Xavier Nicolas

Eric Chénier

Chahinez Tchekiken

Guy Lauriat

Eric Chenier

Revisited analysis of gas convection and heat transfer in micro channels: inuence of viscous stress power at wall on Nusselt number

Keywords: micro-channel, rst-order slip ow, compressible ow, heat transfer, Nusselt number, analytical solution

Revisited analysis of gas convection and heat transfer in micro channels: influence of viscous stress power at wall on Nusselt number

Introduction

Due to the increasing development of Micro Electro Mechanical Systems (MEMS), the study of liquid or gas ows and heat transfer in ducts, heated or not, whose hydraulic diameter, D h , is of the order of a few microns (say 1 to 100 µm), has given rise to a considerable amount of works over the past twenty years. A recent review by Kandlikar et al. [START_REF] Kandlikar | Heat transfer in microchannels -2012 status and research needs[END_REF] is dedicated to them. It is shown that monophasic liquid ows in micro channels have a behavior similar to that observed at the macroscopic scale and the classical continuum mechanics model can be used (Navier-Stokes equations with no slip boundary conditions).

However, for gas ows at the microscopic scale, specic phenomena are observed and require appropriate models [START_REF] Gad-El Hak | Fluid mechanics of microdevices -The Freeman scholar lecture[END_REF][START_REF] Karniadakis | Microows and Nanoows -Fundamentals and Simulation[END_REF]. A slightly rareed ow regime close to the wall, generated by the interaction between the gas molecules and the wall atoms, must be taken into account at the microscopic scale whereas it is negligible at the macroscopic scale or for liquid micro-ows.

More specically, a Knudsen layer whose thickness is of the order of the mean free path of the gas molecules, λ, is formed closed to the wall. In this layer, the velocity magnitudes of the gas molecules considered individually are dierent at a xed distance from the wall, due to their interactions with the wall. In other words, in this layer, the gas is in a state of local thermodynamic non-equilibrium which results in non-linear mean velocity proles and relations between stress and strain rates. From the continuum mechanics point of view, at the micro channel scale, when the Knudsen number is such that 0.001 < Kn = λ/D h < 0.1, these phenomena translate into a slip velocity and a temperature jump at the wall and, possibly, a gas ow driven by the tangential temperature gradient along the wall called thermal creep [START_REF] Sharipov | Data on the velocity slip and temperature jump on a gas-solid interface[END_REF].

The consequences of these phenomena on the macroscopic quantities such as the mass ow rate, the friction factor, the bulk temperature and the wall heat ux can be signicant [START_REF] Lockerby | The usefulness of higher-order constitutive relations for describing the Knudsen layer[END_REF] and must be taken into account in the modeling of the convective heat transfer in MEMS with gas ows. Indeed they may have antagonistic eects on the heat transfer.

Gas micro-ows, possibly with heat transfer, can be found in : micro heat exchangers for the cooling of electronic components or in chemistry [START_REF] Yang | Experimental analysis of the inuence of wall axial conduction on gas-to-gas micro heat exchanger eectiveness[END_REF][START_REF] Dixit | Review of micro-and mini-channel heat sinks and heat exchangers for single phase uids[END_REF], micro pumps and turbines, including the thermal transpiration-driven Knudsen pumps for vacuum pumping applications [810], micro-systems for the species separation in gas mixtures such as the method of gas separation by membranes [START_REF] Nakaye | Thermally enhanced membrane gas separation[END_REF], micro gas analyzers such as micro mass spectrometers and micro-chromatographs [START_REF] Qin | iGC1: An integrated uidic system for gas chromatography including Knudsen pump, preconcentrator, column, and detector microfabricated by a three-mask process[END_REF][START_REF] Qin | iGC2: An architecture for micro gas chromatographs utilizing integrated bi-directional pumps and multi-stage preconcentrators[END_REF], supersonic gas ows in micro-nozzles to control the nano-satellite attitude or the boundary layers in aerodynamics [1418],

articial lungs [START_REF] Potkay | The promise of microuidic articial lungs[END_REF][START_REF] Kovach | In vitro evaluation and in vivo demonstration of a biomimetic, hemocompatible, microuidic articial lung[END_REF], pressure, ow rate and temperature micro-sensors in gas ows [START_REF] Jang | Gaseous slip ow analysis of a micromachined ow sensor for ultra small ow applications[END_REF][START_REF] Vittoriosi | A sensor-equipped microchannel system for the thermal characterization of rareed gas ows[END_REF], etc. This paper investigates the theoretical models available to simulate and analyze the slightly rareed gas micro-ows with heat transfer, when Kn 0.1. We focus on the modeling of forced convection of pure diluted gases in micro channels by a continuous approach based on the Navier-Stokes equations and rst order slip and temperature jump boundary conditions. It appears that simplied models are often used in the literature for this ow type, but without relevant justication and with recurrent errors propagating from one paper to the other, particularly concerning the heat transfer analysis and the energy equation. Our purpose is to provide a consistent model for gas micro-ows and heat transfer and to compare with the vanishing values of the Nusselt number obtained in experiments [START_REF] Demsis | Experimental determination of heat transfer coecient in the slip regime and its anomalously low value[END_REF][START_REF] Demsis | Heat transfer coecient of gas owing in a circular tube under rareed condition[END_REF].

In that aim, the characteristic length scales of the continuum description and the way of modeling the Knudsen layer are reminded in 2. The values of the slip and temperature jump coecients are particularly discussed. The complete model for forced convection in heated micro channels is established and discussed in 3. A dimensional analysis is developed and the analytical solution of the temperature eld given by a simplied asymptotic model for compressible gas convection in an isothermal micro-channel is established in 4. This solution is compared with the numerical solution of the full model obtained from nite volume simulations in 5. Furthermore, from the numerical simulations, the heat ux balances for slip and no slip ows and incompressible and compressible ows are analysed in details. The numerical method to solve the full model is presented in 5.1 and the analytical and numerical solutions are compared in 5.2. The heat ux balances and the very small values of the Nusselt number obtained in the experiments by Demsis et al. [START_REF] Demsis | Experimental determination of heat transfer coecient in the slip regime and its anomalously low value[END_REF][START_REF] Demsis | Heat transfer coecient of gas owing in a circular tube under rareed condition[END_REF] are explained in 5.3.

Length scales of the continuum description and Knudsen layer modeling

The mean free path, λ, is the average distance traveled by the molecules between two successive collisions. It is the main scale to evaluate the rarefaction rate in a gas ow and the validity domain of the continuum description. In this paper, the most standard denition used for ideal gases is retained [START_REF] Gad-El Hak | Fluid mechanics of microdevices -The Freeman scholar lecture[END_REF][START_REF] Lockerby | The usefulness of higher-order constitutive relations for describing the Knudsen layer[END_REF]2628]:

λ = µ p πrT 2 = µ ρ π 2rT = µ π 2pρ (1)
where r is the specic gas constant.

A scale analysis of the breakdown of the continuum description of gas ows was presented by Bird [START_REF] Bird | Molecular gas dynamics and the direct simulation of gas ows[END_REF] and recalled by Gad-El-Hak [START_REF] Gad-El Hak | Fluid mechanics of microdevices -The Freeman scholar lecture[END_REF][START_REF] Gad-El Hak | Comments on "critical view on new results in microuid mechanics[END_REF], Colin [START_REF] Colin | Rarefaction and compressibility eects on steady and transient gas ows in microchannels[END_REF] and Zhang et al. [START_REF] Zhang | A review on slip models for gas microows[END_REF]. For dilute gases, the limit on the range of validity of the continuum equations is rst due to the local thermodynamic non-equilibrium and the presence of very steep gradients in the ow elds (on characteristic lengths, L, of the same order as the mean free path, λ), rather than due to statistical uctuations of the macroscopic variables. It is generally admitted that the thermodynamic equilibrium is satised and the Navier-Stokes equations are valid everywhere for Kn = λ/D h < 0.001 to 0.01, and in the ow core only for 0.01 < Kn < 0.1. In the latter case, in the thin layer close to the wall (the Knudsen layer), the continuum equations are not valid because the gas molecules only "see" a half-space where the nature of the shocks with the wall is dierent from the inter-molecular shocks: a local thermodynamic non-equilibrium is present in this zone.

An accurate description of the thin Knudsen layer whose thickness is between about λ and 3λ [START_REF] Lockerby | The usefulness of higher-order constitutive relations for describing the Knudsen layer[END_REF][START_REF] Zhang | A review on slip models for gas microows[END_REF] is crucial for microuidics applications. Momentum and energy are indeed transferred between the gas and the wall through this layer. An ill description has thus signicant consequences on the evaluations of the mass ow rate and friction factor, or on the maximum bulk temperature and wall heat ux. For 0.001 < Kn < 0.1, a continuous approach coupled with a modeling of the ow and heat transfer in the Knudsen layer is generally considered. The most used model consists in solving the Navier-Stokes equations with slip boundary conditions and semi-empirical coecients, such as the accommodation coecient or the slip length, to model the gas/wall interaction. A similar model is used for heat transfer: the energy equation is solved with a temperature jump boundary condition to mimic the thermal resistance of the Knudsen layer. The choice of the slip and temperature jump coecients is detailed below because inadaquate or unconsistent values of these coecients are regularly used in the literature.

Slip and thermal creep boundary conditions

The rst-order slip boundary condition was rst introduced by Navier in 1823 [START_REF] Navier | Mémoire sur les lois du mouvement des uides[END_REF], then independently by Maxwell in 1879 [START_REF] Maxwell | On stresses in rareed gases arising from inequalities of temperature[END_REF]. The simplied form on an impermeable wall writes:

u g -u w = L s ∂u ∂ - → n g (2)
v g = 0

where u and v are the velocity components of the gas, tangential and normal to the wall respectively, u g is the slip velocity of the gas on the wall, u w is the velocity of the wall (u w = 0 in general), L s is the slip length and -→ n is the direction normal to the wall directed toward the gas with ∂u/∂ -→ n = ∇u. -→ n . Here and in the following, the subscript g is used to denote quantities on the gas side of the wall (slip-related quantities associated with the gas molecules in contact with the wall) and the subscript w is used to denote the quantities on the solid side of the wall.

The slip boundary condition [START_REF] Gad-El Hak | Fluid mechanics of microdevices -The Freeman scholar lecture[END_REF] and the comparison of the true and modeled velocity proles of the gas in the Knudsen layer are illustrated in the left part of Fig. 1. The "true" velocity prole (in red) is non-linear close to the wall and presents a slip speed at the wall denoted u g,true . The Navier and Maxwell model consists in approximating this velocity prole in the Knudsen layer by the blue prole whose slip velocity, u g , is greater than the actual slip velocity, u g,true , by considering that the dierence u g -u w is proportional to the normal velocity gradient along the Knudsen layer,

∂u ∂ -→ n g
. The slip length, L s , in this model is a semi empirical parameter, proportional to the mean free path, λ, and depends on the gas and wall nature, on the wall roughness and, more generally, on the type of gas/wall interaction (diuse, specular or mixed specular and diuse scattering of the gas molecules at the wall). L s must be evaluated so that u g provides a good approximation of the velocity prole outside the Knudsen layer.

