Mec. Ind. (2001) 2, 313–341 2001 Éditions scientifiques et médicales Elsevier SAS. All rights reservedS1296-2139(01)01112-5/REV

Review

Flow physics in MEMS

Mohamed Gad-el-HakDepartment of Aerospace & Mechanical Engineering, University of Notre Dame, Notre Dame, IN 46556, USA

(Received 1 March 2001; accepted 4 May 2001)

Abstract—Interest in microelectromechanical systems (MEMS) has experienced explosive growth during the past few years. Suchsmall devices typically have characteristic size ranging from 1 mm down to 1 micron, and may include sensors, actuators, motors,pumps, turbines, gears, ducts and valves. Microdevices often involve mass, momentum and energy transport. Modeling gas and liquidflows through MEMS may necessitate including slip, rarefaction, compressibility, intermolecular forces and other unconventionaleffects. In this article, I shall provide a methodical approach to flow modeling for a broad variety of microdevices. The continuum-based Navier–Stokes equations—with either the traditional no-slip or slip-flow boundary conditions—work only for a limited rangeof Knudsen numbers above which alternative models must be sought. These include molecular dynamics (MD), Boltzmann equation,Direct Simulation Monte Carlo (DSMC), and other deterministic/probabilistic molecular models. The present paper will broadly surveyavailable methodologies to model and compute transport phenomena within microdevices. 2001 Éditions scientifiques et médicalesElsevier SASflow / microelectromechanical devices

Résumé—Physique des écoulements dans les MEMS. L’intérêt dans les systèmes microélectromécaniques a expérimenté unecroissance explosive pendant les dernières années. Ces très petits systèmes ont des dimensions caractéristiques comprises entre 1mm et 1 micron, peuvent incorporer des capteurs, des moteurs, des pompes, des turbines, des roulements, des conduites et desvalves. Elles font intervenir souvent des phénomènes de transport de masse, quantité de mouvement et énergie. La modélisationde l’écoulement de gaz et des liquides au travers de MEMS peut nécessiter d’incorporer le glissement à la paroi, la raréfaction, lacompressibilité, les forces intermoléculaires et d’autres effets non conventionnels. Dans cet article je me propose de fournir uneapproche méthodique de la modélisation des écoulements dans une large variété de microsystèmes. Les équations de Navier–Stokesappropriées aux milieux continus, avec soit la condition limite d’adhérence traditionnelle ou soit la condition de glissement, estappropriée dans une gamme limitée de nombres de Knudsen au-delà desquels d’autres modèles doivent être utilisés. Ceux-ci incluentla dynamique moléculaire (Molecular dynamics, MD), l’équation de Boltzmann, la Direct Simulation Monte Carlo (DSMC), et d’autresmodèles moléculaires deterministes/probabilistes. Cet article se propose de donner un large aperçu des methodologies disponiblespour modeliser et calculer les phenomènes de transport dans les microsystèmes. 2001 Éditions scientifiques et médicales ElsevierSASécoulements / systèmes microélectromécaniques

1. INTRODUCTION

Manufacturing processes that can create extremelysmall machines have been developed in recent years(Angell et al. [1]; Gabriel et al. [2, 3]; O’Connor [4];Gravesen et al. [5]; Bryzek et al. [6]; Gabriel [7]; Ash-ley [8]; Ho and Tai [9, 10]; Hogan [11]; Ouellette [12];Paula [13]; Robinson et al. [14, 15]; Madou [16];Tien [17]; Amato [18]; Busch-Vishniac [19]; Kovacs[20]; Knight [21]; Epstein [22]; O’Connor and Hutchin-son [23]; Goldin et al. [24]; Chalmers [25]). Electrostatic,magnetic, electromagnetic, pneumatic and thermal actu-ators, motors, valves, gears, cantilevers, diaphragms andtweezers of less than 100-µm size have been fabricated.

E-mail address:gadelhak@nd.edu (M. Gad-el-Hak).

These have been used as sensors for pressure, tempera-ture, mass flow, velocity, sound and chemical composi-tion, as actuators for linear and angular motions, and assimple components for complex systems such as robots,micro-heat-engines and micro-heat-pumps (Lipkin [26];Garcia and Sniegowski [27, 28]; Sniegowski and Gar-cia [29]; Epstein and Senturia [30]; Epstein et al. [31]).

Microelectromechanical systems (MEMS) refer to de-vices that have characteristic length of less than 1 mmbut more than 1 micron, that combine electrical and me-chanical components, and that are fabricated using inte-grated circuit batch-processing technologies. The booksby Madou [32], Kovacs [33] and Gad-el-Hak [34] pro-vide excellent sources for micro science and technology.Current manufacturing techniques for MEMS includesurface silicon micromachining; bulk silicon microma-

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M. Gad-el-Hak

chining; lithography, electrodeposition and plastic mold-ing (or, in its original German, lithographie galvanofor-mung abformung, LIGA); and electrodischarge machin-ing (EDM). MEMS are more than four orders of magni-tude larger than the diameter of the hydrogen atom, butabout four orders of magnitude smaller than the tradi-tional man-made artifacts. Microdevices can have char-acteristics length smaller than the diameter of a humanhair. Nanodevices (some say NEMS) further push the en-velope of electromechanical miniaturization (Roco [35]).

The famed physicist Richard P. Feynman delivered amere two but profound lectures1 on electromechanicalminiaturization: “There’s Plenty of Room at the Bottom”,presented at the annual meeting of the American PhysicalSociety, Pasadena, California, 29 December 1959, and“Infinitesimal Machinery”, presented at the Jet Propul-sion Laboratory on 23 February 1983. He could not see alot of use for micromachines lamenting in 1959 “[Smallbut movable machines] may or may not be useful, butthey surely would be fun to make”, and 24 years later“There is no use for these machines, so I still don’t under-stand why I’m fascinated by the question of making smallmachines with movable and controllable parts”. DespiteFeynman’s demurring regarding the usefulness of smallmachines, MEMS are finding increased applications ina variety of industrial and medical fields, with a poten-tial worldwide market in the billions of dollars ($30 bil-lion by 2004). Accelerometers for automobile airbags,keyless entry systems, dense arrays of micromirrors forhigh-definition optical displays, scanning electron micro-scope tips to image single atoms, micro-heat-exchangersfor cooling of electronic circuits, reactors for separatingbiological cells, blood analyzers and pressure sensors forcatheter tips are but a few of current usage. Microductsare used in infrared detectors, diode lasers, miniature gaschromatographs and high-frequency fluidic control sys-tems. Micropumps are used for ink jet printing, environ-mental testing and electronic cooling. Potential medicalapplications for small pumps include controlled deliveryand monitoring of minute amount of medication, manu-facturing of nanoliters of chemicals and development ofartificial pancreas.

1.1. Present article

In the present article, we review the status of our un-derstanding of fluid flow physics particular to microde-

1 Both talks have been reprinted in the Journal of Microelectro-mechanical Systems 1 (1) (1992) 60–66 and 2 (1) (1993) 4–14. A paperbased on the first talk was published in the book by Gilbert [36].

vices. The paper is an update of the earlier publication bythe same author (Gad-el-Hak [37]). The coverage in hereis broad leaving the details to other articles in this spe-cial volume ofMécanique et Industries. Not all MEMSdevices involve fluid flows of course, but the presentarticle will focus on the ones that do. Microducts, mi-cronozzles, micropumps, microturbines and microvalvesare examples of small devices involving the flow ofliquids and gases. MEMS can also be related to fluidflows in an indirect way. The availability of inexpensive,batch-processing-produced microsensors and microactu-ators provides opportunities for targeting small-scale co-herent structures in macroscopic turbulent shear flows.Flow control using MEMS promises a quantum leap incontrol system performance (Gad-el-Hak [38]). Addi-tionally, the extremely small sensors made possible bymicrofabrication technology allow measurements withspatial and temporal resolutions not achievable before.For example, high-Reynolds-number turbulent flow di-agnosis are now feasible down to the Kolmogorov scales(Löfdahl and Gad-el-Hak [39]).

2. FLOW PHYSICS

The rapid progress in fabricating and utilizing micro-electromechanical systems during the last decade has notbeen matched by corresponding advances in our under-standing of the unconventional physics involved in themanufacture and operation of small devices (Madou [32];Kovacs [33]; Knight [21]). Providing such understandingis crucial to designing, optimizing, fabricating and utiliz-ing improved MEMS devices. The present paper focuseson the physics of fluid flows in microdevices.

Fluid flows in small devices differ from those inmacroscopic machines. The operation of MEMS-basedducts, nozzles, valves, bearings, turbomachines, etc., can-not always be predicted from conventional flow modelssuch as the Navier–Stokes equations with no-slip bound-ary condition at a fluid–solid interface, as routinely andsuccessfully applied for larger flow devices. Many ques-tions have been raised when the results of experimentswith microdevices could not be explained via traditionalflow modeling. The pressure gradient in a long microd-uct was observed to be non-constant and the measuredflowrate was higher than that predicted from the con-ventional continuum flow model. Load capacities of mi-crobearings were diminished and electric currents neededto move micromotors were extraordinarily high. The dy-namic response of micromachined accelerometers oper-ating at atmospheric conditions was observed to be over-damped.

314

Flow physics in MEMS

In the early stages of development of this excitingnew field, the objective was to build MEMS devicesas productively as possible. Microsensors were readingsomething, but not many researchers seemed to knowexactly what. Microactuators were moving, but conven-tional modeling could not precisely predict their motion.After a decade of unprecedented progress in MEMS tech-nology, perhaps the time is now ripe to take stock, slowdown a bit and answer the many questions that arose.The ultimate aim of this long-term exercise is to achieverational-design capability for useful microdevices and tobe able to characterize definitively and with as little em-piricism as possible the operations of microsensors andmicroactuators.

In dealing with fluid flow through microdevices, oneis faced with the question of which model to use,which boundary condition to apply and how to proceedto obtain solutions to the problem at hand. Obviouslysurface effects dominate in small devices. The surface-to-volume ratio for a machine with a characteristiclength of 1 m is 1 m−1, while that for a MEMSdevice having a size of 1 µm is 106 m−1. The million-fold increase in surface area relative to the mass ofthe minute device substantially affects the transport ofmass, momentum and energy through the surface. Thesmall length-scale of microdevices may invalidate thecontinuum approximation altogether. Slip flow, thermalcreep, rarefaction, viscous dissipation, compressibility,intermolecular forces and other unconventional effectsmay have to be taken into account, preferably using onlyfirst principles such as conservation of mass, Newton’ssecond law, conservation of energy, etc.

In this article, we discuss continuum as well asmolecular-based flow models, and the choices to bemade. Computing typical Reynolds, Mach and Knudsennumbers for the flow through a particular device is a goodstart to characterize the flow. For gases, microfluid me-chanics has been studied by incorporating slip bound-ary conditions, thermal creep, viscous dissipation as wellas compressibility effects into the continuum equationsof motion. Molecular-based models have also been at-tempted for certain ranges of the operating parameters.Use is made of the well-developed kinetic theory ofgases, embodied in the Boltzmann equation, and directsimulation methods such as Monte Carlo. Microfluid me-chanics of liquids is more complicated. The moleculesare much more closely packed at normal pressures andtemperatures, and the attractive or cohesive potential be-tween the liquid molecules as well as between the liq-uid and solid ones plays a dominant role if the charac-teristic length of the flow is sufficiently small. In caseswhen the traditional continuum model fails to provide

Figure 1. Molecular and continuum flow models.

accurate predictions or postdictions, expensive moleculardynamics simulations seem to be the only first-principleapproach available to rationally characterize liquid flowsin microdevices. Such simulations are not yet feasible forrealistic flow extent or number of molecules. As a conse-quence, the microfluid mechanics of liquids is much lessdeveloped than that for gases.

3. FLUID MODELING

There are basically two ways of modeling a flow-field. Either as the fluid really is — a collection of mole-cules — or as a continuum where the matter is assumedcontinuous and indefinitely divisible. The former model-ing is subdivided into deterministic methods and prob-abilistic ones, while in the latter approach the velocity,density, pressure, etc., are defined at every point in spaceand time, and conservation of mass, energy and momen-tum lead to a set of nonlinear partial differential equa-tions (Euler, Navier–Stokes, Burnett, etc.). Fluid model-ing classification is depicted schematically infigure 1.

The continuum model, embodied in the Navier–Stokesequations, is applicable to numerous flow situations. Themodel ignores the molecular nature of gases and liquidsand regards the fluid as a continuous medium describ-able in terms of the spatial and temporal variations ofdensity, velocity, pressure, temperature and other macro-scopic flow quantities. For dilute gas flows near equilib-rium, the Navier–Stokes equations are derivable from themolecularly-based Boltzmann equation, but can also bederived independently of that for both liquids and gases.In the case of direct derivation, some empiricism is nec-essary to close the resulting indeterminate set of equa-tions. The continuum model is easier to handle mathe-matically (and is also more familiar to most fluid dynam-

315

M. Gad-el-Hak

icists) than the alternative molecular models. Continuummodels should therefore be used as long as they are ap-plicable. Thus, careful considerations of the validity ofthe Navier–Stokes equations and the like are in order.

