microfluidics (colin/microfluidics) || gaseous microflows

Chapter 2

Gaseous Microflows1

2.1. Continuum model and molecular model

A gas consists of a large number of discrete particles. Ideally, a gaseous flow would be examined by considering the positions, velocities and internal states of all molecules at all times. This examination would rely on modeling interactions between particles and interactions between particles and the walls (if any) that limit the flowfield. In practice, statistical considerations remove the need to actually consider all particles. Nevertheless, this microscopic or molecular model is generally demanding in terms of computational resources.

In many situations, a gas can be regarded as a continuous medium. This is the familiar macroscopic (or continuum) model, presented in most fluid mechanics textbooks. Compared with the molecular model, this one is simpler to use, but its range of validity is more limited.

In this chapter, we explain how microscopic and macroscopic quantities relate to each other. We give a molecular interpretation of concepts, such as pressure, temperature and velocity. We define the conditions under which the continuum model fails and must be replaced by the molecular one. In particular, gaseous flows in microsystems may require a molecular approach or, at least, a correction to the classic continuum model. Then we present the continuum and molecular models and finally give a number of examples, in particular an application to flow rate in a microchannel.

Chapter written by Jean-Claude LENGRAND and Tatiana G. ELIZAROVA.

Microfluidics Edited by Stéphane Colin © 2010 ISTE Ltd. Published 2010 by ISTE Ltd.

26 Microfluidics

The reader interested in situations more complex than those presented in the present chapter (gas mixtures, reactive flows, etc.) or in details of the mathematical developments can read, for example, books by Bird [BIR 98], Kogan [KOG 69] or Brun [BRU 06].

2.1.1. Molecular quantities

A gas consists of particles (molecules, atoms, ions and electrons). For the sake of convenience, they all will be referred to as molecules. In a simple gas, all molecules are identical. A mole consists of N molecules whose molecular mass is m, where N is Avogadro’s number. Therefore, the molar mass is mM = N . The number of molecules per unit volume (or number density) is denoted n and the usual gas density is ρ = n m. The Boltzmann constant k is equivalent to the universal gas constant R, at the molecular level rather than at the molar level: =k R/N = 1.380658 10-23 J K-1. The perfect-gas constant per unit mass /= =R k mR/ M depends on gas nature.

Each molecule moves according to its velocity vector c (see Figure 2.1). For a simple gas, the average velocity of molecules in a small volume around a given point is u = < c >. This is the local macroscopic velocity of the gas, which is considered in the continuum approach. The thermal velocity specific to each molecule is defined as c' = c – u. Its average value is zero: < c' > = < c > – u = u – u = 0.

u

cc'

(a)

u

cc'

(b)

'cuc +=

Figure 2.1. Schematic view of velocity vectors in subsonic (a) and supersonic (b) flows

As will be seen later (equation [2.25]), the most probable value of c' (magnitude of vector c') in an equilibrium gas is: ( ) 2/12' RTc m = , an expression that resembles

that of the speed of sound in a perfect-gas: ( ) 2/1RTa γ= , where γ is the specific

Gaseous Microflows 27

heat ratio and T the thermodynamic temperature. The ratio ( ) 2/12/'/ RTucus m == is called the molecular speed ratio and is close to the Mach number:

( )1/ 2/ /Ma u a u RTγ= = . Schematically, in a subsonic flow, the local velocity of molecules is the sum of a small flow velocity and a large thermal speed (see Figure 2.1a). In contrast, in a supersonic flow it is the sum of a large flow velocity and a small thermal speed (see Figure 2.1b). In a hypersonic flow, all molecules at a given location have nearly identical velocities ( uc ≈ ).

Solving a problem of gas dynamics at a molecular level using a deterministic approach would require handling a tremendous quantity of information (positions, velocities, internal states, etc., of a very large number of molecules). This is far beyond the capabilities of computers. Furthermore, the initial and boundary conditions are not reproducible at a molecular level and cannot be prescribed in a deterministic way. This is why the molecular treatment of gas dynamics relies essentially on statistical or probabilistic approaches rather than on deterministic ones. Let us consider a scalar (respective vector) quantity Q attached to molecules. A distribution function fQ relative to Q is defined as the fraction dn/n of the population whose value of Q is equal to a given value Q0 within a unit interval (respective of a unit volume).

/ with 1Q QQdn n f dQ f dQ= =∫ , [2.1]

The integral covers all possible values of Q. Note that dQfQ is also the

probability that the value of Q for a given molecule will be equal to Q0 within dQ. The average value of any quantity F(Q) that depends on Q is given by:

( ) ( ) / ( ) .QQ QF Q F Q dn n F Q f dQ= =∫ ∫ [2.2]

Thus, the average velocity u of a population of molecules is calculated as

f d= ∫ ccu c c , where fc is the distribution function relative to the velocity-vector c.

Distribution functions usually depend on time t, on location r in physical space, and on the quantity Q under consideration.

2.1.2. Dilute gas

The mean volume available for a molecule is 1/n and the mean molecular spacing is δ = 1/n1/3. The molecular diameter, d, characterizes the range of

28 Microfluidics

intermolecular forces. If molecules are modeled as hard spheres, d is their diameter. A gas that satisfies the condition:

3or 1d n dδ >> << [2.3]

is said to be dilute. The gas itself occupies only a small fraction (typically (d/δ)3) of space. Most of the time, molecules are not submitted to intermolecular forces. They travel at constant speeds along straight lines. Occasionally, they enter the interaction zone of another molecule and a collision takes place. For molecules other than hard spheres, the intermolecular force tends asymptotically to zero when the intermolecular distance tends to infinity. Thus, the interaction zone is theoretically infinite. We will see in section 2.1.3 how it is possible to define a finite value of d for such molecules.

A dilute gas, with well-identified collisions, contrasts with situations where molecules are in constant or frequent interaction with one another, such as a liquid or a real gas at high density. As will be shown later (equation [2.9]) a dilute gas obeys the perfect gas state equation. The concept of dilute gas coincides with the concept of perfect gas defined in thermodynamics. Aerodynamicists frequently call real gas effects manifestations of thermodynamic non-equilibrium or variations of R due to variations of chemical composition in reactive flows. In fact these so-called “real gas effects” are compatible with the concept of a perfect or dilute gas. To avoid confusion, we will speak of a dilute gas rather than of a perfect gas if condition [2.3], and consequently the equation of state [2.9] is satisfied. In the present chapter, we will only consider dilute gases.

1

2

3 σ33

+

-

+

-

c'

c'1

c'2

c'3

(a) (b)

σ32

σ31

σ

Figure 2.2. (a) Stress tensor; and (b) momentum transfer


In the theory of continuous media, we define a local stress tensor (see Figure 2.2a). For a surface element whose normal is oriented in direction i (i = 3 in the figure), σij is the projection on direction j of the stress σ exerted through the unit area from the fluid located on the positive side to the fluid located on the negative side of the surface. In a gas at rest (u = 0, thus c = c'), this stress is due to molecules crossing the surface element in direction i with their momentum m c' (see Figure 2.2b). The transfer in direction i of j-oriented momentum is equal to:

∫ == ' ''/''c jijiij ccmnndnccmnp . [2.4]

pij is the j-component of a force per unit area exerted by the negative side on the positive side. Thus pij is just the negative of σij (pij = -σij). We have given the molecular interpretation of the stress tensor. In a flowing gas ( 0≠u ), we keep

equation [2.4] as a definition of pij. Note that the quantity ji ccmn would include

the momentum associated with the macroscopic flow velocity:

( ) ( ) ,i j i i j j i j ijn m c c n m u c u c u uρ σ′ ′⟨ ⟩ = ⟨ + + ⟩ = − [2.5]

where we recognize an expression that is used when applying the principle of conservation of momentum in fluid mechanics.

The scalar pressure p is defined as the average value of the diagonal terms of tensor p, which is consistent with the definition of continuous media theory:

( ) 2 2 2 2 21 2 3

1 1 1( )

3 3 3p n m c c c n m c cρ′ ′ ′ ′ ′= ⟨ + + ⟩ = ⟨ ⟩ = ⟨ ⟩ . [2.6]

The normal force Fn exerted by the gas to the unit area of a wall is not necessarily equal to the pressure p close to the wall. If the normal is oriented in the i-

direction, Fn is equal to 2icmn , which in the general case differs from the

definition given by equation 2.6. In the case of a full equilibrium between gas and wall, the gas close to the wall is at rest (c = c'), the velocity distribution is isotropic, and reflected molecules seem to come from the continuation of the gas behind the wall. Only in this situation do Fn and p coincide.

The translational temperature trT is a measure of the average translational

energy of molecules, tre . It is defined as:

30 Microfluidics

23 12 2tr trk T m c e′= ⟨ ⟩ = ⟨ ⟩ . [2.7]

Molecules (at least di- or polyatomic molecules) can have one or more forms of internal energy inte (electronic, rotational or vibrational) associated with ζ internal degrees of freedom. If the molecules are completely excited, each of them can be characterized by an internal temperature ,int iT defined by:

, ,1

.2 i int i int ik T eζ = [2.8]

For rotation, ζ is equal to 0, 2 or 3, depending on the mono-, di- or polyatomic character of the molecule, respectively. Here a polyatomic molecule whose atoms are aligned is considered to be a diatomic molecule.

Vibration is usually not fully excited. A vibrational temperature can be defined from the average vibrational energy of molecules distributed over the vibrational energy levels. Then equation [2.8] can be used to define the number of vibrational degrees of freedom. This number is generally a non-integer. If the temperature is sufficiently low, electronic and vibrational energies can be neglected.

From classic thermodynamics, we know that a gas in equilibrium is characterized by a unique temperature T and its energy per unit mass is distributed according to (1 / 2)RT per degree of freedom. The average translational energy of molecules is then (3 / 2) kT and their average rotational energy is: ( / 2)rot RTζ . Thus, the above definitions of and tr intT T are consistent with the classic concept of temperature. More generally, in a non-equilibrium gas, a different temperature can be associated with each form of energy. We can even define a translational temperature specific to each direction in the 3D space. If a gas is in equilibrium, all temperatures coincide with the thermodynamic temperature. If not, their differences are a measure of the various non-equilibriums.

Combining equations [2.6] and [2.7], we obtain:

( ) ( ) ,tr tr trp n k T n m k m T RTρ= = = [2.9]

where we recognize the equation state of the perfect gas. Thus, this equation is still valid in a non-equilibrium situation if the temperature T is replaced by the translational temperature.


An equilibrium situation is used to define the specific heats at constant volume cv and constant pressure cp, by considering the derivative of the mean molecular energy with respect to T. Assuming all forms of energy to be fully excited, ζ is a constant (independent of T) and we get cv and cp. The specific heat ratio γ = cp / cv easily relates to ζ:

( ) 5 5 33 ; ; .

2 3 1p v

vv v

ck m c k mc

c cζ γζ γ ζζ γ

+ + −= + = = = =

+ − [2.10]

2.1.3. Collisions and mean free path

In a dilute gas, all collisions can be considered to be binary. The total collision cross-section is defined by: 2

T dσ π= . In general, it depends on the magnitude

r rc = c of the relative velocity between molecules ( 1 2 1 2r ′ ′= − = −c c c c c ). A collision occurs when the centre of a molecule crosses the cross-section attached to another one (see Figure 2.3). The collision frequency ν is the mean number of collisions suffered by a given molecule during the unit time. It is equal to:

( ) ( )T r r T r rn c c dn n n c cν σ σ= = ⟨ ⟩∫c . [2.11]

d d(b)(a)

σT

σT

σT = π d 2

Figure 2.3. Total collision cross-section and collision frequency

The mean collision time τc = 1/ν is the mean time between collisions of a given molecule. The mean free path characterizes the distance travelled by a molecule between successive collisions in a frame of reference that moves at velocity u:

( )' ' ' ( )c T r rc c c n c cλ τ ν σ= ⟨ ⟩ = ⟨ ⟩ = ⟨ ⟩ ⟨ ⟩ . [2.12]

32 Microfluidics

In a hard-sphere gas, and anticipating that ' 2rc c⟨ ⟩ ⟨ ⟩ = (equations [2.26] and [2.29]), we get:

( )21 2 n dλ π= . [2.13]

Another expression will be given later (equations [2.70] and [2.72]) for a more general gas model. For qualitative considerations, equation [2.13] is sufficient to compare the mean free path λ with the mean molecular spacing δ. Taking into account 13 <<dn , we get 1/ >>δλ , thus the hierarchy λδ <<<<d .

The post-collision velocities of molecules depend on their pre-collision velocities. Let us consider an elastic collision (i.e. a collision with no exchange of internal energy) between molecule 1 and molecule 2. Their initial velocities are 1c and 2c . Their center of mass has the velocity: 1 2( ) / 2m = +c c c . The relative pre-

collision velocity is: 1 2m = −c c c . In the same way, we can define *mc and *

rc

based on post-collision velocities *1c and *

2c . The conservation of momentum and

translational energy results in a vector equality m m= *c c and a scalar

equality *r rc c= . Thus, during an elastic collision, the velocity of the center of mass

is unchanged, as well as the magnitude of the relative velocity. The pre-collision velocities can be expressed as 1 2m r= +c c c and 2 2m r= −c c c . In the center of mass frame of reference, they are antiparallel and define a plane that contains the intermolecular forces. Molecules remain in this plane during the collision. The plane also contains antiparallel post-collision velocities (see Figure 2.4).

χ

b

2/rc−

χ

b

1

2

2/rc

2/rc−

2/rc

Figure 2.4. Intermolecular collision in the center of the mass frame of reference


The miss-distance or impact parameter b between pre-collision velocities is found again between post-collision velocities due to the conservation of angular momentum. During collision, the relative velocity of molecules deviates by an angle χ that depends on the impact parameter b. The function ( )bχ depends on the function ( )F r that relates the intermolecular force F to the distance r between molecules. A frontal collision is characterized by (0)χ π= .

Realistic functions ( )F r tend to zero when r tends to infinity. Thus large values of b result in negligibly small deviations of molecules: ( ) 0bχ → ∞ = . A small limiting angle limχ can be chosen and by convention all collisions resulting in

limχ χ< will be ignored. The corresponding value of b defines a molecular

diameter d by lim( )dχ χ= and a total collision cross-section 2T dσ π= . In

general, both d and Tσ depend on the magnitude of the relative velocity rc .

For hard-sphere molecules, d and Tσ are constant and the relation between b

and χ results from purely geometrical considerations: ( )cos 2b d χ= or:

( ) ( )2 1 cos 2 .b d χ= + [2.14]

In a purely deterministic approach to collisions, each collision would be calculated by equation [2.14], using the particular value of b for that collision. In a probabilistic approach, for a given value of rc , we consider b to be a random parameter in the interval [0,d]. The probability b will be smaller than a given value

0b is equal to 20( / )b d . Thus 2( / )b d is distributed evenly between 0 and 1.