There are many formulations of L s in the literature which are more or less equivalent: some formulations depend on a viscous slip coecient, σ µ , such as L s = σ µ λ; others are function of the Tangential Momentum Accommodation coecient (TMAC) 1 , denoted σ u here, such as 1 TMAC represents the average fraction of the momentum transferred to the wall atoms in the tangential direction by the incident gas molecules impacting the wall : 0 < σu < 1 in case of partial accommodation of the gas molecules with the wall, σu = 1 in case of a diuse reection (or full accommodation) of the gas molecules [START_REF] Loyalka | Some numerical results for the BGK model: thermal creep and viscous slip problems with arbitrary accommodation of the surface[END_REF][START_REF] Cercignani | Mathematical methods in kinetic theory[END_REF]. Recently, following the extensive work by Sharipov [START_REF] Sharipov | Data on the velocity slip and temperature jump on a gas-solid interface[END_REF], the rst formulation seems to settle as a standard [START_REF] Kandlikar | Heat transfer in microchannels -2012 status and research needs[END_REF][START_REF] Zhang | A review on slip models for gas microows[END_REF]. However, the slip boundary condition given by equation ( 2) is very simplied. A more general form, valid for non isothermal ows, can be written :

L s = [ u (2-σ u )/σ u ]λ with u a corrective coecient
u g -u w = σ µ λ ∂u ∂ - → n + ∂v ∂ - → t g + σ θ µr p ∂T ∂ - → t g (4)
where σ µ and σ θ are the viscous and thermal slip coecients (detailed below), µ the dynamic viscosity and -→ t the direction tangential to the wall. Compared with Eq. ( 2), the slip velocity in Eq. ( 4) also depends on the tangential gradients of the normal velocity component, v, and temperature, T . The term ∂v

∂ -→ t g
vanishes if the wall is plane and smooth. However, it must be taken into account in case of high roughness or sharp curvature of the wall, when the radius of curvature is about or less than the mean free path, λ, of the gas molecules [START_REF] Lockerby | Velocity boundary condition at solid walls in rareed gas calculations[END_REF][START_REF] Colin | Gas microows in the slip ow regime: A critical review on convective heat transfer[END_REF].

The last term in Eq. ( 4) is the thermal creep or thermal transpiration term. It reects the presence of a tangential ow along the wall, generated by a longitudinal temperature gradient.

The thermal creep is directed from low temperatures to high temperatures. Thanks to a dimensional analysis, we will show that this phenomenon is only signicant at very small Reynolds numbers. When the thermal creep is negligible, the rst order slip boundary condition (4) on a plane and smooth wall simply writes:

u g -u w = σ µ λ ∂u ∂ - → n g (5)
Sharipov [START_REF] Sharipov | Data on the velocity slip and temperature jump on a gas-solid interface[END_REF] proposed a synthesis of the theoretical and experimental values of the viscous and thermal slip coecients, σ µ and σ θ , and also the thermal jump coecient ξ T (see Eq. ( 6))

available in the literature. The theoretical data in [START_REF] Sharipov | Data on the velocity slip and temperature jump on a gas-solid interface[END_REF] are mainly calculated from the Boltzmann kinetic equation or various kinetic equations, like the BGK kinetic model. Due to the dierent denitions of the mean free path λ between [START_REF] Sharipov | Data on the velocity slip and temperature jump on a gas-solid interface[END_REF] and Eq. ( 1), all the data for these coecients in [START_REF] Sharipov | Data on the velocity slip and temperature jump on a gas-solid interface[END_REF] must be multiplied by the factor 2 √ π to be used with the present formulation. Thus, we get σ µ ≈ 1.1, σ θ ≈ 1.2 and ξ T ≈ 2.2 with the present denition of λ (Eq. ( 1)), in the case of diuse scattering (or full accommodation) of the gas molecules at the wall. To take into and σu → 0 in case of specular reection (no accommodation).

account a non complete accommodation, it is generally assumed that a part, evaluated by the coecient T M AC = σ u , of incident particles is scattered diusely, while the rest of particles, i.e. the (1 -σ u ) part, is reected specularly. Thus, in case of non diuse scattering, only a part of the molecules accommodates with the wall and it is shown that σ µ is greater than 1.1

and diverges for a specular scattering (for σ u → 0), whereas σ θ is smaller than 1.2 and tends to

3 4 × 2 √ π
for a specular scattering. For the sake of simplicity and as, in most of the experiments, the gas/wall interaction is diuse (T M AC = σ u ≈ 1), we will take σ µ = σ θ = 1 and ξ T = 2

in the present numerical applications. Note that, in most of the published numerical works, σ µ = 1 and σ θ = 3/4 are used at the same time what is erroneous since the former expression corresponds to a diuse scattering and the latter to a specular scattering.

Thermal jump boundary condition

Similarly to the Navier and Maxwell slip boundary conditions, Smoluchowski (1898) [START_REF] Smoluchowski | Ueber wärmeleitung in verdünnten gasen[END_REF] introduced an equivalent boundary conditions to characterize the temperature jump observed at the wall in the Knudsen layer. This temperature jump and its modeling are illustrated in the right part of Fig. 1. The boundary conditions equivalent to Eq. ( 5) but for the temperature can be written, at the rst order with Kn [START_REF] Kandlikar | Heat transfer in microchannels -2012 status and research needs[END_REF][START_REF] Sharipov | Data on the velocity slip and temperature jump on a gas-solid interface[END_REF][START_REF] Zhang | A review on slip models for gas microows[END_REF]:

T g -T w = ξ T λ ∂T ∂ - → n g (6)
where T g is the gas temperature at the wall, T w the wall temperature and ξ T the temperature jump coecient such as the thermal jump length in Fig. 1 is L s,T = ξ T λ. According to the bibliographical synthesis [START_REF] Sharipov | Data on the velocity slip and temperature jump on a gas-solid interface[END_REF], with λ given by Eq. ( 1), ξ T ≈ 2.1 to 2.2 for diuse scattering of mono atomic gases (H e , Ar , N e and Xe) and their binary mixtures at the walls.

In the literature, Eq. ( 6) is often considered in the following form:

T g -T w = 2 -σ T σ T 2γ γ + 1 λ P r ∂T ∂ - → n g (7)
where σ T is the thermal accommodation coecient [START_REF] Karniadakis | Microows and Nanoows -Fundamentals and Simulation[END_REF] (equivalent of TMAC for the thermal energy), γ = C p /C v is the isentropic coecient and P r the Prandtl number. However, in the case of a mono-atomic (respectively diatomic) ideal gas, γ = 5/3 (respectively 7/5) and P r ≈ 0.7. So, as σ T = 1 for a diuse gas/wall interaction, the factor 2-σ T σ T 2γ γ+1 1 P r in Eq. (7) is approximately equal to 1.6 to 1.8 for bi and mono atomic gases which is noticeably smaller than ξ T ≈ 2.1 to 2.2 given by Sharipov [START_REF] Sharipov | Data on the velocity slip and temperature jump on a gas-solid interface[END_REF].

Finally note that the temperature jump boundary condition that reects the rarefaction and the presence of the Knudsen layer is valid and must be applied regardless of the type of thermal condition applied: xed temperature, imposed heat ux or heat conduction in the wall.

We insist on this point because some authors did not take into account the temperature jump condition when a heat ux is imposed on a wall to determine the wall temperature.

3 Mathematical model for gas ows in micro channels

Main physical phenomena and modeling issues

For ows in micro channels of large aspect ratio, L/D h , and typical sizes 1 µm ≤ D h ≤ 10 µm et 100 µm ≤ L ≤ 1 mm, submitted to a moderate heating of the walls and to pressure variations between the inlet and the outlet of the channel of the order of one bar to a few bars, the conversion of the mechanical work of the viscous forces into internal energy is very important. the viscous dissipation (heat source) [START_REF] Hong | Heat transfer characteristics of gaseous ows in microchannel with constant heat ux[END_REF][START_REF] Sun | Convective heat transfer in pressure-driven nitrogen slip ows in long microchannels: The eects of pressure work and viscous dissipation[END_REF], the work of the viscous forces at the wall in the presence of dynamic slip [START_REF] Sparrow | Laminar heat transfer in tubes under slip-ow conditions[END_REF][START_REF] Hadjiconstantinou | Dissipation in small gaseous ows[END_REF][START_REF] Hong | Some considerations on thermal boundary condition of slip ow[END_REF],

the variation of the physical properties with temperature [START_REF] Sun | Numerical modeling of pressure-driven nitrogen slip ow in long rectangular microchannels[END_REF][START_REF] Quasi Zade | Heat transfer characteristics of developing gaseous slip-ow in rectangular microchannels with variable physical properties[END_REF],

the dominant thermal and dynamical axial diusions at the inlet and outlet boundaries when Re and P e are less than unity, the heat conduction in the walls (conjugate heat transfer) because they are usually thicker than the channel and more conducting than the gas [START_REF] Sun | Conjugate thermal transport in gas ow in long rectangular microchannel[END_REF].

the relative roughness of the walls, which can be important at these small scales [START_REF] Zhang | A review on slip models for gas microows[END_REF]5256].

A bibliographical review of numerous numerical studies on this subject shows that all these eects are never taken into account simultaneously. This may be justied in some cases when the eect of the omitted term in the model is negligible on the dynamic and thermal behavior of the ow. However, the justication of the simplifying assumptions used in the model is often omitted. One objective of this paper is thus to study the inuence of dierent phenomena described above in the case of gas ows in 2D straight micro channels. The analysis will particularly focus on the thermal aspects: we will analyze the inuence of dierent terms in the energy equation and thermal boundary conditions on the evolution of the temperature eld and Nusselt number.

To this end, we present below full and simplied mathematical models for modeling gas micro-ows in the framework of continuum mechanics. The general conservation equations are presented in 3.2. The viscous stress power at the walls is discussed in 3.3 and its inuence on the sum of the pressure work and viscous dissipation in 3.4. The summary of the governing equations and boundary conditions in their dimensional form is given in 3.5. The dimensionless equations and the main results derived from a dimensional analysis of these micro ows are presented in 3.6.

General conservation equations

In the present study, we consider steady gas ows in isothermal parallel-plate micro channels with the inlet and outlet pressures prescribed at p in and p out . The geometry is presented in Fig. 2. The ow is assumed to be two-dimensional, symmetrical through the horizontal midplane, steady, compressible and laminar. The uid is an ideal gas with potentially variable physical properties. The governing equations are the mass, momentum and energy conservation equations in which the body forces are neglected with respect to the viscous forces. The continuity and Navier-Stokes equations write [START_REF] Bird | Transport phenomena[END_REF]:

∇.(ρ - → v ) = 0 (8) ∇.(ρ - → v ⊗ - → v ) = -∇p + ∇.τ (9)
where the shear stress tensor τ is dened for a Newtonian-Stokes uid by :

τ = µ(∇ - → v + ∇ - → v t ) - 2 3 µ∇. - → v I (10)
By noting h the enthalpy and e c = -→ v 2 /2 the kinetic energy per mass unit of the gas, the total energy conservation equation writes [START_REF] Bird | Transport phenomena[END_REF]:

∇.(ρ - → v h + ρ - → v e c -k∇T -τ . - → v ) = 0 (11)
With the present assumptions, the mechanical energy balance (or equation of change for mechanical energy) can be written:

∇.(ρ - → v e c -τ . - → v ) = -- → v .∇p -τ : ∇ - → v (12)
By subtracting Eq. ( 12) from Eq. ( 11), we get the enthalpic form of the energy equation:

∇.(ρ - → v h -k∇T ) = - → v .∇p + τ : ∇ - → v (13)
The viscous dissipation (V D) is a heat source term in the enthalpy equation ( τ : ∇ -→ v > 0)

whereas the pressure work (P W ) term is a heat sink ( -→ v .∇p < 0) since the velocity vector, -→ v , and the pressure gradient, ∇p, have opposite directions in a channel ow of constant section.

For a compressible gas ow in a constant section duct, both the magnitudes of V D and P W increase downstream due to the gas expansion (acceleration) at the channel outlet and the transformation of the enthalpy/internal energy into kinetic energy.