Basically, the continuum model leads to fairly accu-rate predictions as long as local properties such as den-sity and velocity can be defined as averages over ele-ments large compared with the microscopic structure ofthe fluid but small enough in comparison with the scale ofthe macroscopic phenomena to permit the use of differen-tial calculus to describe them. Additionally, the flow mustnot be too far from thermodynamic equilibrium. The for-mer condition is almost always satisfied, but it is the lat-ter which usually restricts the validity of the continuumequations. As will be seen in the following section, thecontinuum flow equations do not form a determinate set.The shear stress and heat flux must be expressed in termsof lower-order macroscopic quantities such as velocityand temperature, and the simplest (i.e. linear) relationsare valid only when the flow is near thermodynamic equi-librium. Worse yet, the traditional no-slip boundary con-dition at a solid–fluid interface breaks down even beforethe linear stress–strain relation becomes invalid.

To be more specific, we temporarily restrict the dis-cussion to gases where the concept of mean free path iswell defined. Liquids are more problematic and we de-fer their discussion to a later section. For gases, the meanfree pathL is the average distance traveled by moleculesbetween collisions. For an ideal gas modeled as rigidspheres, the mean free path is related to temperatureT

and pressurep as follows:

L= 1√2πnσ 2

= kT√2πpσ 2

(1)

wheren is the number density (number of molecules perunit volume),σ is the molecular diameter, andk is theBoltzmann constant (1.38·10−23 J·K−1·molecule−1).

The continuum model is valid whenL is much smallerthan a characteristic flow dimensionL. As this conditionis violated, the flow is no longer near equilibrium and thelinear relation between stress and rate of strain and theno-slip velocity condition are no longer valid. Similarly,the linear relation between heat flux and temperaturegradient and the no-jump temperature condition at asolid–fluid interface are no longer accurate whenL is notmuch smaller thanL.

The length-scaleL can be some overall dimension ofthe flow, but a more precise choice is the scale of thegradient of a macroscopic quantity, as for example the

densityρ,

L= ρ

|∂ρ/∂y| (2)

The ratio between the mean free path and the character-istic length is known as the Knudsen number

Kn = L

L(3)

and generally the traditional continuum approach is valid,albeit with modified boundary conditions, as long asKn< 0.1.

There are two more important dimensionless parame-ters in fluid mechanics, and the Knudsen number can beexpressed in terms of those two. The Reynolds number isthe ratio of inertial forces to viscous ones

Re= v0L

ν(4)

wherev0 is a characteristic velocity, andν is the kine-matic viscosity of the fluid. The Mach number is the ratioof flow velocity to the speed of sound

Ma = v0

a0(5)

The Mach number is a dynamic measure of fluid com-pressibility and may be considered as the ratio of inertialforces to elastic ones. From the kinetic theory of gases,the mean free path is related to the viscosity as follows

ν = µ

ρ= 1

2L vm (6)

whereµ is the dynamic viscosity, andvm is the meanmolecular speed which is somewhat higher than thesound speeda0,

vm =√

8

πγa0 (7)

where γ is the specific heat ratio (i.e. the isentropicexponent). Combining equations (3)–(7), we reach therequired relation

Kn =√πγ

2

Ma

Re(8)

In boundary layers, the relevant length-scale is theshear-layer thicknessδ, and for laminar flows

δ

L∼ 1√

Re(9)

Kn∼ Ma

Reδ∼ Ma√

Re(10)

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Flow physics in MEMS

Figure 2. Knudsen number regimes.

whereReδ is the Reynolds number based on the free-stream velocityv0 and the boundary layer thicknessδ,andReis based onv0 and the streamwise length-scaleL.

Rarefied gas flows are in general encountered in flowsin small geometries such as MEMS devices and in low-pressure applications such as high-altitude flying andhigh-vacuum gadgets. The local value of Knudsen num-ber in a particular flow determines the degree of rarefac-tion and the degree of validity of the continuum model.The different Knudsen number regimes are determinedempirically and are therefore only approximate for a par-ticular flow geometry. The pioneering experiments in rar-efied gas dynamics were conducted by Knudsen in 1909.In the limit of zero Knudsen number, the transport termsin the continuum momentum and energy equations arenegligible and the Navier–Stokes equations then reduceto the inviscid Euler equations. Both heat conductionand viscous diffusion and dissipation are negligible, andthe flow is then approximately isentropic (i.e. adiabaticand reversible) from the continuum viewpoint while theequivalent molecular viewpoint is that the velocity dis-tribution function is everywhere of the local equilibriumor Maxwellian form. AsKn increases, rarefaction effectsbecome more important, and eventually the continuumapproach breaks down altogether. The different Knudsennumber regimes are depicted infigure 2, and can be sum-marized as follows:

• Euler equations (neglect molecular diffusion):

Kn → 0 (Re→ ∞)

• Navier–Stokes equations with no-slip boundary condi-tions:

Kn< 10−3

• Navier–Stokes equations with slip boundary condi-tions:

10−3 ≤ Kn< 10−1

• Transition regime:

10−1 ≤ Kn< 10

• Free-molecule flow:

Kn≥ 10

As an example, consider air at standard temperature(T = 288 K) and pressure (p = 1.01·105 N·m−2). A cubeone micron on a side contains 2.54·107 molecules sepa-rated by an average distance of 0.0034 µm. The gas isconsidered dilute if the ratio of this distance to the mole-cular diameter exceeds 7, and in the present examplethis ratio is 9, barely satisfying the dilute gas assump-tion. The mean free path computed from equation (1) isL = 0.065 µm. A microdevice with characteristic lengthof 1 µm would haveKn = 0.065, which is in the slip-flow regime. At lower pressures, the Knudsen number in-creases. For example, if the pressure is 0.1 atm and thetemperature remains the same,Kn = 0.65 for the same1 µm device, and the flow is then in the transition regime.There would still be over 2 million molecules in the sameone-micron cube, and the average distance between themwould be 0.0074 µm. The same device at 100 km altitudewould haveKn = 3·104, well into the free-molecule flowregime. Knudsen number for the flow of a light gas likehelium is about 3 times larger than that for air flow atotherwise the same conditions.

Consider a long microchannel where the entrancepressure is atmospheric and the exit conditions are near

317

M. Gad-el-Hak

vacuum. As air goes down the duct, the pressure anddensity decrease while the velocity, Mach number andKnudsen number increase. The pressure drops to over-come viscous forces in the channel. If isothermal con-ditions prevail,2 density also drops and conservation ofmass requires the flow to accelerate down the constant-area tube. The fluid acceleration in turn affects the pres-sure gradient, resulting in a nonlinear pressure drop alongthe channel. The Mach number increases down the tube,limited only by choked-flow conditionMa = 1. Addition-ally, the normal component of velocity is no longer zero.With lower density, the mean free path increases andKncorrespondingly increases. All flow regimes depicted infigure 2 may occur in the same tube: continuum withno-slip boundary conditions, slip-flow regime, transitionregime and free-molecule flow. The air flow may alsochange from incompressible to compressible as it movesdown the microduct. A similar scenario may take place ifthe entrance pressure is, say, 5 atm, while the exit is at-mospheric. This deceivingly simple duct flow may in factmanifest every single complexity discussed in this sec-tion. In the following six sections, we discuss in turn theNavier–Stokes equations, compressibility effects, bound-ary conditions, molecular-based models, liquid flows andsurface phenomena.

4. CONTINUUM MODEL

We recall in this section the traditional conservationrelations in fluid mechanics. A concise derivation ofthese equations can be found in Gad-el-Hak [38]. Inhere, we re-emphasize the precise assumptions needed toobtain a particular form of the equations. A continuumfluid implies that the derivatives of all the dependentvariables exist in some reasonable sense. In other words,local properties such as density and velocity are definedas averages over elements large compared with themicroscopic structure of the fluid but small enough incomparison with the scale of the macroscopic phenomenato permit the use of differential calculus to describe them.As mentioned earlier, such conditions are almost alwaysmet. For such fluids, and assuming the laws of non-relativistic mechanics hold, the conservation of mass,momentum and energy can be expressed at every pointin space and time as a set of partial differential equationsas follows

2 More likely the flow will be somewhere in between isothermaland adiabatic, Fanno flow. In that case both density and temperaturedecrease downstream, the former not as fast as in the isothermal case.None of that changes the qualitative arguments made in the example.

∂ρ

∂t+ ∂

∂xk(ρuk)= 0 (11)

ρ

(∂ui

∂t+ uk ∂ui

∂xk

)= ∂Σki

∂xk+ ρgi (12)

ρ

(∂e

∂t+ uk ∂e

∂xk

)= −∂qk

∂xk+Σki ∂ui

∂xk(13)

where ρ is the fluid density,uk is an instantaneousvelocity component(u, v,w), Σki is the second-orderstress tensor (surface force per unit area),gi is the bodyforce per unit mass,e is the internal energy, andqk is thesum of heat flux vectors due to conduction and radiation.The independent variables are timet and the three spatialcoordinatesx1, x2 andx3 or (x, y, z).

Equations (11)–(13) constitute 5 differential equationsfor the 17 unknownsρ, ui , Σki , e, andqk . Absent anybody couples, the stress tensor is symmetric having onlysix independent components, which reduces the numberof unknowns to 14. Obviously, the continuum flow equa-tions do not form a determinate set. To close the conser-vation equations, relation between the stress tensor anddeformation rate, relation between the heat flux vectorand the temperature field and appropriate equations ofstate relating the different thermodynamic properties areneeded. The stress–rate of strain relation and the heatflux–temperature gradient relation are approximately lin-ear if the flow is not too far from thermodynamic equi-librium. This is a phenomenological result but can be rig-orously derived from the Boltzmann equation for a dilutegas assuming the flow is near equilibrium. For a New-tonian, isotropic, Fourier, ideal gas, for example, thoserelations read

Σki = −pδki +µ(∂ui

∂xk+ ∂uk

∂xi

)+ λ

(∂uj

∂xj

)δki (14)

qi = −κ ∂T∂xi

+ heat flux due to radiation (15)

de= cv dT and p = ρ RT (16)

wherep is the thermodynamic pressure,µ andλ are thefirst and second coefficients of viscosity, respectively,δkiis the unit second-order tensor (Kronecker delta),κ is thethermal conductivity,T is the temperature field,cv is thespecific heat at constant volume, andR is the gas constantwhich is given by the Boltzmann constant divided by themass of an individual moleculek = mR. The Stokes’hypothesis relates the first and second coefficients ofviscosity asλ + 2µ/3 = 0, although the validity of thisassumption for other than dilute, monatomic gases hasoccasionally been questioned (Gad-el-Hak [40]). Withthe above constitutive relations and neglecting radiative

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Flow physics in MEMS

heat transfer, equations (11), (12) and (13) respectivelyread

∂ρ

∂t+ ∂

∂xk(ρuk)= 0 (17)

ρ

(∂ui

∂t+ uk ∂ui

∂xk

)= − ∂p

∂xi+ ρgi + ∂

∂xk

[µ

(∂ui

∂xk+ ∂uk

∂xi

)+ δkiλ∂uj

∂xj

](18)

ρ

(∂e

∂t+ uk ∂e

∂xk

)= ∂

∂xk

(κ∂T

∂xk

)− p∂uk

∂xk+ φ (19)

The three components of the vector equation (18) are theNavier–Stokes equations expressing the conservation ofmomentum for a Newtonian fluid. In the thermal energyequation (19),φ is the always positive dissipation func-tion expressing the irreversible conversion of mechanicalenergy to internal energy as a result of the deformationof a fluid element. The second term on the right-handside of (19) is the reversible work done (per unit time)by the pressure as the volume of a fluid material elementchanges. For a Newtonian, isotropic fluid, the viscousdissipation rate is given by

φ = 1

2µ

(∂ui

∂xk+ ∂uk

∂xi

)2

+ λ(∂uj

∂xj

)2

(20)

There are now six unknowns,ρ, ui , p andT , and thefive coupled equations (17)–(19) plus the equation ofstate relating pressure, density and temperature. Thesesix equations together with sufficient number of initialand boundary conditions constitute a well-posed, albeitformidable, problem. The system of equations (17)–(19) is an excellent model for the laminar or turbulentflow of most fluids such as air and water under manycircumstances, including high-speed gas flows for whichthe shock waves are thick relative to the mean free pathof the molecules.

