minus uN-S (ms)
(b)
Figure 4 Shear-Driven flow Case C (Kn = 10) (a) velocity profile (b) deviation from analytical solution
respectively The results are compared with linear pressuredistribution and slip formulation given by Arkilic et al [26]Figures 5(b) and 6(b) showdeviation betweenMDsimulationresults and analytical solutions Relative pressure differenceis less than 5 for both cases This indicates that adaptedboundary treatment is appropriate for EDMD simulations
Deviation from the linear pressure distribution is given inFigures 5(c) and 6(c) In both cases nonlinearity is slightlylower in EDMD predictions
Local mean free path increases along the channel due topressure gradient This yields an increase in local Knudsennumber (Figure 7) In Case D Kn was between 0080 and
8 Mathematical Problems in Engineering
25
2
15
10 025 05 075 1
N-SLinear MD simulation
PP
e
xL
(a)
0015
0
minus00150 025 05 075 1
N-SMD simulation
xL
(Psim
minusP
N-S)P
N-S
(b)
0 025 05 075 1
N-SMD simulation
xL
(PminusP
lin)P
lin
008
006
004
002
0
(c)
Figure 5 Pressure distribution in Pressure-Driven flow Case D (a) comparison of pressure distribution along 119909-axis (b) deviation betweensimulation and N-S results (c) deviation from linear pressure distribution
5
4
3
2
10 025 05 075 1
N-SLinear MD simulation
PP
e
xL
(a)
0 025 05 075 1
N-SMD simulation
xL
(PminusP
lin)P
lin
01
005
0
(b)
002
0
minus0020 025 05 075 1
N-SMD simulation
xL
(Psim
minusP
N-S)P
N-S
(c)
Figure 6 Pressure distribution in Pressure-Driven flow Case E (a) comparison of pressure distribution along 119909-axis (b) deviation betweensimulation and N-S results (c) deviation from linear pressure distribution
0211 (ie the flow was in slip-to-transition regime) In CaseE flow was in the transitional regime with corresponding Knup to 12
Ratio of the slip velocity to the local average velocity isgiven in Figure 8 Knudsen number values in the abscissa cor-respond to local values formed in Figure 7 along the channel
Slip velocities were linearly extrapolated from outermost binssimilar to Couette flow calculations Simulation results werecompared with analytical solutions with first- and second-order slip boundary conditions (16) linearized Boltzmannsolution andDSMC results [28] It is seen that EDMD resultsare in good agreement with DSMC and linearized Boltzmann
Mathematical Problems in Engineering 9
12
09
06
03
0
Case DCase E
0 025 05 075 1
Kn
xL
Figure 7 Local Kn distribution in Pressure-Driven flow
12
09
06
03
0001
N-S 1st orderN-S 2nd orderLinearized Boltzmann
DSMC
Case D
Case E
Case ECase D
01 1 10
Kn
UsU
Figure 8 Normalized slip velocity for slip-to-transition region
solution On the other hand breakdown of slip boundarymodels starts around Kn = 01 Both first- and second-order models overestimate the velocity slip Note that forboth Cases D and E upstream and downstream slip velocityratios are smaller than in other regions of the computationaldomain Use of Maxwell distribution to sample number
density and velocities in pressure boundary treatment isthe main reason of this behaviour Such implementation ofboundary treatment works better when the rarefaction effectsare small For highly rarefied (highKn) flows Ahmadzadeganet al [29] advise usingChapman-Enskog velocity distributionto improve pressure boundary treatment
10 Mathematical Problems in Engineering
1
075
05
025
0
yH
0 025 05 075 1
uUc
Kn = 0115
xL = 0559
N-S 2nd orderMD simulation
1
075
05
025
0
yH
0 025 05 075 1
uUc
xL = 0128
N-S 2nd orderMD simulation
0085Kn =
1
075
05
025
0
yH
0 025 05 075 1
uUc
xL = 0833
N-S 2nd orderMD simulation
0159Kn =
Figure 9 Velocity profiles in 119910-direction of Pressure-Driven flow Case D
1
075
05
025
0
YH
0 025 05 075 1
UUC
XL = 0814
N-S 2nd orderMD simulation
0686Kn =
1
075
05
025
0
YH
0 025 05 075 1
UUC
XL = 0206
N-S 2nd orderMD simulation
1
075
05
025
0
YH
0 025 05 075 1
UUC
XL = 0579
N-S 2nd orderMD simulation
0457Kn =0308Kn =
Figure 10 Velocity profiles in 119910-direction of Pressure-Driven flow Case E
Normalized velocity profiles (119906119880119888where 119880
119888is local
centreline velocity) in cross-stream direction at three loca-tions along the channel are shown in Figures 9 and 10 forCase D and Case E respectively Note that flow velocityalong the channel increases as pressure decreases Thusfor a given position along the channel normalization wascarried out by corresponding centreline velocity Here resultsof EDMD simulations were compared with second-orderanalytical solution In CaseD where Kn is small curves agreewell whereas in Case E estimation of higher slip velocity inanalytical solution becomes more apparent as Kn increases
5 Conclusion
This paper presents an event-driven molecular dynamics(EDMD) simulation of monatomic gas flows in microchan-nels In this study molecules are considered as hard-sphereswith discrete interaction potentials and interaction times areanalytically predictable Therefore each collision is real andtakes place in its calculated position Postcollisional velocitiesare also analytically calculated according to the conservationofmomentumand energy By its deterministic nature EDMDdiffers from DSMC method in which simulated system is
Mathematical Problems in Engineering 11
represented by a smaller group of molecules and collisionpairs are selected stochastically Additionally by using cellpartitioning technique and use of priority queues for eventsorting EDMD is now a strong alternative to DSMCmethodin terms of not only accuracy but also computational speed
In this study an implicit treatment for flow boundaries isadapted from DSMC method to EDMD for the first time inorder to induce Shear- and Pressure-Driven flows in confinedchannels The number of introduced molecules throughupstream and downstream boundaries and their correspond-ing velocities and locations are determined implicitly fromlocal number density mean flow velocity and temperature
Velocity slip in Pressure-Driven flows is compared withDSMCresults and solution of linearizedBoltzmann equationThe good agreement confirms the robustness of the treatmentand its applicability to EDMD simulations For both Shear-and Pressure-Driven flows velocity slip is also comparedwith the one derived from Navier-Stokes equations of slipboundary conditions For low Knudsen number (Kn) flowsvelocity profiles and calculated slip velocities agree well Butas the flow rarefies slip boundarymodels estimate higher slipvelocities
Additionally simulations of twoflows having the sameKnbut different spacing ratios produce almost identical velocityprofiles in Shear-Driven cases for the investigated range ofKn Since simulations with lower spacing ratios run fasterthis finding can be further extended to scale rarefied flowsand to reduce the computational time
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] T-R Hsu MEMS amp Microsystems Design Manufacture andNanoscale Engineering JohnWiley amp Sons Hoboken NJ USA2nd edition 2008
[2] G A BirdMolecular GasDynamics and theDirect Simulation ofGas Flows Clarendon Press Oxford UK 1994
[3] A Rahman ldquoCorrelations in the motion of atoms in liquidargonrdquo Physical Review vol 136 no 2 pp A405ndashA411 1964
[4] L Verlet ldquoComputer lsquoexperimentsrsquo on classical fluids I Ther-modynamical properties of Lennard-Jones moleculesrdquo PhysicalReview vol 159 no 1 pp 98ndash103 1967
[5] MMareschal and B L Holian ldquoFrom fluid particles to physicalparticles computing hydrodynamicsrdquo in Microscopic Simula-tions of Complex Hydrodynamic Phenomena M Mareschal andB LHolian Eds vol 292 ofNATOASI Series pp 1ndash12 SpringerScience amp Business Media 1992
[6] B J Alder and T E Wainwright ldquoPhase transition for a hardsphere systemrdquo The Journal of Chemical Physics vol 27 no 5pp 1208ndash1209 1957
[7] D C Rapaport ldquoThe event scheduling problem in moleculardynamic simulationrdquo Journal of Computational Physics vol 34no 2 pp 184ndash201 1980
[8] M Marın and P Cordero ldquoAn empirical assessment of priorityqueues in event-driven molecular dynamics simulationrdquo Com-puter Physics Communications vol 92 no 2-3 pp 214ndash2241995
[9] I Kandemir A multicell molecular dynamics method [PhDthesis] CaseWesternReserveUniversity ClevelandOhioUSA1999
[10] W T Tang R S M Goh and I L-J Thng ldquoLadder queuean O(1) priority queue structure for large-scale discrete eventsimulationrdquoACMTransactions onModeling and Computer Sim-ulation vol 15 no 3 pp 175ndash204 2005
[11] G Paul ldquoA Complexity O(1) priority queue for event drivenmolecular dynamics simulationsrdquo Journal of ComputationalPhysics vol 221 no 2 pp 615ndash625 2007
[12] A Donev A L Garcia and B J Alder ldquoStochastic event-drivenmolecular dynamicsrdquo Journal of Computational Physics vol 227no 4 pp 2644ndash2665 2008
[13] M N Bannerman R Sargant and L Lue ldquoDynamO a freeO(N) general event-driven molecular dynamics simulatorrdquoJournal of Computational Chemistry vol 32 no 15 pp 3329ndash3338 2011
[14] J Li D Liao and S Yip ldquoCoupling continuum to molecular-dynamics simulation reflecting particle method and the fieldestimatorrdquo Physical Review E vol 57 no 6 pp 7259ndash7267 1998
[15] F Bao YHuang Y Zhang and J Lin ldquoInvestigation of pressure-driven gas flows in nanoscale channels usingmolecular dynam-ics simulationrdquo Microfluidics and Nanofluidics vol 18 no 5-6pp 1075ndash1084 2014
[16] F Bao Y Huang L Qiu and J Lin ldquoApplicability of moleculardynamics method to the pressure-driven gas flow in finitelength nano-scale slit poresrdquo Molecular Physics vol 113 no 6pp 561ndash569 2015
[17] J R Whitman G L Aranovich and M D DonohueldquoAnisotropic mean free path in simulations of fluids travelingdown a density gradientrdquo Journal of Chemical Physics vol 132no 22 Article ID 224302 2010
[18] Q D To T T Pham G Lauriat and C Leonard ldquoMoleculardynamics simulations of pressure-driven flows and comparisonwith acceleration-driven flowsrdquo Advances in Mechanical Engi-neering vol 2012 Article ID 580763 10 pages 2012
[19] M Lupkowski and F van Swol ldquoComputer simulation of fluidsinteracting with fluctuating wallsrdquo The Journal of ChemicalPhysics vol 93 no 1 pp 737ndash745 1990
[20] C Huang K Nandakumar P Y K Choi and L W KostiukldquoMolecular dynamics simulation of a pressure-driven liquidtransport process in a cylindrical nanopore using two self-adjusting platesrdquo Journal of Chemical Physics vol 124 no 23Article ID 234701 2006
[21] W W Liou and Y C Fang ldquoImplicit boundary conditions fordirect simulation Monte Carlo method in MEMS flow pre-dictionsrdquo Computer Modeling in Engineering and Sciences vol1 no 4 pp 118ndash128 2000
[22] M Wang and Z Li ldquoSimulations for gas flows in microge-ometries using the direct simulation Monte Carlo methodrdquoInternational Journal of Heat and Fluid Flow vol 25 no 6 pp975ndash985 2004
[23] D J Evans and G P Morriss Statistical Mechanics of Nonequi-librium Liquids Cambridge University Press 2nd edition 2008
[24] A L Garcia and W Wagner ldquoGeneration of the Maxwellianinflow distributionrdquo Journal of Computational Physics vol 217no 2 pp 693ndash708 2006
12 Mathematical Problems in Engineering
[25] G Karniadakis A Beskok and N Aluru Microflows andNanoflows Fundamentals and Simulation Springer New YorkNY USA 2006
[26] E B Arkilic M A Schmidt and K S Breuer ldquoGaseous slipflow in long microchannelsrdquo Journal of MicroelectromechanicalSystems vol 6 no 2 pp 167ndash178 1997
[27] A Beskok G E Karniadakis and W Trimmer ldquoRarefactionand compressibility effects in gas microflowsrdquo Journal of FluidsEngineering vol 118 no 3 pp 448ndash456 1996
[28] A Beskok and G E Karniadakis ldquoReport a model for flows inchannels pipes and ducts at micro and nano scalesrdquoMicroscaleThermophysical Engineering vol 3 no 1 pp 43ndash77 1999
[29] A Ahmadzadegan J Wen and M Renksizbulut ldquoThe role ofthe velocity distribution in the DSMC pressure boundary con-dition for gasmixturesrdquo International Journal ofModern PhysicsC vol 23 no 12 Article ID 1250087 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
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OptimizationJournal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 3
velocities were calculated according to mass diameter andprecollisional velocity information of colliding pair by satis-fying conservation equations of energy and momentum
Argon (Ar) was selected as the monatomic gas of thisstudy In order to reduce numerical errors all magnitudeswere nondimensionalized by considering unit Boltzmannconstant (119896
119887= 10) and scaling with the reference variables
temperature (119879ref = 300K) mass (119898 = 662 times 10minus26 kg) and
diameter (119889 = 341times10minus10m) It is worthmentioning that the
results of the study can be extended to any monatomic gas byselecting proper scaling variables
Due to the nature of MD simulations immense compu-tational resources are required Simulation speed decreasesalmost linearly as the number of molecules in the systemincreases In order to speed up the simulations computa-tional domain was partitioned into cubical cells using themethod described in detail by Kandemir [9] A cell has 26neighbours at maximum fewer if it is at an edge of com-putational domain In this method molecules are assignedto the cells according to their current positions An inter-molecular collision is probable only if the distance betweentwo molecules is less than cell width Otherwise moleculeswould encounter cell crossing events beforehand Thus onlythe molecules of same or neighbour cells are considered ascandidate collisional pairs at a given time Implementationof cell partitioning method reduces the computational effortto 119874(119899
119888) for one molecule where 119899
119888is the average number of
molecules in the neighbourhood Since there are119873moleculesin the system total computational effort is about 119874(119873)
assuming that 119874(119899119888) asymp 119874(1) Simulations of the same system
on the basis of single and multicell approaches yield exactlyidentical numerical results using the same seed and randomnumber generator Hence one can conclude that themulticellmethod can guarantee that no collision is missed
Three types of events are possible in event-drivensimulations These are intermolecular collisions molecule-boundary interactions and cell crossings A molecule canbe involved in many future events but only the earliestone is registered as the candidate event for the moleculeThus the number of candidate events is always equal to thenumber of simulated molecules The goals are determinationand execution of the earliest event Following the executioncandidate events of related molecules are invalidated andpossible new candidate events are calculated This processrepeats throughout the simulation
Determination of earliest event by linear search has acomplexity of 119874(119873) and makes EDMD simulations practi-cally impossible for large number of molecules In this studya priority queue method suggested by Paul [11] was imple-mented In his method all events are distributed among timeinterval bins and events belonging to earliest bin are sortedusing a complete binary tree (CBT) structure This approachreduces computational effort approximately to 119874(1) on aver-age
Macroscopic thermodynamic properties of a simulatedsystem can be estimated by ensemble averaging over par-ticlesrsquo behaviour However this would be expensive interms of computational effort because of the requirementof large number of molecules to obtain smooth profiles
The alternative approach is the time averaging for a smallernumber of particles since ensemble average and time averageof a phase variable yield equal values in an ergodic system[23] Since this study deals with large number of moleculesan ensemble averaging scheme was performed by averaging aseries of snapshots of computational domain taken in equalintervals For the sake of profiling computational domainwas partitioned into small subdomains (bins) Eachmoleculeis associated with a certain bin based on its molecular centreAlthough the selection of bin size is arbitrary the averagenumber of molecules inside a bin is an important parameterfor an accurate profiling of system properties Small numberof molecules may demonstrate large fluctuations due tostatistical nature of EDMD simulations On the other handresolution of distribution is reduced for larger bins Inorder to satisfy both criteria computational domain wasdivided into 51 spatial bins along a direction (51 times 51 for 2Ddistributions) in this study Snapshotting was started after atleast 100 collisions per particle (CPP) with a frequency of atleast 1 CPP In order to obtain smoother profiles longer CPPvalues were used for averaging
22 Implicit Treatment for Flow Boundaries From micro-scopic viewpoint in addition to bulk velocity gas moleculesare translated by thermal velocity also In high speed (hyper-sonic) flows the magnitude of the bulk velocity is higherthan that of the thermal velocity For such flows conventionalapproach is the use of vacuum boundary at the exit of thechannel (when a molecule reaches the vacuum boundary itleaves the computational domain permanently and there isno inward flux at that boundary)
In contrast to hypersonic flows themagnitude of thermalmotion is comparable with bulk velocity in low speed flowand inward flux because thermal motion of molecules shouldbe taken into consideration In order to cover a wide rangeof flow regimes Liou and Fang [21] introduced an implicittreatment for flow boundaries in DSMC simulations Thistreatment was adapted for EDMD simulations in this study
In this approach the number of incoming molecules andtheir corresponding velocities are estimated implicitly by thelocal flow properties For either upstream or downstreamboundary molecular flux (119865) into the computational domaincan be determined using the Maxwellian distribution func-tion
119865119895=
119899119895
2radic120587120573119895
exp (minus1199042
119895cos2120601)
+ radic120587119904119895cos120601 [1 + erf (119904
119895cos120601)]
(1)
where
119904119895= 119880119895120573119895
120573119895=
1
119888mp119895=
1
radic2119896119887119879119895119898
(2)
Flux is calculated for each cell face of flow boundary cell 119895Here streamwise velocity and local temperature are denoted
4 Mathematical Problems in Engineering
by119880 and119879 respectively Value of120601 is 0 for upstream and120587 fordownstream boundary 119899