For an ideal gas, the density is given by the equation of state:

ρ(p, T ) = p rT [START_REF] Manzoni | Micro and nanotechnology applications for space micropropulsion[END_REF] where r is the specic gas constant, and the enthalpy per unit of mass is computed by:

h(T ) -h ref (T ref ) = T T ref C p (T ) dT (15)
where

h ref = h(T ref )
is the known reference enthalpy at the reference temperature T ref .

In the previous equations, the thermal conductivity, k, the dynamic viscosity, µ, and the specic heat capacity at constant pressure, C p , are a priori temperature dependent. However, in the present study, since the temperature variations are small due to the used isothermal wall condition, the temperature dependence of the physical properties is negligible and not considered hereafter.

Viscous stress power (V SP ) at impermeable walls

We consider a compressible, non-isothermal gas ow in a channel (as shown in Fig. 2). The boundary of the ow domain, Ω, is limited by the following four surfaces: the inlet section, S in , the outlet section, S out , the surface on the gas side of the impermeable wall, S w,g , and the symmetry surface, S s . -→ n is the inward normal unit vector of the closed surface Σ = S in ∪ S out ∪ S w,g ∪ S s (Fig. 2). Then, by integrating the total energy equation [START_REF] Nakaye | Thermally enhanced membrane gas separation[END_REF] on Ω, applying the Gauss theorem and simplifying, we get:

S in ∪Sout (ρ - → v h + ρ - → v e c -k∇T -τ . - → v ). - → n dS + Sw,g ( -k∇T | g -τ . - → v g ). - → n dS = 0 (16)
in which the power of the viscous stress (τ

. - → v ). - → n g = 0 in case of slipping ow along the wall since - → v g = u g (x) - → e x = - → 0 and (τ . - → v ). - → n g = µ g u g ∂u ∂ - → n g (17)
Therefore, in case of rarefaction and slip conditions, the conservation of the heat ux between the gas and the solid wall implies that the total heat ux density transmitted through the walls,

- → q t,w . - → n = q t,w (x) [W/m 2 ]
, is equal to the sum of the diusion heat ux and the power of the viscous stress:

- → q t,w . - → n = -k∇T | g -τ . - → v g . - → n (18)
Thus, using Eq. ( 17), the total heat ux density transmitted through the wall writes:

q t,w = -k g ∂T ∂ - → n g dif f usion heat f lux - µ g u g ∂u ∂ - → n g viscous stress power≥0 (19)
where k g (x) = k(T g (x)) and µ g (x) = µ(T g (x)) are the gas conductivity and viscosity at the gas temperature close to the walls, T g (x) (see Fig. 1 and Eqs. ( 6) or ( 7)). In Eq. ( 19), the power of the viscous stress at the wall (the last term under the brace) is positive or null. Thus the total heat ux density through the wall, q t,w , is always smaller than the diused or convected heat ux density to the uid and it can be positive or negative according to the relative magnitude of the two R.H.S terms.

The expression [START_REF] Potkay | The promise of microuidic articial lungs[END_REF] of the wall heat ux was presumably introduced by Maslen (1958) [START_REF] Maslen | On heat transfer in slip ow[END_REF] then discussed by Sparrow and Lin (1962) [START_REF] Sparrow | Laminar heat transfer in tubes under slip-ow conditions[END_REF]. Since then, only a few authors have taken into account the power of the viscous stress at the wall in their analysis of heat transfer [START_REF] Miyamoto | Chocked ow and heat transfer of low density gas in a narrow parallel-plate channel with uniformly heating walls[END_REF][START_REF] Hadjiconstantinou | Dissipation in small gaseous ows[END_REF][START_REF] Shi | Chocked ow of low density gas in a narrow parallel-plate channel with adiabatic walls[END_REF].

So, as most of the authors neglected this contribution, it was eventually forgotten. Hong and Asako [START_REF] Hong | Some considerations on thermal boundary condition of slip ow[END_REF] reiterated its importance in a paper in 2010 and, more recently, some papers have used total heat ux expressions similar to Eq. ( 19) to analyse heat transfer in isoux micro channels [START_REF] Ramadan | A Numerical Study of the Extended Graetz Problem in a Microchannel with Constant Wall Heat Flux: Shear Work Eects on Heat Transfer[END_REF][START_REF] Ramadan | Pressure Work and Viscous Dissipation Eects on Heat Transfer in a Parallel-Plate Microchannel Gas Flow[END_REF]. In the present paper, we will analyze the contribution of q t,w on the Nusselt number, in the case of isothermal micro channels. We will show that it allows to recover very low values of the Nusselt number as it was experimentally obtained by Demsis et al. [START_REF] Demsis | Experimental determination of heat transfer coecient in the slip regime and its anomalously low value[END_REF][START_REF] Demsis | Heat transfer coecient of gas owing in a circular tube under rareed condition[END_REF].

It appears that to get a continuum model consistent with the conservation of the total heat ux, it is necessary to consider the power of the viscous stress at the wall when slip is present.

Eq. ( 19) is therefore the consequence of the presence of a slipping ow and this condition is independent of the temperature jump condition, Eqs. ( 6) or [START_REF] Dixit | Review of micro-and mini-channel heat sinks and heat exchangers for single phase uids[END_REF]. So, in case of moderate rarefaction (0.001 ≤ Kn ≤ 0.1), Eqs. ( 19) and ( 6) or ( 7) must be simultaneously satised in the continuum model. Most of the published works consider only one of these two conditions.

Pressure work (P W ) and viscous dissipation (V D) for weakly compressible ows

In numerous studies on gas micro-ows, the contributions of the pressure work and the viscous dissipation in the energy equation ( 13) were either both neglected or the ow was considered quasi-incompressible and only the viscous dissipation was accounted for. We remind here some important theoretical results on these assumptions.

We consider rst the integration of the mechanical energy equation ( 12) on the slice δΩ of the impermeable wall micro-channel of Fig. 2, whose boundary is δΣ = S -∪ S + ∪ δS w,g ∪ δS s and -→ n is the inward normal unit vector. After applying the Gauss theorem and simplifying, we get:

δΩ - → v .∇p + τ : ∇ - → v dΩ P W +V D = S -∪S + (ρ - → v e c -τ . - → v ). - → n dS I S -∪S + ≈0 + δSw,g -(τ . - → v ). - → n dS V SP (20)
where the rst R.H.S integral I S -∪S + is null for fully established incompressible ows and is negligible for weakly compressible ows since, in this case, the velocity prole is (nearly) the same on the inlet (S -) and outlet (S + ) sections of δΩ. Furthermore the last integral of the viscous stress power (V SP ) on the wall (δS w,g ) vanishes for no slipping ows.

Consequently the integral, P W + V D, of the pressure work and viscous dissipation on the slice δΩ is very small (negligible) for fully established no slipping ows with weak streamwise density variations: in this case, the pressure work compensates the viscous dissipation, both locally (on a channel slice) and also on the whole domain, except at the inlet and the outlet where the entrance and expansion eects can be important. It appears thus that it is preferable to neglect P W and V D at the same time instead of one at a time. This is all the more true since the ow is compressible. This is perfectly illustrated in the numerical study of heated compressible ows in 2D micro channels by [START_REF] Hong | Heat transfer characteristics of gaseous ows in microchannel with constant heat ux[END_REF] [START_REF] Hong | Heat transfer characteristics of gaseous ows in microchannel with constant heat ux[END_REF].

For established and moderately rareed slipping ows (0.001 ≤ Kn ≤ 0.1) with small streamwise density variations, it appears from Eq. ( 20) that the rst integral term in the RHS of Eq. ( 20) is approximately equal to zero. Therefore, P W + V D ≈ V SP : the sum of the pressure work and viscous dissipation does not vanish and is approximately equal to the power of the viscous stress at the wall. As a result, P W + V D cannot be neglected in the energy equation if V SP is not negligible at the wall and vice versa.

Governing equations for two-dimensional ows of ideal gases in an isothermally heated at-plate channel

Referring to the coordinate system of Fig. 2, the governing equations and boundary conditions for two-dimensional ows of ideal gases in the x direction, between two isothermal parallel smooth walls of length L, separated by a distance H in the transverse y direction, are summed up below. By denoting (u, v) the velocity components in directions (x, y) of the velocity vector v, the equations ( 8), ( 9) and ( 13) can be written:

∂ρu ∂x + ∂ρv ∂y = 0 (21
C p ∂ρuT ∂x + ∂ρvT ∂y = ∂ ∂x k ∂T ∂x + ∂ ∂y k ∂T ∂y + u ∂p ∂x + v ∂p ∂y + τ : ∇ v (24)
where the viscous dissipation writes:

τ : ∇ v = µ 2 ∂u ∂x 2 + ∂v ∂y 2 + ∂u ∂y + ∂v ∂x 2 - 2 3 ∂u ∂x + ∂v ∂y 2 (25)
The above set of equations is completed with the state equation ( 14) of the ideal gas.

At the inlet section x = 0, uniform pressure and temperature are applied:

p(0, y) = p in , ∂u ∂x (0, y) = 0, v(0, y) = 0, T (0, y) = T in (26)
At the outlet section x = L, the pressure is xed and fully developed ow conditions are applied:

p(L, y) = p out , ∂u ∂x (L, y) = 0, ∂v ∂x (L, y) = 0, ∂T ∂x (L, y) = 0 (27)
The symmetry boundary conditions are considered on the channel axis at y = 0:

∂u ∂y (x, 0) = 0, v(x, 0) = 0, ∂T ∂y (x, 0) = 0 (28)
On the gas side of the impermeable wall at y = H/2 (denoted by the subscript g below), zero normal velocity is applied and, for the tangential velocity, the rarefaction eects are modeled using the rst-order Maxwell slip velocity and thermal creep boundary conditions (Eq. ( 4)):

u g (x) = u(x, H/2) = -σ µ λ g ∂u ∂y g 1 st order slip velocity + σ θ µ g ρ g T g ∂T ∂x g thermal creep (29) v g (x) = v(x, H/2) = 0 ( 30
)
where the mean free path λ g at the wall is viscosity based (Eq. ( 1)):

λ g = µ g p g πrT g 2 = µ g π 2ρ g p g (31)
For a channel with a uniform wall temperature T w , the thermal boundary condition ( 6) is used:

T g (x) = T (x, H/2) = T w - ξ T λ g ∂T ∂y g 1 st order temperature jump (32)
3.6 Dimensionless equations for ows in isothermally-heated, long micro-channels

Reference quantities

In the present study, we consider developed gas ows (i.e. located downstream the thermal entrance zone), in micro-channels isothermally heated at the wall temperature, T w . Thus the following dimensionless variables are used in order to write the conservation equations and boundary conditions in dimensionless form :

X = x/L , Y = y/D h , U = u/u ref , V = v/u ref , P = p -p av ∆p ref , θ = T -T w ∆T ref ρ * = ρ/ρ w , µ * = µ/µ w , C * p = C p /C p,w , k * = k/k w (33)
The subscript w denotes quantities evaluated at the wall temperature, with ρ w = p av /rT w and p av = (p in +p out )/2; u ref , ∆p ref and ∆T ref are reference quantities for the velocity, the pressure and temperature dierences. In macro ows, the pressure gradient is generally considered of the same order as the inertial term, so the reference pressure dierence is taken as ∆p ref = ρ in ū2 in for instance. In macro and micro channels, when the inlet and outlet pressures are the inputs of the problem formulation, the reference pressure is:

∆p ref = p in -p out (34)
In dynamically developed ows in micro and macro channels, as the streamwise pressure gradient balances the viscous term, the equality of these two terms in the Navier-Stokes equations enables us to write the unknown reference velocity, u ref , as:

u ref = D h ∆p ref /µ w (35)
where = D h /L = 2H/L; it is a small parameter in the framework of the long channel approximation. Due to the way it is built, u ref is a relevant reference velocity for the viscous term of the Navier-Stokes equations or for the viscous dissipation term of the energy equation.