Considerable simplification is achieved if the flow isassumed incompressible, usually a reasonable assump-tion provided that the characteristic flow speed is lessthan 0.3 of the speed of sound. The incompressibility as-sumption is readily satisfied for almost all liquid flowsand many gas flows. In such cases, the density is assumedeither a constant or a given function of temperature (orspecies concentration). The governing equations for suchflow are

∂uk

∂xk= 0 (21)

ρ

(∂ui

∂t+ uk ∂ui

∂xk

)= − ∂p

∂xi+ ∂

∂xk

[µ

(∂ui

∂xk+ ∂uk

∂xi

)]+ ρgi (22)

ρcp

(∂T

∂t+ uk ∂T

∂xk

)= ∂

∂xk

(κ∂T

∂xk

)+ φincomp (23)

whereφincompis the incompressible limit of equation (20).These are now five equations for the five dependent vari-ablesui , p andT . Note that the left-hand side of equa-tion (23) has the specific heat at constant pressurecp andnot cv . It is the convection of enthalpy — and not inter-nal energy— that is balanced by heat conduction and vis-cous dissipation. This is the correct incompressible-flowlimit — of a compressible fluid — as discussed in detailin Section 10.9 of Panton [41]; a subtle point perhaps butone that is frequently missed in textbooks.

For both the compressible and the incompressibleequations of motion, the transport terms are neglectedaway from solid walls in the limit of infinite Reynoldsnumber (Kn → 0). The fluid is then approximated as in-viscid and non-conducting, and the corresponding equa-tions read (for the compressible case)

∂ρ

∂t+ ∂

∂xk(ρuk)= 0 (24)

ρ

(∂ui

∂t+ uk ∂ui

∂xk

)= − ∂p

∂xi+ ρgi (25)

ρcv

(∂T

∂t+ uk ∂T

∂xk

)= −p∂uk

∂xk(26)

The Euler equation (25) can be integrated along a stream-line and the resulting Bernoulli’s equation provides a di-rect relation between the velocity and pressure.

5. COMPRESSIBILITY

The issue of whether to consider the continuumflow compressible or incompressible seems to be ratherstraightforward, but is in fact full of potential pitfalls. Ifthe local Mach number is less than 0.3, then the flow of acompressible fluid like air can — according to the con-ventional wisdom — be treated as incompressible. Butthe well-knownMa< 0.3 criterion is only a necessarynot a sufficient one to allow a treatment of the flow asapproximately incompressible. In other words, there aresituations where the Mach number can be exceedinglysmall while the flow is compressible. As is well docu-mented in heat transfer textbooks, strong wall heating orcooling may cause the density to change sufficiently and

319

M. Gad-el-Hak

the incompressible approximation to break down, evenat low speeds. Less known is the situation encounteredin some microdevices where the pressure may stronglychange due to viscous effects even though the speeds maynot be high enough for the Mach number to go above thetraditional threshold of 0.3. Corresponding to the pres-sure changes would be strong density changes that mustbe taken into account when writing the continuum equa-tions of motion. In this section, we systematically explainall situations where compressibility effects must be con-sidered. Let us rewrite the full continuity equation (11) asfollows:

Dρ

Dt+ ρ ∂uk

∂xk= 0 (27)

where D/Dt is the substantial derivative(∂/∂t + uk∂/

∂xk), expressing changes following a fluid element. Theproper criterion for the incompressible approximation tohold is that((1/ρ)Dρ/Dt) is vanishingly small. In otherwords, if density changes following a fluid particle aresmall, the flow is approximately incompressible. Densitymay change arbitrarily from one particle to another with-out violating the incompressible flow assumption. Thisis the case for example in the stratified atmosphere andocean, where the variable-density/temperature/salinityflow is often treated as incompressible.

From the state principle of thermodynamics, we canexpress the density changes of a simple system in termsof changes in pressure and temperature,

ρ = ρ(p,T ) (28)

Using the chain rule of calculus,

1

ρ

Dρ

Dt= αDp

Dt− βDT

Dt(29)

whereα andβ are, respectively, the isothermal compress-ibility coefficient and the bulk expansion coefficient —two thermodynamic variables that characterize the fluidsusceptibility to change of volume — which are definedby the following relations

α(p,T )≡ 1

ρ

∂ρ

∂p

∣∣∣∣T

(30)

β(p,T )≡ − 1

ρ

∂ρ

∂T

∣∣∣∣p

(31)

For ideal gases,α = 1/p andβ = 1/T . Note, however,that in the following arguments it will not be necessaryto invoke the ideal gas assumption. The flow must betreated as compressible if pressure and/or temperaturechanges — following a fluid element — are sufficiently

strong. Equation (29) must of course be properly nondi-mensionalized before deciding whether a term is large orsmall. In here, we follow closely the procedure detailedin Panton [41].

Consider first the case of adiabatic walls. Density isnormalized with a reference valueρ0, velocities witha reference speedv0, spatial coordinates and time withrespectivelyL andL/v0, the isothermal compressibilitycoefficient and bulk expansion coefficient with referencevaluesα0 and β0. The pressure is nondimensionalizedwith the inertial pressure-scaleρ0v

20. This scale is twice

the dynamic pressure, i.e. the pressure change as aninviscid fluid moving at the reference speed is broughtto rest.

Temperature changes for the case of adiabatic wallscan only result from the irreversible conversion of me-chanical energy into internal energy via viscous dissipa-tion. Temperature is therefore nondimensionalized as fol-lows:

T ∗ = T − T0

(µ0v20/κ0)

= T − T0

Pr(v20/cp0)

(32)

whereT0 is a reference temperature,µ0, κ0 andcp0 are,respectively, reference viscosity, thermal conductivityand specific heat at constant pressure, andPr is thereference Prandtl number,(µ0cp0)/κ0.

In the present formulation, the scaling used for pres-sure is based on the Bernoulli’s equation, and thereforeneglects viscous effects. This particular scaling guaran-tees that the pressure term in the momentum equation willbe of the same order as the inertia term. The temperaturescaling assumes that the conduction, convection and dis-sipation terms in the energy equation have the same orderof magnitude. The resulting dimensionless form of equa-tion (29) reads

1

ρ∗Dρ∗

Dt∗= γ0 Ma2

{α∗ Dp∗

Dt∗− PrBβ∗

A

DT ∗

Dt∗

}(33)

where the superscript∗ indicates a nondimensional quan-tity, Ma is the reference Mach number (v0/a0, wherea0is the reference speed of sound), andA andB are di-mensionless constants defined byA ≡ α0ρ0cp0T0 andB ≡ β0T0. If the scaling is properly chosen, the termshaving the∗ superscript in the right-hand side should beof order one, and the relative importance of such termsin the equations of motion is determined by the magni-tude of the dimensionless parameter(s) appearing to theirleft, e.g.,Ma, Pr, etc. Therefore, asMa2 → 0, tempera-ture changes due to viscous dissipation are neglected (un-lessPr is very large, as for example in the case of highly

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Flow physics in MEMS

viscous polymers and oils). Within the same order of ap-proximation, all thermodynamic properties of the fluidare assumed constant.

Pressure changes are also neglected in the limit of zeroMach number. Hence, forMa< 0.3 (i.e. Ma2 < 0.09),density changes following a fluid particle can be ne-glected and the flow can then be approximated as incom-pressible.3 However, there is a caveat in this argument.Pressure changes due to inertia can indeed be neglected atsmall Mach numbers and this is consistent with the waywe nondimensionalized the pressure term above. If on theother hand pressure changes are mostly due to viscouseffects, as is the case for example in a long microductor a micro-gas-bearing, pressure changes may be signif-icant even at low speeds (lowMa). In that case the termDp∗/Dt in equation (33) is no longer of order one, andmay be large regardless of the value ofMa. Density thenmay change significantly and the flow must be treated ascompressible. Had pressure been nondimensionalized us-ing the viscous scale(µ0v0/L) instead of the inertial one(ρ0v

20), the revised equation (33) would haveRe−1 ap-

pearing explicitly in the first term in the right-hand side,accentuating the importance of this term when viscousforces dominate.

A similar result can be gleaned when the Machnumber is interpreted as follows

Ma2 = v20

a20

= v20∂ρ

∂p

∣∣∣∣s

= ρ0v20

ρ0

∂ρ

∂p

∣∣∣∣s

∼ -p

ρ0

-ρ

-p= -ρ

ρ0(34)

where s is the entropy. Again, the above equation as-sumes that pressure changes are inviscid, and thereforesmall Mach number means negligible pressure and den-sity changes. In a flow dominated by viscous effects —such as that inside a microduct— density changes maybe significant even in the limit of zero Mach number.

Identical arguments can be made in the case ofisothermal walls. Here strong temperature changes maybe the result of wall heating or cooling, even if viscousdissipation is negligible. The proper temperature scale inthis case is given in terms of the wall temperatureTw andthe reference temperatureT0 as follows

T = T − T0

Tw − T0(35)

3 With an error of about 10 % atMa = 0.3, 4 % atMa = 0.2, 1 % atMa= 0.1, and so on.

where T is the new dimensionless temperature. Thenondimensional form of equation (29) now reads

1

ρ∗Dρ∗

Dt∗= γ0 Ma2α∗ Dp∗

Dt∗−β∗B

(Tw − T0

T0

)D T

Dt∗(36)

Here we notice that the temperature term is different fromthat in equation (33).Ma is no longer appearing in thisterm, and strong temperature changes, i.e. large(Tw −T0)/T0, may cause strong density changes regardlessof the value of the Mach number. Additionally, thethermodynamic properties of the fluid are not constantbut depend on temperature, and as a result the continuity,momentum and energy equations all couple. The pressureterm in equation (36), on the other hand, is exactly as itwas in the adiabatic case and the same arguments madebefore apply: the flow should be considered compressibleif Ma> 0.3, or if pressure changes due to viscous forcesare sufficiently large.

Experiments in gaseous microducts confirm the abovearguments. For both low- and high-Mach-number flows,pressure gradients in long microchannels are non-constant, consistent with the compressible flow equa-tions. Such experiments were conducted by, among oth-ers, Prud’homme et al. [42], Pfahler et al. [43], van denBerg et al. [44], Liu et al. [45, 46], Pong et al. [47],Harley et al. [48], Piekos and Breuer [49], Arkilic [50]and Arkilic et al. [51–53]. Sample results will be pre-sented in the following section.

There are three additional scenarios in which signifi-cant pressure and density changes may take place with-out inertial, viscous or thermal effects. First is the caseof quasi-static compression/expansion of a gas in, for ex-ample, a piston-cylinder arrangement. The resulting com-pressibility effects are, however, compressibility of thefluid and not of the flow. Two other situations where com-pressibility effects must also be considered are problemswith length-scales comparable to the scale height of theatmosphere and rapidly varying flows as in sound propa-gation (Lighthill [54]).

6. BOUNDARY CONDITIONS

The continuum equations of motion described earlierrequire a certain number of initial and boundary condi-tions for proper mathematical formulation of flow prob-lems. In this section, we describe the boundary conditionsat a fluid–solid interface. Boundary conditions in the in-viscid flow theory pertain only to the velocity componentnormal to a solid surface. The highest spatial derivative

321

M. Gad-el-Hak

of velocity in the inviscid equations of motion is first-order, and only one velocity boundary condition at thesurface is admissible. The normal velocity componentat a fluid–solid interface is specified, and no statementcan be made regarding the tangential velocity component.The normal-velocity condition simply states that a fluid-particle path cannot go through an impermeable wall.Real fluids are of course viscous and the correspondingmomentum equation has second-order derivatives of ve-locity, thus requiring an additional boundary condition onthe velocity component tangential to a solid surface.

Traditionally, the no-slip condition at a fluid–solidinterface is enforced in the momentum equation andan analogous no-temperature-jump condition is appliedin the energy equation. The notion underlying the no-slip/no-jump condition is that within the fluid there can-not be any finite discontinuities of velocity/temperature.Those would involve infinite velocity/temperature gradi-ents and so produce infinite viscous stress/heat flux thatwould destroy the discontinuity in infinitesimal time. Theinteraction between a fluid particle and a wall is simi-lar to that between neighboring fluid particles, and there-fore no discontinuities are allowed at the fluid–solid in-terface either. In other words, the fluid velocity must bezero relative to the surface and the fluid temperature mustequal to that of the surface. But strictly speaking thosetwo boundary conditions are valid only if the fluid flowadjacent to the surface is in thermodynamic equilibrium.This requires an infinitely high frequency of collisionsbetween the fluid and the solid surface. In practice, theno-slip/no-jump condition leads to fairly accurate predic-tions as long asKn< 0.001 (for gases). Beyond that, thecollision frequency is simply not high enough to ensureequilibrium and a certain degree of tangential-velocityslip and temperature jump must be allowed. This is a casefrequently encountered in MEMS flows, and we developthe appropriate relations in this section.