119895is the number density of molecules
in the cell and 119888mp is the most probable speed For a timeinterval 120575119905 the number of molecules (119873
119895) entering domain
from surface area 119860119895of cell 119895 is
119873119895= 119865119895120575119905119860119895 (3)
Incoming molecules are introduced at random positionswithin the corresponding cells touching the outside of thecell surface In case of any overlap (ie centre-to-centredistance of the molecules being less than sum of the radii)molecule is repositioned
Tangential velocity components (V and119908) of an incomingmolecule are independent of streamwise velocity and gener-ated as follows
V = 119881 + 119888mpR119899
radic2
119908 = 119882 + 119888mpR1015840119899
radic2
(4)
Here 119881 and 119882 are mean tangential velocity componentsR119899and R1015840
119899are random numbers generated from normal
distribution with zero mean and unit varianceIf streamwise inlet velocity is zero (119880 = 0) normal
component is generated as follows
119906 = 119888mpradicminus logR119906 (5)
Here R119906is a random number generated from uniform
distribution of interval [0 1) For nonzero streamwise inletvelocities (119880 = 0) Garcia andWagner [24] introduced severalefficient acceptance-rejection methods with the general formof
119906 = 119880 minus 119891(119880
119888mp) 119888mp (6)
Here 119891(119880119888mp) is a random number selected throughan acceptance-rejection method In this study the recom-mended method for low speed flows (minus04119888mp lt 119880 lt 13119888mp)[24] was used
Flow properties of both upstream and downstreamboundaries must be determined in order to be used in (1)ndash(6) For upstream boundary temperature and pressure areknown The number density 119899in is obtained from ideal gasequation
119899in =119875in119896119879in
(7)
Transverse velocities 119881in and 119882in are set to zero Streamwisevelocity (119880in)119895 is extrapolated from neighbour boundary cell119895 where 120588 and 119886 are density and speed of sound respectively[22]
(119880in)119895 = 119880119895+
119875in minus 119875119895
120588119895119886119895
(8)
At downstream boundary only pressure is known Thusnumber density (119899
119890)119895 temperature (119879
119890)119895 and velocity compo-
nents (119880119890)119895 (119881119890)119895 and (119882
119890)119895are extrapolated from neighbour
cell
(119899119890)119895= 119899119895+
119875119890minus 119875119895
1198981198862119895
(119879119890)119895=
119875119890
(119899119890)119895119896
(119880119890)119895= 119880119895+
119875119895minus 119875119890
119898119899119895119886119895
(119881119890)119895= 119881119895
(119882119890)119895= 119882119895
(9)
3 Verifications
Shear-Driven and Pressure-Driven flows in confinedmicrochannels are well-studied problems They are fre-quently used as test cases for the robustness and validations offlow simulations In this section the theoretical backgroundand the simulation setup for such flows are presented Anew implementation of EDMD simulation including cellpartitioning event scheduling and implicit boundary treat-ment was developed in C with serial processing Benefitingfrom object oriented programming (OOP) code can befurther extended to cover different molecular types andboundary treatments On the other hand considering thecomputational performance the parallelization of the code isan essential issue in molecular simulations However thisdeserves a detailed study of its own All simulations werecarried out on a fourth-generation Intel i7 4700CPU com-puter with 16GB of RAM
31 Shear-Driven (Couette) Flow Shear-Driven flow in a con-fined channel is obtained bymoving the top plate at a constantvelocity (Figure 1(a)) Pressure gradient is zero between theupstream and downstream boundaries Initial temperature ofthe argon inside the channel and at the upstream boundaryis 300K Upper wall moves with a velocity of 100ms whilelower wall is stationary Periodicity is applied along 119911-axis Spacing ratios (the ratio of average distance betweenmolecules 119904 to average diameter of molecules 119889) for thisstudy were selected as 40 and 70 Flow configurations werechosen according to Knudsen numbers of 01 05 and 10 inthree cases (Table 1) The number of simulated particles wasaround 200000 for each case
Simulation results were compared with the continuum-based model in which velocity gradient between two wallsof the channel is affected by the Knudsen number and has alinear nature in Shear-Driven flows For a fully diffusive wallnormalized velocity profile is shown as follows where 119880
119908is
wall velocity and 119867 is channel height [25]
119906
119880119908
=1
2Kn + 1(
119910
119867+ Kn) (10)
Mathematical Problems in Engineering 5
Table 1 Flow configurations in Shear-Driven flow
Case A (Kn = 01) Case B (Kn = 05) Case C (Kn = 10)sd = 40 sd = 70 sd = 40 sd = 70 sd = 40 sd = 70
Channel height 119867 (120583m) 0050 0263 0010 0053 0005 0026Global average pressure 119875 1639 kPa (sd = 40) 306 kPa (sd = 70)Global average density 120588 2607 kgm3 (sd = 40) 487 kgm3 (sd = 70)
Boundary conditions Fully diffusive walls 300K along 119910-directionPeriodicity along 119911-direction
Table 2 Flow configurations in Pressure-Driven flow
Case D Case EInlet pressure 119875in 250 kPa 75 kPaPressure ratio PR 25 5Initial global average density (120588) 400 kgm3 120 kgm3
Initial Kn 0080 0268Channel height 119867 04 120583m (1173 molecular diameter)Aspect ratio (119871
119909119871119910) 5
Boundary conditions Fully diffusive walls 300K along 119910-directionPeriodicity along 119911-direction
32 Pressure-Driven Flows Pressure-Driven flow is gener-ated by setting different pressure levels on upstream anddownstream boundaries (Figure 1(b)) Two Pressure-Drivenflows of different pressure ratios (PR = 119875
119894119875119890) in a
microchannel were investigated Flow properties are tabu-lated in Table 2 The number of simulated molecules wasabout 300000 for both cases
For Pressure-Driven flows Arkilic et al defined thepressure distribution along the fully diffusive channel as afunction of Knudsen number at the exit of the channel (Kne)of length 119871 as [26]
119875 (119909)
119875119890
= minus6Kne
+ radic(6Kne)2
+ (1 + 12Kne)119909
119871+ (PR2 + 12KnePR) (1 minus
119909
119871)
(11)
For fully diffusive walls analytical solution of Navier-Stokesequations with second-order slip boundary condition yieldsa normalized velocity profile [27]
119906
119880119888
=[minus (119910119867)
2
+ 119910ℎ + Kn](14 + Kn)
(12)
where 119880119888is centreline velocity Slip-to-centreline velocity
ratio along the channel then becomes
119880119904
119880119888
=Kn
(14 + Kn) (13)
For second-order slip boundary condition velocity profileand corresponding slip-to-centreline ratio are
119906
119880119888
=[minus (119910119867)
2
+ 119910ℎ + Kn (1 + Kn)](14 + Kn (1 + Kn))
(14)
119880119904
119880119888
=Kn (1 + Kn)
(14 + Kn (1 + Kn)) (15)
If slip velocity is normalized with the local average velocity(12) and (15) become [28]
119880119904
119880=
Kn(16 + Kn)
119880119904
119880=
Kn (1 + Kn)(16 + Kn (1 + Kn))
(16)
4 Results and Discussions
In this section we present simulation results of Shear-Driven and Pressure-Driven flows for aforementioned casesResults were comparedwith analytical solutions derived fromNavier-Stokes equations with slip flow boundary conditionsandDSMCresults Simulation configuration (number of sim-ulated molecules spacing ratio boundary types etc) greatlyaffects computational speed As an example simulation ofone million argon molecules in a cubical region when thespacing ratio is 50 takes 450 seconds to reach one collisionper particle on the aforementioned PC configuration
41 Shear-Driven Flow In Cases AndashC six Couette flowconfigurations at different Knudsen numbers were simulatedComparison of the predicted slip velocities between simu-lation results and the analytical solution is given in Table 3
6 Mathematical Problems in Engineering
y
UwTw
H
Tw x
P T
(a) Shear-Driven flow
y
Tw
Tw
H
x
TePin Tin
(b) Pressure-Driven flow
Figure 1 Flow geometries
1
075
05
025
00 25 50 75 100
u (ms)
N-S 1st ordersd = 40
sd = 70
Kn = 01
yH
(a)
1
075
05
025
0minus8 minus4 0 4 8
sd = 40
sd = 70
Kn = 01
yH
u minus uN-S (ms)
(b)
Figure 2 Shear-Driven flow Case A (Kn = 01) (a) velocity profile (b) deviation from analytical solution
Table 3 Comparison of slip velocities
119880119904(ms)
(theory)
119880119904(ms)
(simulation)(sd = 40)
119880119904(ms)
(simulation)(sd = 70)
Case A (Kn = 01) 833 775 756Case B (Kn = 02) 2500 1859 1929Case C (Kn = 05) 3333 2546 2622
On the upper and lower walls slip velocities were estimatedby linearly extrapolating average velocities of neighbour binsKnudsen number was 01 in Case A and the flow was inthe slip-to-transition regime Consequently deviation fromthe analytical solution is reasonably small less than 10 InCases B and C flows were in the transitional regime (01 lt
Kn le 10) Therefore simulated slip velocities were predicted
noticeably lower than the analytical solution Deviation isbetween 213 and 256 in these cases
Predicted velocity profiles and deviations from the ana-lytical solution are presented in Figures 2ndash4 For low Kn flow(Case A) velocity profile is almost linear and correspondingdeviation is insignificant As rarefaction intensifies (Cases Band C) velocity profile deviates particularly at the near wallregions S shaped velocity profile is more apparent in Case C
Another finding is that nearly identical velocity profileswere obtained for different spacing ratios at the same Knud-sen number As the spacing ratio reduces from 70 to 40computational speed quadruples Utilization of this pieceof information can ease the computational cost of largersystems that is EDMD performs faster in smaller spacingratios [9]
42 Pressure-Driven Flow Pressure distribution along thechannel is given in Figures 5(a) and 6(a) for cases D and E
Mathematical Problems in Engineering 7
1
075
05
025
00 25 50 75 100
u (ms)
N-S 1st ordersd = 40
sd = 70
Kn = 05
yH
(a)
1
075
05
025
0minus8 minus4 0 4 8
sd = 40
sd = 70
Kn = 05
yH
u minus uN-S (ms)
(b)
Figure 3 Shear-Driven flow Case B (Kn = 05) (a) velocity profile (b) deviation from analytical solution
1
075
05
025
00 25 50 75 100
u (ms)
N-S 1st ordersd = 40
sd = 70
Kn = 10
yH
(a)
1
075
05
025
0minus8 minus4 0 4 8
sd = 40
sd = 70
Kn = 10
yH
u minus uN-S (ms)
(b)
Figure 4 Shear-Driven flow Case C (Kn = 10) (a) velocity profile (b) deviation from analytical solution
respectively The results are compared with linear pressuredistribution and slip formulation given by Arkilic et al [26]Figures 5(b) and 6(b) showdeviation betweenMDsimulationresults and analytical solutions Relative pressure differenceis less than 5 for both cases This indicates that adaptedboundary treatment is appropriate for EDMD simulations
Deviation from the linear pressure distribution is given inFigures 5(c) and 6(c) In both cases nonlinearity is slightlylower in EDMD predictions
Local mean free path increases along the channel due topressure gradient This yields an increase in local Knudsennumber (Figure 7) In Case D Kn was between 0080 and
8 Mathematical Problems in Engineering
25
2
15
10 025 05 075 1
N-SLinear MD simulation
PP
e
xL
(a)
0015
0
minus00150 025 05 075 1
N-SMD simulation
xL
(Psim
minusP
N-S)P
N-S
(b)
0 025 05 075 1
N-SMD simulation
xL
(PminusP
lin)P
lin
008
006
004
002
0
(c)
Figure 5 Pressure distribution in Pressure-Driven flow Case D (a) comparison of pressure distribution along 119909-axis (b) deviation betweensimulation and N-S results (c) deviation from linear pressure distribution
5
4
3
2
10 025 05 075 1
N-SLinear MD simulation
PP
e
xL
(a)
0 025 05 075 1
N-SMD simulation
xL
(PminusP
lin)P
lin
01
005
0
(b)
002
0
minus0020 025 05 075 1
N-SMD simulation
xL
(Psim
minusP
N-S)P
N-S
(c)
Figure 6 Pressure distribution in Pressure-Driven flow Case E (a) comparison of pressure distribution along 119909-axis (b) deviation betweensimulation and N-S results (c) deviation from linear pressure distribution
0211 (ie the flow was in slip-to-transition regime) In CaseE flow was in the transitional regime with corresponding Knup to 12
Ratio of the slip velocity to the local average velocity isgiven in Figure 8 Knudsen number values in the abscissa cor-respond to local values formed in Figure 7 along the channel
Slip velocities were linearly extrapolated from outermost binssimilar to Couette flow calculations Simulation results werecompared with analytical solutions with first- and second-order slip boundary conditions (16) linearized Boltzmannsolution andDSMC results [28] It is seen that EDMD resultsare in good agreement with DSMC and linearized Boltzmann
Mathematical Problems in Engineering 9
12
09
06
03
0
Case DCase E
0 025 05 075 1
Kn
xL
Figure 7 Local Kn distribution in Pressure-Driven flow
12
09
06
03
0001
N-S 1st orderN-S 2nd orderLinearized Boltzmann
DSMC
Case D
Case E
Case ECase D
01 1 10
Kn
UsU
Figure 8 Normalized slip velocity for slip-to-transition region
solution On the other hand breakdown of slip boundarymodels starts around Kn = 01 Both first- and second-order models overestimate the velocity slip Note that forboth Cases D and E upstream and downstream slip velocityratios are smaller than in other regions of the computationaldomain Use of Maxwell distribution to sample number
density and velocities in pressure boundary treatment isthe main reason of this behaviour Such implementation ofboundary treatment works better when the rarefaction effectsare small For highly rarefied (highKn) flows Ahmadzadeganet al [29] advise usingChapman-Enskog velocity distributionto improve pressure boundary treatment
10 Mathematical Problems in Engineering
1
075
05
025
0
yH
0 025 05 075 1
uUc
Kn = 0115
xL = 0559
N-S 2nd orderMD simulation
1
075
05
025
0
yH
0 025 05 075 1
uUc
xL = 0128
N-S 2nd orderMD simulation
0085Kn =
1
075
05
025
0
yH
0 025 05 075 1
uUc
xL = 0833
N-S 2nd orderMD simulation
0159Kn =
Figure 9 Velocity profiles in 119910-direction of Pressure-Driven flow Case D
1
075
05
025
0
YH
0 025 05 075 1
UUC
XL = 0814
N-S 2nd orderMD simulation
0686Kn =
1
075
05
025
0
YH
0 025 05 075 1
UUC
XL = 0206
N-S 2nd orderMD simulation
1
075
05
025
0
YH
0 025 05 075 1
UUC
XL = 0579
N-S 2nd orderMD simulation
0457Kn =0308Kn =
Figure 10 Velocity profiles in 119910-direction of Pressure-Driven flow Case E
Normalized velocity profiles (119906119880119888where 119880
119888is local
centreline velocity) in cross-stream direction at three loca-tions along the channel are shown in Figures 9 and 10 forCase D and Case E respectively Note that flow velocityalong the channel increases as pressure decreases Thusfor a given position along the channel normalization wascarried out by corresponding centreline velocity Here resultsof EDMD simulations were compared with second-orderanalytical solution In CaseD where Kn is small curves agreewell whereas in Case E estimation of higher slip velocity inanalytical solution becomes more apparent as Kn increases
5 Conclusion
This paper presents an event-driven molecular dynamics(EDMD) simulation of monatomic gas flows in microchan-nels In this study molecules are considered as hard-sphereswith discrete interaction potentials and interaction times areanalytically predictable Therefore each collision is real andtakes place in its calculated position Postcollisional velocitiesare also analytically calculated according to the conservationofmomentumand energy By its deterministic nature EDMDdiffers from DSMC method in which simulated system is
Mathematical Problems in Engineering 11
represented by a smaller group of molecules and collisionpairs are selected stochastically Additionally by using cellpartitioning technique and use of priority queues for eventsorting EDMD is now a strong alternative to DSMCmethodin terms of not only accuracy but also computational speed
In this study an implicit treatment for flow boundaries isadapted from DSMC method to EDMD for the first time inorder to induce Shear- and Pressure-Driven flows in confinedchannels The number of introduced molecules throughupstream and downstream boundaries and their correspond-ing velocities and locations are determined implicitly fromlocal number density mean flow velocity and temperature
Velocity slip in Pressure-Driven flows is compared withDSMCresults and solution of linearizedBoltzmann equationThe good agreement confirms the robustness of the treatmentand its applicability to EDMD simulations For both Shear-and Pressure-Driven flows velocity slip is also comparedwith the one derived from Navier-Stokes equations of slipboundary conditions For low Knudsen number (Kn) flowsvelocity profiles and calculated slip velocities agree well Butas the flow rarefies slip boundarymodels estimate higher slipvelocities
Additionally simulations of twoflows having the sameKnbut different spacing ratios produce almost identical velocityprofiles in Shear-Driven cases for the investigated range ofKn Since simulations with lower spacing ratios run fasterthis finding can be further extended to scale rarefied flowsand to reduce the computational time
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] T-R Hsu MEMS amp Microsystems Design Manufacture andNanoscale Engineering JohnWiley amp Sons Hoboken NJ USA2nd edition 2008
[2] G A BirdMolecular GasDynamics and theDirect Simulation ofGas Flows Clarendon Press Oxford UK 1994
[3] A Rahman ldquoCorrelations in the motion of atoms in liquidargonrdquo Physical Review vol 136 no 2 pp A405ndashA411 1964
[4] L Verlet ldquoComputer lsquoexperimentsrsquo on classical fluids I Ther-modynamical properties of Lennard-Jones moleculesrdquo PhysicalReview vol 159 no 1 pp 98ndash103 1967
[5] MMareschal and B L Holian ldquoFrom fluid particles to physicalparticles computing hydrodynamicsrdquo in Microscopic Simula-tions of Complex Hydrodynamic Phenomena M Mareschal andB LHolian Eds vol 292 ofNATOASI Series pp 1ndash12 SpringerScience amp Business Media 1992
[6] B J Alder and T E Wainwright ldquoPhase transition for a hardsphere systemrdquo The Journal of Chemical Physics vol 27 no 5pp 1208ndash1209 1957
[7] D C Rapaport ldquoThe event scheduling problem in moleculardynamic simulationrdquo Journal of Computational Physics vol 34no 2 pp 184ndash201 1980
[8] M Marın and P Cordero ldquoAn empirical assessment of priorityqueues in event-driven molecular dynamics simulationrdquo Com-puter Physics Communications vol 92 no 2-3 pp 214ndash2241995