Taking into account equation [2.14], cos χ is distributed evenly between -1 and +1. The corresponding solid angle 2 (1 cos )π χΩ = − is distributed evenly between 4π and 0. Thus, for hard-sphere molecules all directions in space have an equal probability to be the direction of the post-collision relative velocity r

*c .

2.1.4. Limits of the continuum model and concept of rarefaction

The continuum model used in the classic theory of gas dynamics consists of locally expressing conservation laws and assuming some expressions for the local transport properties (e.g. viscosity). Thus in a Newtonian fluid, the viscous stress tensor depends linearly on the deformation rate tensor. When the flow is sufficiently rarefied, conservation laws still hold but it is no longer possible to relate transport properties to a few local macroscopic quantities and their derivatives. The

34 Microfluidics

breakdown of the continuum approach is easily demonstrated on the simple example of a Couette flow (see Figure 2.5).

U

u L

xy

A

B-+ σyx

U

u = U/2

(a) (b)

U

(с)

us

Figure 2.5. Velocity profile in a Couette flow: a) continuum regime; b) free molecule regime; c) transition regime

Let us consider a gas between two walls, both normal to direction y. The lower wall ( 0y = ) is at rest, while the upper one ( y L= ) moves with velocity

( , 0, 0)U=U . U is assumed to remain moderate (i.e. the Mach number remains small) for the incompressible approximation to be valid. In the classic formulation of the continuum model, we write:

( )yx xu yσ μ= ∂ ∂ , [2.15]

where the viscosity coefficient μ depends only on local flow parameters. The x-component of local velocity u varies from 0 to U when y varies from 0 to L (see Figure 2.5a).

Let us now decrease the gas density to such an extent that the mean free path (equation [2.13]) becomes much larger than L. Molecules will just travel from one wall to the other, without colliding with each other. This is the free molecule regime. All locations within the flow are equivalent: there is no gradient in the flow. Under some hypotheses relative to the wall interaction mechanism, half of the population of any volume element comes directly from the upper wall and its mean velocity is

xu U= while the other half comes directly from the lower wall with a mean velocity of 0xu = . The velocity vector has a uniform distribution ( )2, 0, 0U=u (see Figure 2.5b). Obviously, equation [2.15] fails. It would result in zero shear stress because 0xu y∂ ∂ = while there is clearly a momentum exchange: the gas exerts a force in x-direction on the lower wall due to the difference in momentum between incident and re-emitted molecules. The concept of viscosity fails in this situation where the gas is far from an equilibrium state. Due to the lack of intermolecular collisions, the gas is just a mixture of two populations whose properties are defined by each wall and which ignore each other.


The failure of the continuum model is due to rarefaction and this example suggests characterizing rarefaction by a Knudsen number, defined as:

Kn Lλ= . [2.16]

When Kn is small (smaller than, say, 0.01), local equilibrium is ensured, which ensures the continuum regime. When Kn is large (larger than, say, 10), intermolecular collisions can be neglected, which ensures the free molecule regime. The free molecule regime is rather simple to analyze, because intermolecular collisions need not be considered. For intermediate values of Kn, the transition regime holds (see Figure 2.5c). The difference between the gas velocity at the walls and the walls’ velocity is the velocity slip. Similarly, a temperature jump is observed in flows with thermal gradients.

To get a qualitative interpretation of velocity slip and temperature jump we note that half of the molecules that are present near a wall come directly from a region located at a distance λ from the wall; a region where macroscopic parameters differ from the wall conditions. A crude estimate yields an expression for the velocity slip:

( )s xu u yλ= ∂ ∂ , [2.17]

an expression already proposed by Maxwell. Expressions that are more accurate will be given later (equations [2.51], [2.53]) and discussed in section 2.5.3.1. In non-rarefied flows, λ is very small compared with the scale length of the gradients, and equation [2.17] is equivalent to the usual boundary condition 0su = .

In moderately rarefied flows (say, 0.01 0.1Kn< < ), we can keep using the continuum approach (e.g. Navier-Stokes equations), provided the classic boundary conditions can be replaced by slip conditions like [2.17]. This range of conditions is called the slip regime. Introducing higher-order dissipative terms is an additional way to extend the validity range of the continuum approach (section 2.3.1.1). However, in more rarefied conditions (say, 0.5Kn > ), a molecular approach must be substituted for the continuum approach.

A classification of flow regimes is given in Table 2.1. It is suitable for a microchannel, if the Knudsen number is based on the smallest transverse dimension H of the channel. The limits between regimes are indicative only. They depend on the accuracy required and on the flow quantity under consideration. They are indicated in terms of Kn, and also in terms of H, assuming the flowing gas to be air under normal conditions, corresponding to a mean free path approximately equal to 0.05 µm. Microfluidic applications are relevant to the continuum regime or to the slip regime if H is larger than 0.5 µm. However, a molecular approach is required for smaller channels or lower densities.

36 Microfluidics

0.01Kn < 5 μm < H Classic continuum regime: Navier-Stokes equations

0.01 0.5Kn< < 0.1 μm < H < 5 μm Slip regime: Navier-Stokes equations with modified boundary conditions

0.5 10Kn< < 0.005 μm < H < 0.1 μm Transition regime: molecular approach required

10 Kn< H < 0.005 μm Free molecule regime: simplified molecular approach

Table 2.1. Classification of rarefaction regimes for a Couette flow or a microchannel

Generalizing this classification to any flow is difficult when the mean free path changes largely throughout the flowfield and/or when the geometry exhibits several characteristic lengths. It is then necessary to return to the physical phenomenon induced by rarefaction, namely whether a local thermodynamic equilibrium exists or not. A local rarefaction parameter is defined by considering the relative variation of a macroscopic quantity Q over the average distance travelled by a molecule between two successive collisions. The gas will be in equilibrium if this variation is small.

In a subsonic flow, the molecular speed is essentially the thermal speed, with random orientation and magnitude. The distance travelled is: cc τ λ′⟨ ⟩ = , the mean free path. The local rarefaction parameter, or local Knudsen number, is therefore:

1 / / / cP c Q Q Q Q Lτ λ λ′= ⟨ ⟩ ∇ = ∇ = [2.18]

by defining a characteristic length /L Q Q= ∇ .

In a supersonic flow, the molecular speed is essentially the oriented velocity u of the flow and the distance travelled is cτu . Introducing θ as the angle between the flow direction and the gradient direction, the relative variation of Q is then:

2

1/ 2

1 1

cos cos

cos cos .2 8

c c mQ

P u s cQ L c L

s P Ma P

θ θλτ τ

π γ πθ θ

⋅ ∇ ′= = =′< >

⎛ ⎞= = ⎜ ⎟⎝ ⎠

u

[2.19]


Here we have anticipated the expressions of the most probable thermal speed mc′ [2.25] and the mean thermal speed c′ [2.26] in an equilibrium gas. 2P can be

regarded as (1/ ) (ln ) /D Q Dtν , often called Bird’s parameter. Depending on the

locally subsonic or supersonic nature of the flow, local equilibrium requires that 1P

or 2P respectively remains small. Some quantities require more collisions than

others to reach equilibrium. This property is characterized by a collision number QZ , whose value varies depending on the equilibrium under consideration. QZ

varies from a few units for translation or rotation to a few thousands for vibration or chemistry. Finally, a unique equilibrium criterion for both subsonic and supersonic cases can be proposed:

QP max(P ,P ) Kn max( ,Ma cos ) / Zθ= <<1 2 1 1 , [2.20]

where Kn and Ma are local quantities. Flow regimes can be classified as indicated in Table 2.2, where the limits between regimes are somewhat arbitrary.

510P −< Equilibrium flow: continuum model is valid, e.g. Navier-Stokes (NS) equations.

510 0.01P− < < So-called non-equilibrium flow: translational and rotational equilibriums are achieved but not vibrational and chemical equilibriums. The continuum model is used (e.g. NS), completed by relaxation equations for vibration and chemistry.

0.01 < P So-called rarefied flow: translational and rotational equilibriums are not achieved. A fortiori, vibrational and chemical equilibriums are also not achieved. Vibrational energy and chemical composition can even be frozen. Different regimes can be defined: 0.01 < P < 0.5 Moderately rarefied flow: the continuum model

can be used with modifications (velocity slip or higher-order dissipative terms).

0.5 < P < 10 Transition regime: molecular model required. 10 < P Collisionless flow: molecular model required (a

simplified one due to the absence of intermolecular collisions).

Table 2.2. Classification of flow regimes according to the local rarefaction parameter

Microfluidic applications with gases are generally characterized by low Mach numbers as well as by the absence of vibrational energy and chemistry. As a rule of

38 Microfluidics

thumb, they can be determined by a continuum model if the local Knudsen number is smaller than 0.1 at any point of the flowfield.

2.2. Molecular description of a flow

2.2.1. Equilibrium gas

2.2.1.1. Distribution functions

In a molecular description, and under the hypothesis of molecular chaos, a flow is entirely determined if the number density and the distribution functions of velocity and internal energies are known as functions of time t and location r. The Boltzmann equation governs the number of molecules that have given velocities and internal energies at time t and location r. It expresses that this number varies due to the displacement and collisions of molecules. Solving the Boltzmann equation would completely determine a flow, whatever its rarefaction level, as it does not require the hypothesis of local equilibrium. However, this approach is severely limited by computational requirements due to the large number of independent variables. In practice, numerically solving the Boltzmann equation is limited to simple geometries and gas models.

When applied to a homogenous gas, the Boltzmann equation describes its evolution towards an equilibrium state. When equilibrium is reached, it can be shown that the distribution function of velocities c or c' is:

3 3/2 2 20 ( / ) exp( ).f f f cβ π β ′= = = −c' c [2.21]

This particular form is the equilibrium or Maxwellian distribution function of velocity. It is denoted 0f and it characterizes the final equilibrium state obtained from any initial state. It depends on the magnitude c' of the thermal speed rather than on the velocity vectors c or c'. This is due to the isotropy of the problem in a frame of reference moving at flow velocity u. In a gas mixture, the Maxwellian distribution function holds for each species. In a gas with internal energy, it still describes the distribution of translational energy.

Parameter β simply relates to the translational temperature trT . We can actually calculate the average translational energy that is mentioned in equation [2.7] by integrating 2 / 2mc′ over all velocity space c'. We find 23 2 3 (4 )trk T m β= . Hence:

( )1/ 2 1/ 21 2 (2 )tr trk T m RTβ = = . [2.22]


Thus in an equilibrium gas, trT determines the expression of the velocity distribution function. In a non-equilibrium situation, trT is only a measure of the average translational energy.

In the same way, the rotational temperature rotT is a measure of the average rotational energy of molecules [2.8]. In an equilibrium gas, it also characterizes the distribution of molecules in discrete rotational energy levels (Boltzmann distribution). An approximation of the latter is given by a continuous distribution function (Hinshelwood distribution):

( )( / 2) 1 exp /( )rote rot rot rotf e e kTζ −∝ − . [2.23]

In an equilibrium gas, the fraction of molecules whose thermal speed c' has a magnitude in an interval dc' around c' is obtained by considering spherical coordinates ( )', ,c θ ϕ in velocity space and integrating 0f over θ and ϕ:

32 2 2 2

0 0 3 / 2 exp( ) sincdn

f dc c c d d dcn

π πϕ θ

β β θ θ ϕπ

′ = =′ ′ ′ ′= = −∫ ∫ .

Hence:

( ) ( )3 2 2 24 / expcf c cβ π β′ ′ ′= − . [2.24]

The distribution function cf ′ presents a maximum for:

( )1/ 21 / 2 ,m trc c RTβ′ ′= = = [2.25]

which is the most probable thermal speed. The mean thermal speed is obtained by:

( ) ( )1/20 2 / 8 /c trc c f dc RTβ π π∞

′′ ′ ′= ∫ = = . [2.26]

As an example, the distribution function cf ′ has been plotted in Figure 2.6 for helium at temperatures of 200 K and 800 K. If we use reduced (non-dimensional)

40 Microfluidics

velocities 1/ 2/(2 )c c RTβ ′ ′= , we can define the distribution function of βc' by / ( )cdn n f d cβ β′ ′= and we obtain:

( ) ( ) ( )2 2 24 / exp ,cf c cβ π β β′ ′ ′= − [2.27]

which is a function independent of the nature of the gas and temperature (see Figure 2.7).

The fraction of molecules whose thermal speed has a component in an interval xdc′ around xc′ is obtained by integrating the two other components from -∞ to +∞.

We find xcf ′ and

xcfβ ′ (see Figure 2.7):

( ) ( ) ( ) ( )2 2 2 2/ exp ; 1/ exp .x xc x c xf c f cββ π β π β′ ′′ ′= − = − [2.28]

The mean relative speed 1 2 1 2r rc c′ ′ ′= = − = −c c c c is obtained by integration over all velocities 1c and 2c :

1 / 23 / 22( ) ( ) 4 tr

r rR T

c c f f d dπβ π

⎛ ⎞⟨ ⟩ = = = ⎜ ⎟

⎝ ⎠∫ ∫1 2

c 1 c 2 1 2c c

c c c c . [2.29]

We note that 0xc′⟨ ⟩ = for symmetry reasons and that: 2r rc c c′ ′⟨ ⟩ = ⟨ ⟩ = ⟨ ⟩ .

0.E+00

5.E-04

1.E-03

0 2000 4000 6000

Dis

tribu

tion

func

tion

Velocity (m/s)

T = 200 KT = 800 K

Figure 2.6. Distribution function of the magnitude of thermal speed fc' (c') in helium (T – temperature)


0.0

0.5

1.0

-3 -2 -1 0 1 2 3

Dis

tribu

tion

func

tion

Reduced velocity

magnitude of c'component of c'

Figure 2.7. Distribution functions of reduced speeds ( )cf cβ β′ ′ and ( )xc xf cβ β′ ′

2.2.1.2. Fluxal quantities in an equilibrium gas

Let Q be a scalar or vector quantity attached to molecules of an equilibrium gas at velocity u. We calculate the flux of Q in the positive x-direction through a unit area surface whose normal is oriented along the x-direction. Let θ be the angle between x (unit-vector along x-axis) and u ( 0 θ π< < ). Let y be the unit-vector defined and oriented by the projection of u on the surface ( 0⋅ ≥u y ). Thus,

cosx xc c u θ′= + , siny yc c u θ′= + and z zc c′= .