For both liquids and gases, the linear Navier boundarycondition empirically relates the tangential velocity slipat the wall-u|w to the local shear

-u|w = ufluid − uwall = Ls∂u

∂y

∣∣∣∣w

(37)

whereLs is the constant slip length, and∂u/∂y|w isthe strain rate computed at the wall. In most practicalsituations, the slip length is so small that the no-slipcondition holds. In MEMS applications, however, thatmay not be the case. Once again we defer the discussionof liquids to a later section and focus for now on gases.

Assuming isothermal conditions prevail, the aboveslip relation has been rigorously derived by Maxwell [55]

from considerations of the kinetic theory of dilute, mona-tomic gases. Gas molecules, modeled as rigid spheres,continuously strike and reflect from a solid surface, justas they continuously collide with each other. For anidealized perfectly smooth wall (at the molecular scale),the incident angle exactly equals the reflected angleand the molecules conserve their tangential momentumand thus exert no shear on the wall. This is termedspecular reflection and results in perfect slip at the wall.For an extremely rough wall, on the other hand, themolecules reflect at some random angle uncorrelatedwith their entry angle. This perfectly diffuse reflectionresults in zero tangential-momentum for the reflectedfluid molecules to be balanced by a finite slip velocityin order to account for the shear stress transmitted to thewall. A force balance near the wall leads to the followingexpression for the slip velocity

ugas− uwall = L∂u

∂y

∣∣∣∣w

(38)

whereL is the mean free path. The right-hand side canbe considered as the first term in an infinite Taylorseries, sufficient if the mean free path is relatively smallenough. The equation above states that significant slipoccurs only if the mean velocity of the molecules variesappreciably over a distance of one mean free path. This isthe case, for example, in vacuum applications and/or flowin microdevices. The number of collisions between thefluid molecules and the solid in those cases is not largeenough for even an approximate flow equilibrium to beestablished. Furthermore, additional (nonlinear) terms inthe Taylor series would be needed asL increases and theflow is further removed from the equilibrium state.

For real walls some molecules reflect diffusively andsome reflect specularly. In other words, a portion ofthe momentum of the incident molecules is lost tothe wall and a (typically smaller) portion is retainedby the reflected molecules. The tangential-momentum-accommodation coefficientσv is defined as the fractionof molecules reflected diffusively. This coefficient de-pends on the fluid, the solid and the surface finish, andhas been determined experimentally to be between 0.2and 0.8 (Thomas and Lord [56]; Seidl and Steiheil [57];Porodnov et al. [58]; Arkilic et al. [53]; Arkilic [50]),the lower limit being for exceptionally smooth surfaceswhile the upper limit is typical of most practical surfaces.The final expression derived by Maxwell for an isother-mal wall reads

ugas− uwall = 2− σvσv

L∂u

∂y

∣∣∣∣w

(39)

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Flow physics in MEMS

For σv = 0, the slip velocity is unbounded, while forσv = 1, equation (39) reverts to (38).

Similar arguments were made for the temperature-jump boundary condition by von Smoluchowski [59].For an ideal gas flow in the presence of wall-normaland tangential temperature gradients, the complete (first-order) slip-flow and temperature-jump boundary condi-tions read

ugas− uwall = 2− σvσv

1

ρ√

2RTgas/πτw

+ 3

4

Pr(γ − 1)

γρ RTgas(−qx)w

= 2− σvσv

L

(∂u

∂y

)w

+ 3

4

µ

ρTgas

(∂T

∂x

)w

(40)

Tgas− Twall = 2− σTσT

[2(γ − 1)

(γ + 1)

]× 1

ρ R√

2RTgas/π(−qy)w

= 2− σTσT

[2γ

(γ + 1)

]L

Pr

(∂T

∂y

)w

(41)

wherex and y are the streamwise and normal coordi-nates,ρ andµ are respectively the fluid density and vis-cosity,R is the gas constant,Tgasis the temperature of thegas adjacent to the wall,Twall is the wall temperature,τwis the shear stress at the wall,Pr is the Prandtl number,γis the specific heat ratio, andqx andqy are respectivelythe tangential and normal heat flux at the wall.

The tangential-momentum-accommodation coeffici-entσv and the thermal-accommodation coefficientσT aregiven by, respectively,

σv = τi − τrτi − τw (42)

σT = dEi − dEr

dEi − dEw(43)

where the subscripts i, r, and w stand for, respectively,incident, reflected and solid wall conditions,τ is atangential momentum flux, and dE is an energy flux.

The second term in the right-hand side of equa-tion (40) is thethermal creepwhich generates slip veloc-ity in the fluid opposite to the direction of the tangentialheat flux, i.e. flow in the direction of increasing temper-ature. At sufficiently high Knudsen numbers, streamwisetemperature gradient in a conduit leads to a measurablepressure gradient along the tube. This may be the case invacuum applications and MEMS devices. Thermal creep

is the basis for the so-called Knudsen pump — a devicewith no moving parts — in which rarefied gas is hauledfrom one cold chamber to a hot one.4 Clearly, such pumpperforms best at high Knudsen numbers, and is typicallydesigned to operate in the free-molecule flow regime.

In dimensionless form, equations (40) and (41) respec-tively read

u∗gas− u∗

wall =2− σvσv

Kn

(∂u∗

∂y∗

)w

+ 3

2π

(γ − 1)

γ

Kn2Re

Ec

(∂T ∗

∂x∗

)w

(44)

T ∗gas− T ∗

wall =2− σTσT

[2γ

(γ + 1)

]Kn

Pr

(∂T ∗

∂y∗

)w

(45)

where the superscript∗ indicates dimensionless quantity,Kn is the Knudsen number,Re is the Reynolds number,andEc is the Eckert number defined by

Ec= v20

cp-T= (γ − 1)

T0

-TMa2 (46)

where v0 is a reference velocity,-T = (Tgas − T0),and T0 is a reference temperature. Note that very lowvalues ofσv andσT lead to substantial velocity slip andtemperature jump even for flows with small Knudsennumber.

The first term in the right-hand side of equation (44)is first-order in Knudsen number, while the thermal creepterm is second-order, meaning that the creep phenom-enon is potentially significant at large values of the Knud-sen number. Equation (45) is first-order inKn. Usingequations (8) and (46), the thermal creep term in equa-tion (44) can be rewritten in terms of-T and Reynoldsnumber. Thus,

u∗gas− u∗

wall =2− σvσv

Kn

(∂u∗

∂y∗

)w

+ 3

4

-T

T0

1

Re

(∂T ∗

∂x∗

)w

(47)

It is clear that large temperature changes along thesurface or low Reynolds numbers lead to significantthermal creep.

The continuum Navier–Stokes equations with no-slip/no-temperature jump boundary conditions are valid

4 The terminology Knudsen pump has been used by, for example,Vargo and Muntz [60], but according to Loeb [61], the originalexperiments demonstrating such pump were carried out by OsborneReynolds.

323

M. Gad-el-Hak

as long as the Knudsen number does not exceed 0.001.First-order slip/temperature-jump boundary conditionsshould be applied to the Navier–Stokes equations inthe range of 0.001< Kn < 0.1. The transition regimespans the range of 0.1 < Kn < 10, and second-orderor higher slip/temperature-jump boundary conditions areapplicable there. Note, however, that the Navier–Stokesequations are first-order accurate inKn as will be shownlater, and are themselves not valid in the transitionregime. Either higher-order continuum equations, e.g.,Burnett equations, should be used there or molecularmodeling should be invoked, abandoning the continuumapproach altogether.

For isothermal walls, Beskok [62] derived a higher-order slip-velocity condition as follows:

ugas− uwall = 2− σvσv

[L

(∂u

∂y

)w

+ L2

2!(∂2u

∂y2

)w

+ L3

3!(∂3u

∂y3

)w

+ · · ·]

(48)

Attempts to implement the above slip condition in nu-merical simulations are rather difficult. Second-order andhigher derivatives of velocity cannot be computed ac-curately near the wall. Based on asymptotic analysis,Beskok [63] and Beskok and Karniadakis [64, 65] pro-posed the following alternative higher-order boundarycondition for the tangential velocity, including the ther-mal creep term,

u∗gas− u∗

wall =2− σvσv

Kn

1− bKn

(∂u∗

∂y∗

)w

+ 3

2π

(γ − 1)

γ

Kn2 Re

Ec

(∂T ∗

∂x∗

)w

(49)

whereb is a high-order slip coefficient determined fromthe presumably known no-slip solution, thus avoiding thecomputational difficulties mentioned above. If this high-order slip coefficient is chosen asb = u′′

w/u′w, where

the prime denotes derivative with respect toy and thevelocity is computed from the no-slip Navier–Stokesequations, equation (49) becomes second-order accuratein Knudsen number. Beskok’s procedure can be extendedto third- and higher-orders for both the slip-velocity andthermal creep terms.

Similar arguments can be applied to the temperature-jump boundary condition, and the resulting Taylor series

Figure 3. Variation of mass flowrate as a function of (p2i − p2

0).Original data acquired by S.A. Tison and plotted by Beskok etal. [70].

reads in dimensionless form (Beskok [63])

T ∗gas− T ∗

wall =2− σTσT

[2γ

(γ + 1)

]1

Pr

[Kn

(∂T ∗

∂y∗

)w

+ Kn2

2!(∂2T ∗

∂y∗2

)w

+ · · ·]

(50)Again, the difficulties associated with computing second-and higher-order derivatives of temperature are alleviatedusing an identical procedure to that utilized for thetangential velocity boundary condition.

Several experiments in low-pressure macroducts orin microducts confirm the necessity of applying slipboundary condition at sufficiently large Knudsen num-bers. Among them are those conducted by Knudsen [66],Pfahler et al. [43], Tison [67], Liu et al. [45, 46], Pong etal. [47], Arkilic et al. [51], Harley et al. [48] and Shih etal. [68, 69]. The experiments are complemented by thenumerical simulations carried out by Beskok [62, 63],Beskok and Karniadakis [64, 65], and Beskok et al. [70].Here we present selected examples of the experimentaland numerical results.

Tison [67] conducted pipe flow experiments at verylow pressures. His pipe has a diameter of 2 mm anda length-to-diameter ratio of 200. Both inlet and outletpressures were varied to yield Knudsen number in therange ofKn = 0−200. Figure 3 shows the variation ofmass flowrate as a function of(p2

i −p20), wherepi is the

324

Flow physics in MEMS

Figure 4. Mass flowrate versus inlet pressure in a microchan-nel. From Shih et al. [68].

inlet pressure andp0 is the outlet pressure.5 The pres-sure drop in this rarefied pipe flow is nonlinear, character-istic of low-Reynolds-number, compressible flows. Threedistinct flow regimes are identified: (1) slip flow regime,0 < Kn < 0.6; (2) transition regime, 0.6 < Kn < 17,where the mass flowrate is almost constant as the pressurechanges; and (3) free-molecule flow,Kn> 17. Note thatthe demarkation between these three regimes is slightlydifferent from that mentioned earlier. As stated, the dif-ferent Knudsen number regimes are determined empiri-cally and are therefore only approximate for a particularflow geometry.

Shih et al. [68] conducted their experiments in a mi-crochannel using helium as a fluid. The inlet pressurevaried but the duct exit was atmospheric. Microsensorswhere fabricated in-situ along their MEMS channel tomeasure the pressure.Figure 4 shows their measuredmass flowrate versus the inlet pressure. The data are com-pared to the no-slip solution and the slip solution us-ing three different values of the tangential-momentum-accommodation coefficient, 0.8, 0.9 and 1.0. The agree-ment is reasonable with the caseσv = 1, indicating per-haps that the channel used by Shih et al. was quite roughon the molecular scale. In a second experiment (Shih etal. [69]), nitrous oxide was used as the fluid. The squareof the pressure distribution along the channel is plotted infigure 5for five different inlet pressures. The experimen-tal data (symbols) compare well with the theoretical pre-dictions (solid lines). Again, the nonlinear pressure dropshown indicates that the gas flow is compressible.

5 The original data in this figure were acquired by S.A. Tison andplotted by Beskok et al. [70].

Figure 5. Pressure distribution of nitrous oxide in a microduct.Solid lines are theoretical predictions. From Shih et al. [69].

Arkilic [50] provided an elegant analysis of the com-pressible, rarefied flow in a microchannel. The results ofhis theory are compared to the experiments of Pong etal. [47] in figure 6. The dotted line is the incompress-ible flow solution, where the pressure is predicted to droplinearly with streamwise distance. The dashed line is thecompressible flow solution that neglects rarefaction ef-fects (assumesKn = 0). Finally, the solid line is the the-oretical result that takes into account both compressibil-ity and rarefaction via slip-flow boundary condition com-puted at the exit Knudsen number ofKn = 0.06. Thattheory compares most favorably with the experimentaldata. In the compressible flow through the constant-areaduct, density decreases and thus velocity increases in thestreamwise direction. As a result, the pressure distrib-ution is nonlinear with negative curvature. A moderateKnudsen number (i.e. moderate slip) actually diminishes,albeit rather weakly, this curvature. Thus, compressibilityand rarefaction effects lead to opposing trends, as pointedout by Beskok et al. [70].