[9] I Kandemir A multicell molecular dynamics method [PhDthesis] CaseWesternReserveUniversity ClevelandOhioUSA1999
[10] W T Tang R S M Goh and I L-J Thng ldquoLadder queuean O(1) priority queue structure for large-scale discrete eventsimulationrdquoACMTransactions onModeling and Computer Sim-ulation vol 15 no 3 pp 175ndash204 2005
[11] G Paul ldquoA Complexity O(1) priority queue for event drivenmolecular dynamics simulationsrdquo Journal of ComputationalPhysics vol 221 no 2 pp 615ndash625 2007
[12] A Donev A L Garcia and B J Alder ldquoStochastic event-drivenmolecular dynamicsrdquo Journal of Computational Physics vol 227no 4 pp 2644ndash2665 2008
[13] M N Bannerman R Sargant and L Lue ldquoDynamO a freeO(N) general event-driven molecular dynamics simulatorrdquoJournal of Computational Chemistry vol 32 no 15 pp 3329ndash3338 2011
[14] J Li D Liao and S Yip ldquoCoupling continuum to molecular-dynamics simulation reflecting particle method and the fieldestimatorrdquo Physical Review E vol 57 no 6 pp 7259ndash7267 1998
[15] F Bao YHuang Y Zhang and J Lin ldquoInvestigation of pressure-driven gas flows in nanoscale channels usingmolecular dynam-ics simulationrdquo Microfluidics and Nanofluidics vol 18 no 5-6pp 1075ndash1084 2014
[16] F Bao Y Huang L Qiu and J Lin ldquoApplicability of moleculardynamics method to the pressure-driven gas flow in finitelength nano-scale slit poresrdquo Molecular Physics vol 113 no 6pp 561ndash569 2015
[17] J R Whitman G L Aranovich and M D DonohueldquoAnisotropic mean free path in simulations of fluids travelingdown a density gradientrdquo Journal of Chemical Physics vol 132no 22 Article ID 224302 2010
[18] Q D To T T Pham G Lauriat and C Leonard ldquoMoleculardynamics simulations of pressure-driven flows and comparisonwith acceleration-driven flowsrdquo Advances in Mechanical Engi-neering vol 2012 Article ID 580763 10 pages 2012
[19] M Lupkowski and F van Swol ldquoComputer simulation of fluidsinteracting with fluctuating wallsrdquo The Journal of ChemicalPhysics vol 93 no 1 pp 737ndash745 1990
[20] C Huang K Nandakumar P Y K Choi and L W KostiukldquoMolecular dynamics simulation of a pressure-driven liquidtransport process in a cylindrical nanopore using two self-adjusting platesrdquo Journal of Chemical Physics vol 124 no 23Article ID 234701 2006
[21] W W Liou and Y C Fang ldquoImplicit boundary conditions fordirect simulation Monte Carlo method in MEMS flow pre-dictionsrdquo Computer Modeling in Engineering and Sciences vol1 no 4 pp 118ndash128 2000
[22] M Wang and Z Li ldquoSimulations for gas flows in microge-ometries using the direct simulation Monte Carlo methodrdquoInternational Journal of Heat and Fluid Flow vol 25 no 6 pp975ndash985 2004
[23] D J Evans and G P Morriss Statistical Mechanics of Nonequi-librium Liquids Cambridge University Press 2nd edition 2008
[24] A L Garcia and W Wagner ldquoGeneration of the Maxwellianinflow distributionrdquo Journal of Computational Physics vol 217no 2 pp 693ndash708 2006
12 Mathematical Problems in Engineering
[25] G Karniadakis A Beskok and N Aluru Microflows andNanoflows Fundamentals and Simulation Springer New YorkNY USA 2006
[26] E B Arkilic M A Schmidt and K S Breuer ldquoGaseous slipflow in long microchannelsrdquo Journal of MicroelectromechanicalSystems vol 6 no 2 pp 167ndash178 1997
[27] A Beskok G E Karniadakis and W Trimmer ldquoRarefactionand compressibility effects in gas microflowsrdquo Journal of FluidsEngineering vol 118 no 3 pp 448ndash456 1996
[28] A Beskok and G E Karniadakis ldquoReport a model for flows inchannels pipes and ducts at micro and nano scalesrdquoMicroscaleThermophysical Engineering vol 3 no 1 pp 43ndash77 1999
[29] A Ahmadzadegan J Wen and M Renksizbulut ldquoThe role ofthe velocity distribution in the DSMC pressure boundary con-dition for gasmixturesrdquo International Journal ofModern PhysicsC vol 23 no 12 Article ID 1250087 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
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Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
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Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
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International Journal of
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Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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Algebra
Discrete Dynamics in Nature and Society
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Decision SciencesAdvances in
Discrete MathematicsJournal of
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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
4 Mathematical Problems in Engineering
by119880 and119879 respectively Value of120601 is 0 for upstream and120587 fordownstream boundary 119899
119895is the number density of molecules
in the cell and 119888mp is the most probable speed For a timeinterval 120575119905 the number of molecules (119873
119895) entering domain
from surface area 119860119895of cell 119895 is
119873119895= 119865119895120575119905119860119895 (3)
Incoming molecules are introduced at random positionswithin the corresponding cells touching the outside of thecell surface In case of any overlap (ie centre-to-centredistance of the molecules being less than sum of the radii)molecule is repositioned
Tangential velocity components (V and119908) of an incomingmolecule are independent of streamwise velocity and gener-ated as follows
V = 119881 + 119888mpR119899
radic2
119908 = 119882 + 119888mpR1015840119899
radic2
(4)
Here 119881 and 119882 are mean tangential velocity componentsR119899and R1015840
119899are random numbers generated from normal
distribution with zero mean and unit varianceIf streamwise inlet velocity is zero (119880 = 0) normal
component is generated as follows
119906 = 119888mpradicminus logR119906 (5)
Here R119906is a random number generated from uniform
distribution of interval [0 1) For nonzero streamwise inletvelocities (119880 = 0) Garcia andWagner [24] introduced severalefficient acceptance-rejection methods with the general formof
119906 = 119880 minus 119891(119880
119888mp) 119888mp (6)
Here 119891(119880119888mp) is a random number selected throughan acceptance-rejection method In this study the recom-mended method for low speed flows (minus04119888mp lt 119880 lt 13119888mp)[24] was used
Flow properties of both upstream and downstreamboundaries must be determined in order to be used in (1)ndash(6) For upstream boundary temperature and pressure areknown The number density 119899in is obtained from ideal gasequation
119899in =119875in119896119879in
(7)
Transverse velocities 119881in and 119882in are set to zero Streamwisevelocity (119880in)119895 is extrapolated from neighbour boundary cell119895 where 120588 and 119886 are density and speed of sound respectively[22]
(119880in)119895 = 119880119895+
119875in minus 119875119895
120588119895119886119895
(8)
At downstream boundary only pressure is known Thusnumber density (119899
119890)119895 temperature (119879
119890)119895 and velocity compo-
nents (119880119890)119895 (119881119890)119895 and (119882
119890)119895are extrapolated from neighbour
cell
(119899119890)119895= 119899119895+
119875119890minus 119875119895
1198981198862119895
(119879119890)119895=
119875119890
(119899119890)119895119896
(119880119890)119895= 119880119895+
119875119895minus 119875119890
119898119899119895119886119895
(119881119890)119895= 119881119895
(119882119890)119895= 119882119895
(9)
3 Verifications
Shear-Driven and Pressure-Driven flows in confinedmicrochannels are well-studied problems They are fre-quently used as test cases for the robustness and validations offlow simulations In this section the theoretical backgroundand the simulation setup for such flows are presented Anew implementation of EDMD simulation including cellpartitioning event scheduling and implicit boundary treat-ment was developed in C with serial processing Benefitingfrom object oriented programming (OOP) code can befurther extended to cover different molecular types andboundary treatments On the other hand considering thecomputational performance the parallelization of the code isan essential issue in molecular simulations However thisdeserves a detailed study of its own All simulations werecarried out on a fourth-generation Intel i7 4700CPU com-puter with 16GB of RAM
31 Shear-Driven (Couette) Flow Shear-Driven flow in a con-fined channel is obtained bymoving the top plate at a constantvelocity (Figure 1(a)) Pressure gradient is zero between theupstream and downstream boundaries Initial temperature ofthe argon inside the channel and at the upstream boundaryis 300K Upper wall moves with a velocity of 100ms whilelower wall is stationary Periodicity is applied along 119911-axis Spacing ratios (the ratio of average distance betweenmolecules 119904 to average diameter of molecules 119889) for thisstudy were selected as 40 and 70 Flow configurations werechosen according to Knudsen numbers of 01 05 and 10 inthree cases (Table 1) The number of simulated particles wasaround 200000 for each case
Simulation results were compared with the continuum-based model in which velocity gradient between two wallsof the channel is affected by the Knudsen number and has alinear nature in Shear-Driven flows For a fully diffusive wallnormalized velocity profile is shown as follows where 119880
119908is
wall velocity and 119867 is channel height [25]
119906
119880119908
=1
2Kn + 1(
119910
119867+ Kn) (10)
Mathematical Problems in Engineering 5
Table 1 Flow configurations in Shear-Driven flow
Case A (Kn = 01) Case B (Kn = 05) Case C (Kn = 10)sd = 40 sd = 70 sd = 40 sd = 70 sd = 40 sd = 70
Channel height 119867 (120583m) 0050 0263 0010 0053 0005 0026Global average pressure 119875 1639 kPa (sd = 40) 306 kPa (sd = 70)Global average density 120588 2607 kgm3 (sd = 40) 487 kgm3 (sd = 70)
Boundary conditions Fully diffusive walls 300K along 119910-directionPeriodicity along 119911-direction
Table 2 Flow configurations in Pressure-Driven flow
Case D Case EInlet pressure 119875in 250 kPa 75 kPaPressure ratio PR 25 5Initial global average density (120588) 400 kgm3 120 kgm3
Initial Kn 0080 0268Channel height 119867 04 120583m (1173 molecular diameter)Aspect ratio (119871
119909119871119910) 5
Boundary conditions Fully diffusive walls 300K along 119910-directionPeriodicity along 119911-direction
32 Pressure-Driven Flows Pressure-Driven flow is gener-ated by setting different pressure levels on upstream anddownstream boundaries (Figure 1(b)) Two Pressure-Drivenflows of different pressure ratios (PR = 119875
119894119875119890) in a
microchannel were investigated Flow properties are tabu-lated in Table 2 The number of simulated molecules wasabout 300000 for both cases
For Pressure-Driven flows Arkilic et al defined thepressure distribution along the fully diffusive channel as afunction of Knudsen number at the exit of the channel (Kne)of length 119871 as [26]
119875 (119909)
119875119890
= minus6Kne
+ radic(6Kne)2
+ (1 + 12Kne)119909
119871+ (PR2 + 12KnePR) (1 minus
119909
119871)
(11)
For fully diffusive walls analytical solution of Navier-Stokesequations with second-order slip boundary condition yieldsa normalized velocity profile [27]
119906
119880119888
=[minus (119910119867)
2
+ 119910ℎ + Kn](14 + Kn)
(12)
where 119880119888is centreline velocity Slip-to-centreline velocity
ratio along the channel then becomes
119880119904
119880119888
=Kn
(14 + Kn) (13)
For second-order slip boundary condition velocity profileand corresponding slip-to-centreline ratio are
119906
119880119888
=[minus (119910119867)
2
+ 119910ℎ + Kn (1 + Kn)](14 + Kn (1 + Kn))
(14)
119880119904
119880119888
=Kn (1 + Kn)
(14 + Kn (1 + Kn)) (15)
If slip velocity is normalized with the local average velocity(12) and (15) become [28]
119880119904
119880=
Kn(16 + Kn)
119880119904
119880=
Kn (1 + Kn)(16 + Kn (1 + Kn))
(16)
4 Results and Discussions
In this section we present simulation results of Shear-Driven and Pressure-Driven flows for aforementioned casesResults were comparedwith analytical solutions derived fromNavier-Stokes equations with slip flow boundary conditionsandDSMCresults Simulation configuration (number of sim-ulated molecules spacing ratio boundary types etc) greatlyaffects computational speed As an example simulation ofone million argon molecules in a cubical region when thespacing ratio is 50 takes 450 seconds to reach one collisionper particle on the aforementioned PC configuration
41 Shear-Driven Flow In Cases AndashC six Couette flowconfigurations at different Knudsen numbers were simulatedComparison of the predicted slip velocities between simu-lation results and the analytical solution is given in Table 3
6 Mathematical Problems in Engineering
y
UwTw
H
Tw x
P T
(a) Shear-Driven flow
y
Tw
Tw
H
x
TePin Tin
(b) Pressure-Driven flow
Figure 1 Flow geometries
1
075
05
025
00 25 50 75 100
u (ms)
N-S 1st ordersd = 40
sd = 70
Kn = 01
yH
(a)
1
075
05
025
0minus8 minus4 0 4 8
sd = 40
sd = 70
Kn = 01
yH
u minus uN-S (ms)
(b)
Figure 2 Shear-Driven flow Case A (Kn = 01) (a) velocity profile (b) deviation from analytical solution
Table 3 Comparison of slip velocities
119880119904(ms)
(theory)
119880119904(ms)
(simulation)(sd = 40)
119880119904(ms)
(simulation)(sd = 70)
Case A (Kn = 01) 833 775 756Case B (Kn = 02) 2500 1859 1929Case C (Kn = 05) 3333 2546 2622
On the upper and lower walls slip velocities were estimatedby linearly extrapolating average velocities of neighbour binsKnudsen number was 01 in Case A and the flow was inthe slip-to-transition regime Consequently deviation fromthe analytical solution is reasonably small less than 10 InCases B and C flows were in the transitional regime (01 lt
Kn le 10) Therefore simulated slip velocities were predicted
noticeably lower than the analytical solution Deviation isbetween 213 and 256 in these cases
Predicted velocity profiles and deviations from the ana-lytical solution are presented in Figures 2ndash4 For low Kn flow(Case A) velocity profile is almost linear and correspondingdeviation is insignificant As rarefaction intensifies (Cases Band C) velocity profile deviates particularly at the near wallregions S shaped velocity profile is more apparent in Case C
Another finding is that nearly identical velocity profileswere obtained for different spacing ratios at the same Knud-sen number As the spacing ratio reduces from 70 to 40computational speed quadruples Utilization of this pieceof information can ease the computational cost of largersystems that is EDMD performs faster in smaller spacingratios [9]
42 Pressure-Driven Flow Pressure distribution along thechannel is given in Figures 5(a) and 6(a) for cases D and E
Mathematical Problems in Engineering 7
1
075
05
025
00 25 50 75 100
u (ms)
N-S 1st ordersd = 40
sd = 70
Kn = 05
yH
(a)
1
075
05
025
0minus8 minus4 0 4 8
sd = 40
sd = 70
Kn = 05
yH
u minus uN-S (ms)
(b)
Figure 3 Shear-Driven flow Case B (Kn = 05) (a) velocity profile (b) deviation from analytical solution
1
075
05
025
00 25 50 75 100
u (ms)
N-S 1st ordersd = 40
sd = 70
Kn = 10
yH
(a)
1
075
05
025
0minus8 minus4 0 4 8
sd = 40
sd = 70
Kn = 10
yH
u minus uN-S (ms)
(b)
Figure 4 Shear-Driven flow Case C (Kn = 10) (a) velocity profile (b) deviation from analytical solution
respectively The results are compared with linear pressuredistribution and slip formulation given by Arkilic et al [26]Figures 5(b) and 6(b) showdeviation betweenMDsimulationresults and analytical solutions Relative pressure differenceis less than 5 for both cases This indicates that adaptedboundary treatment is appropriate for EDMD simulations
Deviation from the linear pressure distribution is given inFigures 5(c) and 6(c) In both cases nonlinearity is slightlylower in EDMD predictions
Local mean free path increases along the channel due topressure gradient This yields an increase in local Knudsennumber (Figure 7) In Case D Kn was between 0080 and
8 Mathematical Problems in Engineering
25
2
15
10 025 05 075 1
N-SLinear MD simulation
PP
e
xL
(a)
0015
0
minus00150 025 05 075 1
N-SMD simulation
xL
(Psim
minusP
N-S)P
N-S
(b)
0 025 05 075 1
N-SMD simulation
xL
(PminusP
lin)P
lin
008
006
004
002
0
(c)
Figure 5 Pressure distribution in Pressure-Driven flow Case D (a) comparison of pressure distribution along 119909-axis (b) deviation betweensimulation and N-S results (c) deviation from linear pressure distribution
5
4
3
2
10 025 05 075 1
N-SLinear MD simulation
PP
e
xL
(a)
0 025 05 075 1
N-SMD simulation
xL
(PminusP
lin)P
lin
01
005
0
(b)
002
0
minus0020 025 05 075 1
N-SMD simulation
xL
(Psim
minusP
N-S)P
N-S
(c)
Figure 6 Pressure distribution in Pressure-Driven flow Case E (a) comparison of pressure distribution along 119909-axis (b) deviation betweensimulation and N-S results (c) deviation from linear pressure distribution
0211 (ie the flow was in slip-to-transition regime) In CaseE flow was in the transitional regime with corresponding Knup to 12
Ratio of the slip velocity to the local average velocity isgiven in Figure 8 Knudsen number values in the abscissa cor-respond to local values formed in Figure 7 along the channel
Slip velocities were linearly extrapolated from outermost binssimilar to Couette flow calculations Simulation results werecompared with analytical solutions with first- and second-order slip boundary conditions (16) linearized Boltzmannsolution andDSMC results [28] It is seen that EDMD resultsare in good agreement with DSMC and linearized Boltzmann
Mathematical Problems in Engineering 9
12
09
06
03
0
Case DCase E
0 025 05 075 1
Kn
xL
Figure 7 Local Kn distribution in Pressure-Driven flow
12
09
06
03
0001
N-S 1st orderN-S 2nd orderLinearized Boltzmann
DSMC
Case D
Case E
Case ECase D
01 1 10
Kn
UsU
Figure 8 Normalized slip velocity for slip-to-transition region
solution On the other hand breakdown of slip boundarymodels starts around Kn = 01 Both first- and second-order models overestimate the velocity slip Note that forboth Cases D and E upstream and downstream slip velocityratios are smaller than in other regions of the computationaldomain Use of Maxwell distribution to sample number
density and velocities in pressure boundary treatment isthe main reason of this behaviour Such implementation ofboundary treatment works better when the rarefaction effectsare small For highly rarefied (highKn) flows Ahmadzadeganet al [29] advise usingChapman-Enskog velocity distributionto improve pressure boundary treatment