On the other hand, the average or bulk velocity remains the adequate reference velocity for the inertial terms and the Reynolds and Mach numbers. Thus, in what follows, a bulk velocity, ūb , dened from the mass ow rate, ṁ, will also be used:

ūb = ṁ/ρ w H (36)
Note that ṁ is a priori unknown for an imposed pressure dierence, p in -p out . It can be estimated from the asymptotic analytical expression Eq. ( 63) for instance.

For a uniform wall temperature, T w , the reference temperature dierence is usually chosen as ∆T ref = T w -T in . Here, we are mainly interested in the fully developed micro-ows downstream the thermal entrance length. Far from the inlet and for isothermal channel walls, the thermal scale can be chosen by assuming that the thermal diusion balances the heat source terms (pressure work and viscous dissipation) in the energy equation [START_REF] Demsis | Heat transfer coecient of gas owing in a circular tube under rareed condition[END_REF] (such an assumption will be justied later on). Then we get:

∆T ref = µ w u 2 ref k w (37)
where u ref is given by Eq. ( 35).

Dimensionless equations and boundary conditions

After introducing the dimensionless variables [START_REF] Lockerby | On the modelling of isothermal gas ows at the microscale[END_REF], taking into account Eqs. ( 34)-( 37), the governing equations ( 21)-( 24) become:

∂ρ * U ∂X + ∂ρ * V ∂Y = 0 (38) ∂ρ * U U ∂X + ∂ρ * U V ∂Y = - 1 Re w ūb u ref ∂P ∂X + 1 Re w ūb u ref ∂ ∂X 2µ * 2 ∂U ∂X - 1 3 2 ∂U ∂X + ∂V ∂Y + ∂ ∂Y µ * ∂U ∂Y + ∂V ∂X (39) ∂ρ * U V ∂X + ∂ρ * V V ∂Y = - 1 Re w ūb u ref ∂P ∂Y + 1 Re w ūb u ref ∂ ∂X µ * ∂U ∂Y + 2 ∂V ∂X + ∂ ∂Y 2µ * ∂V ∂Y - 1 3 ∂U ∂X + ∂V ∂Y (40) C * p ∂ρ * U θ ∂X + ∂ρ * V θ ∂Y = 1 P r w Re w ūb u ref 2 ∂ ∂X k * ∂θ ∂X + ∂ ∂Y k * ∂θ ∂Y + 1 P r w Re w ūb u ref U ∂P ∂X + 1 V ∂P ∂Y + 1 P r w Re w ūb u ref Φ v (41)
where Φ v is the dimensionless viscous dissipation term [START_REF] Bird | Molecular gas dynamics and the direct simulation of gas ows[END_REF]:

Φ v = µ * 2 ∂U ∂X 2 + ∂V ∂Y 2 + ∂U ∂Y + ∂V ∂X 2 - 2 3 ∂U ∂X + ∂V ∂Y 2 (42)
In Eqs. ( 38)-( 42), = D h /L is a small parameter for most micro channels. P r w = µ w C p,w /k w and Re w = ρ w ūb D h /µ w = 2 ṁ/µ w are the Prandtl and Reynolds numbers evaluated at the wall temperature. So the last two terms of the energy equation ( 41), the pressure work (P W ) and viscous dissipation (V D) terms, are both of the order of ūb /(P r w Re w u ref ), considering that, from Eq. ( 38), V ∼ in the P W term. The ratio P W/V D is therefore of the order of -1 since P W is negative and V D is positive.

The dimensionless form of the state equation is:

ρ * = 1 + ∆p ref pav P 1 + ∆T ref Tw θ (43)
The dimensionless forms of the boundary conditions read:

• on the inlet, at X = 0, ∀Y ∈ [0, 1 4 ],

P = P in = 1 2 , ∂U ∂X = V = 0, θ = θ in = Tw ∆T ref T in Tw -1 ;
• on the outlet, at X = 1, ∀Y ∈ [0, 1 4 ],

P = P out = -1 2 , ∂U ∂X = ∂V ∂X = ∂θ ∂X = 0; • on the channel axis, at Y = 0, ∀X ∈ [0, 1], ∂U ∂Y = V = ∂θ ∂Y = 0;
• on the wall, at Y = 1 4 , ∀X ∈ [0, 1], the velocity boundary conditions ( 29) and (30) write:

U g = -σ µ Kn w µ * g ρ * g 1 + ∆p ref pav P g ∂U ∂Y g +σ θ 2 (γ -1) πγ P r w Kn 2 w Re w u ref ūb   µ * g 1 + ∆p ref pav P g   ∂θ ∂X g (44)
V g = 0 [START_REF] Hong | Heat transfer characteristics of gaseous ows in microchannel with constant heat ux[END_REF] where all the variables with subscript g are functions of X at Y = 1 4 , on the gas side.

Equation [START_REF] Miyamoto | Chocked ow and heat transfer of low density gas in a narrow parallel-plate channel with uniformly heating walls[END_REF] shows that the thermal creep (the last R.H.S term) is O( ) and O(Kn 2 w ) that is of second order with Kn w when the slip velocity (the rst R.H.S term) is O(Kn w ).

Finally, in the case of uniform wall temperature T w , the thermal boundary condition [START_REF] Lockerby | Capturing the Knudsen layer in continuum-uid models of nonequilibrium gas ows[END_REF] on the wall at Y = 1 4 writes, ∀X ∈ [0, 1]:

θ g = -ξ T Kn w µ * g ρ * g 1 + ∆p ref pav P g ∂θ ∂Y g ( 46
)
and the resulting total wall heat ux, q t,w , is given by Eq. ( 18) which also writes:

q * t,w = q t,w D h k w ∆T ref = k * g ∂θ ∂Y g + µ * g U g ∂U ∂Y g (47)
Number of independent parameters

In the above dimensionless conservation equations and boundary conditions (Eqs. ( 38)-( 46)), the following set of 12 dimensionless parameters emerges:

• P r w = µ w C p,w k w • Re w = ρ w ūb D h µ w = 2 ṁ µ w • Kn w = λ w D h = π 2 µ w D h √ rT w p av = π 2 µ w D h 1 √ ρ w p av • u ref ūb , ∆p ref pav , ∆T ref Tw , Tw T in • = D h L , σ µ , σ θ , ξ T and γ.
However, using the forthcoming analytical expression of the mass ow rate ṁ (Eq. ( 63)), it can be checked that u ref (Eq. ( 35)) is related to the reference average velocity ūb (Eq. ( 36)) by:

ūb /u ref = (1 + 12σ µ Kn w ) 48 (48)
where Kn w = λ w /D h is the Knudsen number, with the mean free path λ w evaluated from Eq.

(

) 1
Therefore, if the micro channel is heated at constant temperature, the number of independent parameters is reduced to 9. The following set of 9 parameters is used in what follows:

P r w , M a w , Kn w , T w T in , , σ µ , σ θ , ξ T , γ (52)
Note that, if all the physical properties are assumed constant, the Reynolds number is constant in the whole channel due to mass ow rate conservation: Re = Re w = cste. On the other hand, M a and Kn vary along the ow in such a way that, from Eq. ( 49), their ratio is constant:

M a Kn = M a w Kn w = 2 πγ Re = cste (53)
That means that when the compressibility eects increase (M a increases), the rarefaction eects decrease (K n decreases). Furthermore it is not possible to observe compressibility eects (M a ∼ O(1)) and rarefaction eects (O(10 -2 ) Kn O(10 -1 )) at the same time when Re O(1).

4 Dimensional analysis and asymptotic solutions for long micro channels with isothermal walls

A dimensional analysis of the mathematical model is useful to determine which terms are dominant or negligible. The case of developed, subsonic low-rareed gas ows in isothermal long micro-channels of typical hydraulic diameter D h ∼ 1 to 10 µm and length L ∼ 0.1 to 10 mm at P r ∼ 1, M a < 1 and 10 -3 Kn 0.1 is considered here. It is shown that, downstream the entrance zone, the viscous dissipation, pressure work and viscous stress power at the walls cannot be neglected in the mathematical model while the thermal creep is negligible (see Appendix A). Taking into account these results, we can compute an analytical solution of this ow type by solving a simplied model resulting from an asymptotic analysis. This analysis is valid for heated gaseous micro-ows, far from the thermal entrance eects, for long micro channels with isothermal walls. It shows that the Nusselt number tends to zero (and not to the usual value N u ∞ = 7.541) for micro-ows between to at plates. Here we only present and discuss the main results of this analysis for the present study (see Appendix B for more details).

Following the asymptotic analyzes by Arkilic et al. [START_REF] Arkilic | Gaseous slip ow in long microchannels[END_REF] and Cai et al. [START_REF] Cai | Gas ows in microchannels and microtubes[END_REF], considering weakly rareed ows with small temperature variations in long micro-channels ( = D h /L << 1), the dimensionless quantities can be written as asymptotic expansions in :

U = U 0 + U 1 + 2 U 2 + ... ; V = V 1 + 2 V 2 + ... ; θ = θ 1 + 2 θ 2 +
U 0 (X, Y ) = -1 32
p av ∆p ref dP * 0 (X) dX 1 -16Y 2 + A P * 0 (X) = -1 2 dP 0 (X) dX b(X) -Y 2 (54) V 1 (X, Y ) = A 64 p av ∆p ref 1 P * 0 (X)
d 2 P * 0 (X) dX 2 16Y 3 -Y = A 64 1 P * 0 (X) d 2 P 0 (X) dX 2 16Y 3 -Y (55) P * 0 (X) = 1 + ∆p ref p av P 0 (X) = 1 p av p 2 out -p 2 in 1 + 3A 2 X + p in + 3A 2 p av 2 1/2 - 3A 2 (56) θ 1 (X, Y ) = - F (X) 8Y 4 -16b(X)Y 2 + J(X) (57)
with

F (X) = 1 64
dP 0 (X) dX 2 (58) b(X) = 1 16 1 + A P * 0 (X) (59)
J(X) = 1 16

A P * 0 (X) 1 + B P * 0 (X) + 1 32 (60)
A = 8σ µ Kn w

(61) B = 8ξ T Kn w (62)
Approximate analytical expressions of the mass ow rate, ṁ, and total wall Nusselt number, N u t (x), can also be deduced from this asymptotic analysis. At zero order with , ṁ reads as a function of p in and p out :

ṁ = H 3 24Lµ w 1 rT w p 2 in -p 2 out [1 + 12σ µ Kn w ] = ρ w u ref H 1 + 12σ µ Kn w 48 (63)
For a constant wall temperature T w , the total wall Nusselt number is dened by:

N u t (x) = N u dif f + N u V SP = D h k w T w -T (x) -k g ∂T ∂ - → n g dif f usion heat f lux -µ g u g ∂u ∂ - → n g viscous stress power (64)
where the bulk temperatures is:

T (x) = 2 ṁ H/2 0 ρ(x, y)u(x, y)T (x, y)dy (65)
Thus N u t (x) is the sum of two contributions: the diusive heat ux in the wall boundary layer, N u dif f , and the power of the viscous stress at the walls, N u V SP . The asymptotic analysis in Appendix B shows that the viscous stress power at the wall is never equal to zero for Kn w = 0 and is the opposite of the conduction heat ux at the wall, at least at the rst order with , for Re w < O(1) and M a w < O(1). Consequently, under these conditions, N u V SP = -N u dif f and the total wall Nusselt number is zero:

N u t (x) = 0 (66)
It should be emphasized that these results are linked to the approximations used in deriving the analytical solution: the dynamical and thermal entrance eects are indeed neglected. In particular, the convective term and the streamwise variation of the temperature from inlet are not accounted for in the energy equation. This approximation is only justied in the case of long micro-channel because the Reynolds number is generally small and the thermal entrance length is thus very short (a few hydraulic diameters).