7. MOLECULAR-BASED MODELS

In the continuum models discussed thus far, themacroscopic fluid properties are the dependent variableswhile the independent variables are the three spatial co-ordinates and time. The molecular models recognize thefluid as a myriad of discrete particles: molecules, atoms,ions and electrons. The goal here is to determine theposition, velocity and state of all particles at all times.The molecular approach is either deterministic or proba-bilistic (refer tofigure 1). Provided that there is a suffi-cient number of microscopic particles within the smallestsignificant volume of a flow, the macroscopic properties

325

M. Gad-el-Hak

Figure 6. Pressure distribution in a long microchannel. Thesymbols are experimental data while the lines are differenttheoretical predictions. From Arkilic [50].

at any location in the flow can then be computed fromthe discrete-particle information by a suitable averagingor weighted averaging process. The present section dis-cusses molecular-based models and their relation to thecontinuum models previously considered.

The most fundamental of the molecular models is adeterministic one. The motion of the molecules are gov-erned by the laws of classical mechanics, although, at theexpense of greatly complicating the problem, the lawsof quantum mechanics can also be considered in specialcircumstances. The modern molecular dynamics com-puter simulations (MD) have been pioneered by Alderand Wainwright [71–73] and reviewed by Ciccotti andHoover [74], Allen and Tildesley [75], Haile [76] and Ko-plik and Banavar [77]. The simulation begins with a setof N molecules in a region of space, each assigned a ran-dom velocity corresponding to a Boltzmann distributionat the temperature of interest. The interaction between theparticles is prescribed typically in the form of a two-bodypotential energy and the time evolution of the molecu-lar positions is determined by integrating Newton’s equa-tions of motion. Because MD is based on the most basicset of equations, it is valid in principle for any flow extentand any range of parameters. The method is straightfor-ward in principle but there are two hurdles: choosing aproper and convenient potential for particular fluid andsolid combinations, and the colossal computer resourcesrequired to simulate a reasonable flowfield extent.

For purists, the former difficulty is a sticky one. Thereis no totally rational methodology by which a convenientpotential can be chosen. Part of the art of MD is to pick

an appropriate potential and validate the simulation re-sults with experiments or other analytical/computationalresults. A commonly used potential between two mole-cules is the generalized Lennard-Jones 6–12 potential, tobe used in the following section and further discussed inthe section following that.

The second difficulty, and by far the most seriouslimitation of molecular dynamics simulations, is thenumber of moleculesN that can realistically be modeledon a digital computer. Since the computation of anelement of trajectory for any particular molecule requiresconsideration ofall other molecules as potential collisionpartners, the amount of computation required by the MDmethod is proportional toN2. Some saving in computertime can be achieved by cutting off the weak tail of thepotential (seefigure 11) at, say,rc = 2.5σ , and shiftingthe potential by a linear term inr so that the force goessmoothly to zero at the cutoff. As a result, only nearbymolecules are treated as potential collision partners, andthe computation time forN molecules no longer scaleswith N2.

The state of the art of molecular dynamics simulationsin the early 2000 s is such that with a few hours ofCPU time, general purpose supercomputers can handlearound 100 000 molecules. At enormous expense, thefastest parallel machine available can simulate around10 million particles. Because of the extreme diminutionof molecular scales, the above translates into regions ofliquid flow of about 0.02 µm (200 Å) in linear size, overtime intervals of around 0.001 µs, enough for continuumbehavior to set in for simple molecules. To simulate1 s of real time for complex molecular interactions,e.g., including vibration modes, reorientation of polymermolecules, collision of colloidal particles, etc., requiresunrealistic CPU time measured in hundreds of years.

MD simulations are highly inefficient for dilute gaseswhere the molecular interactions are infrequent. Thesimulations are more suited for dense gases and liquids.Clearly, molecular dynamics simulations are reserved forsituations where the continuum approach or the statisticalmethods are inadequate to compute from first principlesimportant flow quantities. Slip boundary conditions forliquid flows in extremely small devices is such a case aswill be discussed in the following section.

An alternative to the deterministic molecular dynam-ics is the statistical approach where the goal is to com-pute the probability of finding a molecule at a particularposition and state. If the appropriate conservation equa-tion can be solved for the probability distribution, im-portant statistical properties such as the mean number,momentum or energy of the molecules within an ele-

326

Flow physics in MEMS

ment of volume can be computed from a simple weightedaveraging. In a practical problem, it is such average quan-tities that concern us rather than the detail for everysingle molecule. Clearly, however, the accuracy of com-puting average quantities, via the statistical approach,improves as the number of molecules in the sampledvolume increases. The kinetic theory of dilute gases iswell advanced, but that for dense gases and liquids ismuch less so due to the extreme complexity of havingto include multiple collisions and intermolecular forcesin the theoretical formulation. The statistical approach iswell covered in books such as those by Kennard [78],Hirschfelder et al. [79], Schaaf and Chambré [80], Vin-centi and Kruger [81], Kogan [82], Chapman and Cowl-ing [83], Cercignani [84, 85] and Bird [86], and reviewarticles such as those by Kogan [87], Muntz [88] andOran et al. [89].

In the statistical approach, the fraction of molecules ina given location and state is the sole dependent variable.The independent variables for monatomic moleculesare time, the three spatial coordinates and the threecomponents of molecular velocity. Those describe a6-dimensional phase space.6 For diatomic or polyatomicmolecules, the dimension of phase space is increased bythe number of internal degrees of freedom. Orientationadds an extra dimension for molecules which are notspherically symmetric. Finally, for mixtures of gases,separate probability distribution functions are requiredfor each species. Clearly, the complexity of the approachincreases dramatically as the dimension of phase spaceincreases. The simplest problems are, for example, thosefor steady, one-dimensional flow of a simple monatomicgas.

To simplify the problem we restrict the discussion hereto monatomic gases having no internal degrees of free-dom. Furthermore, the fluid is restricted to dilute gasesand molecular chaos is assumed. The former restrictionrequires the average distance between moleculesδ to bean order of magnitude larger than their diameterσ . Thatwill almost guarantee that all collisions between mole-cules are binary collisions, avoiding the complexity ofmodeling multiple encounters.7 The molecular chaos re-striction improves the accuracy of computing the macro-scopic quantities from the microscopic information. Inessence, the volume over which averages are computedhas to have sufficient number of molecules to reduce sta-

6 The evolution equation of the probability distribution is considered,hence time is the 7th independent variable.

7 Dissociation and ionization phenomena involve triple collisions andtherefore require separate treatment.

Figure 7. Effective limits of different flow models. FromBird [86].

tistical errors. It can be shown that computing macro-scopic flow properties by averaging over a number ofmolecules will result in statistical fluctuations with astandard deviation of approximately 0.1 % if one millionmolecules are used and around 3 % if one thousand mole-cules are used. The molecular chaos limit requires thelength-scaleL for the averaging process to be at least100 times the average distance between molecules (i.e.typical averaging over at least one million molecules).

Figure 7, adapted from Bird [86], shows the limits ofvalidity of the dilute gas approximation(δ/σ > 7), thecontinuum approach (Kn< 0.1, as discussed previously),and the neglect of statistical fluctuations(L/δ > 100).Using a molecular diameter ofσ = 4·10−10 m as anexample, the three limits are conveniently expressed asfunctions of the normalized gas densityρ/ρ0 or num-ber densityn/n0, where the reference densitiesρ0 andn0 are computed at standard conditions. All three limitsare straight lines in the log-log plot ofL versusρ/ρ0, asdepicted infigure 7. Note the shaded triangular wedge in-side which both the Boltzmann and Navier–Stokes equa-tions are valid. Additionally, the lines describing the threelimits very nearly intersect at a single point. As a conse-quence, the continuum breakdown limit always lies be-tween the dilute gas limit and the limit for molecularchaos. As density or characteristic dimension is reducedin a dilute gas, the Navier–Stokes model breaks down be-

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M. Gad-el-Hak

fore the level of statistical fluctuations becomes signifi-cant. In a dense gas, on the other hand, significant fluctua-tions may be present even when the Navier–Stokes modelis still valid.

The starting point in statistical mechanics is theLiouville equation which expresses the conservation ofthe N -particle distribution function in 6N -dimensionalphase space,8 whereN is the number of particles underconsideration. Considering only external forces whichdo not depend on the velocity of the molecules,9 theLiouville equation for a system ofN mass points reads

∂F

∂t+

N∑k=1

�ξk · ∂F∂ �xk +

N∑k=1

�Fk · ∂F∂�ξk

= 0 (51)

where F is the probability of finding a molecule at aparticular point in phase space,t is time, �ξk is thethree-dimensional velocity vector for thekth molecule,�xk is the three-dimensional position vector for thekthmolecule, and�F is the external force vector. Note thatthe dot product in the above equation is carried out overeach of the three components of the vectors�ξ , �x and �F ,and that the summation is over all molecules. Obviouslysuch an equation is not tractable for realistic number ofparticles.

A hierarchy of reduced distribution functions may beobtained by repeated integration of the Liouville equationabove. The final equation in the hierarchy is for the singleparticle distribution which also involves the two-particledistribution function. Assuming molecular chaos, thatfinal equation becomes a closed one (i.e. one equation inone unknown), and is known as the Boltzmann equation,the fundamental relation of the kinetic theory of gases.That final equation in the hierarchy is the only one whichcarries any hope of obtaining analytical solutions.

A simpler direct derivation of the Boltzmann equationis provided by Bird [86]. For monatomic gas moleculesin binary collisions, the integro-differential Boltzmannequation reads

∂(nf )

∂t+ ξj ∂(nf )

∂xj+ Fj ∂(nf )

∂ξj= J (

f,f ∗),j = 1,2,3 (52)

where nf is the product of the number density andthe normalized velocity distribution function (dn/n =f d�ξ), xj and ξj are respectively the coordinates and

8 Three positions and three velocities foreach molecule of amonatomic gas with no internal degrees of freedom.

9 This excludes Lorentz forces, for example.

speeds of a molecule,10 Fj is a known external force,and J (f,f ∗) is the nonlinear collision integral thatdescribes the net effect of populating and depopulatingcollisions on the distribution function. The collisionintegral is the source of difficulty in obtaining analyticalsolutions to the Boltzmann equation, and is given by

J(f,f ∗) =

∫ ∞

−∞

∫ 4π

0n2(f ∗f ∗

1 − ff1)�ξrσ d8

(d�ξ)1

(53)where the superscript∗ indicates post-collision values,f and f1 represent two different molecules,�ξr is therelative speed between two molecules,σ is the molecularcross-section,8 is the solid angle, and d�ξ = dξ1 dξ2 dξ3.

Once a solution forf is obtained, macroscopic quan-tities such as density, velocity, temperature, etc., can becomputed from the appropriate weighted integral of thedistribution function. For example,

ρ =mn=m∫(nf )d�ξ (54)

ui =∫ξif d�ξ (55)

3

2kT =

∫1

2mξiξif d�ξ (56)

If the Boltzmann equation is nondimensionalized witha characteristic lengthL and characteristic speed[2(k/m)T ]1/2, wherek is the Boltzmann constant,m isthe molecular mass, andT is temperature, the inverseKnudsen number appears explicitly in the right-hand sideof the equation as follows

∂f

∂ t+ ξj ∂f

∂xj+ Fj ∂f

∂ξj= 1

KnJ(f , f ∗),

j = 1,2,3 (57)

where the topping symbolˆ represents a dimensionlessvariable, andf is nondimensionalized using a referencenumber densityn0.

The five conservation equations for the transport ofmass, momentum and energy can be derived by multiply-ing the Boltzmann equation above by, respectively, themolecular mass, momentum and energy, then integratingover all possible molecular velocities. Subject to the re-strictions of dilute gas and molecular chaos stated earlier,the Boltzmann equation is valid for all ranges of Knudsennumber from 0 to∞. Analytical solutions to this equation

10Constituting, together with time, the seven independent variables ofthe single-dependent-variable equation.

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Flow physics in MEMS

for arbitrary geometries are difficult mostly because ofthe nonlinearity of the collision integral. Simple modelsof this integral have been proposed to facilitate analyticalsolutions; see, for example, Bhatnagar et al. [90].