10 Mathematical Problems in Engineering
1
075
05
025
0
yH
0 025 05 075 1
uUc
Kn = 0115
xL = 0559
N-S 2nd orderMD simulation
1
075
05
025
0
yH
0 025 05 075 1
uUc
xL = 0128
N-S 2nd orderMD simulation
0085Kn =
1
075
05
025
0
yH
0 025 05 075 1
uUc
xL = 0833
N-S 2nd orderMD simulation
0159Kn =
Figure 9 Velocity profiles in 119910-direction of Pressure-Driven flow Case D
1
075
05
025
0
YH
0 025 05 075 1
UUC
XL = 0814
N-S 2nd orderMD simulation
0686Kn =
1
075
05
025
0
YH
0 025 05 075 1
UUC
XL = 0206
N-S 2nd orderMD simulation
1
075
05
025
0
YH
0 025 05 075 1
UUC
XL = 0579
N-S 2nd orderMD simulation
0457Kn =0308Kn =
Figure 10 Velocity profiles in 119910-direction of Pressure-Driven flow Case E
Normalized velocity profiles (119906119880119888where 119880
119888is local
centreline velocity) in cross-stream direction at three loca-tions along the channel are shown in Figures 9 and 10 forCase D and Case E respectively Note that flow velocityalong the channel increases as pressure decreases Thusfor a given position along the channel normalization wascarried out by corresponding centreline velocity Here resultsof EDMD simulations were compared with second-orderanalytical solution In CaseD where Kn is small curves agreewell whereas in Case E estimation of higher slip velocity inanalytical solution becomes more apparent as Kn increases
5 Conclusion
This paper presents an event-driven molecular dynamics(EDMD) simulation of monatomic gas flows in microchan-nels In this study molecules are considered as hard-sphereswith discrete interaction potentials and interaction times areanalytically predictable Therefore each collision is real andtakes place in its calculated position Postcollisional velocitiesare also analytically calculated according to the conservationofmomentumand energy By its deterministic nature EDMDdiffers from DSMC method in which simulated system is
Mathematical Problems in Engineering 11
represented by a smaller group of molecules and collisionpairs are selected stochastically Additionally by using cellpartitioning technique and use of priority queues for eventsorting EDMD is now a strong alternative to DSMCmethodin terms of not only accuracy but also computational speed
In this study an implicit treatment for flow boundaries isadapted from DSMC method to EDMD for the first time inorder to induce Shear- and Pressure-Driven flows in confinedchannels The number of introduced molecules throughupstream and downstream boundaries and their correspond-ing velocities and locations are determined implicitly fromlocal number density mean flow velocity and temperature
Velocity slip in Pressure-Driven flows is compared withDSMCresults and solution of linearizedBoltzmann equationThe good agreement confirms the robustness of the treatmentand its applicability to EDMD simulations For both Shear-and Pressure-Driven flows velocity slip is also comparedwith the one derived from Navier-Stokes equations of slipboundary conditions For low Knudsen number (Kn) flowsvelocity profiles and calculated slip velocities agree well Butas the flow rarefies slip boundarymodels estimate higher slipvelocities
Additionally simulations of twoflows having the sameKnbut different spacing ratios produce almost identical velocityprofiles in Shear-Driven cases for the investigated range ofKn Since simulations with lower spacing ratios run fasterthis finding can be further extended to scale rarefied flowsand to reduce the computational time
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] T-R Hsu MEMS amp Microsystems Design Manufacture andNanoscale Engineering JohnWiley amp Sons Hoboken NJ USA2nd edition 2008
[2] G A BirdMolecular GasDynamics and theDirect Simulation ofGas Flows Clarendon Press Oxford UK 1994
[3] A Rahman ldquoCorrelations in the motion of atoms in liquidargonrdquo Physical Review vol 136 no 2 pp A405ndashA411 1964
[4] L Verlet ldquoComputer lsquoexperimentsrsquo on classical fluids I Ther-modynamical properties of Lennard-Jones moleculesrdquo PhysicalReview vol 159 no 1 pp 98ndash103 1967
[5] MMareschal and B L Holian ldquoFrom fluid particles to physicalparticles computing hydrodynamicsrdquo in Microscopic Simula-tions of Complex Hydrodynamic Phenomena M Mareschal andB LHolian Eds vol 292 ofNATOASI Series pp 1ndash12 SpringerScience amp Business Media 1992
[6] B J Alder and T E Wainwright ldquoPhase transition for a hardsphere systemrdquo The Journal of Chemical Physics vol 27 no 5pp 1208ndash1209 1957
[7] D C Rapaport ldquoThe event scheduling problem in moleculardynamic simulationrdquo Journal of Computational Physics vol 34no 2 pp 184ndash201 1980
[8] M Marın and P Cordero ldquoAn empirical assessment of priorityqueues in event-driven molecular dynamics simulationrdquo Com-puter Physics Communications vol 92 no 2-3 pp 214ndash2241995
[9] I Kandemir A multicell molecular dynamics method [PhDthesis] CaseWesternReserveUniversity ClevelandOhioUSA1999
[10] W T Tang R S M Goh and I L-J Thng ldquoLadder queuean O(1) priority queue structure for large-scale discrete eventsimulationrdquoACMTransactions onModeling and Computer Sim-ulation vol 15 no 3 pp 175ndash204 2005
[11] G Paul ldquoA Complexity O(1) priority queue for event drivenmolecular dynamics simulationsrdquo Journal of ComputationalPhysics vol 221 no 2 pp 615ndash625 2007
[12] A Donev A L Garcia and B J Alder ldquoStochastic event-drivenmolecular dynamicsrdquo Journal of Computational Physics vol 227no 4 pp 2644ndash2665 2008
[13] M N Bannerman R Sargant and L Lue ldquoDynamO a freeO(N) general event-driven molecular dynamics simulatorrdquoJournal of Computational Chemistry vol 32 no 15 pp 3329ndash3338 2011
[14] J Li D Liao and S Yip ldquoCoupling continuum to molecular-dynamics simulation reflecting particle method and the fieldestimatorrdquo Physical Review E vol 57 no 6 pp 7259ndash7267 1998
[15] F Bao YHuang Y Zhang and J Lin ldquoInvestigation of pressure-driven gas flows in nanoscale channels usingmolecular dynam-ics simulationrdquo Microfluidics and Nanofluidics vol 18 no 5-6pp 1075ndash1084 2014
[16] F Bao Y Huang L Qiu and J Lin ldquoApplicability of moleculardynamics method to the pressure-driven gas flow in finitelength nano-scale slit poresrdquo Molecular Physics vol 113 no 6pp 561ndash569 2015
[17] J R Whitman G L Aranovich and M D DonohueldquoAnisotropic mean free path in simulations of fluids travelingdown a density gradientrdquo Journal of Chemical Physics vol 132no 22 Article ID 224302 2010
[18] Q D To T T Pham G Lauriat and C Leonard ldquoMoleculardynamics simulations of pressure-driven flows and comparisonwith acceleration-driven flowsrdquo Advances in Mechanical Engi-neering vol 2012 Article ID 580763 10 pages 2012
[19] M Lupkowski and F van Swol ldquoComputer simulation of fluidsinteracting with fluctuating wallsrdquo The Journal of ChemicalPhysics vol 93 no 1 pp 737ndash745 1990
[20] C Huang K Nandakumar P Y K Choi and L W KostiukldquoMolecular dynamics simulation of a pressure-driven liquidtransport process in a cylindrical nanopore using two self-adjusting platesrdquo Journal of Chemical Physics vol 124 no 23Article ID 234701 2006
[21] W W Liou and Y C Fang ldquoImplicit boundary conditions fordirect simulation Monte Carlo method in MEMS flow pre-dictionsrdquo Computer Modeling in Engineering and Sciences vol1 no 4 pp 118ndash128 2000
[22] M Wang and Z Li ldquoSimulations for gas flows in microge-ometries using the direct simulation Monte Carlo methodrdquoInternational Journal of Heat and Fluid Flow vol 25 no 6 pp975ndash985 2004
[23] D J Evans and G P Morriss Statistical Mechanics of Nonequi-librium Liquids Cambridge University Press 2nd edition 2008
[24] A L Garcia and W Wagner ldquoGeneration of the Maxwellianinflow distributionrdquo Journal of Computational Physics vol 217no 2 pp 693ndash708 2006
12 Mathematical Problems in Engineering
[25] G Karniadakis A Beskok and N Aluru Microflows andNanoflows Fundamentals and Simulation Springer New YorkNY USA 2006
[26] E B Arkilic M A Schmidt and K S Breuer ldquoGaseous slipflow in long microchannelsrdquo Journal of MicroelectromechanicalSystems vol 6 no 2 pp 167ndash178 1997
[27] A Beskok G E Karniadakis and W Trimmer ldquoRarefactionand compressibility effects in gas microflowsrdquo Journal of FluidsEngineering vol 118 no 3 pp 448ndash456 1996
[28] A Beskok and G E Karniadakis ldquoReport a model for flows inchannels pipes and ducts at micro and nano scalesrdquoMicroscaleThermophysical Engineering vol 3 no 1 pp 43ndash77 1999
[29] A Ahmadzadegan J Wen and M Renksizbulut ldquoThe role ofthe velocity distribution in the DSMC pressure boundary con-dition for gasmixturesrdquo International Journal ofModern PhysicsC vol 23 no 12 Article ID 1250087 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
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Complex AnalysisJournal of
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OptimizationJournal of
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International Journal of
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Operations ResearchAdvances in
Journal of
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 5
Table 1 Flow configurations in Shear-Driven flow
Case A (Kn = 01) Case B (Kn = 05) Case C (Kn = 10)sd = 40 sd = 70 sd = 40 sd = 70 sd = 40 sd = 70
Channel height 119867 (120583m) 0050 0263 0010 0053 0005 0026Global average pressure 119875 1639 kPa (sd = 40) 306 kPa (sd = 70)Global average density 120588 2607 kgm3 (sd = 40) 487 kgm3 (sd = 70)
Boundary conditions Fully diffusive walls 300K along 119910-directionPeriodicity along 119911-direction
Table 2 Flow configurations in Pressure-Driven flow
Case D Case EInlet pressure 119875in 250 kPa 75 kPaPressure ratio PR 25 5Initial global average density (120588) 400 kgm3 120 kgm3
Initial Kn 0080 0268Channel height 119867 04 120583m (1173 molecular diameter)Aspect ratio (119871
119909119871119910) 5
Boundary conditions Fully diffusive walls 300K along 119910-directionPeriodicity along 119911-direction
32 Pressure-Driven Flows Pressure-Driven flow is gener-ated by setting different pressure levels on upstream anddownstream boundaries (Figure 1(b)) Two Pressure-Drivenflows of different pressure ratios (PR = 119875
119894119875119890) in a
microchannel were investigated Flow properties are tabu-lated in Table 2 The number of simulated molecules wasabout 300000 for both cases
For Pressure-Driven flows Arkilic et al defined thepressure distribution along the fully diffusive channel as afunction of Knudsen number at the exit of the channel (Kne)of length 119871 as [26]
119875 (119909)
119875119890
= minus6Kne
+ radic(6Kne)2
+ (1 + 12Kne)119909
119871+ (PR2 + 12KnePR) (1 minus
119909
119871)
(11)
For fully diffusive walls analytical solution of Navier-Stokesequations with second-order slip boundary condition yieldsa normalized velocity profile [27]
119906
119880119888
=[minus (119910119867)
2
+ 119910ℎ + Kn](14 + Kn)
(12)
where 119880119888is centreline velocity Slip-to-centreline velocity
ratio along the channel then becomes
119880119904
119880119888
=Kn
(14 + Kn) (13)
For second-order slip boundary condition velocity profileand corresponding slip-to-centreline ratio are
119906
119880119888
=[minus (119910119867)
2
+ 119910ℎ + Kn (1 + Kn)](14 + Kn (1 + Kn))
(14)
119880119904
119880119888
=Kn (1 + Kn)
(14 + Kn (1 + Kn)) (15)
If slip velocity is normalized with the local average velocity(12) and (15) become [28]
119880119904
119880=
Kn(16 + Kn)
119880119904
119880=
Kn (1 + Kn)(16 + Kn (1 + Kn))
(16)
4 Results and Discussions
In this section we present simulation results of Shear-Driven and Pressure-Driven flows for aforementioned casesResults were comparedwith analytical solutions derived fromNavier-Stokes equations with slip flow boundary conditionsandDSMCresults Simulation configuration (number of sim-ulated molecules spacing ratio boundary types etc) greatlyaffects computational speed As an example simulation ofone million argon molecules in a cubical region when thespacing ratio is 50 takes 450 seconds to reach one collisionper particle on the aforementioned PC configuration
41 Shear-Driven Flow In Cases AndashC six Couette flowconfigurations at different Knudsen numbers were simulatedComparison of the predicted slip velocities between simu-lation results and the analytical solution is given in Table 3
6 Mathematical Problems in Engineering
y
UwTw
H
Tw x
P T
(a) Shear-Driven flow
y
Tw
Tw
H
x
TePin Tin
(b) Pressure-Driven flow
Figure 1 Flow geometries
1
075
05
025
00 25 50 75 100
u (ms)
N-S 1st ordersd = 40
sd = 70
Kn = 01
yH
(a)
1
075
05
025
0minus8 minus4 0 4 8
sd = 40
sd = 70
Kn = 01
yH
u minus uN-S (ms)
(b)
Figure 2 Shear-Driven flow Case A (Kn = 01) (a) velocity profile (b) deviation from analytical solution
Table 3 Comparison of slip velocities
119880119904(ms)
(theory)
119880119904(ms)
(simulation)(sd = 40)
119880119904(ms)
(simulation)(sd = 70)
Case A (Kn = 01) 833 775 756Case B (Kn = 02) 2500 1859 1929Case C (Kn = 05) 3333 2546 2622
On the upper and lower walls slip velocities were estimatedby linearly extrapolating average velocities of neighbour binsKnudsen number was 01 in Case A and the flow was inthe slip-to-transition regime Consequently deviation fromthe analytical solution is reasonably small less than 10 InCases B and C flows were in the transitional regime (01 lt
Kn le 10) Therefore simulated slip velocities were predicted
noticeably lower than the analytical solution Deviation isbetween 213 and 256 in these cases
Predicted velocity profiles and deviations from the ana-lytical solution are presented in Figures 2ndash4 For low Kn flow(Case A) velocity profile is almost linear and correspondingdeviation is insignificant As rarefaction intensifies (Cases Band C) velocity profile deviates particularly at the near wallregions S shaped velocity profile is more apparent in Case C
Another finding is that nearly identical velocity profileswere obtained for different spacing ratios at the same Knud-sen number As the spacing ratio reduces from 70 to 40computational speed quadruples Utilization of this pieceof information can ease the computational cost of largersystems that is EDMD performs faster in smaller spacingratios [9]
42 Pressure-Driven Flow Pressure distribution along thechannel is given in Figures 5(a) and 6(a) for cases D and E
Mathematical Problems in Engineering 7
1
075
05
025
00 25 50 75 100
u (ms)
N-S 1st ordersd = 40
sd = 70
Kn = 05
yH
(a)
1
075
05
025
0minus8 minus4 0 4 8
sd = 40
sd = 70
Kn = 05
yH
u minus uN-S (ms)
(b)
Figure 3 Shear-Driven flow Case B (Kn = 05) (a) velocity profile (b) deviation from analytical solution
1
075
05
025
00 25 50 75 100
u (ms)
N-S 1st ordersd = 40
sd = 70
Kn = 10
yH
(a)
1
075
05
025
0minus8 minus4 0 4 8
sd = 40
sd = 70
Kn = 10
yH
u minus uN-S (ms)
(b)
Figure 4 Shear-Driven flow Case C (Kn = 10) (a) velocity profile (b) deviation from analytical solution
respectively The results are compared with linear pressuredistribution and slip formulation given by Arkilic et al [26]Figures 5(b) and 6(b) showdeviation betweenMDsimulationresults and analytical solutions Relative pressure differenceis less than 5 for both cases This indicates that adaptedboundary treatment is appropriate for EDMD simulations
Deviation from the linear pressure distribution is given inFigures 5(c) and 6(c) In both cases nonlinearity is slightlylower in EDMD predictions
Local mean free path increases along the channel due topressure gradient This yields an increase in local Knudsennumber (Figure 7) In Case D Kn was between 0080 and
8 Mathematical Problems in Engineering
25
2
15
10 025 05 075 1
N-SLinear MD simulation
PP
e
xL
(a)
0015
0
minus00150 025 05 075 1
N-SMD simulation
xL
(Psim
minusP
N-S)P
N-S
(b)
0 025 05 075 1
N-SMD simulation
xL
(PminusP
lin)P
lin
008
006
004
002
0
(c)
Figure 5 Pressure distribution in Pressure-Driven flow Case D (a) comparison of pressure distribution along 119909-axis (b) deviation betweensimulation and N-S results (c) deviation from linear pressure distribution
5
4
3
2
10 025 05 075 1
N-SLinear MD simulation
PP
e
xL
(a)
0 025 05 075 1
N-SMD simulation
xL
(PminusP
lin)P
lin
01
005
0
(b)
002
0
minus0020 025 05 075 1
N-SMD simulation
xL
(Psim
minusP
N-S)P
N-S
(c)
Figure 6 Pressure distribution in Pressure-Driven flow Case E (a) comparison of pressure distribution along 119909-axis (b) deviation betweensimulation and N-S results (c) deviation from linear pressure distribution
0211 (ie the flow was in slip-to-transition regime) In CaseE flow was in the transitional regime with corresponding Knup to 12
Ratio of the slip velocity to the local average velocity isgiven in Figure 8 Knudsen number values in the abscissa cor-respond to local values formed in Figure 7 along the channel
Slip velocities were linearly extrapolated from outermost binssimilar to Couette flow calculations Simulation results werecompared with analytical solutions with first- and second-order slip boundary conditions (16) linearized Boltzmannsolution andDSMC results [28] It is seen that EDMD resultsare in good agreement with DSMC and linearized Boltzmann
Mathematical Problems in Engineering 9
12
09
06
03
0
Case DCase E
0 025 05 075 1
Kn
xL
Figure 7 Local Kn distribution in Pressure-Driven flow
12
09
06
03
0001
N-S 1st orderN-S 2nd orderLinearized Boltzmann
DSMC
Case D
Case E
Case ECase D
01 1 10
Kn
UsU
Figure 8 Normalized slip velocity for slip-to-transition region
solution On the other hand breakdown of slip boundarymodels starts around Kn = 01 Both first- and second-order models overestimate the velocity slip Note that forboth Cases D and E upstream and downstream slip velocityratios are smaller than in other regions of the computationaldomain Use of Maxwell distribution to sample number
density and velocities in pressure boundary treatment isthe main reason of this behaviour Such implementation ofboundary treatment works better when the rarefaction effectsare small For highly rarefied (highKn) flows Ahmadzadeganet al [29] advise usingChapman-Enskog velocity distributionto improve pressure boundary treatment
10 Mathematical Problems in Engineering
1
075
05
025
0
yH
0 025 05 075 1
uUc
Kn = 0115
xL = 0559
N-S 2nd orderMD simulation
1
075
05
025
0
yH
0 025 05 075 1
uUc
xL = 0128
N-S 2nd orderMD simulation
0085Kn =
1
075
05
025
0
yH
0 025 05 075 1
uUc
xL = 0833
N-S 2nd orderMD simulation