The flux density (per unit time and area) of quantity Q associated with molecules of class c is: ( ) xdQ Q dn Q c dn= ⋅ =c x . By integrating this over all molecules that cross the surface element in the considered direction, we find:

( )0 0

0cos cos .

x x

x y z

x xc c

x x y zc u c c

Q Q c dn n Q c f d

n Q c u f dc dc dcθ θ

> >

+∞ +∞ +∞′ ′ ′=− =−∞ =−∞

= =

′ ′ ′ ′= +

∫ ∫

∫ ∫ ∫

c c [2.30]

The density of molecular flux (number of molecules crossing the surface per unit time and area) is obtained by taking 1Q = . By first integrating yc′ and zc′ , then xc′

we find:

( ) ( ) ( )3 / 2 2 2cos expx x xdN n c u c dcβ π θ β′ ′ ′= + − [2.31]

( / ) ( ) n nN n G sβ= [2.32]

42 Microfluidics

after having defined:

cosns s θ= , 1/ 2/(2 )trs u u RTβ= = ,

( ) ( ){ }( ) ( )2( ) exp 1 erf 2n n n n nG s s s sπ π= − + +

and ( ) ( ) ( )20erf 2 / expns

ns t dtπ= −∫ (error function).

Combining equations [2.31] and [2.32], we obtain the distribution function xcfβ ′

that characterizes the normal velocity component of those molecules that cross the surface element: / ( )

xc xdN N f d cβ β′ ′= . Hence:

( ) ( )2 2expx

x nc x x

n n

c sf c dc

G sββ β

π′

′ + ′ ′= − . [2.33]

That function should not be confused with the distribution function xcfβ ′ [2.28]

of those molecules that are present in a volume element at the considered location.

In the same way, making xQ m c= , we obtain the density of normal momentum flux, a quantity that has the dimension of a stress:

( ) ( )2 2/ ( ) / ( )n p n p nP n m G s G sβ ρ β= = [2.34]

with { }{ } ( )2 2( ) exp( ) 1 erf ( ) 1 2 2p n n n n nG s s s s sπ π⎡ ⎤= − + + +⎢ ⎥⎣ ⎦.

Making yQ m c= , we obtain the density of the tangential momentum flux:

( ) ( )2 2/ ( ) sin / ( ) sint n n n nP nm G s s G s sβ θ ρ β θ= = , [2.35]

which is the product of molecular flux N and the average momentum in y-direction of molecules ( sinm u θ ). The density of momentum flux in the z-direction is zero for reasons of symmetry.


Making 2 2 2( / 2)( )x y zQ m c c c= + + , we obtain the density of translational energy

flux:

( ) ( )3 3, ,/ ( , ) / ( , )tr q tr n q tr nq nm G s s G s sβ ρ β= = [2.36]

with { } { } ( )2 2 2, n

5( , ) 2 4 exp( ) 2 1 erf (s ) 8

2q tr n n nG s s s s s sπ π⎡ ⎤⎧ ⎫= + − + + +⎨ ⎬⎢ ⎥

⎩ ⎭⎣ ⎦.

Like the y-oriented momentum, the internal energy of a molecule is uncoupled from its normal velocity xc . Thus the density of internal energy intq for a gas with ζ internal degrees of freedom is simply the product of the molecular flux N and the average internal energy ( / 2)int inte k Tζ⟨ ⟩ = with (5 3 ) /( 1)ζ γ γ= − − .

The density of total energy flux is the sum of the two previous ones. After algebraic manipulations and under the hypothesis that translational and internal temperatures are equal (as expected in an equilibrium gas), we obtain:

( ) ( )3 3/ ( , ) / ( , )tr int q n q nq q q n m G s s G s sβ ρ β= + = = [2.37]

with:

( )2 2 2n

1 1( , ) 2 exp( ) 2 1 erf (s ) .

1 1 8q n n nG s s s s s sγ γπγ γ π

⎡ ⎤⎛ ⎞ ⎛ ⎞+= + − + + + ×⎢ ⎥⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟− −⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦

Note the behavior and particular values of the G functions introduced before:

2 2

1 1 1 1(0) (0) (0, 0)

4 12 8

1( ) ( ) ( , )

2 2 1

( ) 0 ( ) 0 ( , ) 0.

n p q

nn n p n q

n p q

G G G

sG s G s G s

G G G

γγπ π

γγ

+= = =

−

⎛ ⎞+∞ ≈ +∞ ≈ + +∞ +∞ ≈ +⎜ ⎟⎜ ⎟−⎝ ⎠

−∞ ≈ −∞ ≈ −∞ −∞ ≈

The case ( 0)ns s= = corresponds to a gas at rest. The case ( 0, 0)ns s= ≠ corresponds to a flow whose macroscopic velocity is parallel to the surface ( / 2θ π= ). The limiting case ns → +∞ corresponds to a surface facing a hypersonic flow: all molecules cross the surface with ≈c u . The limiting case

44 Microfluidics

ns → −∞ corresponds to molecules travelling backwards in a hypersonic flow. The number of these molecules is very small when the flow velocity is much larger than the typical thermal speed. The corresponding fluxes are therefore very small.

The estimation of the fluxes mentioned in the present section, completed by a model for gas surface interaction, is the basis of aerothermodynamic calculations in the free molecule regime.

2.2.2. Molecular methods in rarefied gas dynamics

2.2.2.1. Free molecule calculations

Let us consider a body located in a uniform equilibrium gas flow, characterized by a number density n∞ , a temperature T∞ and a velocity ∞u . If molecules just reflected by the body travel a large distance (compared with the body size) before experiencing a first collision, they have a negligible probability of colliding with the body again. Their collision partners have also a negligible probability of colliding with the body. If the body is convex, it can only be hit by molecules coming from the free stream. Applying equations [2.33] to [2.37] is sufficient to estimate the momentum and energy exchange due to incoming molecules. If we can describe how incoming molecules are reflected by the wall, it will also be possible to estimate the momentum and energy exchange due to reflected molecules. Combining incoming and reflected fluxes relative to each surface element of the body will allow an estimation of the dynamic and thermal effects of the gas on the body. This will, for example, result in the distribution of normal stress, shear stress and heat transfer coefficients along the wall. With the most usual models for gas-surface interaction (section 2.4.2), the reflection of molecules is also based on the properties of an equilibrium gas. The corresponding fluxes are still given by equations [2.33] to [2.37] and everything is known to calculate the exchanges between the wall and the body.

The treatment of a flow in a channel or microchannel is also based on equations [2.33] to [2.37] when the mean free path of molecules is large compared with the transverse dimensions of the channel. Gas–gas collisions can be disregarded and the flow is governed by gas–surface collisions, whose descriptions are based on equilibrium gas properties.

2.2.2.2. Direct simulation Monte Carlo method

The direct simulation Monte Carlo (DSMC) method is a powerful tool by which to analyze flows in the transition regime, i.e. flows that require a molecular approach but are not sufficiently rarefied to be analyzed by free molecule methods. The DSMC method was first proposed by Bird in 1963, and then improved by Bird


himself and a number of other workers. A detailed description of the method can be found in [BIR 98].

2.2.2.2.1. Principle

Basically, the method is a numerical experiment consisting of tracking a limited number of molecules, each of them being representative of a large number W of real molecules. W is called the weighting factor. The coordinates, velocity components, internal energy (or energies) of each tracked molecule is stored and constantly updated as molecules move and collide with one another or with a wall.

Considering an unsteady problem, the initial state is set to represent the actual initial state of the gas, i.e. quantities associated with molecules are sampled from distribution functions that correspond to the initial state. Every time they are analyzed, the set of tracked molecules is representative of the actual macroscopic state, except for statistical fluctuations due to the limited sample size. The uncertainties due to these fluctuations can be reduced by repeating the experiment, starting from different initial microscopic states, all being representative of the same actual macroscopic state. This increases the sample size from which useful results will be extracted.

Considering a steady problem, the computation starts from an arbitrary initial state. The system evolves spontaneously towards a steady state that is representative of the solution. When this steady state is achieved, the computation continues and information is periodically extracted from the flowfield. The successive states observed are different microscopic states, all of them being representative of the solution. This method allows us to increase the size of the samples. The calculation is stopped when the samples reach a sufficient number.

For both steady and unsteady flows in the presence of a wall, exchanges between the gas and wall are recorded every time a gas–surface collision occurs. At the end of the computation, this information is used to get the dynamic and thermal loads exerted by the gas on the surface.

Here we briefly describe the DSMC method for a simple gas. However, the DSMC method can be used for gas mixtures and even reactive flows.

2.2.2.2.2. Discretization

The analysis of collisions experienced by simulation molecules is designed to be representative of collisions experienced by real molecules (in terms of collision frequency, selection of partners, modeling of energy and momentum exchange, etc.). The DSMC method is therefore a correct simulation of the physical flow, except for

46 Microfluidics

statistical fluctuations. However, as in most numerical methods, an approximation is introduced due to discretization of time and space.

A time step δt is introduced to govern the progress of the calculation. A network of cells is created in the physical space. Let δx be a typical cell dimension. Cells are used to sample the gas properties and to select collision partners. At each time step, two series of operations are carried out. They correspond to an uncoupled analysis of molecular motion and intermolecular collisions, respectively.

Molecular motion

Molecules are injected randomly into the computational domain through fluid boundaries. Usually the gas close to those boundaries is assumed to be in equilibrium with known properties. The number flux of injected molecules is given by equation [2.32]. Their velocity components and internal energy are prescribed randomly from the distribution functions relative to an equilibrium gas. For instance, the velocity normal to the boundary is governed by equation [2.33].

All molecules are moved by a quantity tδc .

During this displacement, a molecule can hit a wall. It is then re-emitted according to a gas–surface interaction model (section 2.4.2). During the collision, its velocity and its energy change. The corresponding exchange of momentum and energy with the wall is recorded.

Some molecules exit the computational domain. They are removed from the memory.

Molecules are sorted according to the index of the cell in which they are located.

Intermolecular collisions

Intermolecular collisions are calculated. To keep them representative of real collisions, the collision partners are chosen within the same cell and with a probability proportional to T rcσ . It is even possible to divide the cells into sub-cells and to chose partners within a sub-cell, while the cells remain a basis for sampling flow properties.

A simulation molecule must experience the same collision frequency ν as a real molecule. This determines the number of collisions to be calculated in each cell during each time step. Let V be the volume of the cell, N the (fluctuating) number of simulated molecules present in the cell at the time considered and W their weighting factor. For each molecule, the number of collisions to compute is tν δ and we must


compute ( ) ( )2 2 T rN t N n c tν δ σ δ= ⟨ ⟩ collisions. Taking /n N W V= into account, the average number of collisions in a given cell during δt should be:

2( / 2) [( ) /(2 )]T r T rN n c t Vn W c tσ δ σ δ⟨ ⟩ ⟨ ⟩ = ⟨ ⟩

Computing T rcσ in each cell at each time step would require the calculation of rc and the corresponding Tσ for all pairs of molecules in the cell. Bird’s NTC (no time counter) algorithm is less demanding. It consists of calculating:

max[ ( 1) /(2 )] ( )t T rN W N N V c tσ δ= − ,

where max( )T rcσ is an overestimation of the maximum value of T rcσ . Then Nt pairs of molecules are randomly chosen with uniform probabilities. Each time a pair is selected, the corresponding quantity T rcσ is computed and we decide, with a probability of max( ) / ( ) ,T r T rc cσ σ to actually compute the collision. This algorithm ensures that collisional pairs are selected with a probability proportional to .T rcσ The mean number of collisions is equal to:

[ ( 1) /(2 )] T rW N N V c tσ δ⟨ − ⟩ ⟨ ⟩ .

The fluctuation of N obeys a Poisson distribution, for which the relation 2( 1)N N N⟨ − ⟩ = ⟨ ⟩ holds. Taking 2 2( / )N nV W⟨ ⟩ = into account, the average

number of collisions in a given cell becomes 2[( ) /(2 )] T rV n W c tσ δ⟨ ⟩ , which is the correct value determined above.

Computing a collision consists ofn estimating post-collision velocities, 1*c and

2*c , and post-collision internal energies, *

,1inte and *,2inte , from their pre-collision

values. The details of this calculation depend on the model retained to describe intermolecular collisions. A popular one is the VHS (variable hard sphere) model, associated with the Larsen-Borgnakke one for internal energy exchange (section 2.4.1.2). When all collisions have been calculated in all cells, the computation moves to the next time step.

2.2.2.2.3. Sampling

To determine macroscopic quantities, we gather information by periodically sampling the flow field during the simulation.

48 Microfluidics

For a steady flow, sampling begins once the steady state has been reached, as evidenced, for example, by an approximately constant number of simulated molecules in the computational domain. For each cell, the following quantities are cumulated in variables:

– the number of molecules observed in the cell N∑ ;

– their velocity components , ,x y zc c c∑ ∑ ∑ ;

– the squared values of their velocity components 2 2 2, ,x y zc c c∑ ∑ ∑ ; and

– their internal energies inte∑ .

When the size of the samples is considered to be sufficiently large, the calculation stops. The cells have been sampled N0 times. The local macroscopic flow quantities are obtained readily:

– number density ( ) /( )on N W N V= ∑ ;

– components of macroscopic velocity = ⟨ ⟩u c , with

x xu c N= ∑ ∑ , y yu c N= ∑ ∑ and z zu c N= ∑ ∑ ;

– translational temperature ( ) 2/(3 )trT m k c′= ⟨ ⟩ , with

( )22 2 2 2 2 2 22 ( ) ( )x y zc c u c c c N u′⟨ ⟩ = ⟨ − ⟩ = ⟨ ⟩ − ⋅ ⟨ ⟩ + = + + −∑ ∑ ∑ ∑c u u c ;

– internal temperature ( ) ( )2 /( ) /int int intT k e Nζ= ∑ ∑ .

If a body is present in the flow, its surface is divided into the elements of an area δA. On each element, we gather the changes in velocity and energy experienced by molecules that are reflected by this element, i.e.:

– incident re-emitted( )−∑ c c ; and

– 2 2

incident re-emitted

1 12 2int intmc e mc e

⎧ ⎫⎛ ⎞ ⎛ ⎞⎪ ⎪+ − +⎨ ⎬⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠⎪ ⎪⎩ ⎭

∑ .

At the end of the calculation, the distribution of mechanical and thermal loads is obtained as:

– normal and tangential stress: incident re-emitted( )mWA tδ

= −Δ

∑σ c c ; and


– density of energy flux:

2 2

incident re-emitted

1 12 2int int

Wq mc e mc e

A tδ

⎧ ⎫⎛ ⎞ ⎛ ⎞⎪ ⎪= + − +⎨ ⎬⎜ ⎟ ⎜ ⎟ Δ⎝ ⎠ ⎝ ⎠⎪ ⎪⎩ ⎭∑ ;

where Δt is the duration of the simulation, in terms of physical time. The weighting factor W is introduced here to convert the results from simulated to real molecules.

For unsteady flows, sampling occurs from the beginning of the computation. Information is stored separately for each time considered and cumulated over a large number of different runs (ensemble averaging rather than time averaging).

Thus the DSMC method allows us to obtain the distribution of flow parameters in the flowfield and the exchanges between gas and walls. These results are affected by statistical scatter, however, due to the limited size of the samples. In principle, fluctuations can be made arbitrarily small by repeating the simulation a sufficiently large number of times (unsteady flows) or by continuing the simulation over a sufficiently long period (steady flows). In practice, however, statistical fluctuations are a problem in flows at very low Mach number, because the macroscopic velocity is difficult to extract from thermal speed and statistical fluctuations.