There are two important asymptotes to equation (57).First, asKn → ∞, molecular collisions become unim-portant. This is the free-molecule flow regime depictedin figure 2 for Kn> 10, where the only important col-lision is that between a gas molecule and the solid sur-face of an obstacle or a conduit. Analytical solutions arethen possible for simple geometries, and numerical sim-ulations for complicated geometries are straightforwardonce the surface-reflection characteristics are accuratelymodeled. Secondly, asKn → 0, collisions become im-portant and the flow approaches the continuum regimeof conventional fluid dynamics. The Second Law speci-fies a tendency for thermodynamic systems to revert toequilibrium state, smoothing out any discontinuities inmacroscopic flow quantities. The number of molecularcollisions in the limitKn → 0 is so large that the flow ap-proaches the equilibrium state in a time short compared tothe macroscopic time-scale. For example, for air at stan-dard conditions (T = 288 K;p = 1 atm), each moleculeexperiences, on the average, 10 collisions per nanosec-ond and travels 1 micron in the same time period. Sucha molecule has alreadyforgottenits previous state after1 ns. In a particular flowfield, if the macroscopic quan-tities vary little over a distance of 1 µm or over a timeinterval of 1 ns, the flow of STP air is near equilibrium.

At Kn = 0, the velocity distribution function is every-where of the local equilibrium or Maxwellian form:

f (0) = n

n0π−3/2 exp

[−(ξ − u)2] (58)

wheref andu are, respectively, the dimensionless speedsof a molecule and of the flow. In this Knudsen num-ber limit, the velocity distribution of each element of thefluid instantaneously adjusts to the equilibrium thermo-dynamic state appropriate to the local macroscopic prop-erties as this molecule moves through the flowfield. Fromthe continuum viewpoint, the flow is isentropic and heatconduction and viscous diffusion and dissipation vanishfrom the continuum conservation relations.

The Chapman–Enskog theory attempts to solve theBoltzmann equation by considering a small perturbationof f from the equilibrium Maxwellian form. For smallKnudsen numbers, the distribution function can be ex-panded in terms ofKn in the form of a power series

f = f (0)+ Kn f (1) + Kn2f (2) + · · · (59)

By substituting the above series in the Boltzmann equa-tion (57) and equating terms of equal order, the followingrecurrent set of integral equations result:

J(f (0), f (0)

) = 0

J(f (0), f (1)

) = ∂f

∂ t+ ξj ∂f

(0)

∂xj+ Fj ∂f

(0)

∂ξj

(60)

The first integral is nonlinear and its solution is the lo-cal Maxwellian distribution, equation (58). The distri-bution functionsf (1), f (2), etc., each satisfies an in-homogeneous linear equation whose solution leads tothe transport terms needed to close the continuum equa-tions appropriate to the particular level of approximation.The continuum stress tensor and heat flux vector can bewritten in terms of the distribution function, which inturn can be specified in terms of the macroscopic veloc-ity and temperature and their derivatives (Kogan [87]).The zeroth-order equation yields the Euler equations, thefirst-order equation results in the linear transport termsof the Navier–Stokes equations, the second-order equa-tion gives the nonlinear transport terms of the Burnettequations, and so on. Keep in mind, however, that theBoltzmann equation as developed in this section is for amonatomic gas. This excludes the all important air whichis composed largely of diatomic nitrogen and oxygen.

As discussed earlier, the Navier–Stokes equations canand should be used up to a Knudsen number of 0.1.Beyond that, the transition flow regime commences(0.1 < Kn < 10). In this flow regime, the molecularmean free path for a gas becomes significant relativeto a characteristic distance for important flow-propertychanges to take place. The Burnett equations can beused to obtain analytical/numerical solutions for at leasta portion of the transition regime for a monatomic gas,although their complexity have precluded much resultsfor realistic geometries (Agarwal et al. [91]). There isalso a certain degree of uncertainty about the properboundary conditions to use with the continuum Burnettequations, and experimental validation of the results havebeen very scarce. Additionally, as the gas flow furtherdeparts from equilibrium, the bulk viscosity(= λ+2µ/3,whereλ is the second coefficient of viscosity) is no longerzero, and the Stokes’ hypothesis no longer holds (seeGad-el-Hak [40] for an interesting summary of the issueof bulk viscosity).

In the transition regime, the molecularly-based Boltz-mann equation cannot easily be solved either, unless thenonlinear collision integral is simplified. So, clearly thetransition regime is one of dire need of alternative meth-ods of solution. MD simulations as mentioned earlier are

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M. Gad-el-Hak

not suited for dilute gases. The best approach for the tran-sition regime right now is the direct simulation MonteCarlo (DSMC) method developed by Bird [86, 92–95]and briefly described below. Some recent reviews ofDSMC include those by Muntz [88], Cheng [96], Chengand Emmanuel [97] and Oran et al. [89]. The mechan-ics as well as the history of the DSMC approach and itsancestors are well described in the book by Bird [86].

Unlike molecular dynamics simulations, DSMC is astatistical computational approach to solving rarefied gasproblems. Both approaches treat the gas as discrete par-ticles. Subject to the dilute gas and molecular chaos as-sumptions, the direct simulation Monte Carlo method isvalid for all ranges of Knudsen number, although it be-comes quite expensive forKn < 0.1. Fortunately, thisis the continuum regime where the Navier–Stokes equa-tions can be used analytically or computationally. DSMCis therefore ideal for the transition regime(0.1< Kn<10), where the Boltzmann equation is difficult to solve.The Monte Carlo method is, like its name sake, a ran-dom number strategy based directly on the physics of theindividual molecular interactions. The idea is to track alarge number of randomly selected, statistically represen-tative particles, and to use their motions and interactionsto modify their positions and states. The primary approxi-mation of the direct simulation Monte Carlo method is touncouple the molecular motions and the intermolecularcollisions over small time intervals. A significant advan-tage of this approximation is that the amount of compu-tation required is proportional toN , in contrast toN2 formolecular dynamics simulations. In essence, particle mo-tions are modeled deterministically while collisions aretreated probabilistically, each simulated molecule repre-senting a large number of actual molecules. Typical com-puter runs of DSMC in the 1990 s involve tens of millionsof intermolecular collisions and fluid–solid interactions.

The DSMC computation is started from some initialcondition and followed in small time steps that can berelated to physical time. Colliding pairs of molecules ina small geometric cell in physical space are randomlyselected after each computational time step. Complexphysics such as radiation, chemical reactions and speciesconcentrations can be included in the simulations withoutthe necessity of nonequilibrium thermodynamic assump-tions that commonly afflict nonequilibrium continuum-flow calculations. DSMC is more computationally in-tensive than classical continuum simulations, and shouldtherefore be used only when the continuum approach isnot feasible.

The DSMC technique is explicit and time march-ing, and therefore always produces unsteady flow simu-

lations. For macroscopically steady flows, Monte Carlosimulation proceeds until a steady flow is established,within a desired accuracy, at sufficiently large time. Themacroscopic flow quantities are then the time average ofall values calculated after reaching the steady state. Formacroscopically unsteady flows, ensemble averaging ofmany independent Monte Carlo simulations is carried outto obtain the final results within a prescribed statisticalaccuracy.

8. LIQUID FLOWS

From the continuum point of view, liquids and gasesare both fluids obeying the same equations of motion. Forincompressible flows, for example, the Reynolds num-ber is the primary dimensionless parameter that deter-mines the nature of the flowfield. True, water, for exam-ple, has density and viscosity that are, respectively, threeand two orders of magnitude higher than those for air, butif the Reynolds number and geometry are matched, liq-uid and gas flows should be identical.11 For MEMS ap-plications, however, we anticipate the possibility of non-equilibrium flow conditions and the consequent invalidityof the Navier–Stokes equations and the no-slip boundaryconditions. Such circumstances can best be researchedusing the molecular approach. This was discussed forgases earlier, and the corresponding arguments for liquidswill be given in the present section. The literature on non-Newtonian fluids in general and polymers in particular isvast (for example, the bibliographic survey by Nadolinkand Haigh [98] cites over 4 900 references on polymerdrag reduction alone) and provides a rich source of infor-mation on the molecular approach for liquid flows.

Solids, liquids and gases are distinguished merely bythe degree of proximity and the intensity of motionsof their constituent molecules. In solids, the moleculesare packed closely and confined, each hemmed in byits neighbors (Chapman and Cowling [83]). Only rarelywould one solid molecule slip from its neighbors to joina new set. As the solid is heated, molecular motion be-comes more violent and a slight thermal expansion takesplace. At a certain temperature that depends on ambientpressure, sufficiently intense motion of the molecules en-ables them to pass freely from one set of neighbors to an-other. The molecules are no longer confined but are nev-ertheless still closely packed, and the substance is now

11Barring phenomena unique to liquids such as cavitation, free surfaceflows, etc.

330

Flow physics in MEMS

considered a liquid. Further heating of the matter eventu-ally releases the molecules altogether, allowing them tobreak the bonds of their mutual attractions. Unlike solidsand liquids, the resulting gas expands to fill any volumeavailable to it.

Unlike solids, both liquids and gases cannot resist fi-nite shear force without continuous deformation; that isthe definition of a fluid medium. In contrast to the re-versible, elastic, static deformation of a solid, the con-tinuous deformation of a fluid resulting from the appli-cation of a shear stress results in an irreversible workthat eventually becomes random thermal motion of themolecules; that is viscous dissipation. There are around25-million molecules of STP air in a one-micron cube.The same cube would contain around 34-billion mole-cules of water. So, liquid flows are continuum even inextremely small devices through which gas flows wouldnot. The average distance between molecules in the gasexample is one order of magnitude higher than the diam-eter of its molecules, while that for the liquid phase ap-proaches the molecular diameter. As a result, liquids arealmost incompressible. Their isothermal compressibilitycoefficientα and bulk expansion coefficientβ are muchsmaller compared to those for gases. For water, for exam-ple, a hundred-fold increase in pressure leads to less than0.5 % decrease in volume. Sound speeds through liquidsare also high relative to those for gases, and as a resultmost liquid flows are incompressible.12 The exceptionbeing propagation of ultra-high-frequency sound wavesand cavitation phenomena.

The mechanism by which liquids transport mass,momentum and energy must be very different from thatfor gases. In dilute gases, intermolecular forces playno role and the molecules spend most of their time infree flight between brief collisions at which instancesthe molecules’ direction and speed abruptly change. Therandom molecular motions are responsible for gaseoustransport processes. In liquids, on the other hand, themolecules are closely packed though not fixed in oneposition. In essence, the liquid molecules are always ina collision state. Applying a shear force must create avelocity gradient so that the molecules move relativeto one another,ad infinitum as long as the stress isapplied. For liquids, momentum transport due to therandom molecular motion is negligible compared to thatdue to the intermolecular forces. The straining betweenliquid molecules causes some to separate from their

12Note that we distinguish between a fluid and a flow being compress-ible/incompressible. For example, theflow of the highly compressibleair can be either compressible or incompressible.

original neighbors, bringing them into the force field ofnew molecules. Across the plane of the shear stress, thesum of all intermolecular forces must, on the average,balance the imposed shear. Liquids at rest transmit onlynormal force, but when a velocity gradient occurs, the netintermolecular force would have a tangential component.

The incompressible Navier–Stokes equations describeliquid flows under most circumstances. Liquids, however,do not have a well advanced molecular-based theory asthat for dilute gases. The concept of mean free pathis not very useful for liquids and the conditions underwhich a liquid flow fails to be in quasi-equilibrium stateare not well defined. There is no Knudsen number forliquid flows to guide us through the maze. We do notknow, from first principles, the conditions under whichthe no-slip boundary condition becomes inaccurate, orthe point at which the stress–rate of strain relation or theheat flux–temperature gradient relation fails to be linear.Certain empirical observations indicate that those simplerelations that we take for granted occasionally fail toaccurately model liquid flows. For example, it has beenshown in rheological studies (Loose and Hess [99]) thatnon-Newtonian behavior commences when the strain rateapproximately exceeds twice the molecular frequency-scale

γ = ∂u

∂y≥ 2T−1 (61)

where the molecular time-scaleT is given by

T=[mσ 2

ε

]1/2

(62)

where m is the molecular mass, andσ and ε arerespectively the characteristic length- and energy-scalefor the molecules. For ordinary liquids such as water,this time-scale is extremely small and the thresholdshear rate for the onset of non-Newtonian behavioris therefore extraordinarily high. For high-molecular-weight polymers, on the other hand,m andσ are bothmany orders of magnitude higher than their respectivevalues for water, and the linear stress–strain relationbreaks down at realistic values of the shear rate.

The moving contact line when a liquid spreads ona solid substrate is an example where slip flow mustbe allowed to avoid singular or unrealistic behavior inthe Navier–Stokes solutions (Dussan and Davis [100];Dussan [101, 102]; Thompson and Robbins [103]). Otherexamples where slip-flow must be admitted includecorner flows (Moffatt [104]; Koplik and Banavar [77])and extrusion of polymer melts from capillary tubes(Pearson and Petrie [105]; Richardson [106]; Den [107]).