0159Kn =
Figure 9 Velocity profiles in 119910-direction of Pressure-Driven flow Case D
1
075
05
025
0
YH
0 025 05 075 1
UUC
XL = 0814
N-S 2nd orderMD simulation
0686Kn =
1
075
05
025
0
YH
0 025 05 075 1
UUC
XL = 0206
N-S 2nd orderMD simulation
1
075
05
025
0
YH
0 025 05 075 1
UUC
XL = 0579
N-S 2nd orderMD simulation
0457Kn =0308Kn =
Figure 10 Velocity profiles in 119910-direction of Pressure-Driven flow Case E
Normalized velocity profiles (119906119880119888where 119880
119888is local
centreline velocity) in cross-stream direction at three loca-tions along the channel are shown in Figures 9 and 10 forCase D and Case E respectively Note that flow velocityalong the channel increases as pressure decreases Thusfor a given position along the channel normalization wascarried out by corresponding centreline velocity Here resultsof EDMD simulations were compared with second-orderanalytical solution In CaseD where Kn is small curves agreewell whereas in Case E estimation of higher slip velocity inanalytical solution becomes more apparent as Kn increases
5 Conclusion
This paper presents an event-driven molecular dynamics(EDMD) simulation of monatomic gas flows in microchan-nels In this study molecules are considered as hard-sphereswith discrete interaction potentials and interaction times areanalytically predictable Therefore each collision is real andtakes place in its calculated position Postcollisional velocitiesare also analytically calculated according to the conservationofmomentumand energy By its deterministic nature EDMDdiffers from DSMC method in which simulated system is
Mathematical Problems in Engineering 11
represented by a smaller group of molecules and collisionpairs are selected stochastically Additionally by using cellpartitioning technique and use of priority queues for eventsorting EDMD is now a strong alternative to DSMCmethodin terms of not only accuracy but also computational speed
In this study an implicit treatment for flow boundaries isadapted from DSMC method to EDMD for the first time inorder to induce Shear- and Pressure-Driven flows in confinedchannels The number of introduced molecules throughupstream and downstream boundaries and their correspond-ing velocities and locations are determined implicitly fromlocal number density mean flow velocity and temperature
Velocity slip in Pressure-Driven flows is compared withDSMCresults and solution of linearizedBoltzmann equationThe good agreement confirms the robustness of the treatmentand its applicability to EDMD simulations For both Shear-and Pressure-Driven flows velocity slip is also comparedwith the one derived from Navier-Stokes equations of slipboundary conditions For low Knudsen number (Kn) flowsvelocity profiles and calculated slip velocities agree well Butas the flow rarefies slip boundarymodels estimate higher slipvelocities
Additionally simulations of twoflows having the sameKnbut different spacing ratios produce almost identical velocityprofiles in Shear-Driven cases for the investigated range ofKn Since simulations with lower spacing ratios run fasterthis finding can be further extended to scale rarefied flowsand to reduce the computational time
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] T-R Hsu MEMS amp Microsystems Design Manufacture andNanoscale Engineering JohnWiley amp Sons Hoboken NJ USA2nd edition 2008
[2] G A BirdMolecular GasDynamics and theDirect Simulation ofGas Flows Clarendon Press Oxford UK 1994
[3] A Rahman ldquoCorrelations in the motion of atoms in liquidargonrdquo Physical Review vol 136 no 2 pp A405ndashA411 1964
[4] L Verlet ldquoComputer lsquoexperimentsrsquo on classical fluids I Ther-modynamical properties of Lennard-Jones moleculesrdquo PhysicalReview vol 159 no 1 pp 98ndash103 1967
[5] MMareschal and B L Holian ldquoFrom fluid particles to physicalparticles computing hydrodynamicsrdquo in Microscopic Simula-tions of Complex Hydrodynamic Phenomena M Mareschal andB LHolian Eds vol 292 ofNATOASI Series pp 1ndash12 SpringerScience amp Business Media 1992
[6] B J Alder and T E Wainwright ldquoPhase transition for a hardsphere systemrdquo The Journal of Chemical Physics vol 27 no 5pp 1208ndash1209 1957
[7] D C Rapaport ldquoThe event scheduling problem in moleculardynamic simulationrdquo Journal of Computational Physics vol 34no 2 pp 184ndash201 1980
[8] M Marın and P Cordero ldquoAn empirical assessment of priorityqueues in event-driven molecular dynamics simulationrdquo Com-puter Physics Communications vol 92 no 2-3 pp 214ndash2241995
[9] I Kandemir A multicell molecular dynamics method [PhDthesis] CaseWesternReserveUniversity ClevelandOhioUSA1999
[10] W T Tang R S M Goh and I L-J Thng ldquoLadder queuean O(1) priority queue structure for large-scale discrete eventsimulationrdquoACMTransactions onModeling and Computer Sim-ulation vol 15 no 3 pp 175ndash204 2005
[11] G Paul ldquoA Complexity O(1) priority queue for event drivenmolecular dynamics simulationsrdquo Journal of ComputationalPhysics vol 221 no 2 pp 615ndash625 2007
[12] A Donev A L Garcia and B J Alder ldquoStochastic event-drivenmolecular dynamicsrdquo Journal of Computational Physics vol 227no 4 pp 2644ndash2665 2008
[13] M N Bannerman R Sargant and L Lue ldquoDynamO a freeO(N) general event-driven molecular dynamics simulatorrdquoJournal of Computational Chemistry vol 32 no 15 pp 3329ndash3338 2011
[14] J Li D Liao and S Yip ldquoCoupling continuum to molecular-dynamics simulation reflecting particle method and the fieldestimatorrdquo Physical Review E vol 57 no 6 pp 7259ndash7267 1998
[15] F Bao YHuang Y Zhang and J Lin ldquoInvestigation of pressure-driven gas flows in nanoscale channels usingmolecular dynam-ics simulationrdquo Microfluidics and Nanofluidics vol 18 no 5-6pp 1075ndash1084 2014
[16] F Bao Y Huang L Qiu and J Lin ldquoApplicability of moleculardynamics method to the pressure-driven gas flow in finitelength nano-scale slit poresrdquo Molecular Physics vol 113 no 6pp 561ndash569 2015
[17] J R Whitman G L Aranovich and M D DonohueldquoAnisotropic mean free path in simulations of fluids travelingdown a density gradientrdquo Journal of Chemical Physics vol 132no 22 Article ID 224302 2010
[18] Q D To T T Pham G Lauriat and C Leonard ldquoMoleculardynamics simulations of pressure-driven flows and comparisonwith acceleration-driven flowsrdquo Advances in Mechanical Engi-neering vol 2012 Article ID 580763 10 pages 2012
[19] M Lupkowski and F van Swol ldquoComputer simulation of fluidsinteracting with fluctuating wallsrdquo The Journal of ChemicalPhysics vol 93 no 1 pp 737ndash745 1990
[20] C Huang K Nandakumar P Y K Choi and L W KostiukldquoMolecular dynamics simulation of a pressure-driven liquidtransport process in a cylindrical nanopore using two self-adjusting platesrdquo Journal of Chemical Physics vol 124 no 23Article ID 234701 2006
[21] W W Liou and Y C Fang ldquoImplicit boundary conditions fordirect simulation Monte Carlo method in MEMS flow pre-dictionsrdquo Computer Modeling in Engineering and Sciences vol1 no 4 pp 118ndash128 2000
[22] M Wang and Z Li ldquoSimulations for gas flows in microge-ometries using the direct simulation Monte Carlo methodrdquoInternational Journal of Heat and Fluid Flow vol 25 no 6 pp975ndash985 2004
[23] D J Evans and G P Morriss Statistical Mechanics of Nonequi-librium Liquids Cambridge University Press 2nd edition 2008
[24] A L Garcia and W Wagner ldquoGeneration of the Maxwellianinflow distributionrdquo Journal of Computational Physics vol 217no 2 pp 693ndash708 2006
12 Mathematical Problems in Engineering
[25] G Karniadakis A Beskok and N Aluru Microflows andNanoflows Fundamentals and Simulation Springer New YorkNY USA 2006
[26] E B Arkilic M A Schmidt and K S Breuer ldquoGaseous slipflow in long microchannelsrdquo Journal of MicroelectromechanicalSystems vol 6 no 2 pp 167ndash178 1997
[27] A Beskok G E Karniadakis and W Trimmer ldquoRarefactionand compressibility effects in gas microflowsrdquo Journal of FluidsEngineering vol 118 no 3 pp 448ndash456 1996
[28] A Beskok and G E Karniadakis ldquoReport a model for flows inchannels pipes and ducts at micro and nano scalesrdquoMicroscaleThermophysical Engineering vol 3 no 1 pp 43ndash77 1999
[29] A Ahmadzadegan J Wen and M Renksizbulut ldquoThe role ofthe velocity distribution in the DSMC pressure boundary con-dition for gasmixturesrdquo International Journal ofModern PhysicsC vol 23 no 12 Article ID 1250087 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
6 Mathematical Problems in Engineering
y
UwTw
H
Tw x
P T
(a) Shear-Driven flow
y
Tw
Tw
H
x
TePin Tin
(b) Pressure-Driven flow
Figure 1 Flow geometries
1
075
05
025
00 25 50 75 100
u (ms)
N-S 1st ordersd = 40
sd = 70
Kn = 01
yH
(a)
1
075
05
025
0minus8 minus4 0 4 8
sd = 40
sd = 70
Kn = 01
yH
u minus uN-S (ms)
(b)
Figure 2 Shear-Driven flow Case A (Kn = 01) (a) velocity profile (b) deviation from analytical solution
Table 3 Comparison of slip velocities
119880119904(ms)
(theory)
119880119904(ms)
(simulation)(sd = 40)
119880119904(ms)
(simulation)(sd = 70)
Case A (Kn = 01) 833 775 756Case B (Kn = 02) 2500 1859 1929Case C (Kn = 05) 3333 2546 2622
On the upper and lower walls slip velocities were estimatedby linearly extrapolating average velocities of neighbour binsKnudsen number was 01 in Case A and the flow was inthe slip-to-transition regime Consequently deviation fromthe analytical solution is reasonably small less than 10 InCases B and C flows were in the transitional regime (01 lt
Kn le 10) Therefore simulated slip velocities were predicted
noticeably lower than the analytical solution Deviation isbetween 213 and 256 in these cases
Predicted velocity profiles and deviations from the ana-lytical solution are presented in Figures 2ndash4 For low Kn flow(Case A) velocity profile is almost linear and correspondingdeviation is insignificant As rarefaction intensifies (Cases Band C) velocity profile deviates particularly at the near wallregions S shaped velocity profile is more apparent in Case C
Another finding is that nearly identical velocity profileswere obtained for different spacing ratios at the same Knud-sen number As the spacing ratio reduces from 70 to 40computational speed quadruples Utilization of this pieceof information can ease the computational cost of largersystems that is EDMD performs faster in smaller spacingratios [9]
42 Pressure-Driven Flow Pressure distribution along thechannel is given in Figures 5(a) and 6(a) for cases D and E
Mathematical Problems in Engineering 7
1
075
05
025
00 25 50 75 100
u (ms)
N-S 1st ordersd = 40
sd = 70
Kn = 05
yH
(a)
1
075
05
025
0minus8 minus4 0 4 8
sd = 40
sd = 70
Kn = 05
yH
u minus uN-S (ms)
(b)
Figure 3 Shear-Driven flow Case B (Kn = 05) (a) velocity profile (b) deviation from analytical solution
1
075
05
025
00 25 50 75 100
u (ms)
N-S 1st ordersd = 40
sd = 70
Kn = 10
yH
(a)
1
075
05
025
0minus8 minus4 0 4 8
sd = 40
sd = 70
Kn = 10
yH
u minus uN-S (ms)
(b)
Figure 4 Shear-Driven flow Case C (Kn = 10) (a) velocity profile (b) deviation from analytical solution
respectively The results are compared with linear pressuredistribution and slip formulation given by Arkilic et al [26]Figures 5(b) and 6(b) showdeviation betweenMDsimulationresults and analytical solutions Relative pressure differenceis less than 5 for both cases This indicates that adaptedboundary treatment is appropriate for EDMD simulations
Deviation from the linear pressure distribution is given inFigures 5(c) and 6(c) In both cases nonlinearity is slightlylower in EDMD predictions
Local mean free path increases along the channel due topressure gradient This yields an increase in local Knudsennumber (Figure 7) In Case D Kn was between 0080 and
8 Mathematical Problems in Engineering
25
2
15
10 025 05 075 1
N-SLinear MD simulation
PP
e
xL
(a)
0015
0
minus00150 025 05 075 1
N-SMD simulation
xL
(Psim
minusP
N-S)P
N-S
(b)
0 025 05 075 1
N-SMD simulation
xL
(PminusP
lin)P
lin
008
006
004
002
0
(c)
Figure 5 Pressure distribution in Pressure-Driven flow Case D (a) comparison of pressure distribution along 119909-axis (b) deviation betweensimulation and N-S results (c) deviation from linear pressure distribution
5
4
3
2
10 025 05 075 1
N-SLinear MD simulation
PP
e
xL
(a)
0 025 05 075 1
N-SMD simulation
xL
(PminusP
lin)P
lin
01
005
0
(b)
002
0
minus0020 025 05 075 1
N-SMD simulation
xL
(Psim
minusP
N-S)P
N-S
(c)
Figure 6 Pressure distribution in Pressure-Driven flow Case E (a) comparison of pressure distribution along 119909-axis (b) deviation betweensimulation and N-S results (c) deviation from linear pressure distribution
0211 (ie the flow was in slip-to-transition regime) In CaseE flow was in the transitional regime with corresponding Knup to 12
Ratio of the slip velocity to the local average velocity isgiven in Figure 8 Knudsen number values in the abscissa cor-respond to local values formed in Figure 7 along the channel
Slip velocities were linearly extrapolated from outermost binssimilar to Couette flow calculations Simulation results werecompared with analytical solutions with first- and second-order slip boundary conditions (16) linearized Boltzmannsolution andDSMC results [28] It is seen that EDMD resultsare in good agreement with DSMC and linearized Boltzmann
Mathematical Problems in Engineering 9
12
09
06
03
0
Case DCase E
0 025 05 075 1
Kn
xL
Figure 7 Local Kn distribution in Pressure-Driven flow
12
09
06
03
0001
N-S 1st orderN-S 2nd orderLinearized Boltzmann
DSMC
Case D
Case E
Case ECase D
01 1 10
Kn
UsU
Figure 8 Normalized slip velocity for slip-to-transition region
solution On the other hand breakdown of slip boundarymodels starts around Kn = 01 Both first- and second-order models overestimate the velocity slip Note that forboth Cases D and E upstream and downstream slip velocityratios are smaller than in other regions of the computationaldomain Use of Maxwell distribution to sample number
density and velocities in pressure boundary treatment isthe main reason of this behaviour Such implementation ofboundary treatment works better when the rarefaction effectsare small For highly rarefied (highKn) flows Ahmadzadeganet al [29] advise usingChapman-Enskog velocity distributionto improve pressure boundary treatment
10 Mathematical Problems in Engineering
1
075
05
025
0
yH
0 025 05 075 1
uUc
Kn = 0115
xL = 0559
N-S 2nd orderMD simulation
1
075
05
025
0
yH
0 025 05 075 1
uUc
xL = 0128
N-S 2nd orderMD simulation
0085Kn =
1
075
05
025
0
yH
0 025 05 075 1
uUc
xL = 0833
N-S 2nd orderMD simulation
0159Kn =
Figure 9 Velocity profiles in 119910-direction of Pressure-Driven flow Case D
1
075
05
025
0
YH
0 025 05 075 1
UUC
XL = 0814
N-S 2nd orderMD simulation
0686Kn =
1
075
05
025
0
YH
0 025 05 075 1
UUC
XL = 0206
N-S 2nd orderMD simulation
1
075
05
025
0
YH
0 025 05 075 1
UUC
XL = 0579
N-S 2nd orderMD simulation
0457Kn =0308Kn =
Figure 10 Velocity profiles in 119910-direction of Pressure-Driven flow Case E
Normalized velocity profiles (119906119880119888where 119880
119888is local
centreline velocity) in cross-stream direction at three loca-tions along the channel are shown in Figures 9 and 10 forCase D and Case E respectively Note that flow velocityalong the channel increases as pressure decreases Thusfor a given position along the channel normalization wascarried out by corresponding centreline velocity Here resultsof EDMD simulations were compared with second-orderanalytical solution In CaseD where Kn is small curves agreewell whereas in Case E estimation of higher slip velocity inanalytical solution becomes more apparent as Kn increases
5 Conclusion
This paper presents an event-driven molecular dynamics(EDMD) simulation of monatomic gas flows in microchan-nels In this study molecules are considered as hard-sphereswith discrete interaction potentials and interaction times areanalytically predictable Therefore each collision is real andtakes place in its calculated position Postcollisional velocitiesare also analytically calculated according to the conservationofmomentumand energy By its deterministic nature EDMDdiffers from DSMC method in which simulated system is
Mathematical Problems in Engineering 11
represented by a smaller group of molecules and collisionpairs are selected stochastically Additionally by using cellpartitioning technique and use of priority queues for eventsorting EDMD is now a strong alternative to DSMCmethodin terms of not only accuracy but also computational speed
In this study an implicit treatment for flow boundaries isadapted from DSMC method to EDMD for the first time inorder to induce Shear- and Pressure-Driven flows in confinedchannels The number of introduced molecules throughupstream and downstream boundaries and their correspond-ing velocities and locations are determined implicitly fromlocal number density mean flow velocity and temperature
Velocity slip in Pressure-Driven flows is compared withDSMCresults and solution of linearizedBoltzmann equationThe good agreement confirms the robustness of the treatmentand its applicability to EDMD simulations For both Shear-and Pressure-Driven flows velocity slip is also comparedwith the one derived from Navier-Stokes equations of slipboundary conditions For low Knudsen number (Kn) flowsvelocity profiles and calculated slip velocities agree well Butas the flow rarefies slip boundarymodels estimate higher slipvelocities
Additionally simulations of twoflows having the sameKnbut different spacing ratios produce almost identical velocityprofiles in Shear-Driven cases for the investigated range ofKn Since simulations with lower spacing ratios run fasterthis finding can be further extended to scale rarefied flowsand to reduce the computational time
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] T-R Hsu MEMS amp Microsystems Design Manufacture andNanoscale Engineering JohnWiley amp Sons Hoboken NJ USA2nd edition 2008
[2] G A BirdMolecular GasDynamics and theDirect Simulation ofGas Flows Clarendon Press Oxford UK 1994
[3] A Rahman ldquoCorrelations in the motion of atoms in liquidargonrdquo Physical Review vol 136 no 2 pp A405ndashA411 1964
[4] L Verlet ldquoComputer lsquoexperimentsrsquo on classical fluids I Ther-modynamical properties of Lennard-Jones moleculesrdquo PhysicalReview vol 159 no 1 pp 98ndash103 1967
[5] MMareschal and B L Holian ldquoFrom fluid particles to physicalparticles computing hydrodynamicsrdquo in Microscopic Simula-tions of Complex Hydrodynamic Phenomena M Mareschal andB LHolian Eds vol 292 ofNATOASI Series pp 1ndash12 SpringerScience amp Business Media 1992