The number of molecules in a cell should be not less than approximately 10. Optimal efficiency is obtained when the samples observed in all cells are about the same size, which implies that the average number of molecules Nsim is approximately the same in all cells. Thus the optimal value of the weighting factor is

/ simW nV N= , with, for example 10simN = . At the beginning of the simulation, n is unknown and it is difficult to set the optimal value of W. We can solve this by performing a first simulation with a rough estimate of n and W. Its result gives a better estimate of n, which allows us to adjust W for a second and final simulation.

Introducing a weighting factor that varies from cell to cell requires a special treatment when a molecule moves from one cell to another: it must be cloned a number of times if its weighting factor decreases and it must receive have a probability of vanishing if its weighting factor increases.

2.2.2.2.4. Validity conditions

If physical models for gas–gas and gas–surface collisions are reasonably correct and if the above-mentioned algorithms are applied correctly, the essential limit to the validity of the DSMC method is related to space and time discretization.

50 Microfluidics

Space discretization

The cell size must be small compared to the scale length of the macroscopic gradients for two reasons:

– Cells serve as a basis for sampling flow properties. Thus, their dimensions determine the limit of spatial resolution.

– Collision partners are chosen within the same cell, but without considering their exact location within the cell. Thus, the distribution functions or macroscopic quantities must present only small variations within a cell.

In regions where the characteristic lengths of the gradients are of the order of the mean free path (e.g. in a shock wave or in a Knudsen layer), the cell size in the direction of the gradients must be (much) smaller than the mean free path. Elsewhere, the requirement xδ λ is not justified. Time discretization

The time step tδ must fulfill two requirements:

– Uncoupling motion and collisions of molecules is only valid if a molecule experiences zero or one collisions when it moves during tδ . Therefore tδ must be smaller than the mean collision time ( 1 /ctδ τ ν< = ).

– Furthermore, collisions do not take place at their actual locations but they are postponed to the end of the molecular displacement, in the arrival cell. For the collision partners to be representative of real ones, the distribution functions or macroscopic quantities must present only a small variation in molecule displacement. Reasoning in the same way as in section 2.1.4 leads to a different condition, depending on whether the flow is locally subsonic or supersonic, i.e. whether the molecule displacement is governed essentially by thermal speed or flow velocity. The resulting condition is:

( )1 / max / , . /t c Q Q Q Qδ ′< ⟨ ⟩ ∇ ∇u ,

Q denoting a macroscopic quantity of the flow.

The requirement ctδ τ< becomes increasingly severe as we move closer to the continuum regime. Computing times increase as δt decreases. This makes the practical application of the DSMC method to non-rarefied flows impossible or expensive. However, the progress in computers constantly pushes the limit.

Although the DSMC method has become very popular for analyzing problems in the transition regime, it is not the only existing method. A review of other methods is given by Bird [BIR 98]: moment methods, model equations, direct solution of the


Boltzmann equation, molecular dynamics, lattice gas cellular automata, discrete velocity method, etc.

A popular model is the so-called BGK equation proposed by Bhatnagar, Gross and Krook in 1954. It leads to the correct solution in both limiting cases (equilibrium flow and free molecule flow). Although its validity in the intermediate range is questionable, it has been successfully applied to isothermal microfluidic applications, where the flow remains close to equilibrium, even for relatively large values of the Knudsen number.

2.3 Continuum description of a flow

2.3.1. Equation system for gas dynamics

Gas dynamics equations can formally be obtained from molecular considerations: an approximate form of the distribution function f is chosen and we integrate the Boltzmann equation after having multiplied it successively by collisional invariants. It is also possible to retrieve them by using continuum considerations if a number of hypotheses are introduced, such as the Fourier law, a Newtonian behavior and Stokes hypothesis.

2.3.1.1. Invariant forms of equations

A set of equations for a gaseous flow consists of three partial derivative equations that express, respectively:

– mass conservation (continuity equation):

0ii Jt

ρ∂+ ∇ =

∂ [2.38]

– momentum conservation:

( )ki k k ik

i iu

J u pt

ρ∂+ ∇ + ∇ = ∇ ∏

∂ [2.39]

– energy conservation:

( ) ( )i

i ik ki i i

E JE p q u

t ρ∂

+ ∇ + + ∇ = ∇ ∏∂

. [2.40]

To close the system, the mass flux vector iJ , stress tensor ik∏ and energy flux vector iq must be expressed in terms of the local macroscopic flow properties:

52 Microfluidics

density ρ, velocity iu and pressure p. For Navier-Stokes equations [LOI 66, PEY 96, SCH 79], we get:

i iJ uρ= [2.41]

(2 / 3)ik ik k i i k ik jNS ju u g uμ ⎡ ⎤∏ = ∏ = ∇ + ∇ − ∇⎣ ⎦ [2.42]

i iq Tκ= − ∇ . [2.43]

Here, i∇ and i∇ are the co- and contravariant derivatives, respectively. ikg is the metric tensor, μ and κ are the dynamic viscosity and heat conductivity coefficients, respectively, /( ( 1))pε ρ γ= − is the internal energy per unit mass, and

ikNS∏ is the Navier-Stokes stress tensor.

The system [2.38] to [2.40] can be closed in other ways. The first one, based on kinetic theory considerations [ELI 01, SHE 97, SHE 00] is written:

( [ ( ) ])i i i j ijJ u u u p

τρ ρρ

= − ∇ + ∇ [2.44]

ik ik i j k k ik j jNS j j ju u u p g u p p uτ ρ τ γ⎡ ⎤ ⎡ ⎤∏ = ∏ + ∇ + ∇ + ∇ + ∇⎣ ⎦ ⎣ ⎦ [2.45]

[ (1 )]i i i j jj jq T u u p uκ τ ρ ε ρ= − ∇ − ∇ + ∇ [2.46]

Equations [2.38] to [2.40], completed by equations [2.44] to [2.46], have been called quasi-gas dynamic (QGD) equations and describe flows of dilute viscous gases.

A second variant for closing the system of equations [2.38] to [2.40] is more general and also applies to non-dilute gases [SHE 97, SHE 00]:

( )i i iJ u wρ= − [2.47]

ik ik i kNS u wρ∏ = ∏ + [2.48]

i iq Tκ= − ∇ [2.49]

where ( )k j k kjw u u p

τ ρρ

= ∇ + ∇ .


This system has been called quasi-hydrodynamic (QHD) equations.

QGD and QHD systems differ from the Navier-Stokes one by terms of order O(τ), where τ is a coefficient that has the dimension of a time. It is equal to

2( / )( /( ))Sc aτ γ μ ρ= , where Sc is the Schmidt number, which is close to 1. For steady flows, additional terms in QGD or QHD equations are formally of order O(τ2) when 0τ → . QGD and QHD equations are compatible with the second law of thermodynamics. Introducing the boundary layer approximation into them leads to the classic Prandtl equations. Dissipative terms in QGD and QHD equations act as regularization terms that would be added to Navier-Stokes equations, just like artificial viscosity.

In practice, the same formulation can be used for a purely numerical purpose. In this case, τ has no physical meaning and can then be chosen according to the time-space discretization of the numerical problem. This allows us to build stable and efficient numerical algorithms.

For a dilute gas, assuming 1Sc = , and taking p RTρ= and 1/2( )a RTγ= into account, parameter τ reduces to the Maxwell relaxation time ( / pτ μ= ), which relates to the mean collision time. It keeps its physical meaning and the difference with Navier-Stokes equations comes from a different approximation of the distribution function close to equilibrium. If equations are non-dimensionalized, the additional terms are of order O(Kn2) when 0Kn → .

2.3.1.2. Navier-Stokes equations for a plane or axisymmetric configuration

The equation system [2.38] to [2.43] for bidimensional unsteady flows is written:

1 ( )0

kr z

kr u u

t r zrρ∂ ∂ ∂+ + =

∂ ∂ ∂ ,2( ) 1 ( ) ( ) 1 ( ) NSk k NS NS

r r z r rr zrk k

u r u u u p rk

t r z r r z rr rϕϕρ ρ ρ ∏∂ ∂ ∂ ∂ ∂ ∏ ∂∏

+ + + = + −∂ ∂ ∂ ∂ ∂ ∂

2( ) 1 ( ) ( ) 1 ( )k k NS NSz z r z rz zz

k ku r u u u p rt r z z r zr r

ρ ρ ρ∂ ∂ ∂ ∂ ∂ ∏ ∂∏+ + + = +

∂ ∂ ∂ ∂ ∂ ∂ ,1 ( ) ( ) 1 ( )

1[ ( )] ( ) .

k k NS NSr z r z

k k

k NS NS NS NSrr r rz z zr r zz zk k

E r u H u H r q qt r z r zr r

r u u u uzr r

ρ ρ∂ ∂ ∂ ∂ ∂+ + + +

∂ ∂ ∂ ∂ ∂∂ ∂

= ∏ + ∏ + ∏ + ∏∂∂

54 Microfluidics

Here k is equal to zero in the plane case and equal to one in the axisymmetric one. In the latter case, ϕ denotes the azimuthal angle. Closing relationships for a dilute gas are obtained by expressing pressure, total volumic energy and total mass energy as:

2 2 2 2, , .

2 1 2 1r z r zu u p u u p

p RT E Hγρ ρ

γ γ ρ+ +

= = + = +− −

The components of the viscous stress tensor are given by:

22 div

3NS rrr

ur

μ μ∂∏ = −

∂u , ( )NS NS r z

rz zru uz r

μ ∂ ∂∏ = ∏ = +

∂ ∂,

22 div

3NS zzz

uz

μ μ∂∏ = −

∂u , 2

2 div3

NS rurϕϕ μ μ∏ = − u ,

where:

1div ( )k z

rku

r ur zr∂ ∂

= +∂ ∂

u .

The components of the thermal flux vector are: ,NS NSr z

T Tq q

r zκ κ∂ ∂

= − = −∂ ∂

.

The temperature dependence of dynamic viscosity μ and thermal conductivity κ is given, for example, by a power-law or a Sutherland law (section 2.4.1). The complete form of Navier-Stokes equations was obtained by Stokes in 1845.

The set of continuum flow equations, completed by QGD regularization [2.44]-[2.46] has been successfully used to solve different problems in a wide range of Knudsen numbers in both steady and unsteady situations [ELI 01, ELI 09].

2.3.1.3. Boundary conditions

The systems presented above must be completed by boundary conditions. In general, these conditions are determined by the concrete problem under consideration and by the form of the equations. For Navier-Stokes equations, the conditions on, for example, a solid wall at temperature Tw, we write:

( )0 , 0 , (imposed wall temperature)

0 (adiabatic wall)n s s w

w

u u T TT n

= = =∂ ∂ = [2.50]


where nu and su are respectively the normal and tangential components of the velocity vector u, sT is the temperature of the gas at the wall and n the distance from the wall.

For any Knudsen number, however small it is, a domain exists along the wall whose thickness is of the order of the mean free path and in which the gas is not in equilibrium. This domain is called the Knudsen layer. To account for it in Navier-Stokes calculations, conditions of velocity slip and temperature jump are set for the gas along the wall. Different variants of these conditions can be found in the literature [BIR 98, KOG 69, LIF 63]. They all involve local derivatives of velocity and temperature, as well as coefficients that characterize gas–surface interaction (section 2.4.2). These expressions often look like:

s wu ( u / n)λ ∂ ∂∼ and s w wT T ( T / n)λ− ∂ ∂∼ .

They differ essentially by numerical coefficients that remain close to 1. Assuming full accommodation at the wall, expressions proposed by Kogan [KOG 69] are written:

1/21.012 2 0.84xs

w w

u TuRT n T x

μμρ ρ

⎛ ⎞∂ ∂⎛ ⎞⎛ ⎞= + ⎜ ⎟⎜ ⎟⎜ ⎟ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠, [2.51]

1/ 20.586 11 2s w

w

TT TPr RT n

γ π μγ ρ

∂⎛ ⎞ ⎛ ⎞− = ⎜ ⎟ ⎜ ⎟− ∂⎝ ⎠ ⎝ ⎠, [2.52]

where direction x is that of the gas velocity u close to the wall. Other expressions have been proposed, for example by Deissler [DEI 64], for velocity:

2 2

22 9

4 2u

su w w

a u uu

a n yλλ

⎛ ⎞− ∂ ∂⎛ ⎞ ⎜ ⎟= +⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠, [2.53]

an expression where ua is the accommodation coefficient of tangential momentum.

In microfluidic applications, conditions at the wall of an axisymmetric channel can be written, in (z,r) co-ordinates, as:

0, ,r z s su u u T T= = = . [2.54]

The conditions at the channel entrance are:

( ) ( ) , ( ) ( ) , ( ) ( ) , ( ) ( )z ze r re e eu r u r u r u r p r p r T r T r= = = = , [2.55]

56 Microfluidics

where quantities eu , ep and eT are assumed to be uniform or known as functions of r. At the channel exit, soft conditions can be used:

0, 0, 0, 0z ru u p Tz z z z

∂ ∂ ∂ ∂= = = =

∂ ∂ ∂ ∂. [2.56]

Finally, symmetry conditions hold along the channel axis:

0, 0, 0, 0zr

u p Tu

r r r∂ ∂ ∂

= = = =∂ ∂ ∂

. [2.57]

These conditions are easy to adapt to a microchannel with a rectangular cross-section. Aubert and Colin [AUB 01] have studied the role of second-order terms in the expressions of su and sT .

2.3.2. Simplified forms of Navier-Stokes equations

2.3.2.1. Prandtl’s approximation

The basis of the laminar boundary layer theory was set in 1904 by Prandtl [AND 84, LOI 66, SCH 79]. Consider a case when the thickness of the viscous layer along a wall is much smaller than the size of the body in the direction of the flow. Taking into account the order of magnitude of the Navier-Stokes equation terms allows us to reduce the number of equations and simplify them. More generally, such simplifications apply to flows with a preferential direction (jets, channels and tubes). Prandtl’s equation set for the boundary layer along a wall parallel to the x-direction is:

( )( )0yx uu

t x y

ρρ ρ ∂∂ ∂+ + =

∂ ∂ ∂,

2 ( )( ) ( ) x yx x xu uu u p ut x y x y y

ρρ ρ μ∂ ⎛ ⎞∂ ∂ ∂ ∂ ∂

+ + + = ⎜ ⎟⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠, 0

py∂

=∂

,

2 2 2

2 2 2x x x

x yu u p u p

u ut x y

ρ ε ρ ε ρ ερ ρ

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞∂ ∂ ∂⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎜ ⎟ ⎜ ⎟ ⎜ ⎟+ + + + + + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥ ⎢ ⎥ ⎢ ⎥∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎣ ⎦ ⎣ ⎦ ⎣ ⎦

xx

u Tu

y y y yμ κ

⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂= +⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠

,


where the unknowns are ρ , xu , yu , p and T . The closing relations are

p RTρ= and /( ( 1))pε ρ γ= − .