On the opposite, when convection is dominant while the pressure work and viscous dissi- This theoretical analysis could then explain the very small values of the Nusselt numbers obtained by Demsis et al. [START_REF] Demsis | Experimental determination of heat transfer coecient in the slip regime and its anomalously low value[END_REF][START_REF] Demsis | Heat transfer coecient of gas owing in a circular tube under rareed condition[END_REF] in their experiments. The authors used a standard tubein-tube heat exchanger in which a rareed cold gas ows in the interior tube and hot water in the exterior annular tube. The thermal conditions for the rareed gas correspond to ows in a nearly isothermal pipe. The total wall heat transfer coecient h is determined through the measurements of the inlet and outlet gas and water temperatures of the tube-in-tube heat exchanger. For four rareed gas ows (nitrogen, oxygen, argon and helium) in the range of Knudsen number 0.0022 -0.032 and range of Reynolds numbers 0.13 -14.7, the obtained Nusselt numbers vary between 6.2×10 -5 and 2.8×10 -2 . This is two to ve orders of magnitude smaller than the corresponding values in the continuum regime since in an isothermal duct the fully-developed Nusselt number at large Péclet numbers is N u ∞ = 3.66.

Till now, these very small values of the Nusselt number had never been explained theoretically because only the contribution of the diusion heat ux at the walls was considered while the contribution of the viscous stress power was always omitted in the wall total heat ux. In what follows, we are going to check the validity of the asymptotic solution, based on an approximate model, by numerically solving the full system of governing equations for compressible micro-ows in parallel plate micro-channels.

5 Analysis of the heat transfer balance from numerical simulations 5.1 Numerical method.

An in-house nite volume code has been developed to solve the steady Navier-Stokes and energy equations with rst-order slip boundary conditions (Eqs. ( 21)-(32) in 3.5) on unstructured meshes. Details upon the discretization of the dierent terms can be found in [6466].

Only a few points are reminded here. A second-order centered scheme is used for the diffusive and convective terms because the maximum Reynolds number is Re max < 15 and cell Reynolds number is Re = u∆x ν < 1 ; the same inequalities also apply for the Péclet number since P r ≤ O(1). The discrete nonlinear steady equations are solved in a coupled way by Newton's algorithm. Since the channel geometry is here reduced to a rectangular domain, a Cartesian grid is used. The mesh size is based on the half channel width, with symmetry conditions (see Fig. 2). Mesh sizes are provided in each of the next subsections, according to the studied problem. For each case, a particular attention will be paid to the relevance of the results by checking the sensitivity of the solutions to the mesh renement. The code has already been validated by numerous comparisons with experimental and numerical results [6770].

Comparison of the analytical and numerical solutions

In this section, we compare the numerical and analytical solutions (Eqs. ( 54)-( 62) in 4).

The comparisons are carried out on three characteristics micro-ows, slipping or not, compressible or not. The number of control volume on each space direction is N x × N y = 12000 × 60 for a 100-aspect ratio physical domain. The mesh is uniform on x-direction and rened near the wall on y-direction, with a size ratio between two successive cells equal to 0.975.

Studied cases

The numerical simulations are performed for a nitrogen ow at T in = 270 K in a microchannel of length L = 300 µm and height H = 3 µm, with walls at T w = 300 K. The physical properties of the gas are assumed constant and their values at T w are gathered in Table 1. The three simulated cases are presented in Table 2:

• Case (a) is at low pressure and corresponds to a quasi incompressible ow (M a < 0.09) but with a strong slip velocity all along the channel walls because the Knudsen number values are relatively large, both at the inlet and outlet sections (0.02 < Kn < 0.1);

• Case (b) is at a large pressure dierence between inlet and outlet. It corresponds to a strongly compressible ow since M a out = 0.6428 and it presents a strong slip at the channel outlet since Kn out = 0.09354;

• Case (c) is at high pressure but small relative dierence in the inlet and outlet pressures.

It corresponds to an incompressible ow with a very low slip since M a < 0.03 and Kn < 0.0023 all along the channel. Tab. 1: Nitrogen physical properties at T w = 300 K. Relative error on ṁ and L 2 -norm of the relative errors on u, p, and T between the analytical and numerical solutions.

k [W/m.K] C p [J/kg.K] µ × 10 5 [P a.s] P r r [J/kg.K] γ = C p /C v σ µ σ θ ξ T 0.

Comparisons of the velocity and temperature elds

The comparison of the analytical and numerical solutions for the streamwise, u, and transverse, v, velocity components and the temperature, T , are presented in Figures 345for the cases (a)-(c) respectively. Table 3 also gives the L 2 -norm of the relative errors on u, p, and T between the analytical and numerical solutions. The values and absolute errors on v being both very small, the relative errors on v are not signicant and then not presented here. Globally, the analytical and numerical solutions for u, p and T are in a good accordance since the maximum relative error is 6.9% on u in the case (b). The graphical comparisons show a good agreement between the solutions, except at the outlet in the compressible case (b) (Fig. 4) and at the inlet for the temperature eld. In the former case, the Reynolds and Mach numbers are relatively large (Re = 7.8 and M a out = 0.64): the upper limit of the validity domain of the asymptotic analysis is reached (Re w O(1), M a w < O(1)). In the later case, the inlet and wall temperatures dier from each other in the numerical simulations, what produces a thermal entrance length which is not considered in the used analytical model. To a lesser extent, the entrance region also aects the v-eld in Fig 5 but, in this case, the v-value remains very small.

Comparisons of the Nusselt number values

The wall total Nusselt number, N u t (x) = N u dif f (x) + N u V SP (x), is dened by Eq. ( 64).

The integral average of N u t (x) along the wall is denoted by N u t and the numerical values for the cases (a) to (c) are reported in Table 2. They are very low (N u t ∼ 0.1) compared to the standard value N u ∞ = 7.541 calculated for a fully-established incompressible ow at large Péclet numbers between two isothermal parallel plates. The computed N u t -values are in qualitative agreement with the zero value of the total Nusselt number predicted analytically (Eq. ( 99)), by neglecting the thermal entrance length and keeping the leading terms of the -expansion. Furthermore, by comparing the cases (c) and (a) in Table 2, it appears that N u t decreases when Kn increases.

The proles of N u t , N u dif f and N u V SP , computed from the numerical simulations, are plotted with respect to x/L on Figs. 6-8 for the cases (a)-(c), respectively. From these proles and from Figs. 345, it appears that the length of the thermal entrance is approximately equal to 5H in the low Reynolds number case (a) (Fig. 6) and to 10H at maximum in the higher Reynolds number case (c) (Fig. 8). The total local Nusselt number, N u t , is very small in the whole channel except at the entrance where the ow is not thermally-established, and close to the outlet section due to the gas expansion and cooling associated with the pressure work.

Downstream the thermal entrance zone, N u dif f and -N u V SP vary between ∼ 0.2 in case (c) (Fig. 8) and ∼ 1.5 in case (a) (Fig. 6), while N u varies between less than 0.002 in case (a) and less than 0.03 in case (b) (Fig. 7). Thus even though N u dif f and N u V SP take non negligible values, N u t = N u dif f +N u V SP nearly vanishes because N u V SP is nearly the opposite of N u dif f downstream the thermal entrance zone (note that it is -N u dif f that is plotted on Figs. 678).

These Nusselt number behaviors are investigated in more details in the next section.

Detailed analysis of the Nusselt number

The aim of this section is to understand why and how N u t tends to very small values in the preceding cases (a)-(c). We rst investigate the classical case of an incompressible no slip ow between two innite plates for which the standard Heat Transfer textbooks give N u ∞ = 7.541.

Nusselt number for incompressible no slip ows

The numerical simulations are for a nitrogen ow at T in = 270 K and T w = 300 K (see Table 1). The full numerical model is solved again. The pressure work (P W ) and viscous dissipation (V D) source terms are kept in the energy equation. However, the Knudsen number is xed here at Kn = 0 (no slip) and the density is xed at ρ av = pav rTw , where p av = p in +pout 2 (incompressible ow). The pressure dierence between the inlet and the outlet is very small (∆p ref = 0.01 bar) while the average pressure is high (p av = 4.995 bar). Furthermore the channel aspect ratio is set at L/H = 50 and the channel height varies between H = 3 µm and H = 96 µm (6 µm ≤ D h ≤ 192 µm). The size of the uniform mesh is N x × N y = 3000 × 60, except when it is stated that N x × N y = 12000 × 240.

The Nusselt number and bulk temperature of incompressible no slip ows are dened by:

N u(x) = -k ∂T ∂y w k T w -T (x) /D h (67) T (x) = 2 uH H/2 0 u(x, y)T (x, y)dy (68)
The N u proles are plotted with respect to x/L on Fig. Let us rst reformulate the enthalpy equation ( 13) in its integral form as follows:

δΣ(x) -ρC p T - → v . - → n dS =CON V ×∆x + δΣ(x) k∇T. - → n dS =DIF F ×∆x = δΩ(x) - → v .∇pdω =P W ×∆x + δΩ(x) τ : ∇ - → v dω =V D×∆x (69)
where -→ n is the inward unit vector, δΩ(x) is a channel slice of width ∆x at the mean abscissa x, and δΣ(x) = S -(x)∪S + (x)∪δS w,g (x)∪δS s (x) is its surface (see Fig. 2). The notations CON V , DIF F , P W and V D refer to the average values per unit length ∆x of the CONVective and DIFFusive heat uxes transferred through the surface δΣ(x), and the average Pressure Work and Viscous Dissipation produced into the slice volume δΩ(x), respectively. The streamwise variations of these terms are plotted on Fig. 10 in the case of the narrowest channel of Fig. 9, (D h = 6 µm). As the incompressible ow is dynamically fully-established from the inlet to the outlet of the channel, Fig. 10 shows that the P W and V D source terms are constant and strictly opposite all along the channel length: P W (x) + V D(x) = 0 2 . Since the ow tends to be thermally fully established far from the channel entrance ( ∂T ∂x → 0 and ∂T ∂x → 0), the average convective term CON V also tends to zero, as well as the average diusion heat ux DIF F = P W +V D-CON V → 0. Considering both this later equality, the streamwise diusion heat ux which tends to zero, and the symmetry condition ( ∂T ∂y s = 0), the diusion heat ux at the wall necessarily tends towards zero too. This result is illustrated on Fig. 11 where both the numerator, -k ∂T ∂y w , and the denominator, k(T w -T )/D h , of the Nusselt number expression (Eq. ( 67)) are plotted. It is worth noticing that the diusion heat ux at the wall tends to zero with the grid renement: past x/L ∼ 0.15, its value is reduced by 4 2 by dividing the Cartesian cell sizes by 4, from N x × N y = 3000 × 60 to 12000 × 240. The factor is equal to 4 2 because the mesh size ratio is equal to four in each space direction and the space convergence order of the code is O(∆x 2 , ∆y 2 ). This is highlighted by the black arrow between the red dashed line and the red square symbols. On the other hand, the Nusselt denominator, k(T w -T )/D h , is insensitive to the mesh size and keeps a non-zero constant value for x/L 0.15. This temperature dierence is related to the transverse thermal gradient induced by the source terms P W and V D in the integral energy equation [START_REF] Tchekiken | A propos de la modélisation numérique de la convection de gaz en micro conduites[END_REF]. Indeed, though P W and V D counterbalance each other (Fig. 10), they contribute to the local thermal unbalance, acting close to the wall for the viscous dissipation and near the symmetry plane for the pressure work. Thus, as soon as the source terms dominate the convective and diusive contributions for x/L 0.75 (Fig. 10), a constant transverse thermal gradient exists in the core ow. Consequently, the reference heat ux k(T w -T )/D h becomes constant and the Nusselt number cancels out (see the encapsulated graphic in Fig. 11).