[6] B J Alder and T E Wainwright ldquoPhase transition for a hardsphere systemrdquo The Journal of Chemical Physics vol 27 no 5pp 1208ndash1209 1957
[7] D C Rapaport ldquoThe event scheduling problem in moleculardynamic simulationrdquo Journal of Computational Physics vol 34no 2 pp 184ndash201 1980
[8] M Marın and P Cordero ldquoAn empirical assessment of priorityqueues in event-driven molecular dynamics simulationrdquo Com-puter Physics Communications vol 92 no 2-3 pp 214ndash2241995
[9] I Kandemir A multicell molecular dynamics method [PhDthesis] CaseWesternReserveUniversity ClevelandOhioUSA1999
[10] W T Tang R S M Goh and I L-J Thng ldquoLadder queuean O(1) priority queue structure for large-scale discrete eventsimulationrdquoACMTransactions onModeling and Computer Sim-ulation vol 15 no 3 pp 175ndash204 2005
[11] G Paul ldquoA Complexity O(1) priority queue for event drivenmolecular dynamics simulationsrdquo Journal of ComputationalPhysics vol 221 no 2 pp 615ndash625 2007
[12] A Donev A L Garcia and B J Alder ldquoStochastic event-drivenmolecular dynamicsrdquo Journal of Computational Physics vol 227no 4 pp 2644ndash2665 2008
[13] M N Bannerman R Sargant and L Lue ldquoDynamO a freeO(N) general event-driven molecular dynamics simulatorrdquoJournal of Computational Chemistry vol 32 no 15 pp 3329ndash3338 2011
[14] J Li D Liao and S Yip ldquoCoupling continuum to molecular-dynamics simulation reflecting particle method and the fieldestimatorrdquo Physical Review E vol 57 no 6 pp 7259ndash7267 1998
[15] F Bao YHuang Y Zhang and J Lin ldquoInvestigation of pressure-driven gas flows in nanoscale channels usingmolecular dynam-ics simulationrdquo Microfluidics and Nanofluidics vol 18 no 5-6pp 1075ndash1084 2014
[16] F Bao Y Huang L Qiu and J Lin ldquoApplicability of moleculardynamics method to the pressure-driven gas flow in finitelength nano-scale slit poresrdquo Molecular Physics vol 113 no 6pp 561ndash569 2015
[17] J R Whitman G L Aranovich and M D DonohueldquoAnisotropic mean free path in simulations of fluids travelingdown a density gradientrdquo Journal of Chemical Physics vol 132no 22 Article ID 224302 2010
[18] Q D To T T Pham G Lauriat and C Leonard ldquoMoleculardynamics simulations of pressure-driven flows and comparisonwith acceleration-driven flowsrdquo Advances in Mechanical Engi-neering vol 2012 Article ID 580763 10 pages 2012
[19] M Lupkowski and F van Swol ldquoComputer simulation of fluidsinteracting with fluctuating wallsrdquo The Journal of ChemicalPhysics vol 93 no 1 pp 737ndash745 1990
[20] C Huang K Nandakumar P Y K Choi and L W KostiukldquoMolecular dynamics simulation of a pressure-driven liquidtransport process in a cylindrical nanopore using two self-adjusting platesrdquo Journal of Chemical Physics vol 124 no 23Article ID 234701 2006
[21] W W Liou and Y C Fang ldquoImplicit boundary conditions fordirect simulation Monte Carlo method in MEMS flow pre-dictionsrdquo Computer Modeling in Engineering and Sciences vol1 no 4 pp 118ndash128 2000
[22] M Wang and Z Li ldquoSimulations for gas flows in microge-ometries using the direct simulation Monte Carlo methodrdquoInternational Journal of Heat and Fluid Flow vol 25 no 6 pp975ndash985 2004
[23] D J Evans and G P Morriss Statistical Mechanics of Nonequi-librium Liquids Cambridge University Press 2nd edition 2008
[24] A L Garcia and W Wagner ldquoGeneration of the Maxwellianinflow distributionrdquo Journal of Computational Physics vol 217no 2 pp 693ndash708 2006
12 Mathematical Problems in Engineering
[25] G Karniadakis A Beskok and N Aluru Microflows andNanoflows Fundamentals and Simulation Springer New YorkNY USA 2006
[26] E B Arkilic M A Schmidt and K S Breuer ldquoGaseous slipflow in long microchannelsrdquo Journal of MicroelectromechanicalSystems vol 6 no 2 pp 167ndash178 1997
[27] A Beskok G E Karniadakis and W Trimmer ldquoRarefactionand compressibility effects in gas microflowsrdquo Journal of FluidsEngineering vol 118 no 3 pp 448ndash456 1996
[28] A Beskok and G E Karniadakis ldquoReport a model for flows inchannels pipes and ducts at micro and nano scalesrdquoMicroscaleThermophysical Engineering vol 3 no 1 pp 43ndash77 1999
[29] A Ahmadzadegan J Wen and M Renksizbulut ldquoThe role ofthe velocity distribution in the DSMC pressure boundary con-dition for gasmixturesrdquo International Journal ofModern PhysicsC vol 23 no 12 Article ID 1250087 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 7
1
075
05
025
00 25 50 75 100
u (ms)
N-S 1st ordersd = 40
sd = 70
Kn = 05
yH
(a)
1
075
05
025
0minus8 minus4 0 4 8
sd = 40
sd = 70
Kn = 05
yH
u minus uN-S (ms)
(b)
Figure 3 Shear-Driven flow Case B (Kn = 05) (a) velocity profile (b) deviation from analytical solution
1
075
05
025
00 25 50 75 100
u (ms)
N-S 1st ordersd = 40
sd = 70
Kn = 10
yH
(a)
1
075
05
025
0minus8 minus4 0 4 8
sd = 40
sd = 70
Kn = 10
yH
u minus uN-S (ms)
(b)
Figure 4 Shear-Driven flow Case C (Kn = 10) (a) velocity profile (b) deviation from analytical solution
respectively The results are compared with linear pressuredistribution and slip formulation given by Arkilic et al [26]Figures 5(b) and 6(b) showdeviation betweenMDsimulationresults and analytical solutions Relative pressure differenceis less than 5 for both cases This indicates that adaptedboundary treatment is appropriate for EDMD simulations
Deviation from the linear pressure distribution is given inFigures 5(c) and 6(c) In both cases nonlinearity is slightlylower in EDMD predictions
Local mean free path increases along the channel due topressure gradient This yields an increase in local Knudsennumber (Figure 7) In Case D Kn was between 0080 and
8 Mathematical Problems in Engineering
25
2
15
10 025 05 075 1
N-SLinear MD simulation
PP
e
xL
(a)
0015
0
minus00150 025 05 075 1
N-SMD simulation
xL
(Psim
minusP
N-S)P
N-S
(b)
0 025 05 075 1
N-SMD simulation
xL
(PminusP
lin)P
lin
008
006
004
002
0
(c)
Figure 5 Pressure distribution in Pressure-Driven flow Case D (a) comparison of pressure distribution along 119909-axis (b) deviation betweensimulation and N-S results (c) deviation from linear pressure distribution
5
4
3
2
10 025 05 075 1
N-SLinear MD simulation
PP
e
xL
(a)
0 025 05 075 1
N-SMD simulation
xL
(PminusP
lin)P
lin
01
005
0
(b)
002
0
minus0020 025 05 075 1
N-SMD simulation
xL
(Psim
minusP
N-S)P
N-S
(c)
Figure 6 Pressure distribution in Pressure-Driven flow Case E (a) comparison of pressure distribution along 119909-axis (b) deviation betweensimulation and N-S results (c) deviation from linear pressure distribution
0211 (ie the flow was in slip-to-transition regime) In CaseE flow was in the transitional regime with corresponding Knup to 12
Ratio of the slip velocity to the local average velocity isgiven in Figure 8 Knudsen number values in the abscissa cor-respond to local values formed in Figure 7 along the channel
Slip velocities were linearly extrapolated from outermost binssimilar to Couette flow calculations Simulation results werecompared with analytical solutions with first- and second-order slip boundary conditions (16) linearized Boltzmannsolution andDSMC results [28] It is seen that EDMD resultsare in good agreement with DSMC and linearized Boltzmann
Mathematical Problems in Engineering 9
12
09
06
03
0
Case DCase E
0 025 05 075 1
Kn
xL
Figure 7 Local Kn distribution in Pressure-Driven flow
12
09
06
03
0001
N-S 1st orderN-S 2nd orderLinearized Boltzmann
DSMC
Case D
Case E
Case ECase D
01 1 10
Kn
UsU
Figure 8 Normalized slip velocity for slip-to-transition region
solution On the other hand breakdown of slip boundarymodels starts around Kn = 01 Both first- and second-order models overestimate the velocity slip Note that forboth Cases D and E upstream and downstream slip velocityratios are smaller than in other regions of the computationaldomain Use of Maxwell distribution to sample number
density and velocities in pressure boundary treatment isthe main reason of this behaviour Such implementation ofboundary treatment works better when the rarefaction effectsare small For highly rarefied (highKn) flows Ahmadzadeganet al [29] advise usingChapman-Enskog velocity distributionto improve pressure boundary treatment
10 Mathematical Problems in Engineering
1
075
05
025
0
yH
0 025 05 075 1
uUc
Kn = 0115
xL = 0559
N-S 2nd orderMD simulation
1
075
05
025
0
yH
0 025 05 075 1
uUc
xL = 0128
N-S 2nd orderMD simulation
0085Kn =
1
075
05
025
0
yH
0 025 05 075 1
uUc
xL = 0833
N-S 2nd orderMD simulation
0159Kn =
Figure 9 Velocity profiles in 119910-direction of Pressure-Driven flow Case D
1
075
05
025
0
YH
0 025 05 075 1
UUC
XL = 0814
N-S 2nd orderMD simulation
0686Kn =
1
075
05
025
0
YH
0 025 05 075 1
UUC
XL = 0206
N-S 2nd orderMD simulation
1
075
05
025
0
YH
0 025 05 075 1
UUC
XL = 0579
N-S 2nd orderMD simulation
0457Kn =0308Kn =
Figure 10 Velocity profiles in 119910-direction of Pressure-Driven flow Case E
Normalized velocity profiles (119906119880119888where 119880
119888is local
centreline velocity) in cross-stream direction at three loca-tions along the channel are shown in Figures 9 and 10 forCase D and Case E respectively Note that flow velocityalong the channel increases as pressure decreases Thusfor a given position along the channel normalization wascarried out by corresponding centreline velocity Here resultsof EDMD simulations were compared with second-orderanalytical solution In CaseD where Kn is small curves agreewell whereas in Case E estimation of higher slip velocity inanalytical solution becomes more apparent as Kn increases
5 Conclusion
This paper presents an event-driven molecular dynamics(EDMD) simulation of monatomic gas flows in microchan-nels In this study molecules are considered as hard-sphereswith discrete interaction potentials and interaction times areanalytically predictable Therefore each collision is real andtakes place in its calculated position Postcollisional velocitiesare also analytically calculated according to the conservationofmomentumand energy By its deterministic nature EDMDdiffers from DSMC method in which simulated system is
Mathematical Problems in Engineering 11
represented by a smaller group of molecules and collisionpairs are selected stochastically Additionally by using cellpartitioning technique and use of priority queues for eventsorting EDMD is now a strong alternative to DSMCmethodin terms of not only accuracy but also computational speed
In this study an implicit treatment for flow boundaries isadapted from DSMC method to EDMD for the first time inorder to induce Shear- and Pressure-Driven flows in confinedchannels The number of introduced molecules throughupstream and downstream boundaries and their correspond-ing velocities and locations are determined implicitly fromlocal number density mean flow velocity and temperature
Velocity slip in Pressure-Driven flows is compared withDSMCresults and solution of linearizedBoltzmann equationThe good agreement confirms the robustness of the treatmentand its applicability to EDMD simulations For both Shear-and Pressure-Driven flows velocity slip is also comparedwith the one derived from Navier-Stokes equations of slipboundary conditions For low Knudsen number (Kn) flowsvelocity profiles and calculated slip velocities agree well Butas the flow rarefies slip boundarymodels estimate higher slipvelocities
Additionally simulations of twoflows having the sameKnbut different spacing ratios produce almost identical velocityprofiles in Shear-Driven cases for the investigated range ofKn Since simulations with lower spacing ratios run fasterthis finding can be further extended to scale rarefied flowsand to reduce the computational time
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] T-R Hsu MEMS amp Microsystems Design Manufacture andNanoscale Engineering JohnWiley amp Sons Hoboken NJ USA2nd edition 2008
[2] G A BirdMolecular GasDynamics and theDirect Simulation ofGas Flows Clarendon Press Oxford UK 1994
[3] A Rahman ldquoCorrelations in the motion of atoms in liquidargonrdquo Physical Review vol 136 no 2 pp A405ndashA411 1964
[4] L Verlet ldquoComputer lsquoexperimentsrsquo on classical fluids I Ther-modynamical properties of Lennard-Jones moleculesrdquo PhysicalReview vol 159 no 1 pp 98ndash103 1967
[5] MMareschal and B L Holian ldquoFrom fluid particles to physicalparticles computing hydrodynamicsrdquo in Microscopic Simula-tions of Complex Hydrodynamic Phenomena M Mareschal andB LHolian Eds vol 292 ofNATOASI Series pp 1ndash12 SpringerScience amp Business Media 1992
[6] B J Alder and T E Wainwright ldquoPhase transition for a hardsphere systemrdquo The Journal of Chemical Physics vol 27 no 5pp 1208ndash1209 1957
[7] D C Rapaport ldquoThe event scheduling problem in moleculardynamic simulationrdquo Journal of Computational Physics vol 34no 2 pp 184ndash201 1980
[8] M Marın and P Cordero ldquoAn empirical assessment of priorityqueues in event-driven molecular dynamics simulationrdquo Com-puter Physics Communications vol 92 no 2-3 pp 214ndash2241995
[9] I Kandemir A multicell molecular dynamics method [PhDthesis] CaseWesternReserveUniversity ClevelandOhioUSA1999
[10] W T Tang R S M Goh and I L-J Thng ldquoLadder queuean O(1) priority queue structure for large-scale discrete eventsimulationrdquoACMTransactions onModeling and Computer Sim-ulation vol 15 no 3 pp 175ndash204 2005
[11] G Paul ldquoA Complexity O(1) priority queue for event drivenmolecular dynamics simulationsrdquo Journal of ComputationalPhysics vol 221 no 2 pp 615ndash625 2007
[12] A Donev A L Garcia and B J Alder ldquoStochastic event-drivenmolecular dynamicsrdquo Journal of Computational Physics vol 227no 4 pp 2644ndash2665 2008
[13] M N Bannerman R Sargant and L Lue ldquoDynamO a freeO(N) general event-driven molecular dynamics simulatorrdquoJournal of Computational Chemistry vol 32 no 15 pp 3329ndash3338 2011
[14] J Li D Liao and S Yip ldquoCoupling continuum to molecular-dynamics simulation reflecting particle method and the fieldestimatorrdquo Physical Review E vol 57 no 6 pp 7259ndash7267 1998
[15] F Bao YHuang Y Zhang and J Lin ldquoInvestigation of pressure-driven gas flows in nanoscale channels usingmolecular dynam-ics simulationrdquo Microfluidics and Nanofluidics vol 18 no 5-6pp 1075ndash1084 2014
[16] F Bao Y Huang L Qiu and J Lin ldquoApplicability of moleculardynamics method to the pressure-driven gas flow in finitelength nano-scale slit poresrdquo Molecular Physics vol 113 no 6pp 561ndash569 2015
[17] J R Whitman G L Aranovich and M D DonohueldquoAnisotropic mean free path in simulations of fluids travelingdown a density gradientrdquo Journal of Chemical Physics vol 132no 22 Article ID 224302 2010
[18] Q D To T T Pham G Lauriat and C Leonard ldquoMoleculardynamics simulations of pressure-driven flows and comparisonwith acceleration-driven flowsrdquo Advances in Mechanical Engi-neering vol 2012 Article ID 580763 10 pages 2012
[19] M Lupkowski and F van Swol ldquoComputer simulation of fluidsinteracting with fluctuating wallsrdquo The Journal of ChemicalPhysics vol 93 no 1 pp 737ndash745 1990
[20] C Huang K Nandakumar P Y K Choi and L W KostiukldquoMolecular dynamics simulation of a pressure-driven liquidtransport process in a cylindrical nanopore using two self-adjusting platesrdquo Journal of Chemical Physics vol 124 no 23Article ID 234701 2006
[21] W W Liou and Y C Fang ldquoImplicit boundary conditions fordirect simulation Monte Carlo method in MEMS flow pre-dictionsrdquo Computer Modeling in Engineering and Sciences vol1 no 4 pp 118ndash128 2000
[22] M Wang and Z Li ldquoSimulations for gas flows in microge-ometries using the direct simulation Monte Carlo methodrdquoInternational Journal of Heat and Fluid Flow vol 25 no 6 pp975ndash985 2004
[23] D J Evans and G P Morriss Statistical Mechanics of Nonequi-librium Liquids Cambridge University Press 2nd edition 2008
[24] A L Garcia and W Wagner ldquoGeneration of the Maxwellianinflow distributionrdquo Journal of Computational Physics vol 217no 2 pp 693ndash708 2006
12 Mathematical Problems in Engineering
[25] G Karniadakis A Beskok and N Aluru Microflows andNanoflows Fundamentals and Simulation Springer New YorkNY USA 2006
[26] E B Arkilic M A Schmidt and K S Breuer ldquoGaseous slipflow in long microchannelsrdquo Journal of MicroelectromechanicalSystems vol 6 no 2 pp 167ndash178 1997
[27] A Beskok G E Karniadakis and W Trimmer ldquoRarefactionand compressibility effects in gas microflowsrdquo Journal of FluidsEngineering vol 118 no 3 pp 448ndash456 1996
[28] A Beskok and G E Karniadakis ldquoReport a model for flows inchannels pipes and ducts at micro and nano scalesrdquoMicroscaleThermophysical Engineering vol 3 no 1 pp 43ndash77 1999
[29] A Ahmadzadegan J Wen and M Renksizbulut ldquoThe role ofthe velocity distribution in the DSMC pressure boundary con-dition for gasmixturesrdquo International Journal ofModern PhysicsC vol 23 no 12 Article ID 1250087 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
8 Mathematical Problems in Engineering
25
2
15
10 025 05 075 1
N-SLinear MD simulation
PP
e
xL
(a)
0015
0
minus00150 025 05 075 1
N-SMD simulation
xL
(Psim
minusP
N-S)P
N-S
(b)
0 025 05 075 1
N-SMD simulation
xL
(PminusP
lin)P
lin
008
006
004
002
0
(c)
Figure 5 Pressure distribution in Pressure-Driven flow Case D (a) comparison of pressure distribution along 119909-axis (b) deviation betweensimulation and N-S results (c) deviation from linear pressure distribution
5
4
3
2
10 025 05 075 1
N-SLinear MD simulation
PP
e
xL
(a)
0 025 05 075 1
N-SMD simulation
xL
(PminusP
lin)P
lin
01
005
0
(b)
002
0
minus0020 025 05 075 1
N-SMD simulation
xL
(Psim
minusP
N-S)P
N-S
(c)
Figure 6 Pressure distribution in Pressure-Driven flow Case E (a) comparison of pressure distribution along 119909-axis (b) deviation betweensimulation and N-S results (c) deviation from linear pressure distribution
0211 (ie the flow was in slip-to-transition regime) In CaseE flow was in the transitional regime with corresponding Knup to 12
Ratio of the slip velocity to the local average velocity isgiven in Figure 8 Knudsen number values in the abscissa cor-respond to local values formed in Figure 7 along the channel
Slip velocities were linearly extrapolated from outermost binssimilar to Couette flow calculations Simulation results werecompared with analytical solutions with first- and second-order slip boundary conditions (16) linearized Boltzmannsolution andDSMC results [28] It is seen that EDMD resultsare in good agreement with DSMC and linearized Boltzmann