2.3.2.2. Parabolized Navier-Stokes equations

Consider a steady flow characterized by a preferential direction and a non-viscous supersonic domain. The full Navier-Stokes set can be simplified by eliminating part of the derivatives in the transverse direction. The resulting equations are called parabolized. They were proposed in 1968 by Rudman and Rubin. Their derivation is not as rigorous as that of Prandtl’s equations and we can find different variants of them [AND 84, LOI 66, SCH 79]. For example, Kovenia et al. [KOV 90, p.54] write:

( )( )0yx uu

x y

ρρ ∂∂+ =

∂ ∂,

2 ( )( )( )x yx xu uu p u

x y x y y

ρρ μ∂∂ ∂ ∂ ∂

+ + =∂ ∂ ∂ ∂ ∂

,

2( ) ( ) 4( )

3x y y yu u u upx y y y y

ρ ρμ

∂ ∂ ∂∂ ∂+ + =

∂ ∂ ∂ ∂ ∂,

( )( ) 4( ) ( ) ( )

3y yx x

x yu H uu H T u

u ux y y y y y y y

ρρ κ μ μ∂ ∂∂ ∂ ∂ ∂ ∂ ∂

+ = + +∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

,

where:

2 2

2 1x yu u p

Hγ

γ ρ+

= +−

.

The non-divergent form derived from the above equations is written:

( )( )0yx uu

x y

ρρ ∂∂+ =

∂ ∂, ( )x x x

x yu u p u

u ux y x y y

ρ ρ μ∂ ∂ ∂ ∂ ∂+ + =

∂ ∂ ∂ ∂ ∂,

4( )

3y y y

x yu u up

u ux y y y y

ρ ρ μ∂ ∂ ∂∂ ∂

+ + =∂ ∂ ∂ ∂ ∂

2 243

y yx xx y

u uu T uu u p

x y x y y y y yε ερ ρ κ μ μ

∂ ∂⎛ ⎞ ⎛ ⎞ ⎛ ⎞ ⎛ ⎞∂ ∂ ∂ ∂ ∂ ∂+ + + = + +⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠

.

2.3.2.3. Incompressible flows

In flows characterized by a small Mach number and small temperature gradients, the incompressible approximation ( cteρ = ) is valid, which simplifies the Navier-

58 Microfluidics

Stokes equations. For a bidimensional plane ( 0k = ) or axisymmetric ( 1k = ) flow, in the absence of external forces:

1 ( )0

kr z

kr u u

r zr∂ ∂

+ =∂ ∂

[2.58]

2

2

1 ( ) ( ) 1

2 2( ) [ ( )] ,

kr r z r

k

k r z r rk

u r u u u pt r z rr

u u u v ur v v k

r r z r zr r

ρρ

∂ ∂ ∂ ∂+ + +

∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂

= + + −∂ ∂ ∂ ∂ ∂

[2.59]

21 ( ) 1

1[ ( )] 2 ( ) ,

kz z r z

k

k z r zk

u r u u u pt r z zr

u u ur v v

r r z z zr

ρρ

∂ ∂ ∂ ∂+ + +

∂ ∂ ∂ ∂∂ ∂ ∂ ∂ ∂

= + +∂ ∂ ∂ ∂ ∂

[2.60]

1 1( ) ( ) ( )k k

r zk kT T T

r u T u T rt r z r r z zr r

κ κρ ρ

∂ ∂ ∂ ∂ ∂ ∂ ∂+ + = +

∂ ∂ ∂ ∂ ∂ ∂ ∂ [2.61]

where ν and κ are the coefficients of kinematic viscosity (ν μ ρ= ) and thermal conductivity, respectively. In this case, ν and κ are assumed to be constant. Equation [2.61] in T is not necessary to solve the dynamic problem and it can be omitted if we are not interested in the thermal problem. To build regularizators of the Navier-Stokes system, we can use QHD equations [2.38] to [2.40] and [2.47] to [2.49] rather than the QGD system, which is suited to compressible flows. In particular, the regularizators that need to be added to the right-hand side members of [2.58] to [2.61] are, respectively:

1 ( )kr z

kr w w

Rr zr

ρ∂ ∂

= +∂ ∂

[2.62]

2 ( ) ( ) ( )kr r r z z r

r kr u w u w u w

Rr z zr

∂ ∂ ∂= + +

∂ ∂ ∂ [2.63]

1 ( ) ( ) 1 ( )2

k kz r z z r z

z k kr u w u w r u w

Rr z rr r

∂ ∂ ∂= + +

∂ ∂ ∂ [2.64]


1( ) ( )k

T r zkR r w T w Tr zr∂ ∂

= +∂ ∂

[2.65]

where 1

( )r rr r z

u u pw u u

r z rτ

ρ∂ ∂ ∂

= + +∂ ∂ ∂

and 1

( )z zz r z

u u pw u u

r z zτ

ρ∂ ∂ ∂

= + +∂ ∂ ∂

.

Then the pressure field is derived from the velocity field through the Poisson equation:

2

21 1 1 1 ( )

( )

1( ) ( ),

kk r z

k k

k r r z zr z r zk

p p r u ur

r r r zr z r

u u u ur u u u u

r r z z r zr

ρ τ

⎡ ⎤ ⎛ ⎞∂ ∂ ∂ ∂ ∂⎜ ⎟+ = +⎢ ⎥⎜ ⎟∂ ∂ ∂ ∂∂⎢ ⎥⎣ ⎦ ⎝ ⎠

∂ ∂ ∂ ∂ ∂ ∂⎛ ⎞− + − +⎜ ⎟∂ ∂ ∂ ∂ ∂ ∂⎝ ⎠

[2.66]

which is equivalent to equation [2.58] with regularizator [2.62] when csteτ = . Boundary conditions must be added to the system [2.58]-[2.61], completed by [2.62]-[2.65]. Along a solid wall, we can write:

0, 0, / 0,r z wu u p n T T= = ∂ ∂ = = . [2.67]

This approach, completed by taking external forces into account, has been successfully applied to problems of thermal and thermocapillary convection in liquids [ELI 01]. For microchannel applications, QGD and QHD equations simplify and become identical. A number of results have been obtained [ELI 03, ELI 07, ELI 09] and used in section 2.5.3.

2.4. Physical modeling

A gas flow is entirely governed by the way molecules collide with one another (gas–gas collisions) and collide with walls (gas–surface collisions). The corresponding models are naturally molecular ones that can be used as such in molecular approaches. However, the same models appear under an integrated form in macroscopic approaches. For example, a law ( )F r for the interaction force between two molecules corresponds, at a macroscopic level, to a viscosity law ( )Tμ that can be used in a continuum approach.

60 Microfluidics

2.4.1. Gas models

2.4.1.1. Gas–gas interaction and transport properties

2.4.1.1.1. Power-law for molecular interaction

In this model, we consider that two molecules whose centers are separated by a distance r exert a repulsive force F C r η−= on each other. This is a two-parameter model ( C and η ). It is convenient to introduce 2 /( 1)α η= − and (1 / 2)ω α= + . From the laws of classic mechanics, the value of the impact of parameter b that corresponds to the limiting deviation limχ varies as: 2 /( 1)

rc η− − . This defines the molecular diameter d (section 2.1.3) and the total collision cross-section is written:

( ) ( ) ( )( )4 /( 1)2 2, 4 2T ref r r ref ref ref rd c c k T m c

αησ π σ σ α

− −= = = − , [2.68]

where a reference temperature refT has been introduced, defined by:

2,(2 ) (1/ 4)ref r refk T m cα− = .

In place of C and η , we can characterize the model by α and a value refσ (or 1/ 2( / )ref refd σ π= ), associated with some arbitrary temperature Tref.

Hard-sphere molecules can be considered as a particular case of this model, with 0α = . Maxwell molecules are another particular case, corresponding to 1 2α = .

Real molecules can often be modeled by an intermediate value ( 0 1 2α< < , or 1 2 1ω< < ).

In an equilibrium gas, the collision frequency T rn cν σ= is obtained by

averaging T rcσ . This is done similarly to estimating rc (equation [2.29]), but

here we integrate 1 2rc α− instead of rc . The solution involves Euler’s gamma

function Γ. We find that:

( ) ( ) ( ) ( )1 / 2 14 2 2 / /ref refn R T T T n Tαα ων α α σ π −= − Γ − ∝ [2.69]

The mean free path 1 / 2/ (8 / ) /c R Tλ ν π ν′= = is equal to:


( ) ( )

1/ 21

2 2 2 refref

T T TT n nn

α α ω

αλα α σ

−⎛ ⎞⎜ ⎟= ∝ =⎜ ⎟− Γ − ⎝ ⎠

. [2.70]

with 0 or 1 / 2α ω= = , equation [2.70] reduces to the hard-sphere expression [2.13].

2.4.1.1.2. Variable hard spheres (VHS)

The VHS model was proposed by Bird for applications to the DSMC method. For a VHS gas, the probability of collision between two molecules is the same as for a repulsive power-law force. Thus, all of the formulae in section 2.4.1.1.1 remain valid. However, the collisions themselves are treated as being between hard spheres. In particular, cos χ is distributed uniformly between -1 and +1.

The Chapman-Enskog method allows us to relate the viscosity coefficient to the parameters of the molecular model. Applied to VHS molecules, it yields:

( )( ) ( )

1/ 2(1 / 2)15

.8 2 4 refref

m R T TT T

T

αα ω

απ

μα α σ

+⎛ ⎞⎜ ⎟= ∝ =⎜ ⎟− Γ − ⎝ ⎠

[2.71]

Thus, the VHS model takes the best parts of two classic models: while reproducing a rather realistic power law for the viscosity-temperature dependence, it keeps the calculation of the collision as simple as between hard spheres. VHS molecules can be regarded as hard spheres whose diameters adapt to the relative velocity of the collision partners. Similarly to the power-law model, it includes two parameters. It can be fitted to a power law viscosity-temperature dependence. The coefficient of thermal conductivity is already determined and cannot be adjusted independently.

Eliminating refσ between [2.70] and [2.71], then between [2.69] and [2.71], we obtain expressions for λ and ν , respectively, as functions of macroscopic flow parameters:

( )

( ) ( )

1/ 2( ) 2 (7 2 )(5 2 )

( ) with ( )152

307 2 5 2

TK K

RTp

μ ω ωλ ω ωρ π

νμ ω ω

− −= × =

=− −

[2.72]

62 Microfluidics

For a hard-sphere gas ( 0 or 1 / 2α ω= = ) the expression reduces to:

( )1/ 216 ( )

,5 2

T

RT

μλρ π

=

which differs slightly for Chapman’s formula, also derived for hard spheres, but with a different definition of the mean free path:

1/ 2( )2

TRT

μ πλρ

⎛ ⎞= ⎜ ⎟⎝ ⎠

.

Introducing the magnitude of the flow velocity u, we relate the Knudsen number Kn Lλ= to the Reynolds number Re u Lρ μ= and to the Mach number

( )1/ 2Ma u RTγ= or to the molecular speed ratio ( )1/ 22s u R T= :

1/ 2

1/ 2 1/ 22(7 2 )(5 2 ) 2(7 2 )(5 2 )

215 15

s MaKn

Re Reω ω γ ω ωπ π

− − − −⎛ ⎞= = ⎜ ⎟⎝ ⎠

[2.73]

These VHS expressions for λ , ν and Kn are more accurate than those that are generally used and derived under the hard-sphere hypothesis ( 1 2ω = ). By extension, they can be used for any gas whose viscosity-temperature relationship is known by fitting a VHS power law. This is done by considering the values 1μ and

2μ of viscosity at temperatures 1T and 2T respectively, or by considering the variation of viscosity around 1T :

( ) ( )1 1T T T ωμ μ= with ( )( )

2 1

2 1

ln

ln T Tμ μ

ω = or 1

1

1

dd T

TTμω

μ⎛ ⎞= ⎜ ⎟⎝ ⎠

. [2.74]

This power law can be used as such in a continuum approach. It can also be used to fit a VHS model for a molecular approach: ω is taken from equation [2.74] and

refσ is obtained from equation [2.71]. It is generally impossible to find a power law

or VHS model that reproduces the viscosity variation in a large range of temperatures. If a perfect simulation is not possible, the model must be fitted to yield the correct viscosity at the temperature level that plays an essential role in the problem, usually the wall temperature.


Other models (variable soft spheres, generalized hard spheres, Lennard-Jones) are described and commented upon by Bird [BIR 98]. At moderate temperatures, air or nitrogen viscosity is well represented by Sutherland’s law, completed by a linear relationship at low temperature:

3 / 2ATB T

μ =+

if T B> and 1/ 22

ATB

μ = if T B≤ . [2.75]

For air, 6 -1/21.458 10 Pa s KA −= ⋅ and B = 110.4K

For nitrogen, 6 -1/21.374 10 Pa s KA −= ⋅ and 100 KB = .

The log-derivative ( ) ( )d dT Tω μ μ= varies monotonically from one (for T B< ) to 1/2 (for T → ∞ ). It is the exponent of a local power-law viscosity-temperature relationship that can be used in equations [2.71] to [2.73]. This is equivalent to locally fitting a VHS model in order to estimate the mean free path, the collision frequency and the Knudsen number.

In molecular approaches, the thermal behavior of a gas is determined by the parameters of the intermolecular collision model. In continuum approaches, the gas model includes not only viscosity but also the thermal conductivity κ or Prandtl number pPr cμ κ= . If the latter are unknown, we can use Eucken’s

approximation that relates Pr to γ:

( )4 9 5Pr γ γ= − . [2.76]

2.4.1.2. Internal energy exchange

There are circumstances (section 2.1.4) when differences can be observed between translation temperature and the temperature(s) associated with internal degree(s) of freedom (rotation and vibration). In a macroscopic approach, different levels of approximation can be used to treat this non-equilibrium:

– it is neglected in the classic one-temperature Navier-Stokes formulation;

– it is taken into account by one or more relaxation equation(s), such as:

i tr i

i

DT T TDt τ

−= [2.77]

where iτ is a relaxation time characteristic of the energy under consideration;

64 Microfluidics

– it is taken into account by a set of relaxation equations for the populations of the different rotational and/or vibrational energy levels.

In equation [2.77], iτ is generally fitted to experimental results by functions defined by:

( )i trp F Tτ = or ( )i i c trZ G Tτ τ= = . [2.78]

Both formulations are equivalent if we consider the expression of cτ . For example, with a VHS gas, equation [2.72] can be used to estimate 1/cτ ν= and we obtain:

( ) ( ) ( )( )( ) ( )30 30

7 2 5 2 (7 2 )(5 2 )tri i

i trc tr tr

F TpZ G T

T Tτ ττ μ ω ω μ ω ω

= = = =− − − −

,

which relates ( ) and G( )tr trF T T .