331

M. Gad-el-Hak

Existing experimental results of liquid flow in mi-crodevices are contradictory. This is not surprising giventhe difficulty of such experiments and the lack of a guid-ing rational theory. Pfahler et al. [43, 108], Pfahler [109]and Bau [110] summarize the relevant literature. Forsmall-length-scale flows, a phenomenological approachfor analyzing the data is to define anapparentviscos-ity µa calculated so that if it were used in the traditionalno-slip Navier–Stokes equations instead of the fluid vis-cosity µ, the results would be in agreement with ex-perimental observations. Israelachvili [111] and Gee etal. [112] found thatµa = µ for thin-film flows as long asthe film thickness exceeds 10 molecular layers (≈5 nm).For thinner films,µa depends on the number of molec-ular layers and can be as much as 105 times larger thanµ. Chan and Horn’s [113] results are somewhat different:the apparent viscosity deviates from the fluid viscosityfor films thinner than 50 nm.

In polar-liquid flows through capillaries, Migun andProkhorenko [114] report thatµa increases for tubessmaller than 1 µm in diameter. In contrast, Debye andCleland [115] reportµa smaller thanµ for paraffin flowin porous glass with average pore size several times largerthan the molecular length-scale. Experimenting with mi-crochannels ranging in depths from 0.5 µm to 50 µm,Pfahler et al. [43] found thatµa is consistently smallerthanµ for both liquid (isopropyl alcohol; silicone oil)and gas (nitrogen; helium) flows in microchannels. Forliquids, the apparent viscosity decreases with decreasingchannel depth. Other researchers using small capillariesreport thatµa is about the same asµ (Anderson andQuinn [116]; Tukermann and Pease [117, 118]; Tucker-mann [119]; Guvenc [120]; Nakagawa et al. [121]).

Very recently, Sharp [122] and Sharp et al. [123]asserted that, despite the significant inconsistencies in theliterature regarding liquid flows in microchannels, suchflows are well predicted by macroscale continuum theory.A case can be made to the contrary, however, as will beseen at the end of this section, and the final verdict on thiscontroversy is yet to come.

The above contradictory results point to the need forreplacing phenomenological models by first-principlesones. The lack of molecular-based theory of liquids —despite extensive research by the rheology and polymercommunities — leaves molecular dynamics simulationsas the nearest weapon to first-principles arsenal. MD sim-ulations offer a unique approach to checking the validityof the traditional continuum assumptions. However, aswas pointed out earlier, such simulations are limited toexceedingly minute flow extent.

Figure 8. Velocity profiles in a Couette flow geometry atdifferent interfacial parameters. All three profiles are for U =σ T−1, and h = 24.57σ . The dashed line is the no-slip Couette-flow solution. From Thompson and Troian [124].

Thompson and Troian [124] provide molecular dy-namics simulations to quantify the slip-flow boundarycondition dependence on shear rate. Recall the linearNavier boundary condition introduced earlier

-u|w = ufluid − uwall = Ls∂u

∂y

∣∣∣∣w

(63)

where Ls is the constant slip length, and∂u/∂y|wis the strain rate computed at the wall. The goal ofThompson and Troian’s simulations was to determinethe degree of slip at a solid–liquid interface as theinterfacial parameters and the shear rate change. In theirsimulations, a simple liquid underwent planar shear ina Couette cell as shown infigure 8. The typical cellmeasured 12.51× 7.22× h, in units of molecular length-scaleσ , where the channel depthh varied in the rangeof 16.71σ−24.57σ , and the corresponding number ofmolecules simulated ranged from 1.152 to 1.728. Theliquid is treated as an isothermal ensemble of sphericalmolecules. A shifted Lennard-Jones 6–12 potential isused to model intermolecular interactions, with energy-

332

Flow physics in MEMS

and length-scalesε andσ , and cut-off distancerc = 2.2σ :

V (r)= 4ε

[(r

σ

)−12

−(r

σ

)−6

−(rc

σ

)−12

+(rc

σ

)−6](64)

The truncated potential is set to zero forr > rc.

The fluid–solid interaction is also modeled with a trun-cated Lennard-Jones potential, with energy- and length-scalesεwf andσwf , and cut-off distancerc. The equilib-rium state of the fluid is a well-defined liquid phase char-acterized by number densityn = 0.81σ−3 and tempera-tureT = 1.1ε/k, wherek is the Boltzmann constant.

The steady state velocity profiles resulting fromThompson and Troian’s [124] MD simulations are de-picted in figure 8 for different values of the interfacialparametersεwf, σwf andnw. Those parameters, shownin units of the corresponding fluid parametersε, σ andn, characterize, respectively, the strength of the liquid–solid coupling, the thermal roughness of the interface andthe commensurability of wall and liquid densities. Themacroscopic velocity profiles recover the expected flowbehavior from continuum hydrodynamics with boundaryconditions involving varying degrees of slip. Note thatwhen slip exists, the shear rateγ no longer equalsU/h.The degree of slip increases (i.e. the amount of momen-tum transfer at the wall-fluid interface decreases) as therelative wall densitynw increases or the strength of thewall-fluid couplingσwf decreases; in other words whenthe relative surface energy corrugation of the wall de-creases. Conversely, the corrugation is maximized whenthe wall and fluid densities are commensurate and thestrength of the wall–fluid coupling is large. In this case,the liquid feels the corrugations in the surface energyof the solid owing to the atomic close-packing. Con-sequently, there is efficient momentum transfer and theno-slip condition applies, or in extreme cases, a ‘stick’boundary condition takes hold.

Variations of the slip lengthLs and viscosityµ asfunctions of shear rateγ are shown in parts (a) and (b) offigure 9, for five different sets of interfacial parameters.For Couette flow, the slip length is computed from itsdefinition, -u|w/γ = (U/γ − h)/2. The slip length,viscosity and shear rate are normalized in the figure usingthe respective molecular scales for lengthσ , viscosityεTσ−3, and inverse timeT −1. The viscosity of the fluidis constant over the entire range of shear rates (figure 9b),indicating Newtonian behavior. As indicated earlier, non-Newtonian behavior is expected forγ ≥ 2T−1, well

Figure 9. Variation of slip length and viscosity as functions ofshear rate. From Thompson and Troian [124].

above the shear rates used in Thompson and Troian’ssimulations.

At low shear rates, the slip length behavior is con-sistent with the Navier model, i.e. is independent of theshear rate. Its limiting valueL0

s ranges from 0 to∼17σfor the range of interfacial parameters chosen (figure 9a).In general, the amount of slip increases with decreasingsurface energy corrugation. Most interestingly, at highshear rates the Navier condition breaks down as the sliplength increases rapidly withγ . The critical shear-ratevalue for the slip length to diverge,γc, decreases as thesurface energy corrugation decreases. Surprisingly, theboundary condition is nonlinear even though the liquidis still Newtonian. In dilute gases, the linear slip condi-tion and the Navier–Stokes equations, with their linearstress–strain relation, are both valid to the same order ofapproximation in Knudsen number. In other words, devi-ation from linearity is expected to take place at the samevalue ofKn = 0.1. In liquids, in contrast, the slip lengthappears to become nonlinear and to diverge at a criticalvalue of shear rate well below the shear rate at which thelinear stress–strain relation fails. Moreover, the bound-ary condition deviation from linearity is not gradual but

333

M. Gad-el-Hak

Figure 10. Universal relation of slip length as a function ofshear rate. From Thompson and Troian [124].

is rather catastrophic. The critical value of shear rateγcsignals the point at which the solid can no longer impartmomentum to the liquid. This means that the same liquidmolecules sheared against different substrates will expe-rience varying amounts of slip and vice versa.

Based on the above results, Thompson and Troian[124] suggest a universal boundary condition at a solid–liquid interface. Scaling the slip lengthLs by its asymp-totic limiting valueL0

s and the shear rateγ by its criticalvalueγc, collapses the data in the single curve shown infigure 10. The data points are well described by the rela-tion

Ls = L0s

[1− γ

γc

]−1/2

(65)

The nonlinear behavior close to a critical shear rate sug-gests that the boundary condition can significantly affectflow behavior at macroscopic distances from the wall.Experiments with polymers confirm this observation (At-wood and Schowalter [125]). The rapid change in the sliplength suggests that for flows in the vicinity ofγc, smallchanges in surface properties can lead to large fluctua-tions in the apparent boundary condition. Thompson andTroian [124] conclude that the Navier slip condition isbut the low-shear-rate limit of a more generalized univer-sal relationship which is nonlinear and divergent. Theirrelation provides a mechanism for relieving the stress sin-gularity in spreading contact lines and corner flows, as itnaturally allows for varying degrees of slip on approachto regions of higher rate of strain.

To place the above results in physical terms, considerwater13 at a temperature ofT = 288 K. The energy-scalein the Lennard-Jones potential is thenε = 3.62·10−21 J.For water,m = 2.99·10−26 kg, σ = 2.89·10−10 m, andat standard temperaturen = 3.35·1028 molecules·m−3.The molecular time-scale can thus be computed,T =[mσ 2/ε]1/2 = 8.31·10−13 s. For the third case depicted infigure 10(the open squares),γcT = 0.1, and the criticalshear rate at which the slip condition diverges is thusγc =1.2·1011 s−1. Such an enormous rate of strain14 may befound in extremely small devices having extremely highspeeds. On the other hand, the conditions to achieve ameasurable slip of 17σ (the solid circles infigure 9) arenot difficult to encounter in microdevices: density of solidfour times that of liquid, and energy-scale for wall–fluidinteraction that is one fifth of energy-scale for liquid.

The limiting value of slip length is independent of theshear rate and can be computed for water asL0

s = 17σ =4.91·10−9 m. Consider a water microbearing having ashaft diameter of 100 µm and rotation rate of 20 000 rpmand a minimum gap ofh = 1 µm. In this case,U =0.1 m·s−1 and the no-slip shear rate isU/h = 105 s−1.When slip occurs at the limiting value just computed,the shear rate and the wall slip-velocity are computed asfollows

γ = U

h+ 2L0s

= 9.90·104 s−1 (66)

∆u|w = γ Ls = 4.87·10−4 m·s−1 (67)

As a result of the Navier slip, the shear rate is reducedby 1 % from its no-slip value, and the slip velocity at thewall is about 0.5 % ofU , small but not insignificant.

9. SURFACE PHENOMENA

The surface-to-volume ratio for a machine with acharacteristic length of 1 m is 1 m−1, while that for aMEMS device having a size of 1 µm is 106 m−1. Themillion-fold increase in surface area relative to the massof the minute device substantially affects the transport

13Water molecules are complex ones, forming directional, short-range covalent bonds. Thus requiring a more complex potential thanthe Lennard-Jones to describe the intermolecular interactions. For thepurpose of the qualitative example described here, however, we use thecomputational results of Thompson and Troian [124] who employed theLennarg-Jones potential.14Note however thatγc for high-molecular-weight polymers would

be many orders of magnitude smaller than the value developed here forwater.

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Flow physics in MEMS

of mass, momentum and energy through the surface.Obviously surface effects dominate in small devices.The surface boundary conditions in MEMS flows havealready been discussed earlier. In microdevices, it hasbeen shown that it is possible to have measurable slip-velocity and temperature jump at a solid–fluid interface.In this section, we illustrate other ramifications of thelarge surface-to-volume ratio unique to MEMS, andprovide a molecular viewpoint to surface forces.

In microdevices, both radiative and convective heatloss/gain are enhanced by the huge surface-to-volume ra-tio. Consider a device having a characteristic lengthLs.Use of the lumped capacitance method to compute therate of convective heat transfer, for example, is justifiedif the Biot number (≡ hLs/κs, whereh is the convectiveheat transfer coefficient of the fluid andκs is the ther-mal conductivity of the solid) is less than 0.1. SmallLsimplies small Biot number, and a nearly uniform temper-ature within the solid. Within this approximation, the rateat which heat is lost to the surrounding fluid is given by

ρsL3scs

dT

dt= −hL2

s(Ts − T∞) (68)

whereρs andcs are respectively the density and specificheat of the solid,Ts is its (uniform) temperature, andT∞ is the ambient fluid temperature. Solution of theabove equation is trivial, and the temperature of a hotsurface drops exponentially with time from an initialtemperatureTi ,

Ts(t)− T∞Ti − T∞

= exp

[− t

T

](69)

where the time constantT is given by

T= ρsL3scs

hL2s

(70)

For small devices, the time it takes the solid to cooldown is proportionally small. Clearly, the million-foldincrease in surface-to-volume ratio implies a proportionalincrease in the rate at which heat escapes. Identicalscaling arguments can be made regarding mass transfer.

Another effect of the diminished scale is the increasedimportance of surface forces and the waning importanceof body forces. Based on biological studies, Went [126]concludes that the demarkation length-scale is around1 mm. Below that, surface forces dominate over gravi-tational forces. A 10-mm piece of paper will fall downwhen gently placed on a smooth, vertical wall, while a0.1-mm piece will stick. Try it!Stictionis a major prob-lem in MEMS applications. Certain structures such as

long, thin polysilicon beams and large, thin comb driveshave a propensity to stick to their substrates and thus failto perform as designed (Mastrangelo and Hsu [127]; Tanget al. [128]).