Nusselt number for incompressible slipping ows

For an incompressible slipping ow, the Nusselt number behavior is similar to that of the incompressible no slip ow. This is illustrated by simulating a ow in a micro-channel at D h = 6 µm, using the same conditions as in the previous subsection, except the Knudsen num- ) and N u denominator (k(T w -T )/D h ), for the incompressible no slip ow at D h = 6 µm, p in = 5 bar and p out = 4.99 bar (Fig. 9) and for two uniform Cartesian meshes of sizes N x × N y = 3000 × 60 and 12000 × 240.

p in = 0.1 bar and p out = 0.09 bar. The computed ow parameters are gathered in Table 4. For a dynamically fully-developed incompressible ow, the integral of the total energy equation on the same channel slice δΩ(x) as in the previous subsection, of surface δΣ(x) at the mean abscissa x (see Fig. 2), writes (see Eq. ( 11)):

δΣ(x) -ρC p T - → v . - → n dS =CON V ×∆x + δΣ(x) k∇T. - → n dS =DIF F ×∆x = δΣ(x) -(τ . - → v ). - → n dS =V SP ×∆x (70)
where the V SP term is the average power of the viscous stress on the slice surface δΣ(x) per unit length. Assuming a fully-developed ow, this last term writes:

V SP = 1 ∆x δSw,g(x) -(τ . - → v ). - → n dS = -(τ . - → v ) y=H/2 . - → n = µ g u g ∂u ∂y g (71)
As the thermal energy balance (Eq. ( 69)) is still valid for an incompressible slipping ow, the thermal and mechanical energy balances write in a compact form:

CON V + DIF F = P W + V D = V SP (72)
The ve terms of these equations are plotted on Fig. 12. It can eectively be checked that P W + V D = CON V + DIF F = V SP = 0. When the ow tends to be thermally fullyestablished, that is when ∂T ∂x → 0, the convective term tends towards zero and the integral of the diusive heat ux density tends towards the integral of the power of the viscous term at the wall: if CON V → 0 then DIF F → V SP = 0. From a numerical point of view, the zoom in Fig. 12 shows that DIF F → V SP with a second order space convergence.

- 4). The zoom shows that DIF F → V SP for two uniform Cartesian meshes of sizes N x × N y = 3000 × 60 and 12000 × 240.

As a result, the total wall heat ux must tend to zero when the ow tends to be thermally fully-established because it is the sum of two opposite contributions at the walls: the diusive ux and the power of the viscous stress (see Eq. ( 18)). Then the total local Nusselt number dened by N u t (x) = N u dif f (x) + N u V SP (x) (Eq. ( 64)) must also tends towards zero. Indeed, as in the no slip case, the dierence (T w -T ) at the N u t denominator still tends towards a non zero constant value (the P W and V D source terms do not cancel out and do not compensate each other locally). This is conrmed in Fig. 13 which presents the streamwise proles of N u t , N u dif f and N u V SP for the incompressible slipping ow at D h = 6 µm (see Table 4), for two uniform Cartesian meshes of dierent sizes. N u dif f and N u V SP tend towards opposite values after the thermal entrance zone and N u t tends towards zero. Once again, the second-order space convergence to a zero value of the total Nusselt number, N u t , is checked on Fig. 13.

Nusselt number for compressible slipping ows

We now consider compressible slipping ows of nitrogen in a micro-channel of aspect ratio

L/H = 50, with H = 3 µm (D h = 6 µm). Not any simplication in the full model of 3.5 is invoked. The inlet and wall temperatures are T in = 270 K and T w = 300 K. An example of the energy balance is presented in Fig. 14 for the micro-ow at p in = 0.2 bar and p out = 0.1 bar (5 th row of Table 5). Here the ow is nearly incompressible since the maximum Mach number is M a out ≈ 0.03. However, the full compressible model is used for the simulation and weak compressible eects are highlighted in the energy balance. In the case of compressible micro-ows, the thermal energy balance CON V + DIF F = P W + V D is still valid but P W + V D = V SP . The simplied mechanical energy balance for incompressible ows (Eqs. ( 70) and (72)) is not valid because, due to the streamwise density variation, the velocity prole is not the same between two dierent channel sections. As a result, the kinetic energy and the power of the viscous stress vary between two channel sections and the ow is neither dynamically nor thermally fully-developed. Thus, from Eq. ( 20), the sum of the source terms, P W + V D, and the sum CON V + DIF F are not zero and are never equal to V SP . Consequently, contrary to the previously considered incompressible cases, DIF F does not converge to V SP with the grid size after the thermal entrance zone (see the zoom in Fig. 14).

- 5).

In conclusion, for compressible ows in isothermal wall channels, the total wall heat ux is not null because the power of the viscous stress does not compensate for the diusive ux.

Therefore N u t = N u dif f +N u V SP is not zero everywhere along the walls. This is illustrated for instance in Fig. 7, for the compressible case (b) of Table 2, where the total local Nusselt number increases downstream due to the gas expansion and its cooling in the channel core. However, when the hydraulic diameter is very small and the pressure dierence between the inlet and the outlet of the channel is moderate, the Mach number generally remains small. The source term variation has almost no eect on the Nusselt number, except in the thermal entrance zone and just upstream the outlet where the gas expansion can be locally more important. This is illustrated on Fig. 15 which presents the streamwise proles of the total Nusselt number, N u t , for the weakly compressible ows of Table 5. Since the temperature dierence at the inlet is T w -T in = 30 K, the Nusselt number reaches at rst a plateau for 0.01

x L 0.05, where 2 N u 4. This rst plateau corresponds to the zone where the CON V and DIF F terms of the energy balance dominate the P W and V D source terms (see Fig. 14). Then N u t steeply decreases for 0.05

x L 0.2 to reach a second plateau where N u t < 5 × 10 -4 , except close to the channel outlet where the cooling due to gas expansion (at p av = 0.1 bar) and the outlet boundary conditions disrupt the Nusselt number variation. This second plateau corresponds to the zone where the CON V and DIF F terms reach very small values, i.e. much smaller than the P W and V D values in the energy balance (see Fig. 14). Therefore, when D h (or Re Dh ) is small, N u t is not null but keeps very small values along the walls, at least in the central zone of the channel, far enough from the inlet and outlet sections.

2 × 10 -2 while they vary between 0.16 and 0.3 in our simulations (Fig. 16). It is clear from Fig. 15 that our values of N u t are highly inuenced by the thermal entrance zone of length x = 0.1L = 5H = 2.5D h where the Nusselt number is larger. If the channel aspect ratio was much larger than the one used (L/H = 50), the N u t numerical values would be much smaller.

However in the experiments [START_REF] Demsis | Experimental determination of heat transfer coecient in the slip regime and its anomalously low value[END_REF][START_REF] Demsis | Heat transfer coecient of gas owing in a circular tube under rareed condition[END_REF], the channel aspect ratio is also relatively small (L/D ≈ 40,

with D the tube diameter). Therefore, it looks like as only the central part of the duct with low Nusselt numbers be selected for the calculations of the average Nusselt numbers in [START_REF] Demsis | Experimental determination of heat transfer coecient in the slip regime and its anomalously low value[END_REF][START_REF] Demsis | Heat transfer coecient of gas owing in a circular tube under rareed condition[END_REF]. This could be an explanation for the small Nusselt number values obtained. In these experiments, the temperature probes were indeed shifted 1.4D downstream the inlet and 1.4D upstream the outlet of the inner tube of the heat exchanger. Furthermore, due to the heat exchanger design, the small values of the Reynolds number and the likely presence of inverse diusion, the inlet and outlet thermal conditions were maybe not perfectly controlled (read the criticism in [START_REF] Morini | A critical review of the measurement techniques for the analysis of gas microows through microchannels[END_REF]).

Conclusion

The thermal aspects of the modeling of weakly rareed gaseous ows (0.001 < Kn ≤ 0.1)

with rst-order slip and temperature jump models have been discussed in details thanks to a dimensional analysis, an asymptotic analysis and numerical simulations. This model has been analyzed in the case of the forced convection of a cold gas owing in long at micro channels isothermally heated. The order of magnitude of the pressure work (P W ) and viscous dissipation (V D) in the bulk ow and the order of magnitude of the power of the viscous stress (V SP ) and thermal creep (T C) at the walls have been determined with respect to the ow parameters (Re w , M a w , Kn w ) and = D h /L.

The main remarks and tricky points highlighted are:

in the wall boundary conditions, the values of the viscous, thermal slip and temperature jump coecients must be all compatible with the same characteristics: a diusive or fully accommodating wall for instance (see [START_REF] Sharipov | Data on the velocity slip and temperature jump on a gas-solid interface[END_REF]);

modeling the Knudsen layer requires to introduce a temperature jump, even if a heat ux is imposed at the wall; computing the total heat ux at the wall requires to take into account the power of the viscous stress at the wall (V SP ); the integral P W + V D of the pressure work and viscous dissipation in a channel slice is approximately equal to the power of the viscous stress at the wall. Therefore, V SP cannot be neglected in the energy ux if P W + V D is not negligible; for the fully-developed weakly rareed gas ows in isothermally heated micro-channels considered in the present paper, the viscous dissipation, the pressure work and the viscous stress power at the walls cannot be neglected in the mathematical model while the thermal creep is negligible far from the entrance.

In the second part of the paper, the analytical asymptotic solutions of the ow and thermal elds established by Arkilic et al. [START_REF] Arkilic | Gaseous slip ow in long microchannels[END_REF] and Cai et al. [START_REF] Cai | Gas ows in microchannels and microtubes[END_REF] for forced convection of gas in isothermal wall micro-channels of large aspect ratio are extended to compressible ows and the Nusselt number computation (see Eqs. ( 54 This numerical solution takes into account the rst order slip and thermal jump conditions at the walls, the power of the viscous stress at the wall, the pressure work, the viscous dissipation and the thermal creep term. In the simulated cases reported in Table 2, the analytical and numerical solutions agree well for a wide range of ow parameters and ow types.

Finally, a thorough analysis of the streamwise distribution of the dierent contributions to the thermal and mechanical energy balances and to the total heat ux is carried out in the cases of non slipping or slipping, incompressible and compressible ows in isothermal wall micro-channels. It is shown that the contribution of the power of the viscous stress at the walls must not be neglected because it is opposite to and of the order of magnitude of the diusive ux at the walls. Furthermore, the total wall heat ux and total Nusselt number tend towards zero all the more so as the Knudsen number is large (signicant rarefaction eects) and the Reynolds number is small (short thermal entrance zone). This is coherent with the Nusselt number values obtained in the experiments by Demsis et al. [START_REF] Demsis | Experimental determination of heat transfer coecient in the slip regime and its anomalously low value[END_REF][START_REF] Demsis | Heat transfer coecient of gas owing in a circular tube under rareed condition[END_REF].

Appendix A: order of magnitude of the dierent terms linked with heat transfer in long micro channels

The order of magnitude of the dierent terms linked with heat transfer in the conservation equations and boundary conditions are evaluated as a function of the rarefaction and compressibility intensities, represented by Kn w and M a w , respectively. The study is limited to subsonic low rareed gas ows: P r ∼ 1, M a w < 1 and 10 -3 Kn w 0.1. The typical micro-ows considered concern gas ows through long micro channels of hydraulic diameter D h ∼ 1 to 10 µm and length L ∼ 0.1 to 10 mm. The inverse of the longitudinal aspect ratio is therefore a small parameter: 10 -4 = D h L 0.1.