In a molecular approach, the interaction model includes some probability that translational energy can be transferred to or from internal energy modes. Larsen and Borgnakke consider that only a fraction if of collisions are inelastic (i.e. allows for internal energy transfer). During inelastic collisions, the total energy cE available in the collision is gathered and redistributed between the colliding molecules and their different energy modes. This redistribution obeys probability rules that are suggested by the properties of an equilibrium gas. Thus, each collision tends to make the gas closer to a translational/internal equilibrium state. The parameter if is adjusted on experimental data or on existing macroscopic relaxation rates.

We can theoretically relate if and iZ . A constant value of if corresponds to a constant value of iZ . The relationship 1i if Z= is often used. It is, however, only an approximation and should be used for qualitative considerations only. A probability ( )i cf E that depends on the collisional invariant cE satisfies the principle of detailed balancing and allows us to reproduce the dependence ( )i trZ T observed experimentally. If the exact functions ( )i cf E or ( )i trZ T are unknown, constant values can be taken, valid for the temperature range that plays an essential role in the problem, usually the wall temperature.


2.4.2. Gas–surface interaction models

At a macroscopic scale, gas–surface interaction is only involved through the accommodation coefficients that appear in the boundary conditions (velocity slip and temperature jump).

At a molecular scale, the gas–surface interaction model is directly involved in the description of molecular motion. It allows us to obtain the velocity 2c and the internal energies ,2ie of a molecule after its reflection on a wall at temperature wT if this molecule has impinged the wall with velocity 1c and internal energies ,1ie . Different models have been proposed and the most popular of them are briefly described in the following.

2.4.2.1. Specular reflection

The molecule reflects on the wall like a light ray on a mirror. Only the normal velocity component is affected (it changes sign). The molecule exchanges neither tangential momentum nor energy with the wall. Wall temperature has no influence. This simple model is unrealistic for industrial gases and surfaces.

2.4.2.2. Diffuse reflection with full accommodation

The incoming molecule stays on the wall for a sufficiently long time to get in equilibrium with it. The molecule is reflected as if it came from a gas at rest at wall temperature. It completely forgets its initial state. We can imagine a virtual gas at rest ( 2 =u 0 ) at temperature 2 wT T= behind the wall that emits a molecular flux whose properties are estimated by the methods presented in section 2.2.1. The number density 2n of this gas is determined by the condition that the reflected molecular flux is equal to the incoming one.

2.4.2.3. Maxwell and Knudsen models

In the Maxwell model, each incident molecule has a probability a of being diffusely reflected with perfect accommodation and has a probability (1 a− ) of being reflected specularly. This model reproduces the scatter of velocity (in direction and magnitude) of reflected molecules, while ensuring a preferential direction (symmetric to the incoming direction with respect to the wall normal). Whereas the previous models do not include parameters, the Maxwell model includes a parameter a that can be fitted to experimental data.

The Knudsen model is identical to the Maxwell model, except for the temperature of the virtual gas, which is supposed to emit molecules. Its temperature

66 Microfluidics

2T is not necessarily equal to wT , which offers an additional parameter for fitting to experimental data.

2.4.2.4. Hurlbut-Sherman-Nocilla model

In the Nocilla model, all molecules are reflected as if coming from a virtual gas at temperature 2T and non-zero velocity 2u . The direction of 2u is symmetrical to the incoming direction with respect to the wall normal. Its magnitude and 2T are adjustable. This two-parameter model includes, as particular cases, the specular reflection ( 2 1u c= , 2 0T = ) and diffuse reflection with full accommodation

2 2( 0, )wu T T= = .

Hurlbut and Sherman make 2u and 2T depend on the angle of incidence. It has been actually observed experimentally that a low-angled incidence results in a close-to-specular reflection, while a normal incidence results in a close-to-diffuse reflection.

2.4.2.5. Cercignani-Lampis-Lord model

This model is a good compromise between theoretical validity (detailed balancing is satisfied), reproduction of experimental data and ease of use [LOR 91]. Let us consider a wall with a unit-normal n oriented towards the gas. A local frame of reference ( , , )x y z is defined with the x-direction along n. Each post-collision molecular quantity (2) is a random quantity whose distribution function depends on the corresponding pre-collision quantity (1) and on an adjustable parameter. The probability a molecule with velocity 1c will be reflected with 2c is given by:

1 2 ,1 ,2 ,1 ,2 ,1 ,2( ) ( , ) ( , ) ( , )x x y y z zP c c c c c c→ = Φ Ψ Ψc c ,

where functions Φ and Ψ involve parameters na and ta respectively. The latter can be interpreted as accommodation coefficients of normal and tangential translational energy respectively. Similarly, the probability a molecule with rotational energy 1e will be reflected with rotational energy 2e is function 1 2( , )P e e that involves a rotational energy accommodation coefficient and the number of rotational degrees of freedom. All functions P, Φ and Ψ satisfy normalization conditions and detailed balancing.

2.4.2.6. Accommodation coefficients

The general definition of the accommodation coefficient Qa relative to a

molecular quantity Q is a dimensionless number defined as:


( ) ( )Q i r i wa Q Q Q Q= − − ,

which is equivalent to (1 )r Q i Q wQ a Q a Q= − + .

iQ is the density of the incoming flux of quantity Q and rQ the density of its

actually reflected flux. wQ is the density of the flux that would be reflected if molecules were diffusely reflected with full accommodation, i.e. if they were emitted by a virtual gas behind the wall, with zero velocity, number density wn and temperature Tw. The density wn would then be chosen to ensure equality between the incident number flux iN and the reflected number flux rN . Mass conservation (each incident molecule is reflected) means that Qa can also be expressed by

replacing the fluxes Q by the average values of Q over incident and reflected molecules.

Accommodation coefficients relate more or less directly to the models’ parameters. In particular, for the Maxwell model, the reflected flux of any quantity Q is written (1 )r Q i Q wQ a Q a Q= − + and the unique model parameter a is the

accommodation coefficient of any quantity Q (normal, tangential or total translational energy, rotational energy, normal and tangential momentum).

The choice of a gas–surface interaction model and of its imbedded parameters may greatly affect the results of calculations. Different models, even corresponding to identical accommodation coefficients, do not necessarily lead to identical results.

2.5. Examples of microflows

Gaseous microflows may not satisfy the validity criteria of classic fluid mechanics (the continuum approach) and therefore present rarefaction effects.

2.5.1. Couette flow in a free molecule regime

Through this example, we illustrate the unusual behavior of a gas flow due to rarefaction and show how the methods described in section 2.2.2.1 can be applied in practice.

Let us consider a gas of molecular mass m contained between two parallel planes of respective temperatures 1T and 2T , separated by a length L much smaller than the

68 Microfluidics

gas mean free path. Wall 2 moves in its own plane with velocity U. Molecules are assumed to diffusely reflect on both walls with full accommodation. A virtual gas of number density 1n and temperature 1T seems to emit molecules from wall 1. Its velocity is zero. Similarly a virtual gas characterized by 2n , 2T and a macroscopic velocity U , is associated with wall 2. The gas present between the walls is the superposition of two populations:

– gas 1, from which half of molecules have been removed (those who fly from 2 to 1);

– gas 2, from which half of molecules have been removed (those who fly from 1 to 2).

Its concentration is therefore: 1 2( ) 2n n n= + . Considering the unit area on the wall, the molecular flux from 1 to 2 is equal to the molecular flux from 2 to 1. Therefore, according to equation [2.32]:

1 / 2 1 / 21 1 ,1 2 2 ,2(2 ) ( ) (2 ) ( )n n n nn k T m G s n k T m G s= ,

with ,1 0ns = because gas 1 is at rest and ,2 2 cos 0ns s θ= = because 2θ π= (the wall normal is perpendicular to the velocity of the wall, and hence to the velocity of gas 2). Therefore:

1/ 21 2 2 1( )n n T T=

when using the expression of n:

[ ]( )1/ 21 1 22 1n n T T= + and [ ]( )1/ 2

2 2 12 1n n T T= + .

2.5.1.1. Tangential stress

The tangential momentum received by wall 1 is the same as if the wall was placed in gas 2, i.e. for a unit-area (equation [2.35]):

( )22 2 ,2 2( ) sint n nP n m G s sβ θ= .

Here 1/ 22 2(2 / )s U k T m= and 2θ π= (the normal to wall 1 is

perpendicular to the macroscopic velocity of gas 2). Thus ,2 0ns = and 1 / 2

,2( ) 1 (2 )n nG s π= , which results in:


1/ 2 1/ 22 2

2 1/ 22 1

22 21 ( )

tR T n mU R T

P n mUT Tπ π

⎡ ⎤ ⎡ ⎤= =⎢ ⎥ ⎢ ⎥

+⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦.

The tangential momentum emitted by wall 1 is zero because emitted molecules have no preferential tangential direction. Finally, wall 1 experiences a tangential stress equal to tP . This is to be compared with the classic result:

/ /u y U Lμ μ∂ ∂ ≈ .

In the continuum regime, the shear stress is independent of the number density and inversely proportional to the distance between the walls. In the free molecule regime, it is independent of the distance between the walls and proportional to the number density.

2.5.1.2. Energy flux

Similarly, the densities of energy flux associated with molecules flying from 2 to 1 and from 1 to 2, respectively (equation [2.37]) are:

( )32 1 2 2 2( , 0)qq n m G sβ→ = and ( )3

1 2 1 1 (0, 0)qq n m Gβ→ = .

Finally the density of net thermal flux that transfers from wall 2 to wall 1 is:

( )1/ 2 1 / 21 2 2

2 1 1 2 2 11 / 2 1 / 21 2

1( )

2 1

T TRq q n m R T T U

T Tγ

π γ→ →⎛ ⎞ ⎛ ⎞+

− = − +⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ −+ ⎝ ⎠⎝ ⎠.

If wall 2 is at rest ( 0U = ), this expression can be compared with the classic approach 2 1( ) /q T y T T Lκ κ= − ∂ ∂ ≈ − (Fourier’s law).

In the continuum regime, energy exchange is independent of number density and inversely proportional to the distance between the walls. In the free molecule regime, it is independent of the distance between the walls and proportional to number density.

2.5.2. Micro-orifice

Although this is a limiting case rarely encountered in practice, the present example is a good illustration of the simultaneous effects of pressure and temperature gradients on a microflow.

70 Microfluidics

Let A and B be two reservoirs that communicate through a small orifice whose dimension is much smaller than the mean free path in both reservoirs. The wall between the reservoirs is supposed to be infinitely thin. The gas in reservoirs A and B is in equilibrium at temperatures AT and BT , respectively. AT and BT can differ from one another.

2.5.2.1. Case 1

Assume reservoirs A and B are at the same temperature T at the initial time and contain molecules of different species with respective molecular masses 1m and 2m and number densities 1An and 2Bn .

The initial pressures are 1A Ap n k T= and 2B Bp n k T= in reservoirs A and B, respectively. Considering the unit area, the number flux from A to B is, according to equation [2.32]:

( )1/ 21 1 12 / ( )A n nN n k T m G s= ,

with 0ns = because the gas is macroscopically at rest. Similarly, the number flux from B to A is:

( )1/ 22 2 22 / (0)B nN n k T m G= .

The ratio between the fluxes is:

( ) ( ) ( ) ( )1/ 2 1/ 21 2 1 2 2 1 2 1A B A BN N n n m m p p m m= = .

The net molecular flux is not necessarily from the high-pressure reservoir to the low-pressure reservoir. For example, if reservoir A contains helium (m1 = 6.65·10-27 kg) at pressure pA = 50 kPa and reservoir B contains nitrogen (m2 = 46.5·10-27 kg) at pressure 100 kPa, we find N / N .=1 2 1 32 The helium molecular flux is larger than the nitrogen molecular flux. The total number of molecules in reservoir A decreases and its pressure ( p n k T= ) decreases although it was already lower.

2.5.2.2. Case 2

Suppose now that reservoir A contains a mixture of molecules 1 and 2 and that the pressure in reservoir B is kept sufficiently low to make the flux B A→ negligible. The flux of molecules 1 exiting reservoir A is:


( )1/ 21 1 12 / ( )A n nN n k T m G s= with 0ns =

and the flux of exiting molecules 2 is:

( )1/ 22 2 22 / ( )A n nN n k T m G s= with 0ns = .

The ratio of these fluxes is:

( ) ( )1/ 21 2 1 2 2 1A AN N n n m m= .

If concentrations are equal in the initial state ( 1 2A An n= ), the molecular flux of the lighter gas is larger than that of the heavier gas. Its mole fraction in reservoir A decreases. This phenomenon is exploited for isotope separation by diffusion process.

2.5.2.3. Case 3

Consider now a gas consisting of identical molecules and two reservoirs kept at different temperatures ( )A BT T> . The equilibrium state corresponds to equal molecular fluxes A B B AN N→ →= , which are written:

( ) ( )1/ 2 1/ 22 (0) 2 (0)A A n B B nn k T m G n k T m G= or

1/ 2( )A B B An n T T= and

1/ 2/ ( ) /( ) ( / ) 1A B A A B B A Bp p n T n T T T= = > .

Equilibrium does not correspond to pressure equality. Number density is higher in the cold reservoir but the pressure is higher in the hot reservoir.

A similar phenomenon appears in a tube subject to a temperature difference between both ends: molecules accumulate near the cold end.

Suppose now that an equal pressure is imposed in both reservoirs, for example if they also communicate with each other through a large orifice, the molecular fluxes through the small orifice cannot equilibrate. We obtain:

A Bp p= or A A B Bn T n T= or / /A B B An n T T= ,

and the ratio of molecular fluxes is equal to:

72 Microfluidics

( )1/ 21/ 2 1/ 2/ ( ) ( ) 1A B B A A A B B B AN N n T n T T T→ → = = < .

The device operates like a compressor (so-called Knudsen compressor) that pumps gas from the cold reservoir to the hot one through the small orifice (see section 8.2.1.2).

2.5.3. Flow rate through a rectangular or circular microchannel

The flow rate through a microchannel will now be examined in detail because it is encountered in practical applications. Let us first consider two reservoirs connected by a rectangular microchannel of length L, width D and depth H, where L D H . The gas in both reservoirs and the channel walls are kept at the same temperature T. A problem of interest consists of estimating the mass flow rate induced by a pressure difference between the reservoirs. After having successively presented how the flow rate can be calculated in the continuum and in the free molecule regimes, we will propose an expression that covers the intermediate regime.