Mathematical Problems in Engineering 9
12
09
06
03
0
Case DCase E
0 025 05 075 1
Kn
xL
Figure 7 Local Kn distribution in Pressure-Driven flow
12
09
06
03
0001
N-S 1st orderN-S 2nd orderLinearized Boltzmann
DSMC
Case D
Case E
Case ECase D
01 1 10
Kn
UsU
Figure 8 Normalized slip velocity for slip-to-transition region
solution On the other hand breakdown of slip boundarymodels starts around Kn = 01 Both first- and second-order models overestimate the velocity slip Note that forboth Cases D and E upstream and downstream slip velocityratios are smaller than in other regions of the computationaldomain Use of Maxwell distribution to sample number
density and velocities in pressure boundary treatment isthe main reason of this behaviour Such implementation ofboundary treatment works better when the rarefaction effectsare small For highly rarefied (highKn) flows Ahmadzadeganet al [29] advise usingChapman-Enskog velocity distributionto improve pressure boundary treatment
10 Mathematical Problems in Engineering
1
075
05
025
0
yH
0 025 05 075 1
uUc
Kn = 0115
xL = 0559
N-S 2nd orderMD simulation
1
075
05
025
0
yH
0 025 05 075 1
uUc
xL = 0128
N-S 2nd orderMD simulation
0085Kn =
1
075
05
025
0
yH
0 025 05 075 1
uUc
xL = 0833
N-S 2nd orderMD simulation
0159Kn =
Figure 9 Velocity profiles in 119910-direction of Pressure-Driven flow Case D
1
075
05
025
0
YH
0 025 05 075 1
UUC
XL = 0814
N-S 2nd orderMD simulation
0686Kn =
1
075
05
025
0
YH
0 025 05 075 1
UUC
XL = 0206
N-S 2nd orderMD simulation
1
075
05
025
0
YH
0 025 05 075 1
UUC
XL = 0579
N-S 2nd orderMD simulation
0457Kn =0308Kn =
Figure 10 Velocity profiles in 119910-direction of Pressure-Driven flow Case E
Normalized velocity profiles (119906119880119888where 119880
119888is local
centreline velocity) in cross-stream direction at three loca-tions along the channel are shown in Figures 9 and 10 forCase D and Case E respectively Note that flow velocityalong the channel increases as pressure decreases Thusfor a given position along the channel normalization wascarried out by corresponding centreline velocity Here resultsof EDMD simulations were compared with second-orderanalytical solution In CaseD where Kn is small curves agreewell whereas in Case E estimation of higher slip velocity inanalytical solution becomes more apparent as Kn increases
5 Conclusion
This paper presents an event-driven molecular dynamics(EDMD) simulation of monatomic gas flows in microchan-nels In this study molecules are considered as hard-sphereswith discrete interaction potentials and interaction times areanalytically predictable Therefore each collision is real andtakes place in its calculated position Postcollisional velocitiesare also analytically calculated according to the conservationofmomentumand energy By its deterministic nature EDMDdiffers from DSMC method in which simulated system is
Mathematical Problems in Engineering 11
represented by a smaller group of molecules and collisionpairs are selected stochastically Additionally by using cellpartitioning technique and use of priority queues for eventsorting EDMD is now a strong alternative to DSMCmethodin terms of not only accuracy but also computational speed
In this study an implicit treatment for flow boundaries isadapted from DSMC method to EDMD for the first time inorder to induce Shear- and Pressure-Driven flows in confinedchannels The number of introduced molecules throughupstream and downstream boundaries and their correspond-ing velocities and locations are determined implicitly fromlocal number density mean flow velocity and temperature
Velocity slip in Pressure-Driven flows is compared withDSMCresults and solution of linearizedBoltzmann equationThe good agreement confirms the robustness of the treatmentand its applicability to EDMD simulations For both Shear-and Pressure-Driven flows velocity slip is also comparedwith the one derived from Navier-Stokes equations of slipboundary conditions For low Knudsen number (Kn) flowsvelocity profiles and calculated slip velocities agree well Butas the flow rarefies slip boundarymodels estimate higher slipvelocities
Additionally simulations of twoflows having the sameKnbut different spacing ratios produce almost identical velocityprofiles in Shear-Driven cases for the investigated range ofKn Since simulations with lower spacing ratios run fasterthis finding can be further extended to scale rarefied flowsand to reduce the computational time
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] T-R Hsu MEMS amp Microsystems Design Manufacture andNanoscale Engineering JohnWiley amp Sons Hoboken NJ USA2nd edition 2008
[2] G A BirdMolecular GasDynamics and theDirect Simulation ofGas Flows Clarendon Press Oxford UK 1994
[3] A Rahman ldquoCorrelations in the motion of atoms in liquidargonrdquo Physical Review vol 136 no 2 pp A405ndashA411 1964
[4] L Verlet ldquoComputer lsquoexperimentsrsquo on classical fluids I Ther-modynamical properties of Lennard-Jones moleculesrdquo PhysicalReview vol 159 no 1 pp 98ndash103 1967
[5] MMareschal and B L Holian ldquoFrom fluid particles to physicalparticles computing hydrodynamicsrdquo in Microscopic Simula-tions of Complex Hydrodynamic Phenomena M Mareschal andB LHolian Eds vol 292 ofNATOASI Series pp 1ndash12 SpringerScience amp Business Media 1992
[6] B J Alder and T E Wainwright ldquoPhase transition for a hardsphere systemrdquo The Journal of Chemical Physics vol 27 no 5pp 1208ndash1209 1957
[7] D C Rapaport ldquoThe event scheduling problem in moleculardynamic simulationrdquo Journal of Computational Physics vol 34no 2 pp 184ndash201 1980
[8] M Marın and P Cordero ldquoAn empirical assessment of priorityqueues in event-driven molecular dynamics simulationrdquo Com-puter Physics Communications vol 92 no 2-3 pp 214ndash2241995
[9] I Kandemir A multicell molecular dynamics method [PhDthesis] CaseWesternReserveUniversity ClevelandOhioUSA1999
[10] W T Tang R S M Goh and I L-J Thng ldquoLadder queuean O(1) priority queue structure for large-scale discrete eventsimulationrdquoACMTransactions onModeling and Computer Sim-ulation vol 15 no 3 pp 175ndash204 2005
[11] G Paul ldquoA Complexity O(1) priority queue for event drivenmolecular dynamics simulationsrdquo Journal of ComputationalPhysics vol 221 no 2 pp 615ndash625 2007
[12] A Donev A L Garcia and B J Alder ldquoStochastic event-drivenmolecular dynamicsrdquo Journal of Computational Physics vol 227no 4 pp 2644ndash2665 2008
[13] M N Bannerman R Sargant and L Lue ldquoDynamO a freeO(N) general event-driven molecular dynamics simulatorrdquoJournal of Computational Chemistry vol 32 no 15 pp 3329ndash3338 2011
[14] J Li D Liao and S Yip ldquoCoupling continuum to molecular-dynamics simulation reflecting particle method and the fieldestimatorrdquo Physical Review E vol 57 no 6 pp 7259ndash7267 1998
[15] F Bao YHuang Y Zhang and J Lin ldquoInvestigation of pressure-driven gas flows in nanoscale channels usingmolecular dynam-ics simulationrdquo Microfluidics and Nanofluidics vol 18 no 5-6pp 1075ndash1084 2014
[16] F Bao Y Huang L Qiu and J Lin ldquoApplicability of moleculardynamics method to the pressure-driven gas flow in finitelength nano-scale slit poresrdquo Molecular Physics vol 113 no 6pp 561ndash569 2015
[17] J R Whitman G L Aranovich and M D DonohueldquoAnisotropic mean free path in simulations of fluids travelingdown a density gradientrdquo Journal of Chemical Physics vol 132no 22 Article ID 224302 2010
[18] Q D To T T Pham G Lauriat and C Leonard ldquoMoleculardynamics simulations of pressure-driven flows and comparisonwith acceleration-driven flowsrdquo Advances in Mechanical Engi-neering vol 2012 Article ID 580763 10 pages 2012
[19] M Lupkowski and F van Swol ldquoComputer simulation of fluidsinteracting with fluctuating wallsrdquo The Journal of ChemicalPhysics vol 93 no 1 pp 737ndash745 1990
[20] C Huang K Nandakumar P Y K Choi and L W KostiukldquoMolecular dynamics simulation of a pressure-driven liquidtransport process in a cylindrical nanopore using two self-adjusting platesrdquo Journal of Chemical Physics vol 124 no 23Article ID 234701 2006
[21] W W Liou and Y C Fang ldquoImplicit boundary conditions fordirect simulation Monte Carlo method in MEMS flow pre-dictionsrdquo Computer Modeling in Engineering and Sciences vol1 no 4 pp 118ndash128 2000
[22] M Wang and Z Li ldquoSimulations for gas flows in microge-ometries using the direct simulation Monte Carlo methodrdquoInternational Journal of Heat and Fluid Flow vol 25 no 6 pp975ndash985 2004
[23] D J Evans and G P Morriss Statistical Mechanics of Nonequi-librium Liquids Cambridge University Press 2nd edition 2008
[24] A L Garcia and W Wagner ldquoGeneration of the Maxwellianinflow distributionrdquo Journal of Computational Physics vol 217no 2 pp 693ndash708 2006
12 Mathematical Problems in Engineering
[25] G Karniadakis A Beskok and N Aluru Microflows andNanoflows Fundamentals and Simulation Springer New YorkNY USA 2006
[26] E B Arkilic M A Schmidt and K S Breuer ldquoGaseous slipflow in long microchannelsrdquo Journal of MicroelectromechanicalSystems vol 6 no 2 pp 167ndash178 1997
[27] A Beskok G E Karniadakis and W Trimmer ldquoRarefactionand compressibility effects in gas microflowsrdquo Journal of FluidsEngineering vol 118 no 3 pp 448ndash456 1996
[28] A Beskok and G E Karniadakis ldquoReport a model for flows inchannels pipes and ducts at micro and nano scalesrdquoMicroscaleThermophysical Engineering vol 3 no 1 pp 43ndash77 1999
[29] A Ahmadzadegan J Wen and M Renksizbulut ldquoThe role ofthe velocity distribution in the DSMC pressure boundary con-dition for gasmixturesrdquo International Journal ofModern PhysicsC vol 23 no 12 Article ID 1250087 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 9
12
09
06
03
0
Case DCase E
0 025 05 075 1
Kn
xL
Figure 7 Local Kn distribution in Pressure-Driven flow
12
09
06
03
0001
N-S 1st orderN-S 2nd orderLinearized Boltzmann
DSMC
Case D
Case E
Case ECase D
01 1 10
Kn
UsU
Figure 8 Normalized slip velocity for slip-to-transition region
solution On the other hand breakdown of slip boundarymodels starts around Kn = 01 Both first- and second-order models overestimate the velocity slip Note that forboth Cases D and E upstream and downstream slip velocityratios are smaller than in other regions of the computationaldomain Use of Maxwell distribution to sample number
density and velocities in pressure boundary treatment isthe main reason of this behaviour Such implementation ofboundary treatment works better when the rarefaction effectsare small For highly rarefied (highKn) flows Ahmadzadeganet al [29] advise usingChapman-Enskog velocity distributionto improve pressure boundary treatment
10 Mathematical Problems in Engineering
1
075
05
025
0
yH
0 025 05 075 1
uUc
Kn = 0115
xL = 0559
N-S 2nd orderMD simulation
1
075
05
025
0
yH
0 025 05 075 1
uUc
xL = 0128
N-S 2nd orderMD simulation
0085Kn =
1
075
05
025
0
yH
0 025 05 075 1
uUc
xL = 0833
N-S 2nd orderMD simulation
0159Kn =
Figure 9 Velocity profiles in 119910-direction of Pressure-Driven flow Case D
1
075
05
025
0
YH
0 025 05 075 1
UUC
XL = 0814
N-S 2nd orderMD simulation
0686Kn =
1
075
05
025
0
YH
0 025 05 075 1
UUC
XL = 0206
N-S 2nd orderMD simulation
1
075
05
025
0
YH
0 025 05 075 1
UUC
XL = 0579
N-S 2nd orderMD simulation
0457Kn =0308Kn =
Figure 10 Velocity profiles in 119910-direction of Pressure-Driven flow Case E
Normalized velocity profiles (119906119880119888where 119880
119888is local
centreline velocity) in cross-stream direction at three loca-tions along the channel are shown in Figures 9 and 10 forCase D and Case E respectively Note that flow velocityalong the channel increases as pressure decreases Thusfor a given position along the channel normalization wascarried out by corresponding centreline velocity Here resultsof EDMD simulations were compared with second-orderanalytical solution In CaseD where Kn is small curves agreewell whereas in Case E estimation of higher slip velocity inanalytical solution becomes more apparent as Kn increases
5 Conclusion
This paper presents an event-driven molecular dynamics(EDMD) simulation of monatomic gas flows in microchan-nels In this study molecules are considered as hard-sphereswith discrete interaction potentials and interaction times areanalytically predictable Therefore each collision is real andtakes place in its calculated position Postcollisional velocitiesare also analytically calculated according to the conservationofmomentumand energy By its deterministic nature EDMDdiffers from DSMC method in which simulated system is
Mathematical Problems in Engineering 11
represented by a smaller group of molecules and collisionpairs are selected stochastically Additionally by using cellpartitioning technique and use of priority queues for eventsorting EDMD is now a strong alternative to DSMCmethodin terms of not only accuracy but also computational speed
In this study an implicit treatment for flow boundaries isadapted from DSMC method to EDMD for the first time inorder to induce Shear- and Pressure-Driven flows in confinedchannels The number of introduced molecules throughupstream and downstream boundaries and their correspond-ing velocities and locations are determined implicitly fromlocal number density mean flow velocity and temperature
Velocity slip in Pressure-Driven flows is compared withDSMCresults and solution of linearizedBoltzmann equationThe good agreement confirms the robustness of the treatmentand its applicability to EDMD simulations For both Shear-and Pressure-Driven flows velocity slip is also comparedwith the one derived from Navier-Stokes equations of slipboundary conditions For low Knudsen number (Kn) flowsvelocity profiles and calculated slip velocities agree well Butas the flow rarefies slip boundarymodels estimate higher slipvelocities
Additionally simulations of twoflows having the sameKnbut different spacing ratios produce almost identical velocityprofiles in Shear-Driven cases for the investigated range ofKn Since simulations with lower spacing ratios run fasterthis finding can be further extended to scale rarefied flowsand to reduce the computational time
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] T-R Hsu MEMS amp Microsystems Design Manufacture andNanoscale Engineering JohnWiley amp Sons Hoboken NJ USA2nd edition 2008
[2] G A BirdMolecular GasDynamics and theDirect Simulation ofGas Flows Clarendon Press Oxford UK 1994
[3] A Rahman ldquoCorrelations in the motion of atoms in liquidargonrdquo Physical Review vol 136 no 2 pp A405ndashA411 1964
[4] L Verlet ldquoComputer lsquoexperimentsrsquo on classical fluids I Ther-modynamical properties of Lennard-Jones moleculesrdquo PhysicalReview vol 159 no 1 pp 98ndash103 1967
[5] MMareschal and B L Holian ldquoFrom fluid particles to physicalparticles computing hydrodynamicsrdquo in Microscopic Simula-tions of Complex Hydrodynamic Phenomena M Mareschal andB LHolian Eds vol 292 ofNATOASI Series pp 1ndash12 SpringerScience amp Business Media 1992
[6] B J Alder and T E Wainwright ldquoPhase transition for a hardsphere systemrdquo The Journal of Chemical Physics vol 27 no 5pp 1208ndash1209 1957
[7] D C Rapaport ldquoThe event scheduling problem in moleculardynamic simulationrdquo Journal of Computational Physics vol 34no 2 pp 184ndash201 1980
[8] M Marın and P Cordero ldquoAn empirical assessment of priorityqueues in event-driven molecular dynamics simulationrdquo Com-puter Physics Communications vol 92 no 2-3 pp 214ndash2241995
[9] I Kandemir A multicell molecular dynamics method [PhDthesis] CaseWesternReserveUniversity ClevelandOhioUSA1999
[10] W T Tang R S M Goh and I L-J Thng ldquoLadder queuean O(1) priority queue structure for large-scale discrete eventsimulationrdquoACMTransactions onModeling and Computer Sim-ulation vol 15 no 3 pp 175ndash204 2005
[11] G Paul ldquoA Complexity O(1) priority queue for event drivenmolecular dynamics simulationsrdquo Journal of ComputationalPhysics vol 221 no 2 pp 615ndash625 2007
[12] A Donev A L Garcia and B J Alder ldquoStochastic event-drivenmolecular dynamicsrdquo Journal of Computational Physics vol 227no 4 pp 2644ndash2665 2008
[13] M N Bannerman R Sargant and L Lue ldquoDynamO a freeO(N) general event-driven molecular dynamics simulatorrdquoJournal of Computational Chemistry vol 32 no 15 pp 3329ndash3338 2011
[14] J Li D Liao and S Yip ldquoCoupling continuum to molecular-dynamics simulation reflecting particle method and the fieldestimatorrdquo Physical Review E vol 57 no 6 pp 7259ndash7267 1998
[15] F Bao YHuang Y Zhang and J Lin ldquoInvestigation of pressure-driven gas flows in nanoscale channels usingmolecular dynam-ics simulationrdquo Microfluidics and Nanofluidics vol 18 no 5-6pp 1075ndash1084 2014
[16] F Bao Y Huang L Qiu and J Lin ldquoApplicability of moleculardynamics method to the pressure-driven gas flow in finitelength nano-scale slit poresrdquo Molecular Physics vol 113 no 6pp 561ndash569 2015
[17] J R Whitman G L Aranovich and M D DonohueldquoAnisotropic mean free path in simulations of fluids travelingdown a density gradientrdquo Journal of Chemical Physics vol 132no 22 Article ID 224302 2010
[18] Q D To T T Pham G Lauriat and C Leonard ldquoMoleculardynamics simulations of pressure-driven flows and comparisonwith acceleration-driven flowsrdquo Advances in Mechanical Engi-neering vol 2012 Article ID 580763 10 pages 2012
[19] M Lupkowski and F van Swol ldquoComputer simulation of fluidsinteracting with fluctuating wallsrdquo The Journal of ChemicalPhysics vol 93 no 1 pp 737ndash745 1990
[20] C Huang K Nandakumar P Y K Choi and L W KostiukldquoMolecular dynamics simulation of a pressure-driven liquidtransport process in a cylindrical nanopore using two self-adjusting platesrdquo Journal of Chemical Physics vol 124 no 23Article ID 234701 2006
[21] W W Liou and Y C Fang ldquoImplicit boundary conditions fordirect simulation Monte Carlo method in MEMS flow pre-dictionsrdquo Computer Modeling in Engineering and Sciences vol1 no 4 pp 118ndash128 2000
[22] M Wang and Z Li ldquoSimulations for gas flows in microge-ometries using the direct simulation Monte Carlo methodrdquoInternational Journal of Heat and Fluid Flow vol 25 no 6 pp975ndash985 2004
[23] D J Evans and G P Morriss Statistical Mechanics of Nonequi-librium Liquids Cambridge University Press 2nd edition 2008