Using Eqs. ( 49), ( 50) and ( 51), the order of magnitude of Re w , 39)-( 41), state equation ( 43) and boundary conditions ( 44)-( 47) can be computed with respect to the orders of magnitude of M a w and Kn w . The orders of magnitude of all these linked parameters are given in Table 6.

They are expressed as a function of only. To this end, it was considered that ūref u ref ∼ 1/2 for 10 -3 10 -2 since, from Eq. ( 48), (c) Order of magnitude of 5 to 10 × T C in Eq. ( 44): ∂P 0 ∂Y = 0 ⇐⇒ P 0 = P 0 (X)

1 48 ≤ ūref /u ref ≤ 1 22 for 0 ≤ Kn w ≤ 0.1 and σ µ = 1. M a w Re w 2 1 Kn w 2 1 -1 -2 1 -1 1/2 1.5 0.5 -0.5 (
P rwKn 2 w Rew u ref ūref ∼ 1/2 M awKnw M a w ∆T ref Tw 2 1 Kn w 2 3 -1 3 -1 1/2 3 -1 (d)
For Re w O( -1 ), we also get P 1 = P 1 (X). At rst order, the continuity equation ( 38) is:

∂ρ * 0 U 0 ∂X + ∂ρ * 0 V 1 ∂Y = 0 (74)
and by noting P * 0 (X) = 1 + ∆p ref pav P 0 (X), the ideal gas law (43) at the zero order with reads:

ρ * 0 (X) = 1 + ∆p ref p av P 0 (X) = P * 0 (X) (75)
Then Eq. ( 74) at rst order with can also be written:

∂P * 0 U 0 ∂X + ∂P * 0 V 1 ∂Y = 0 (76)
For Re w O(1) and M a w < O(1), at the lowest order with , the streamwise momentum equation ( 39) becomes:

- ∂P 0 ∂X + ∂ ∂Y µ * 0 ∂U 0 ∂Y = 0 (77)
and, taking into account the continuity equation ( 74), the energy equation ( 41) writes:

∂ ∂Y k * 0 ∂θ 1 ∂Y = - 1 U 0 ∂P 0 ∂X + µ * 0 ∂U 0 ∂Y 2 (78)
Thus the analytical solutions of micro-ows at Re w O(1) and M a w < O(1) can be computed from the asymptotic continuity equation (76), streamwise Navier-Stokes equation (77) and energy equation (78). The boundary conditions for U 0 , V 0 , V 1 , P 0 and θ 1 are the same as for U, V, P and θ, with a negligible thermal creep term in Eq. ( 44) (see Appendix A):

• on the entrance, at X = 0, ∀Y ∈ [0, 1 4 ], P 0,in = 1 2 , ∂U 0

∂X = 0, V 1,in = 0 and θ 1,in = T in -Tw ∆T ref ; • on the outlet, at X = 1, ∀Y ∈ [0, 1 4 ], P 0,out = -1 2 , ∂U 0 ∂X = ∂V 1 ∂X = ∂θ 1 ∂X = 0 ; • on the channel axis, at Y = 0, ∀X ∈ [0, 1], ∂U 0 ∂Y = V 1 = ∂θ 1 ∂Y = 0 ;
• on the wall, at Y = 1 4 , ∀X ∈ [0, 1], the velocity boundary conditions are:

U 0,g = -σ µ Kn w µ * 0,g 1 + ∆p ref pav P 0,g ∂U 0 ∂Y g (79) V 1,g = 0 (80)
The thermal boundary conditions in the case of a uniform wall temperature T w read:

θ 1,g = -ξ T Kn w µ * 0,g 1 + ∆p ref p in P 0,g ∂θ 1 ∂Y g (81)
and, from Eq. ( 47), the dimensionless total wall heat ux writes:

q * t,w = q w D h k w ∆T ref = k * 0,g ∂θ 1 ∂Y g + µ * 0,g U 0,g ∂U 0 ∂Y g (82)
The analytical solution of this asymptotic model can easily be computed if the physical properties are considered constant and equal to their values at T w : µ * 0 = 1 and k * 0 = 1. In this case, for a uniform wall temperature T w , this analytical solution, adapted from Arkilic et al. [START_REF] Arkilic | Gaseous slip ow in long microchannels[END_REF] and Cai et al. [START_REF] Cai | Gas ows in microchannels and microtubes[END_REF], is given by the equations ( 54)-(62) in 4.

Temperature jump at the wall: θ 1,g (X)

The temperature jump at the wall is equal to θ g ≈ θ 1,g with:

θ 1,g (X) = θ 1 (X, Y = 1 4 ) = -A.B 16
F (X) P * 2 0 (X) = - σ µ ξ T 16 Kn 2 w 1 P * 0 (X) dP 0 (X) dX 2 (83)
The uid temperature close to the wall is thus dierent from the wall temperature (θ 1,g = 0)

when Kn w = 0.

Average streamwise velocity: U 0 (X)

The average velocity is dened by: U 0 (X) = 4 The expression (89) is similar to that derived by Arkilic et al. [START_REF] Arkilic | Gaseous slip ow in long microchannels[END_REF] for an isothermal weakly compressible ow. It can be deduced from Eqs. ( 88) and (89) that:

4ρ * 0 (X)

1/4 Y =0
U 0 (X, Y )dY = ρ * 0 (X)U 0 (X) = P * 0 (X)U 0 (X) = 1 + 12σ µ Kn w 48

(90)

Wall Nusselt number: N u t (X)

For a constant wall temperature T w , the total wall Nusselt number can be dened as:

N u t (x) = q t,w D h k w T w -T (x) = -q t,w D h k w ∆T ref θ(X) (91)
where q t,w is given by Eq. [START_REF] Potkay | The promise of microuidic articial lungs[END_REF]. The dimensionless form of the bulk temperature (Eq. ( 65 At the rst order of the temperature, θ(X) ≈ θ 1 (X). Then, using Eq. (82) with constant physical properties, the total local Nusselt number, dened by Eq. ( 64), also writes:

N u t (x) = N u dif f + N u V SP = -1 θ 1 (X)   ∂θ 1 ∂Y g + U 0,g ∂U 0 ∂Y g   (93)
The three terms involved in Eq. ( 93) are calculated below:

• The temperature rst derivative at the walls is obtained from Eq. ( 57) for θ 1 (X, Y ).

Thus, using Eq. ( 59), the conductive heat ux close to the walls is:

∂θ 1 ∂Y g = F (X) 16b(x) -1 2 = F (X) 2 A P * 0 (X) (94)
• The viscous stress power at the walls is computed from Eq. ( 54) for U 0 (X, Y ):

U 0,g ∂U 0 ∂Y g = -1 2 dP 0 (X) dX 2 b(X) -Y 2 Y g = -1 128
dP 0 (X) dX 2 [16b(X) -1]
From the denitions of F (X) and b(X) (Eqs. ( 58) and ( 59)), we obtain:

U 0,g ∂U 0 ∂Y g = - F (X) 2 A P * 0 (X) (95)
• From the denition (92) of the average bulk temperature, using Eqs. ( 86) and (90), the rst order approximation of θ(X) becomes:

θ 1 (X) = 4 U 0 (X) 1/4 Y =0 U 0 (X)θ 1 (X, Y )dY = 192 48b(X) -1 1/4 Y =0 b(X) -Y 2 θ 1 (X, Y )dY (96)
Using Eq. ( 57) for θ 1 (X, Y ) leads to:

θ 1 (X) = -192 48b(X) -1 F (X) 1/4 Y =0 b(X) -Y 2 8Y 4 -16b(X)Y 2 + J(X) dY (97)
and, after integration, we get:

θ 1 (X) = F (X)
17920b(X) 2 -1008b(X) + 15 + J(X) (1120 -53760b(X)) 1120 (48b(X) -1)

For Kn w = 0 (no slip and no rarefaction case ), b = 1 16 , J = 1 32 and θ 1 (X) = 3F (X) 140 .

The viscous stress power at the wall (Eq. ( 95)) is thus the opposite of the conduction heat ux at the wall (Eq. ( 94)). Consequently, using Eq. (98), the total wall Nusselt number (93) is:

N u t (X) = -560 (48b(X) -1) 17920b(X) 2 -1008b(X) + 15 + J(X) (1120 -53760b(X)) A P * 0 (X)

- A P * 0 (X) = 0 (99)
Since b(X) > 0 and J(X) > 0 whatever X ∈ [0, 1], the Nusselt number denominator never tends towards zero and N u t (X) = 0 everywhere at the rst order with .

From Eqs. ( 94) and ( 95), it appears that the conduction heat ux and the viscous stress power at the walls both vanish when Kn w = 0 since then A = 0 (see Eq. ( 61)). On the other hand, for Kn w = 0, the conduction heat ux at the walls is dierent from zero but the total heat ux at the walls and thus the total Nusselt number tend everywhere towards zero, at least at the rst order with . These results are linked to the approximations used in deriving the analytical solution: the dynamical and thermal entrance eects are indeed neglected. In particular, the convective term and the streamwise variation of the temperature from inlet are not accounted for in the energy equation (78). This approximation is justied in the case of long micro-channel because the Reynolds number is generally small and the thermal entrance length is thus very short.

Fig. 1 :

 1 Fig. 1: Schematic representation of the true (in red) and approximated (in blue) velocity and temperature proles in the Knudsen layer and (in green) of the slip length, L s , and temperature jump length, L s,T .

Fig. 2 :

 2 Fig. 2: Considered geometry for gas ows in 2D isothermal micro channels

  pation are neglected in the energy equation, the Nusselt number can reach a nite value in a channel at constant wall temperature. For instance, for laminar incompressible ows between two innite plates at constant wall temperature, it is well established in standard Heat Transfer textbooks that N u(X → ∞) = N u ∞ = 7.541 if Kn w = 0 provided that the axial diusion is neglected. Considering now both the viscous dissipation and the pressure work in the energy equation, our rst order asymptotic analysis clearly shows that N u(X) = 0 for Re w < O[START_REF] Kandlikar | Heat transfer in microchannels -2012 status and research needs[END_REF] and M a w < O(1).

Fig. 3 :

 3 Fig. 3: Comparison of the analytical solution (black isolines) and numerical solution (color elds) for u (top, left), v (top, right) and T (bottom), in case (a). Symmetry is taken into account and a half channel is presented.

Fig. 4 :

 4 Fig. 4: Same as Figure 3 but for case (b) and with a zoom on the outlet zone for each eld (see coordinates on the abscissa axes).

Fig. 5 :

 5 Fig. 5: Same as Figure 3 but for case (c).

Fig. 6 :Fig. 7 :Fig. 8 :

 678 Fig.6: Proles of N u t , N u V SP and -N u dif f , with respect to x/L, in the simulated case (a).A zoom on the N u t prole is included.

Fig. 9 :

 9 Fig. 9: Proles of the wall Nusselt number, N u, with respect to X = x/L, for incompressible no slip ows at various hydraulic diameters and Reynolds numbers (p in = 5 bar, p out = 4.99 bar, L/H = 50). A zoom on the N u proles at the channel entrance is included.

Fig. 10 :

 10 Fig. 10: Streamwise variation of each term of the energy equation integrated on a channel slice (convective (C ON V ) and diusive (DI F F ) uxes and pressure work (P W ) and viscous dissipation (V D) source terms), and total energy balance (C ON V +DIF F-P W-V D), for the no slip incompressible ow at D h = 6 µm, p in = 5 bar, p out = 4.99 bar of Fig. 9.