2.5.3.1. Continuum regime

A natural expression for the flow rate in the continuum regime is obtained by multiplying the classic Poiseuille expression, by a polynomial expression that accounts for rarefaction as the Knudsen number increases:

23 22out

1 out 2 out1( / 2)

13

prD H pq A Kn A Kn

RTLμ

⎡ ⎤− ⎡ ⎤⎢ ⎥= × × + +⎣ ⎦⎢ ⎥⎣ ⎦. [2.79]

The first factor is the Poiseuille flow rate. The subscripts in and out refer to upstream and downstream reservoirs, respectively, and pr is the pressure ratio

in out/p p between the channel ends. The gas viscosity μ is estimated at temperature T, R is the perfect-gas constant per unit-mass, and outKn is the Knudsen number

out / Hλ . Coefficients 1A and 2A introduce, respectively, first- and second-order corrections to the Poiseuille flow rate.

2.5.3.1.1. First-order correction

The first-order correction ( )1 outA Kn originates in the non-zero velocity of the

gas at the wall. A simple estimation of this velocity slip is ( )s xu u yλ= ∂ ∂


(equation [2.17]), but detailed kinetic theory considerations lead to slightly different expressions. Barber and Emerson [BAR 06] write:

1 12 2

with =1.016 1.1466uw

u w

a uu

a yα λ α

π

⎛ ⎞ ⎛ ⎞− ∂= × × × ≅⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ∂⎝ ⎠⎝ ⎠

. [2.80]

Here au is the accommodation coefficient for tangential momentum. The numerical coefficient 1α comes from kinetic theory and is a source of difficulty:

– factor 1.016 is replaced by 0.98737 by some authors due to a different approach to solving the Boltzmann equation;

– it has been derived for perfect accommodation ( 1ua = ) and should depend on

ua for partial accommodation;

– it has been obtained for a hard-sphere gas, i.e. for a gas whose viscosity varies as 1/2Tμ ∝ ;

– it has been obtained for a plane wall and should be different for a curved wall, e.g. for a serpentine channel or a circular section whose radius is not large compared with the mean free path.

A number of authors just take 1 1α = . The accommodation coefficient is often derived by fitting a flow rate formula to experimental results. Therefore, any change in 1α leads to a different conclusion as to the value of the accommodation coefficient. Barber and Emerson [BAR 06] show that the experimental results of Colin et al. [COL 04] interpret the accommodation coefficient as 0.93ua = if we take 1 1α = or 0.998ua = if we take 1 0.998α = .

The definition of the mean free path (and consequently of the Knudsen number) that appears in equation [2.79] is also a source of confusion. As discussed in section 2.4.1.1.2, a number of authors calculate the mean free path with expressions only valid for hard spheres ( 1/2Tμ ∝ ), whatever the viscosity law. A VHS expression (equation [2.72]) would be more correct. The resulting values of the mean free path may differ by a factor as high as 1.6 in the worst case. Therefore, when using literature data that are expressed in terms of λ, it is essential to know how the mean free path has been calculated.

There is clearly a need for a theoretical expression of the velocity slip that would be valid for a gas other than a hard-sphere, for incomplete accommodation and for a curved wall.

74 Microfluidics

If equation [2.80] is used, we calculate the velocity profile in the channel as a function of the local gradient /dp dx , assuming the flow to be bidimensional (i.e. neglecting side-wall effects). Then we integrate from one end to the other, we assume the pressures at the ends of the channel are equal to the reservoir pressures. This hypothesis is legitimate if the channel is long and the velocity is small (compression and expansion at inlet/outlet are neglected). Coefficient A1 in equation [2.79] is then found:

11

12 21

u

p u

aA

r aα ⎡ ⎤−

= × ⎢ ⎥+ ⎢ ⎥⎣ ⎦

. [2.81]

2.5.3.1.2. Second-order correction

Different authors proposed introducing an additional second-order term in the expression of velocity slip. Assuming full accommodation of the gas at the wall, the general expression is:

22

1 2 2 -sw w

u uu

y yα λ α λ

⎛ ⎞⎛ ⎞∂ ∂⎜ ⎟= × ×⎜ ⎟⎜ ⎟ ⎜ ⎟∂ ∂⎝ ⎠ ⎝ ⎠

. [2.82]

Barber and Emerson [BAR 06] and Karniadakis et al. [KAR 05] present tables with values of α1 and α2 proposed by different authors. Proposed values of α2 range from -0.5 to 1.309, with a possible dependence on the Prandtl number and specific heat ratio. By integrating the velocity profile over the channel, the value of α2 determines the value of A2 in equation [2.79]:

22 224 ln( ) / ( 1)p pA r rα= × − . [2.83]

For example, the value 9/8 proposed by Deissler [DEI 64] leads to: 2

2 27 ln( ) / ( 1)p pA r r= × − . This value results in a good agreement between the flow

rates calculated by equation [2.79] and experimental results by Aubert and Colin [AUB 01] if Kn remains smaller than approximately 0.5.

To avoid a second-order derivative in the expression of velocity slip, Karniadakis et al. [KAR 05] propose to replace equation [2.82] with:

21

us

u w

a Kn uu

a b Kn y

⎡ ⎤⎛ ⎞− ∂⎢ ⎥= × ⎜ ⎟⎜ ⎟− ∂⎢ ⎥⎝ ⎠⎣ ⎦

, [2.84]

where b is a generalized slip coefficient.


Hadjiconstantinou [HAD 03] reminds us of the meaning of the velocity slip: the value of us is the boundary condition that must be introduced into Navier-Stokes equations to make their solution correct in the flow region where they are valid, but not necessarily in the immediate vicinity of the wall. In the Knudsen layer, whose thickness is of the order of λ, along the wall, the gas is in strong non-equilibrium and the Navier-Stokes solution does not correspond to the physical velocity profile (see Figure 2.8). Therefore, the flow rate obtained by integrating the NS velocity profile across the channel section is in error. The error is of second-order in Kn and does not affect the first-order correction. Hadjiconstantinou demonstrates that this phenomenon requires us to replace 2α by 2( )α ξ− in equation [2.83], with

0.296ξ = for a hard-sphere gas, hence:

22 224 ( ) ln( ) / ( 1)p pA r rα ξ= − × − . [2.85]

Thus, determining second-order coefficients from experimental data results in a value for 2( )α ξ− rather than for 2α .

u physical

u NS

us

Figure 2.8. Velocity slip at the wall (NS – Navier-Stokes)

A number of questions remain as to the second-order formulation of the velocity slip:

– How must 2α be modified in the case of a partial accommodation? Karniadakis et al. put the expression (2 ) /u ua a− in the factor of both terms on the right-hand side of equation [2.82].

– What is the value of ξ for a gas other than a hard-sphere gas?

76 Microfluidics

– What is the physical meaning of associating a second-order boundary condition to an equation set which is only first-order accurate?

Another interpretation of the second-order correction in equation [2.79] is proposed by Elizarova and Sheretov [ELI 03]. They apply the QHD equations described in section 2.3.1.1 to the flow through a microchannel, using a first-order expression of velocity slip. Their resulting expression for the flow rate is similar to equation [2.79], but the second-order term has its origin in the additional dissipative terms that are present in QHD equations but not in Navier-Stokes equations. Assuming a VHS gas, they find:

2 2 2

ln( )48,

( ) 1p

p

rA

K Sc rπ

ω

⎡ ⎤= ×⎢ ⎥

−⎢ ⎥⎣ ⎦ [2.86]

where Sc is the Schmidt number that can be estimated by 5 / (7 2 )Sc ω= − [BIR 98]. Although based on a different physical interpretation, it turns out that equation [2.86] results in a numerical value of A2, close to that given by equation [2.83] with Deissler’s value of α2, and therefore also has good agreement with the experiment.

1.40 1.60 1.80 2.000.00

1.00

2.00

3.00

4.00Q*10 kg/s

p / p1 2

13

1

2

3

4

Figure 2.9. Helium flow rate in a microchannel1: Navier-Stokes (NS) equations with no-slip, 2: NS equations with 1st-order slip, 3: NS equations

with 2nd-order slip 4: QHD equations with 1st-order slip, Full/open symbols: range of experimental data [LAL 01]


This is illustrated in Figure 2.9, taken from [ELI 03], where the experimental mass flow rate obtained by Lalonde [LAL 01] and also available in Colin et al. [COL 04] is compared with the Poiseuille solution and with different variants of correction terms, assuming full accommodation. The example plotted is relative to helium through a rectangular microchannel with H = 0.54 μm, D = 50 μm, L = 5 mm. The outlet pressure was 2 0.75 barp = and the Knudsen number was in the range 0.23-0.47.

The second-order correction accounts for different phenomena and it is probably unrealistic to get its theoretical expression from considerations based on small perturbations. This is because in the range of Kn, where the corrections have an interest, the sum of first-order and second-order corrections is larger than the Poiseuille term. The second-order term can be regarded as phenomenological and be written as:

( )22 ( , ) ln( ) / 1u p pA B a r rω= × − [2.87]

where B is a function of ω and au to be fitted to experimental or simulation data.

The present discussion relative to a rectangular channel can be repeated for a circular channel whose radius is denoted H. The equivalent of equation [2.79] becomes:

24 22out

1 out 2 out1

18 2

prH pq A Kn A Kn

RTLπμ

⎡ ⎤− ⎡ ⎤⎢ ⎥= × + +⎣ ⎦⎢ ⎥⎣ ⎦, [2.88]

with 11

8 21

u

p u

aA

r aα −

= ×+

.

The general form of 2A is given by equation [2.87]. If QHD equations are applied to the problem, they result in equation [2.88], with the same expression for

1A and with:

2 2 2

ln( )32

( ) 1p

p

rA

K Sc rπ

ω= ×

−. [2.89]

78 Microfluidics

Finally, equations [2.79] or [2.88], completed by expressions [2.81] and [2.87], can give a correct estimate of the flow rate for Knudsen numbers smaller than approximately 0.5. For the first-order term, the value of 1α in [2.81] must be consistent with the definition of the mean free path. Otherwise, the accommodation coefficient present looses its physical meaning.

In both cases (rectangular or circular channel), the flow rate is proportional to the difference between the squared pressures (at least for the Poiseuille term). It is also inversely proportional to the length of the channel: 2 2

in out( ) /q p p L∝ − .

We remind ourselves here that the above expressions only hold under the conditions of isothermicity, low velocity and “long channel”.

2.5.3.2. Free molecule regime

The free molecule regime applies when the mean free paths in both reservoirs are large compared with the transverse dimension of the channel. The gases in the upstream and downstream reservoirs are at rest and in equilibrium. They are characterized by pressures and temperatures ( in in,p T ) and ( out out,p T ), respectively. In the channel, gas molecules collide with the channel walls, but not with one another. The local properties of the gas in the channel can be defined, but they do not play any role in the process. Each molecule behaves independently of the others.

As illustrated in Figure 2.10, a molecule that enters the channel at one end experiences zero or more collisions with the walls before exiting at the other end (with probability P) or through the same end (with probability 1 – P).

Figure 2.10. Schematic path of molecules in a microchannel in a free molecule regime

The density of molecular flux that enters the channel through the upstream end is

given by equation [2.32]: ( ) ( )1/2 1/22 2in in inN n RT π= . It contributes to the mass

flow rate by inmN P× . Similarly, the density of molecular flux that enters the


channel through the downstream end contributes to a reverse flow by

( ) ( )1/2 1/22 2out outm n RT Pπ × . For reasons of symmetry, the probability P is the

same in both directions. The net mass flow rate is:

( )( ) ( )( )

( ) ( ) ( )

1/21/2ML in in out out1/2

in out1/2 1/2 1/2

in out

2

,2

P Aq RT RT

P A p p

RT RT

ρ ρπ

π

= −

⎛ ⎞⎜ ⎟= −⎜ ⎟⎝ ⎠

[2.90]

where A denotes the area of the channel section. Equation [2.90] is valid for any shape of channel section. In particular, A HD= for a rectangular channel of depth H and width D, and 2A Hπ= for a circular channel of radius H.

In the isothermal case ( in outT T T= = ), the flow rate is written:

( )( )out

ML 1/ 2

1

2

pp rq P A

RTπ

× −= × × . [2.91]

It is proportional to the pressure difference, rather than to the difference of squared pressures, as in the Poiseuille expression.

The estimation of P is conveniently done by a Monte Carlo procedure described in [BIR 98] or [LEN 06]. A large number of molecules are successively injected into the channel through one of its ends. Each molecule is tracked as it moves. When it collides with a wall, it is re-emitted according to the gas–surface interaction model. Assuming Maxwell’s model, it has a probability a of being re-emitted in a random direction and a probability (1 – a) of being reflected specularly. a is the accommodation coefficient of any quantity. Finally, the molecule exits through one or the other end. After a large number of molecules have been tracked, the transmission probability P is obtained. The result does not depend on wall temperature. It depends only on channel geometry and a, for example

( / , )P P L H a= for a circular or 2D plane channel or ( / , / , )P P L H L D a= for a rectangular channel. This is illustrated in Table 2.3 and in Figure 2.11 for a circular channel.

80 Microfluidics

L/H a = 1 a = 0.5 a = 0.2 a = 0.1 a = 0.01 1.00E-02 9.95E-01 9.98E-01 9.99E-01 1.00E+00 1.00E+00 3.00E-02 9.85E-01 9.93E-01 9.97E-01 9.98E-01 1.00E+00 1.00E-01 9.53E-01 9.75E-01 9.90E-01 9.95E-01 9.99E-01 3.00E-01 8.70E-01 9.28E-01 9.68E-01 9.82E-01 9.98E-01 1.00E+00 6.72E-01 7.96E-01 8.93E-01 9.37E-01 9.90E-01 3.00E+00 4.21E-01 5.88E-01 7.56E-01 8.41E-01 9.71E-01 1.00E+01 1.91E-01 3.39E-01 5.35E-01 6.67E-01 9.21E-01 3.00E+01 7.68E-02 1.65E-01 3.22E-01 4.59E-01 8.32E-01 1.00E+02 2.52E-02 6.11E-02 1.45E-01 2.41E-01 6.64E-01 3.00E+02 8.75E-03 2.24E-01 5.88E-02 1.09E-01 4.61E-01 1.00E+03 2.67E-03 7.09E-03 1.96E-02 3.86E-02 2.45E-01 3.00E+03 8.87E-04 2.37E-03 6.75E-03 1.38E-02 1.11E-01 1.00E+04 2.58E-04 7.07E-04 2.10E-03 4.29E-03 3.98E-02

Table 2.3. Probability P(L/H, a) that a molecule will pass through a circular microchannel of length L and radius H. a is the accommodation coefficient at the wall

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00

1.E-01 1.E+00 1.E+01 1.E+02 1.E+03 1.E+04

P

L/H

a=1a=0.5a=0.2a=0.1a=0.01

Figure 2.11. Probability P(L/H,a) that a molecule will pass through a circular microchannel of length L and radius H. a is the accommodation coefficient at the wall


In contrast with the continuum formulation presented before, this calculation in the free molecule regime does not require L/H to be large. If L/H is very large ( 410> ), the probability P and the flow rate tend towards zero as 1( / )L H − , a behavior similar to that of the continuum regime.