Conventional dry friction between two solids in rel-ative motion is proportional to the normal force whichis usually a component of the moving device weight.The friction is independent of the contact-surface areabecause the van der Waals cohesive forces are negligi-ble relative to the weight of the macroscopic device. InMEMS applications, the cohesive intermolecular forcesbetween two surfaces are significant and the stiction isindependent of the device mass but is proportional toits surface area. The first micromotor did not move —despite large electric current through it — until the con-tact area between the 100-µm rotor and the substrate wasreduced significantly by placing dimples on the rotor’ssurface (Fan et al. [129, 130], Tai and Muller [131]).

One last example of surface effects that to my knowl-edge has not been investigated for microflows is the ad-sorbed layer in gaseous wall-bounded flows. It is wellknown (Brunauer [132]; Lighthill [54]) that when a gasflows in a duct, the gas molecules are attracted to thesolid surface by the van der Waals and other forces of co-hesion. The potential energy of the gas molecules dropson reaching the surface. The adsorbed layer partakes thethermal vibrations of the solid, and the gas molecules canonly escape when their energy exceeds the potential en-ergy minimum. In equilibrium, at least part of the solidwould be covered by a monomolecular layer of adsorbedgas molecules. Molecular species with significant partialpressure — relative to their vapor pressure — may locallyform layers two or more molecules thick. Consider, forexample, the flow of a mixture of dry air and water vaporat STP. The energy of adsorption of water is much largerthan that for nitrogen and oxygen, making it more dif-ficult for water molecules to escape the potential energytrap. It follows that the life time of water molecules in theadsorbed layer significantly exceeds that for the air mole-cules (by 60 000 folds, in fact) and, as a result, the thinsurface layer would be mostly water. For example, if theproportion of water vapor in the ambient air is 1:1 000(i.e. very low humidity level), the ratio of water to air inthe adsorbed layer would be 60:1. Microscopic rough-ness of the solid surface causes partial condensation ofthe water along portions having sufficiently strong con-cave curvature. So, surfaces exposed to non-dry air flowsare mainly liquid water surfaces. In most applications,this thin adsorbed layer has little effect on the flow dy-namics, despite the fact that the density and viscosity ofliquid water are far greater than those for air. In MEMSapplications, however, the layer thickness may not be an

335

M. Gad-el-Hak

insignificant portion of the characteristic flow dimensionand the water layer may have a measurable effect on thegas flow. A hybrid approach of molecular dynamics andcontinuum flow simulations or MD-Monte Carlo simula-tions may be used to investigate this issue.

It should be noted that recently, Majumdar and Mezic[133, 134] have studied the stability and rupture intodroplets of thin liquid films on solid surfaces. They pointout that the free energy of a liquid film consists of asurface tension component as well as highly nonlinearvolumetric intermolecular forces resulting from van derWaals, electrostatic, hydration and elastic strain interac-tions. For water films on hydrophilic surfaces such as sil-ica and mica, Majumdar and Mezic [133] estimate theequilibrium film thickness to be about 0.5 nm (2 mono-layers) for a wide range of ambient-air relative humidi-ties. The equilibrium thickness grows very sharply, how-ever, as the relative humidity approaches 100 %.

Majumdar and Mezic’s [133, 134] results open manyquestions. What are the stability characteristics of theirwater film in the presence of air flow above it? Wouldthis water film affect the accommodation coefficient formicroduct air flow? In a modern Winchester-type harddisk, the drive mechanism has a read/write head thatfloats 50 nm above the surface of the spinning platter. Thehead and platter together with the air layer in betweenform a slider bearing. Would the computer performancebe affected adversely by the high relative humidity on aparticular day when the adsorbed water film is no longer“thin”? If a microduct hauls liquid water, would the waterfilm adsorbed by the solid walls influence the effectiveviscosity of the water flow? Electrostatic forces canextend to almost 1 µm (the Debye length), and that lengthis known to be highly pH-dependent. Would the waterflow be influenced by the surface and liquid chemistry?Would this explain the contradictory experimental resultsof liquid flows in microducts discussed earlier?

The few examples above illustrate the importance ofsurface effects in small devices. From the continuumviewpoint, forces at a solid–fluid interface are the limitof pressure and viscous forces acting on a parallel ele-mentary area displaced into the fluid, when the displace-ment distance is allowed to tend to zero. From the mole-cular point of view, all macroscopic surface forces areultimately traced to intermolecular forces, which subjectis extensively covered in the book by Israelachvilli [135]and references therein. Here we provide a very brief in-troduction to the molecular viewpoint. The four forcesin nature are (1) the strong and (2) weak forces describ-ing the interactions between neutrons, protons, electrons,etc.; (3) the electromagnetic forces between atoms and

molecules; and (4) gravitational forces between masses.The range of action of the first two forces is around 10−5

nm, and hence neither concerns us overly in MEMS ap-plications. The electromagnetic forces are effective over amuch larger though still small distance on the order of theinteratomic separations (0.1–0.2 nm). Effects over longerrange — several orders of magnitude longer — can anddo rise from the short range intermolecular forces. Forexample, the rise of liquid column in capillaries and theaction of detergent molecules in removing oily dirt fromfabric are the result of intermolecular interactions. Grav-itational forces decay with the distance to second power,while intermolecular forces decay much quicker, typi-cally with the seventh power. Cohesive forces are there-fore negligible once the distance between molecules ex-ceeds few molecular diameters, while massive bodies likestars and planets are still strongly interacting, via gravity,over astronomical distances.

Electromagnetic forces are the source of all intermole-cular interactions and the cohesive forces holding atomsand molecules together in solids and liquids. They canbe classified into (1) purely electrostatic arising from theCoulomb force between charges, interactions betweencharges, permanent dipoles, quadrupoles, etc.; (2) po-larization forces arising from the dipole moments in-duced in atoms and molecules by the electric field ofnearby charges and permanent dipoles; and (3) quantummechanical forces that give rise to covalent or chemi-cal bonding and to repulsive steric or exchange inter-actions that balance the attractive forces at very shortdistances. The Hellman–Feynman theorem of quantummechanics states that once the spatial distribution ofthe electron clouds has been determined by solving theappropriate Schrödinger equation, intermolecular forcesmay be calculated on the basis of classical electrostatics,in effect reducing all intermolecular forces to Coulom-bic forces. Note however that intermolecular forces ex-ist even when the molecules are totally neutral. Solutionsof the Schrödinger equation for general atoms and mole-cules are not easy of course, and alternative modeling aresought to represent intermolecular forces. The van derWaals attractive forces are usually represented with a po-tential that varies as the inverse-sixth power of distance,while the repulsive forces are represented with either apower or an exponential potential.

A commonly used potential between two molecules isthe generalized Lennard-Jones 6–12 pair potential givenby

Vij (r)= 4ε

[cij

(r

σ

)−12

− dij(r

σ

)−6](71)

336

Flow physics in MEMS

whereVij is the potential energy between two particlesiand j , r is the distance between the two molecules,ε

andσ are respectively characteristic energy- and length-scales, andcij anddij are parameters to be chosen forthe particular fluid and solid combinations under con-sideration. The first term in the right-hand side is thestrong repulsive force that is felt when two moleculesare at extremely close range comparable to the molecularlength-scale. That short-range repulsion prevents over-lap of the molecules in physical space. The second termis the weaker, van der Waals attractive force that com-mences when the molecules are sufficiently close (severaltimesσ ). That negative part of the potential representsthe attractive polarization interaction of neutral, spheri-cally symmetric particles. The power of 6 associated withthis term is derivable from quantum mechanics consider-ations, while the power of the repulsive part of the poten-tial is found empirically. The Lennard-Jones potential iszero at very large distances, has a weak negative peak atr

slightly larger thanσ , is zero atr = σ , and is infinite asr → 0.

The force field resulting from this potential is given by

Fij (r)= −∂Vij∂r

= 48ε

σ

[cij

(r

σ

)−13

− dij

2

(r

σ

)−7](72)

A typical Lennard-Jones 6–12 potential and force fieldare shown infigure 11, for c = d = 1. The minimumpotential Vmin = −ε corresponds to the equilibriumposition (zero force) and occurs atr = 1.12σ . Theattractive van der Waals contribution to the minimumpotential is−2ε, while the repulsive energy contributionis +ε. Thus, the inverse 12th-power repulsive forceterm decreases the strength of the binding energy atequilibrium by 50 %.

The Lennard-Jones potential is commonly used inmolecular dynamics simulations to model intermolecularinteractions between dense gas or liquid molecules andbetween fluid and solid molecules. As mentioned earlier,such potential is not accurate for complex substancessuch as water whose molecules form directional covalentbonds. As a result, MD simulations for water are muchmore involved.

10. PARTING REMARKS

The forty-year-old vision of Richard Feynman ofbuilding minute machines is now a reality. Microelectro-

Figure 11. Typical Lennard-Jones 6–12 potential and theintermolecular force field resulting from it. Only a small portionof the potential function is shown for clarity.

mechanical systems have witnessed explosive growthduring the last decade and are finding increased appli-cations in a variety of industrial and medical fields. Thephysics of fluid flows in microdevices has been exploredin this paper. While we now know considerably morethan we did just few years ago, much physics remainsto be explored so that rational tools can be developed forthe design, fabrication and operation of MEMS devices.

The traditional Navier–Stokes model of fluid flowswith no-slip boundary conditions works only for a certainrange of the governing parameters. This model basicallydemands two conditions:

(1) The fluid is a continuum, which is almost alwayssatisfied as there are usually more than 1 million mole-cules in the smallest volume in which appreciable macro-scopic changes take place. This is the molecular chaosrestriction; and

(2) The flow is not too far from thermodynamic equi-librium, which is satisfied if there is sufficient numberof molecular encounters during a time period small com-pared to the smallest time-scale for flow changes. Duringthis time period the average molecule would have moveda distance small compared to the smallest flow length-scale.

For gases, the Knudsen number determines the degreeof rarefaction and the applicability of traditional flowmodels. As Kn → 0, the time- and length-scales ofmolecular encounters are vanishingly small comparedto those for the flow, and the velocity distribution ofeach element of the fluid instantaneously adjusts to theequilibrium thermodynamic state appropriate to the localmacroscopic properties as this molecule moves throughthe flowfield. From the continuum viewpoint, the flowis isentropic and heat conduction and viscous diffusion

337

M. Gad-el-Hak

and dissipation vanish from the continuum conservationrelations, leading to the Euler equations of motion. Atsmall but finiteKn, the Navier–Stokes equations describenear-equilibrium, continuum flows.

Slip flow must be taken into account forKn> 0.001.The slip boundary condition is at first linear in Knudsennumber then nonlinear effects take over beyond a Knud-sen number of 0.1. At the same transition regime, i.e.0.1< Kn< 10, the linear stress–rate of strain and heatflux–temperature gradient relations — needed to close theNavier–Stokes equations — also break down, and alter-native continuum equations (e.g., Burnett or higher-orderequations) or molecular-based models must be invoked.In the transition regime, provided that the dilute gas andmolecular chaos assumptions hold, solutions to the dif-ficult Boltzmann equation are sought, but physical sim-ulations such as Monte Carlo methods are more readilyexecuted in this range of Knudsen number. In the free-molecule flow regime, i.e.Kn> 10, the nonlinear col-lision integral is negligible and the Boltzmann equationis drastically simplified. Analytical solutions are possi-ble in this case for simple geometries and numerical inte-gration of the Boltzmann equation is straightforward forarbitrary geometries, provided that the surface-reflectioncharacteristics are accurately modeled.

Gaseous flows are often compressible in microdeviceseven at low Mach numbers. Viscous effects can causesufficient pressure drop and density changes for theflow to be treated as compressible. In a long, constant-area microduct, all Knudsen number regimes may beencountered and the degree of rarefaction increases alongthe tube. The pressure drop is nonlinear and the Machnumber increases downstream, limited only by choked-flow condition.

Similar deviation and breakdown of the traditionalNavier–Stokes equations occur for liquids as well, butthere the situation is more murky. Existing experimentsare contradictory. There is no kinetic theory of liquids,and first-principles prediction methods are scarce. Mole-cular dynamics simulations can be used, but they are lim-ited to extremely small flow extents. Nevertheless, mea-surable slip is predicted from MD simulations at realisticshear rates in microdevices.

Much non-traditional physics is still to be learnedand many exciting applications of microdevices are yetto be discovered. The future is bright for this emergingfield of science and technology. Richard Feynman wasright about the possibility of building mite-size machines,but was somewhat cautious in forecasting that suchmachines, while “would be fun to make”, may or maynot be useful.

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