[24] A L Garcia and W Wagner ldquoGeneration of the Maxwellianinflow distributionrdquo Journal of Computational Physics vol 217no 2 pp 693ndash708 2006
12 Mathematical Problems in Engineering
[25] G Karniadakis A Beskok and N Aluru Microflows andNanoflows Fundamentals and Simulation Springer New YorkNY USA 2006
[26] E B Arkilic M A Schmidt and K S Breuer ldquoGaseous slipflow in long microchannelsrdquo Journal of MicroelectromechanicalSystems vol 6 no 2 pp 167ndash178 1997
[27] A Beskok G E Karniadakis and W Trimmer ldquoRarefactionand compressibility effects in gas microflowsrdquo Journal of FluidsEngineering vol 118 no 3 pp 448ndash456 1996
[28] A Beskok and G E Karniadakis ldquoReport a model for flows inchannels pipes and ducts at micro and nano scalesrdquoMicroscaleThermophysical Engineering vol 3 no 1 pp 43ndash77 1999
[29] A Ahmadzadegan J Wen and M Renksizbulut ldquoThe role ofthe velocity distribution in the DSMC pressure boundary con-dition for gasmixturesrdquo International Journal ofModern PhysicsC vol 23 no 12 Article ID 1250087 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
10 Mathematical Problems in Engineering
1
075
05
025
0
yH
0 025 05 075 1
uUc
Kn = 0115
xL = 0559
N-S 2nd orderMD simulation
1
075
05
025
0
yH
0 025 05 075 1
uUc
xL = 0128
N-S 2nd orderMD simulation
0085Kn =
1
075
05
025
0
yH
0 025 05 075 1
uUc
xL = 0833
N-S 2nd orderMD simulation
0159Kn =
Figure 9 Velocity profiles in 119910-direction of Pressure-Driven flow Case D
1
075
05
025
0
YH
0 025 05 075 1
UUC
XL = 0814
N-S 2nd orderMD simulation
0686Kn =
1
075
05
025
0
YH
0 025 05 075 1
UUC
XL = 0206
N-S 2nd orderMD simulation
1
075
05
025
0
YH
0 025 05 075 1
UUC
XL = 0579
N-S 2nd orderMD simulation
0457Kn =0308Kn =
Figure 10 Velocity profiles in 119910-direction of Pressure-Driven flow Case E
Normalized velocity profiles (119906119880119888where 119880
119888is local
centreline velocity) in cross-stream direction at three loca-tions along the channel are shown in Figures 9 and 10 forCase D and Case E respectively Note that flow velocityalong the channel increases as pressure decreases Thusfor a given position along the channel normalization wascarried out by corresponding centreline velocity Here resultsof EDMD simulations were compared with second-orderanalytical solution In CaseD where Kn is small curves agreewell whereas in Case E estimation of higher slip velocity inanalytical solution becomes more apparent as Kn increases
5 Conclusion
This paper presents an event-driven molecular dynamics(EDMD) simulation of monatomic gas flows in microchan-nels In this study molecules are considered as hard-sphereswith discrete interaction potentials and interaction times areanalytically predictable Therefore each collision is real andtakes place in its calculated position Postcollisional velocitiesare also analytically calculated according to the conservationofmomentumand energy By its deterministic nature EDMDdiffers from DSMC method in which simulated system is
Mathematical Problems in Engineering 11
represented by a smaller group of molecules and collisionpairs are selected stochastically Additionally by using cellpartitioning technique and use of priority queues for eventsorting EDMD is now a strong alternative to DSMCmethodin terms of not only accuracy but also computational speed
In this study an implicit treatment for flow boundaries isadapted from DSMC method to EDMD for the first time inorder to induce Shear- and Pressure-Driven flows in confinedchannels The number of introduced molecules throughupstream and downstream boundaries and their correspond-ing velocities and locations are determined implicitly fromlocal number density mean flow velocity and temperature
Velocity slip in Pressure-Driven flows is compared withDSMCresults and solution of linearizedBoltzmann equationThe good agreement confirms the robustness of the treatmentand its applicability to EDMD simulations For both Shear-and Pressure-Driven flows velocity slip is also comparedwith the one derived from Navier-Stokes equations of slipboundary conditions For low Knudsen number (Kn) flowsvelocity profiles and calculated slip velocities agree well Butas the flow rarefies slip boundarymodels estimate higher slipvelocities
Additionally simulations of twoflows having the sameKnbut different spacing ratios produce almost identical velocityprofiles in Shear-Driven cases for the investigated range ofKn Since simulations with lower spacing ratios run fasterthis finding can be further extended to scale rarefied flowsand to reduce the computational time
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] T-R Hsu MEMS amp Microsystems Design Manufacture andNanoscale Engineering JohnWiley amp Sons Hoboken NJ USA2nd edition 2008
[2] G A BirdMolecular GasDynamics and theDirect Simulation ofGas Flows Clarendon Press Oxford UK 1994
[3] A Rahman ldquoCorrelations in the motion of atoms in liquidargonrdquo Physical Review vol 136 no 2 pp A405ndashA411 1964
[4] L Verlet ldquoComputer lsquoexperimentsrsquo on classical fluids I Ther-modynamical properties of Lennard-Jones moleculesrdquo PhysicalReview vol 159 no 1 pp 98ndash103 1967
[5] MMareschal and B L Holian ldquoFrom fluid particles to physicalparticles computing hydrodynamicsrdquo in Microscopic Simula-tions of Complex Hydrodynamic Phenomena M Mareschal andB LHolian Eds vol 292 ofNATOASI Series pp 1ndash12 SpringerScience amp Business Media 1992
[6] B J Alder and T E Wainwright ldquoPhase transition for a hardsphere systemrdquo The Journal of Chemical Physics vol 27 no 5pp 1208ndash1209 1957
[7] D C Rapaport ldquoThe event scheduling problem in moleculardynamic simulationrdquo Journal of Computational Physics vol 34no 2 pp 184ndash201 1980
[8] M Marın and P Cordero ldquoAn empirical assessment of priorityqueues in event-driven molecular dynamics simulationrdquo Com-puter Physics Communications vol 92 no 2-3 pp 214ndash2241995
[9] I Kandemir A multicell molecular dynamics method [PhDthesis] CaseWesternReserveUniversity ClevelandOhioUSA1999
[10] W T Tang R S M Goh and I L-J Thng ldquoLadder queuean O(1) priority queue structure for large-scale discrete eventsimulationrdquoACMTransactions onModeling and Computer Sim-ulation vol 15 no 3 pp 175ndash204 2005
[11] G Paul ldquoA Complexity O(1) priority queue for event drivenmolecular dynamics simulationsrdquo Journal of ComputationalPhysics vol 221 no 2 pp 615ndash625 2007
[12] A Donev A L Garcia and B J Alder ldquoStochastic event-drivenmolecular dynamicsrdquo Journal of Computational Physics vol 227no 4 pp 2644ndash2665 2008
[13] M N Bannerman R Sargant and L Lue ldquoDynamO a freeO(N) general event-driven molecular dynamics simulatorrdquoJournal of Computational Chemistry vol 32 no 15 pp 3329ndash3338 2011
[14] J Li D Liao and S Yip ldquoCoupling continuum to molecular-dynamics simulation reflecting particle method and the fieldestimatorrdquo Physical Review E vol 57 no 6 pp 7259ndash7267 1998
[15] F Bao YHuang Y Zhang and J Lin ldquoInvestigation of pressure-driven gas flows in nanoscale channels usingmolecular dynam-ics simulationrdquo Microfluidics and Nanofluidics vol 18 no 5-6pp 1075ndash1084 2014
[16] F Bao Y Huang L Qiu and J Lin ldquoApplicability of moleculardynamics method to the pressure-driven gas flow in finitelength nano-scale slit poresrdquo Molecular Physics vol 113 no 6pp 561ndash569 2015
[17] J R Whitman G L Aranovich and M D DonohueldquoAnisotropic mean free path in simulations of fluids travelingdown a density gradientrdquo Journal of Chemical Physics vol 132no 22 Article ID 224302 2010
[18] Q D To T T Pham G Lauriat and C Leonard ldquoMoleculardynamics simulations of pressure-driven flows and comparisonwith acceleration-driven flowsrdquo Advances in Mechanical Engi-neering vol 2012 Article ID 580763 10 pages 2012
[19] M Lupkowski and F van Swol ldquoComputer simulation of fluidsinteracting with fluctuating wallsrdquo The Journal of ChemicalPhysics vol 93 no 1 pp 737ndash745 1990
[20] C Huang K Nandakumar P Y K Choi and L W KostiukldquoMolecular dynamics simulation of a pressure-driven liquidtransport process in a cylindrical nanopore using two self-adjusting platesrdquo Journal of Chemical Physics vol 124 no 23Article ID 234701 2006
[21] W W Liou and Y C Fang ldquoImplicit boundary conditions fordirect simulation Monte Carlo method in MEMS flow pre-dictionsrdquo Computer Modeling in Engineering and Sciences vol1 no 4 pp 118ndash128 2000
[22] M Wang and Z Li ldquoSimulations for gas flows in microge-ometries using the direct simulation Monte Carlo methodrdquoInternational Journal of Heat and Fluid Flow vol 25 no 6 pp975ndash985 2004
[23] D J Evans and G P Morriss Statistical Mechanics of Nonequi-librium Liquids Cambridge University Press 2nd edition 2008
[24] A L Garcia and W Wagner ldquoGeneration of the Maxwellianinflow distributionrdquo Journal of Computational Physics vol 217no 2 pp 693ndash708 2006
12 Mathematical Problems in Engineering
[25] G Karniadakis A Beskok and N Aluru Microflows andNanoflows Fundamentals and Simulation Springer New YorkNY USA 2006
[26] E B Arkilic M A Schmidt and K S Breuer ldquoGaseous slipflow in long microchannelsrdquo Journal of MicroelectromechanicalSystems vol 6 no 2 pp 167ndash178 1997
[27] A Beskok G E Karniadakis and W Trimmer ldquoRarefactionand compressibility effects in gas microflowsrdquo Journal of FluidsEngineering vol 118 no 3 pp 448ndash456 1996
[28] A Beskok and G E Karniadakis ldquoReport a model for flows inchannels pipes and ducts at micro and nano scalesrdquoMicroscaleThermophysical Engineering vol 3 no 1 pp 43ndash77 1999
[29] A Ahmadzadegan J Wen and M Renksizbulut ldquoThe role ofthe velocity distribution in the DSMC pressure boundary con-dition for gasmixturesrdquo International Journal ofModern PhysicsC vol 23 no 12 Article ID 1250087 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Mathematical Problems in Engineering 11
represented by a smaller group of molecules and collisionpairs are selected stochastically Additionally by using cellpartitioning technique and use of priority queues for eventsorting EDMD is now a strong alternative to DSMCmethodin terms of not only accuracy but also computational speed
In this study an implicit treatment for flow boundaries isadapted from DSMC method to EDMD for the first time inorder to induce Shear- and Pressure-Driven flows in confinedchannels The number of introduced molecules throughupstream and downstream boundaries and their correspond-ing velocities and locations are determined implicitly fromlocal number density mean flow velocity and temperature
Velocity slip in Pressure-Driven flows is compared withDSMCresults and solution of linearizedBoltzmann equationThe good agreement confirms the robustness of the treatmentand its applicability to EDMD simulations For both Shear-and Pressure-Driven flows velocity slip is also comparedwith the one derived from Navier-Stokes equations of slipboundary conditions For low Knudsen number (Kn) flowsvelocity profiles and calculated slip velocities agree well Butas the flow rarefies slip boundarymodels estimate higher slipvelocities
Additionally simulations of twoflows having the sameKnbut different spacing ratios produce almost identical velocityprofiles in Shear-Driven cases for the investigated range ofKn Since simulations with lower spacing ratios run fasterthis finding can be further extended to scale rarefied flowsand to reduce the computational time
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
References
[1] T-R Hsu MEMS amp Microsystems Design Manufacture andNanoscale Engineering JohnWiley amp Sons Hoboken NJ USA2nd edition 2008
[2] G A BirdMolecular GasDynamics and theDirect Simulation ofGas Flows Clarendon Press Oxford UK 1994
[3] A Rahman ldquoCorrelations in the motion of atoms in liquidargonrdquo Physical Review vol 136 no 2 pp A405ndashA411 1964
[4] L Verlet ldquoComputer lsquoexperimentsrsquo on classical fluids I Ther-modynamical properties of Lennard-Jones moleculesrdquo PhysicalReview vol 159 no 1 pp 98ndash103 1967
[5] MMareschal and B L Holian ldquoFrom fluid particles to physicalparticles computing hydrodynamicsrdquo in Microscopic Simula-tions of Complex Hydrodynamic Phenomena M Mareschal andB LHolian Eds vol 292 ofNATOASI Series pp 1ndash12 SpringerScience amp Business Media 1992
[6] B J Alder and T E Wainwright ldquoPhase transition for a hardsphere systemrdquo The Journal of Chemical Physics vol 27 no 5pp 1208ndash1209 1957
[7] D C Rapaport ldquoThe event scheduling problem in moleculardynamic simulationrdquo Journal of Computational Physics vol 34no 2 pp 184ndash201 1980
[8] M Marın and P Cordero ldquoAn empirical assessment of priorityqueues in event-driven molecular dynamics simulationrdquo Com-puter Physics Communications vol 92 no 2-3 pp 214ndash2241995
[9] I Kandemir A multicell molecular dynamics method [PhDthesis] CaseWesternReserveUniversity ClevelandOhioUSA1999
[10] W T Tang R S M Goh and I L-J Thng ldquoLadder queuean O(1) priority queue structure for large-scale discrete eventsimulationrdquoACMTransactions onModeling and Computer Sim-ulation vol 15 no 3 pp 175ndash204 2005
[11] G Paul ldquoA Complexity O(1) priority queue for event drivenmolecular dynamics simulationsrdquo Journal of ComputationalPhysics vol 221 no 2 pp 615ndash625 2007
[12] A Donev A L Garcia and B J Alder ldquoStochastic event-drivenmolecular dynamicsrdquo Journal of Computational Physics vol 227no 4 pp 2644ndash2665 2008
[13] M N Bannerman R Sargant and L Lue ldquoDynamO a freeO(N) general event-driven molecular dynamics simulatorrdquoJournal of Computational Chemistry vol 32 no 15 pp 3329ndash3338 2011
[14] J Li D Liao and S Yip ldquoCoupling continuum to molecular-dynamics simulation reflecting particle method and the fieldestimatorrdquo Physical Review E vol 57 no 6 pp 7259ndash7267 1998
[15] F Bao YHuang Y Zhang and J Lin ldquoInvestigation of pressure-driven gas flows in nanoscale channels usingmolecular dynam-ics simulationrdquo Microfluidics and Nanofluidics vol 18 no 5-6pp 1075ndash1084 2014
[16] F Bao Y Huang L Qiu and J Lin ldquoApplicability of moleculardynamics method to the pressure-driven gas flow in finitelength nano-scale slit poresrdquo Molecular Physics vol 113 no 6pp 561ndash569 2015
[17] J R Whitman G L Aranovich and M D DonohueldquoAnisotropic mean free path in simulations of fluids travelingdown a density gradientrdquo Journal of Chemical Physics vol 132no 22 Article ID 224302 2010
[18] Q D To T T Pham G Lauriat and C Leonard ldquoMoleculardynamics simulations of pressure-driven flows and comparisonwith acceleration-driven flowsrdquo Advances in Mechanical Engi-neering vol 2012 Article ID 580763 10 pages 2012
[19] M Lupkowski and F van Swol ldquoComputer simulation of fluidsinteracting with fluctuating wallsrdquo The Journal of ChemicalPhysics vol 93 no 1 pp 737ndash745 1990
[20] C Huang K Nandakumar P Y K Choi and L W KostiukldquoMolecular dynamics simulation of a pressure-driven liquidtransport process in a cylindrical nanopore using two self-adjusting platesrdquo Journal of Chemical Physics vol 124 no 23Article ID 234701 2006
[21] W W Liou and Y C Fang ldquoImplicit boundary conditions fordirect simulation Monte Carlo method in MEMS flow pre-dictionsrdquo Computer Modeling in Engineering and Sciences vol1 no 4 pp 118ndash128 2000
[22] M Wang and Z Li ldquoSimulations for gas flows in microge-ometries using the direct simulation Monte Carlo methodrdquoInternational Journal of Heat and Fluid Flow vol 25 no 6 pp975ndash985 2004
[23] D J Evans and G P Morriss Statistical Mechanics of Nonequi-librium Liquids Cambridge University Press 2nd edition 2008
[24] A L Garcia and W Wagner ldquoGeneration of the Maxwellianinflow distributionrdquo Journal of Computational Physics vol 217no 2 pp 693ndash708 2006
12 Mathematical Problems in Engineering
[25] G Karniadakis A Beskok and N Aluru Microflows andNanoflows Fundamentals and Simulation Springer New YorkNY USA 2006
[26] E B Arkilic M A Schmidt and K S Breuer ldquoGaseous slipflow in long microchannelsrdquo Journal of MicroelectromechanicalSystems vol 6 no 2 pp 167ndash178 1997
[27] A Beskok G E Karniadakis and W Trimmer ldquoRarefactionand compressibility effects in gas microflowsrdquo Journal of FluidsEngineering vol 118 no 3 pp 448ndash456 1996
[28] A Beskok and G E Karniadakis ldquoReport a model for flows inchannels pipes and ducts at micro and nano scalesrdquoMicroscaleThermophysical Engineering vol 3 no 1 pp 43ndash77 1999
[29] A Ahmadzadegan J Wen and M Renksizbulut ldquoThe role ofthe velocity distribution in the DSMC pressure boundary con-dition for gasmixturesrdquo International Journal ofModern PhysicsC vol 23 no 12 Article ID 1250087 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
12 Mathematical Problems in Engineering
[25] G Karniadakis A Beskok and N Aluru Microflows andNanoflows Fundamentals and Simulation Springer New YorkNY USA 2006
[26] E B Arkilic M A Schmidt and K S Breuer ldquoGaseous slipflow in long microchannelsrdquo Journal of MicroelectromechanicalSystems vol 6 no 2 pp 167ndash178 1997
[27] A Beskok G E Karniadakis and W Trimmer ldquoRarefactionand compressibility effects in gas microflowsrdquo Journal of FluidsEngineering vol 118 no 3 pp 448ndash456 1996
[28] A Beskok and G E Karniadakis ldquoReport a model for flows inchannels pipes and ducts at micro and nano scalesrdquoMicroscaleThermophysical Engineering vol 3 no 1 pp 43ndash77 1999
[29] A Ahmadzadegan J Wen and M Renksizbulut ldquoThe role ofthe velocity distribution in the DSMC pressure boundary con-dition for gasmixturesrdquo International Journal ofModern PhysicsC vol 23 no 12 Article ID 1250087 2012
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical Problems in Engineering
Hindawi Publishing Corporationhttpwwwhindawicom
Differential EquationsInternational Journal of
Volume 2014
Applied MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Mathematical PhysicsAdvances in
Complex AnalysisJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Operations ResearchAdvances in
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of Mathematics and Mathematical Sciences
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Algebra
Discrete Dynamics in Nature and Society
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Decision SciencesAdvances in
Discrete MathematicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom
Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Stochastic AnalysisInternational Journal of