The above free molecule calculation can be used for Knudsen numbers larger than approximately 10.

2.5.3.3. Transition regime

The objective of the transition regime is to estimate the flow rate through a microchannel when the conditions are too rarefied for a continuum approach but not sufficiently rarefied for the free molecule regime.

Although an exact method could be used, such as the DSMC method or solving a model Boltzmann equation, we present a bridging method here, similar to that used for aerothermodynamic problems. Basically the idea consists of finding an expression of the flow rate that coincides with the continuum formula when Kn → ∞ and reduces to the free molecule formula when Kn → ∞ .

Elizarova [ELI 07] proposes to introduce a denominator into the last term of equations [2.79] and [2.88].

Equation [2.79], valid for a rectangular channel, becomes:

23 2 2out 2 out

1 outout

1( / 2)1

3 1prD H p A Kn

q A KnRTL Knμ β

⎡ ⎤ ⎡ ⎤−⎢ ⎥= × × + +⎢ ⎥⎢ ⎥ +⎢ ⎥⎣ ⎦⎣ ⎦

, [2.92]

with 11

12 21

u

p u

aA

r aα −

= ×+

and 2 2

ln( )( , )

1p

up

rA B a

rω= ×

−.

Equation [2.88], valid for a circular channel, becomes:

24 2 2out 2 out

1 outout

11

8 2 1prH p A Kn

q A KnRTL Kn

πμ β

⎡ ⎤ ⎡ ⎤−⎢ ⎥= × × + +⎢ ⎥⎢ ⎥ +⎢ ⎥⎣ ⎦⎣ ⎦

. [2.93],

with 11

8 21

u

p u

aA

r aα −

= ×+

and 2 2

ln( )( , )

1p

up

rA B a

rω= ×

−.

82 Microfluidics

This denominator ( out1 Knβ+ ) does not affect the validity of the equations at low Knudsen numbers. The free parameter β is adjusted to ensure compatibility with the free molecule regime. While Elizarova uses the solution to the BGK model equation as a free molecular reference, the determination of β presented here uses equation [2.91] as a free molecular reference.

Using equation [2.72] to relate viscosity and mean free path (or Knudsen number) and taking:

( )1

1ln( )( , ) ( / , ) 224 ( ) 21

pu u

up

rB a P L H a L aK H ar

ω αβω

−⎡ ⎤⎛ ⎞⎛ ⎞ −

= × × × − ×⎢ ⎥⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟− ⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦ [2.94]

for a rectangular channel or:

( )1

1ln( )( , ) ( / , ) 216 ( ) 21

pu u

up

rB a P L H a L aK H ar

ω αβω

−⎡ ⎤⎛ ⎞⎛ ⎞ −

= × × × − ×⎢ ⎥⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟− ⎢ ⎥⎝ ⎠ ⎝ ⎠⎣ ⎦ [2.95]

for a circular channel, it can be shown that equations [2.92] and [2.93] reduce to the free molecule flow rate [2.91] when Kn → ∞ .

These formulae look complicated but they do not require more input than the continuum approach and the free molecular approach. For example, B can be taken

from the QHD formulation: [ ] 2 48 ( )B K Scπ ω⎡ ⎤= ⎣ ⎦ or [ ] 2 32 ( )B K Scπ ω⎡ ⎤= ⎣ ⎦

for rectangular or circular channels, respectively. P can be taken from Table 2.3 for a circular channel or be calculated as indicated for other geometries. ω is obtained simply from the viscosity law.

For the sake of illustration, the flow rate through a circular microchannel is plotted in Figure 2.12 for the conditions defined hereafter. It has been non-dimensionalized by the Poiseuille value. The working gas is air at T = 300 K (μ = 1.85x10-5 Pa.s, ω = 0.769, R = 287 J.kg-1.K-1). The channel is characterized by a radius H = 4 μm and a length L = 12 mm. Perfect accommodation is assumed ( 1ua a= = ). The pressure ratio is 2pr = . We have chosen 1 1.146α = , resulting

in 1 3.056A = and the QHD value 17.28B = , resulting in 2 3.99A = . Exit pressure outp varies from 1 to 100,000 Pa to make the Knudsen number vary from

35.45 10 to 545−× . The ratio L/H is equal to 3,000 and the probability P is equal to 0.00875.


The curve noted q1 corresponds to the first-order correction. It starts to depart from the Poiseuille value for out 0.02Kn = , approximately.

The curve noted q2 corresponds to the second-order correction. It departs from the previous one for out 0.1Kn = .

The curve noted qgen corresponds to the generalized expression [2.93] associated with [2.95] for β. It follows the previous curve up to out 0.3Kn = , approximately, and smoothly joins the free molecule curve qfm for out 2Kn = . Thus, it fills the gap in the transition regime, while being consistent with both the continuum and free molecule results where they are valid.

1

10

100

1000

1.E-02 1.E-01 1.E+00 1.E+01 1.E+02

Red

uced

flow

rate

Kn.out

q1

q2

qfm

qgen

Figure 2.12. Comparison of approaches for the flow through a microchannel

2.5.3.4. Experimental data and generalization to other situations

Many investigators have studied the flow rate through a microchannel. A number of them have carried out experiments. Well-documented experimental results contribute to clarifying a number of questions that remain open, e.g. the expression for slip coefficients or the modeling of gas–surface accommodation.

84 Microfluidics

Examples of such studies are presented by Colin et al. [COL 01, COL 04], who investigated nitrogen and helium flows in both circular and rectangular microchannels. Maurer et al. [MAU 03] and Ewart et al. [EWA 06] obtain experimental values of the flow rate through microchannels and fit their results by a second-order polynomial in the form of equation [2.79]. All authors discuss slip and accommodation coefficients. Ewart et al. find that the apparent accommodation coefficient depends on the tube radius. This is likely to be due to a variation in

1α when the tube radius is not large compared to the mean free path (section 2.5.3.1.1).

Based on dimensional analysis, Lengrand et al. [LEN 06] write a dimensionless form of equation [2.79]:

( )2out out1/2

out

1 ( ) 1 1 ( , , , )48 p p u

H Ks r f Kn r aL Kn

ω ωπ

⎡ ⎤= × × × − +⎣ ⎦ , [2.96]

where outs is the molecular speed ratio in the outlet conditions. The major physical uncertainties are included in a function out( , , , )p uf Kn r a ω that depends on a limited

number of variables. They suggest carrying out experiments or numerical simulations in conditions that correspond to realistic values of variables:

out , , and p uKn r a ω . Other parameters can differ from realistic ones and be chosen to

facilitate the experiment or the calculation, e.g. smaller values of /L H , lower pressure, larger channel size or greater velocity.

While the flow through a microchannel may be induced by a pressure difference, as presented in detail above, it can also be induced by a temperature difference between the reservoirs and/or a temperature gradient in the channel walls. This temperature-induced flow is known as thermal creep. It is easily modeled in the free molecule regime by equation [2.90]. It is clear that a non-zero flow rate exists if, for example in outT T≠ and in outp p= . However, temperature-driven flows can be observed even in conditions that are not sufficiently rarefied for the free molecule regime to apply. They are the basis of the so-called Knudsen compressor. Han and Muntz [HAN 09] use the DSMC method to model a Knudsen compressor and they mention previous works devoted to this application.

Based on macroscopic and phenomenological considerations, Sharipov [SHA 05] models the combination of pressure- and temperature-driven flows in a microchannel oriented in x-direction. He obtains the resulting mass flow rate:


1/ 2(2 ) ( ), with and .p p T T p TH dp H dT

q Ap RT G Gp dx T dx

ξ ξ ξ ξ= − + = = [2.97]

Here A is the area of the channel and H the radius or depth of the channel. Tables allow us to find coefficients and p TG G as functions of accommodation coefficients

and rarefaction level. Note that Sharipov characterizes rarefaction by a parameter:

1

1/2 1/2( )

(2 ) ( / 2)

pH TH

RT RTμδ

μ ρ

−⎛ ⎞

= = × ⎜ ⎟⎜ ⎟× ×⎝ ⎠. [2.98]

Considering equation [2.72], δ appears to be equal to 1/H Knλ −= except for a numerical factor. Using δ rather than Kn to characterize rarefaction avoids the before-mentioned difficulty in the definition and determination of the mean free path.

Reviews on pressure-driven and temperature-driven microflows can be found in [COL 05] and [SHA 98].

2.6. Bibliography

[AND 84] ANDERSON D.A., TANNEHILL J.C., PLETCHER R.H., Computational Fluid Mechanics and Heat Transfer, New York, Hemisphere Publishing Corporation, 1984.

[AUB 01] AUBERT C., COLIN S., “High-order boundary conditions for gaseous flows in rectangular microchannels”, Microscale Thermophysical Engineering, vol. 5, no. 1, pp. 41-54, 2001.

[BAR 06] BARBER R.W., EMERSON D.R., “Challenges in modeling gas-phase flow in microchannels: from slip to transition”, Heat Transfer Engineering, vol. 27, no. 4, pp. 3-12, 2006.

[BIR 98] BIRD G.A., Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Oxford, Clarendon Press, 1998.

[BRU 06] BRUN R., Introduction à la Dynamique des Gaz Réactifs, Toulouse, Cépaduès, 2006.

[COL 98] COLIN, S., AUBERT, C., & CAEN, R., “Unsteady gaseous flows in rectangular microchannels: frequency response of one or two pneumatic lines connected in series”, European Journal of Mechanics. B, Fluids, vol. 17, no. 1, pp. 79-104, 1998.

[COL 04] COLIN, S., LALONDE, P., CAEN, R., “Validation of a second-order slip flow model in rectangular microchannels”, Heat Transfer Engineering, vol. 25, no. 3, pp. 23-30, 2004.

86 Microfluidics

[COL 05] COLIN, S., “Rarefaction and compressibility effects on steady and transient gas flows in microchannels”, Microfluidics and Nanofluidics, vol. 1, no. 3, pp. 268-279, 2005.

[COL 06] COLIN, S., “Single-phase gas flow in microchannels”, in: Heat Transfer and Fluid Flow in Minichannels and Microchannels, S. G. Kandlikar, et al., Eds, Elsevier, pp. 9-86, 2006.

[DEI 64] DEISSLER R.G., “An analysis of second-order slip flow and temperature-jump boundary conditions for rarefied gases”, International Journal of Heat and Mass Transfer, vol. 7, pp. 681-694, 1964.

[ELI 01] ELIZAVORA T.G., SHERETOV YU.V., “Theroretical and numerical investigation of quasi-gasdynamic and quasi-hydrodynamic equations”, Computational Mathematics and Mathematical Physics, vol. 41, no. 2, pp. 219-234, 2001.

[ELI 03] ELIZAROVA T.G., SHERETOV YU.V., “Analyse du problème de l'écoulement gazeux dans les microcanaux par les équations quasi-hydrodynamiques”, La Houille Blanche, no. 5-2003, pp. 66 -72, 2003.

[ELI 07] ELIZAROVA T.G., “Knudsen effect and a unified formula for mass flow rate in microchannels”, Proc. of the 25th International Symposium on Rarefied Gas Dynamics, M.S. Ivanov and A.K. Rebrov, Eds, Siberian Branch of RAS, Novosibirsk, pp.1164-1169, 2007.

[ELI 09] ELIZAROVA T.G., Quasi Gas Dynamic Equations, Springer, 2009.

[EWA 06] EWART T., PERRIER P., GRAUR I.A., MÉOLANS J.G., “Mass flow rate measurements in gas micro flows”, Experiments in Fluids, vol. 43, no. 3, pp.487-498, 2006.

[HAD 03] HADJICONSTANTINOU, N.G., “Comments on Cercignani’s second-order slip coefficient”, Physics of Fluids, vol. 15, pp.2352-2354, 2003.

[HAN 09] HAN Y.-L., MUNTZ E.P., “Implications of imposing working gas temperature change limits on thermal creep driven flows”, Proc. of the 26th International Symposium on Rarefied Gas Dynamics, Kyoto, Japan, July 20-25, 2008, Edited by American Institute of Physics, pp. 305-310, 2009.

[KAR 05] KARNIADAKIS, G., BESKOK A., ALURU N., Microflows and Nanoflows. Fundamentals and Simulation, Springer, 2005.

[KOG 69] KOGAN M.N., Rarefied Gas Dynamics, New York, Plenum Press, 1969.

[KOV 90] KOVENIA V.M., TARNAVSKI L.P., CHERNIY S.G., Implementation of the Splitting Method for the Aerodynamic Problems, Novosibirsk, Nauka, 1990.

[LAL 01] LALONDE, P., “Etude expérimentale d’écoulements gazeux dans les microsystèmes à fluides”, Ph.D. Thesis, Institut National des Sciences Appliquées, Toulouse, France, 2001.

[LEN 06] LENGRAND, J.C., ELIZAROVA, T.G., SHIROKOV, I.A., “Calcul de l'écoulement visqueux compressible d’un gaz dans un microcanal”, La Houille Blanche, no. 1-2006, pp.40-46, 2006.


[LIF 63] LIFSHIZ E.M., PITAEVSKY L.P., Physical Kinetics, London, Pergamon Press, 1963.

[LOI 66] LOITSYANSKII L.G., Mechanics of Liquids and Gases, London, Pergamon Press, 1966.

[LOR 91] LORD R.G., “Some extensions to the Cercignani-Lampis gas surface scattering kernel”, Physics of Fluids A, vol. 3, no. 4, pp. 706-710, 1991.

[MAU 2003] MAURER J., TABELING P., JOSEPH P., WILLAIME H., “Second-order slip laws in microchannels for helium and nitrogen”, Phys. Fluids, vol. 15, no. 2613-2621, 2003.

[PEY 96] PEYRET R., Handbook of Computational Fluid Mechanics, London, Academic Press, 1996.

[SCH 79] SCHLICHTING H., Boundary-layer Theory, New York, McGraw-Hill, 1979.

[SHA 98] SHARIPOV F., SELEZNEV V., “Data on internal rarefied gas flows”, Journal of Physical and Chemical Reference Data, vol. 27, no. 3, pp. 657-706, 1998

[SHA 05] SHARIPOV F., “Recent results of rarefied gas dynamics and their applications in microflows”, Proc. of the 3rd International Conference on Microchannels and Minichannels, Toronto, Canada, June 13-15, 2005, ASME, 2005.

[SHE 97] SHERETOV YU.V., “Quasihydrodynamic equations as a model for viscous compressible heat conductive flows”, in: Implementation of Functional Analysis in the Theory of Approaches, Tver, Tver University Ed., pp. 127-155, 1997.

[SHE 00] SHERETOV YU.V., Mathematical Modeling of Gas and Liquid Flows Based on Quasihydrodynamic and Quasigasdynamic Equations, Tver, Tver University Ed., 2000 (in Russian).

microfluidics (colin/microfluidics) || gaseous microflows

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