Fig. 11 :

 11 Fig. 11: Streamwise proles of the Nusselt number N u (enclosed gure), N u numerator (-k ∂T ∂y w

Tab. 4 :

 4 Imposed inlet conditions and computed outlet conditions for the incompressible slipping ow in a micro-channel at D h = 6 µm.

Fig. 12 :

 12 Fig. 12: Streamwise proles of each term of Eq. (72) and energy balance (C ON V + DIF F -P W -V D), for the incompressible slipping ow at D h = 6 µm, p in = 0.1 bar and p out = 0.09 bar (see Table4). The zoom shows that DIF F → V SP for two uniform Cartesian meshes of sizes N x × N y = 3000 × 60 and 12000 × 240.

Tab. 5 :

 5 Imposed inlet conditions and computed outlet conditions of the compressible ows simulated with the full model in a micro-channel at D h = 6 µm and ∆p ref = 0.1 bar.

Fig. 14 :

 14 Fig.14: Streamwise proles of the thermal energy balance (C ON V + DIF F -P W -V D), of its dierent terms and of the power of the viscous stress at the wall (V SP ), for the compressible slipping ow at D h = 6 µm, p in = 0.2 bar, p out = 0.1 bar (see Table5).

  pav that appear in the conservation equations (

a) Rew = πγ 2 M

 2 aw/Knw ∼ M aw/Knw (Eq. (49)) Order of magnitude of P W and V D in Eq. (41):

Tab. 6 :

 6 Orders of magnitude of the dimensionless groups that appear in the equations and boundary conditions of the model, with respect to M a w and Kn w expressed as a function of .

ṁ 4 YU 0 p 2 in -p 2 out [ 1 +

 40221 The inlet and outlet pressures, p in and p out , being inputs of the problem formulation, the mass ow rate, ṁ, is an output of the calculations. An approximate analytical expression of ṁ as a function of p in and p out can be established, at zero order with , starting from its denition: y)u(x, y)dy = 4ρ w u ref H1/=0 ρ * (X, Y )U (X, Y )dY(87)Then, using Eq. (75), ṁ can be written:ṁ = 4ρ w u ref Hρ * 0 (X) (X, Y )dY = ρ w u ref HP * 0 (X)U 0 (X)(88)From Eqs. (35), (56) and (85), the following expression is derived: 12σ µ Kn w ] = ρ w u ref H1 + 12σ µ Kn w 48(89)

  , y)u(x, y)θ(x, y)dy(92)

NomenclatureBrS

  Brinkman number, Br = µū 2 k∆T ref C p , C v specic heat at constant pressure and volume respectively [J.kg -1 .K -1 ] CON V integral mean of the convective term on a channel slice d mean molecule diameter [m] D h hydraulic diameter, D h = 2H [m] DIF F integral mean of the diusive term on a channel slice e c kinetic energy per mass unit, e c = -→ v 2 /2 [J.kg -3 ] Ec Eckert number, Ec = ū2 Cp∆T ref f body force vector [N.m -3 ] g gravity acceleration vector [m.s -2 ] h enthalpy per mass unit [J.kg -1 ]h heat transfer coecient [W.m -2 .K -1 ] H channel height [m] k thermal conductivity [W.m -1 .K -1 ] k B Boltzmann constant, k B = 1.38065 × 10 -23 [J.K -1 ] Kn Knudsen number, Kn = λdepth unit, ṁ = ρūH [kg.s -1 .m -1 ] M molar mass [kg.mol -1 ] Avogadro number, N a = 6.02214 × 10 23 [mol -1 ] N x , N y cell number in the streamwise and transverse directions respectively N u Nusselt number, N u = hD h k p pressure [P a] P dimensionless pressure, P = p-p in ∆p ref P e Péclet number, P e = ReP r = ūD h α P r Prandlt number, P r = µCp k P W Pressure Work, -→ v .∇p [W.m -3 ], or integral of this term q w heat ux density at the wall [W.m -2 ]r specic gas constant, r = R M [J.kg -1 .K -1 ] R ideal gas constant, R = N A k B = 8.31446 [J.mol -1 .K -1 ] Re Reynolds number, Re = ρūD h surface [m 2 ] t time [s] or tangential direction to the wall [m] T temperature [K ]

  Br = µū 2 /(k∆T ), can reach or exceed the unit and the Reynolds number, Re = ūD h /ν, and the Péclet number, P e = ūD h /α, can vary between 10 -2 and 10 2 . Accordingly, modeling this

	It can also be shown that the Mach number, M a = ū/ √	γrT , and the Brinkman number,
	type of ow requires to take into account a wide range of physical phenomena:
	the local thermodynamic non equilibrium and rarefaction eects in the Knudsen layer as
	described in the previous sections (dynamic slip, temperature jump and thermal creep),
	the compressibility of the gas (the volume expansion and cooling of the gas associated
	with the pressure work) [4346],

  )

	∂ρuu ∂x	+	∂ρuv ∂y	= -	∂p ∂x	+	∂ ∂x	2µ	∂u ∂x	-	1 3	∂u ∂x	+	∂v ∂y	+	∂ ∂y	µ	∂u ∂y	+	∂v ∂x	(22)
	∂ρuv ∂x	+	∂ρvv ∂y	= -	∂p ∂y	+	∂ ∂x	µ	∂u ∂y	+	∂v ∂x	+	∂ ∂y	2µ	∂v ∂y	-	1 3	∂u ∂x	+	∂v ∂y	(23)

  at T w and at the average pressure p av = (p in + p out )/2. Furthermore, by introducing the Mach number M a 2 w = ρ w ū2 b /γp av = ū2 b /γrT w , it can be shown that:

				1 Re w	=	2 πγ	Kn w M a w	(49)
	∆p ref p av	=	γ M a 2 w Re w	u ref ūb	when ∆p ref is given by Eqs. (34)	(50)
	∆T ref
	T

w = (γ -1)P r w M a w u ref ūb 2

when ∆T ref is given by Eq.

[START_REF] Maxwell | On stresses in rareed gases arising from inequalities of temperature[END_REF]

  ... ;P = P 0 + P 1 + 2 P 2 + ...

	and the analytical solution of the asymptotic model for constant physical properties reads (see
	Appendix B):

Table 3

 3

	Case	p in	p out	∆p ref	Re in =	Kn in	Kn out	M a in	M a out	ṁ × 10 6	N u t
		[bar]	[bar]	[bar]	Re out					[kg/m.s]
	(a)	0.5	0.1	0.4	1.038 0.019620 0.1032	0.01373 0.08848	8.608	0.1082
	(b)	1.5	0.1	1.4	7.833 0.006539 0.09354 0.03453 0.64280	64.91	0.1472
	(c)	5	4.6	0.4	12.05 0.001962 0.002248 0.01595 0.02716	99.97	0.2215
	Tab. 2:	Imposed inlet and outlet pressures for each simulated case. The others quantities,
		except Kn in , result from the simulations.
	the maximum relative error is 1.2% for the case (b).
			Case	p in [bar] p out [bar] e ṁ [%] e u [%]	e p [%]	e T [%]
			(a)	0.5	0.1	0.06	0.32	0.040	0.43
			(b)	1.5	0.1	1.2	6.9	0.68	0.53
			(c)	5	4.6	0.04	0.68	4.6 × 10 -4	0.54
	Tab. 3:

gives the relative errors on the mass ow rate, e ṁ, between the analytical solution of Eq. (89) and the numerical solutions in Table

2

. An excellent agreement is observed since

  . However, when D h (or Re D h ) decreases, a dierent behavior is observed. The nite N u-value, reached after its sudden decrease close to the inlet section, tends towards zero further downstream. It is then possible to distinguish two entrance lengths: the rst one is dened from the inlet section to the rst Nusselt plateau and the second one to the fully established region where N u ∼ 0. The length of the rst plateau decreases with D h (or Re D h ) as well as the length from the inlet to N u ∼ 0. The explanation of N u ∼ 0 is detailed below.

	Nu	15 20 25 30	8.2 8 7.8 7.6 7.4	0	0.05	0.1	0.15	0.2	D h =6 µm (Re D h =0.680) D h =12 µm (Re D h =2.72) D h =24 µm (Re D h =10.9) D h =48 µm (Re D h =43.5) D h =96 µm (Re D h =174)
		10						D h =192 µm (Re D h =697)
	5 7.541
		0	0	0.1	0.2	0.3	0.4	0.5 x/L	0.6	0.7	0.8	0.9	1

9, for various hydraulic diameters or Reynolds numbers. The Reynolds number, Re D h , is proportional to D 2 h because the analytical average velocity, u, is proportional to D h in simulations at xed ∆p ref and L/H values. One can note that, when D h and Re D h are large enough (D h = 192 µm and Re D h = 697), N u decreases at the channel entrance and tends towards the classical N u value, N u ∞ = 7.541, downstream

  The physical properties are given in Table1and the ow parameters in Table5. The pressure dierence Fig.13: Streamwise proles of the total Nusselt number N u t (x), its diusive part, N u dif f (x), and its part due to V SP , N u V SP (x), for the incompressible slipping ow at D h = 6 µm, p in = 0.1 bar and p out = 0.09 bar (see Table4) and for two uniform Cartesian meshes of size N x × N y = 3000 × 60 and 12000 × 240. A zoom on the N u t proles is included.between the inlet and the outlet is xed at ∆p ref = 0.1 bar but the average pressure decreases from one case to another one and, therefore, the average Knudsen number increases. The size of the uniform mesh is N x × N y = 12000 × 240. It was checked to be enough ne to provide accurate solutions.

			10 12.5 15		0.01 0.1 1 10							Nu VSP (N y =240) Nu diff (N y =240) Nu t (N y =240)
		Nu	5 7.5		0.001 1e-05 0.0001	0	0.2	0.4	0.6 ×4 2	0.8	1	Nu t (N y =60) Nu VSP (N y =60) Nu diff (N y =60)
			2.5
			0
			-2.5
			-5	0	0.1		0.2	0.3	0.4	0.5 x/L	0.6	0.7	0.8	0.9	1
	p in	p out	Re in =		Kn in	Kn out	M a in	M a out	ṁ × 10 6	N u t
	[bar]	[bar]		Re out								[kg/m.s]
	2	1.9	2.5402 0.00490 0.00544 0.01247 0.01370	21.07	0.2932
	1	0.9	1.3156 0.00981 0.01149 0.01268 0.01467	10.91	0.2598
	0.6	0.5	0.8257 0.01635 0.02068 0.01299 0.01613	6.848	0.2384
	0.3	0.2	0.4582 0.03270 0.05169 0.01380 0.02090	3.800	0.2048
	0.2	0.1	0.3357 0.04904 0.10336 0.01466 0.02864	2.785	0.1804
	0.15	0.05	0.2745 0.06539 0.20689 0.01555 0.04381	2.277	0.1611

  )-(66) in 4 and Appendix B). It is shown that the local total Nusselt number, N u t (x), tends towards zero, at least at rst order with , far enough from the thermal entrance zone, provided that the contribution of the power of the viscous stress at the wall is accounted for. This analytical solution, valid at low Reynolds and Mach numbers (Re w O(1) and M a w < O(1)), is compared with the numerical solution of the full model by a nite volume method.

This result has already been obtained and discussed at

3.4 where the L.H.S. term of Eq. (20) is equal to zero for dynamically fully-developed and incompressible no slip ows.

The integral mean of the total Nusselt number along the whole wall, N u t , and along the wall part corresponding to the second N u-plateau, N u t,plateau#2 , are plotted on Fig. 16 with respect to the average Knudsen number, Kn av = (Kn in +Kn out )/2, for the weakly compressible microows of Table 5. Due to the inuence of the rst N u-plateau close to the entrance, the N u t values are at least three orders of magnitude higher than the N u Fig. 16: Integral mean of the total Nusselt number along the whole wall (N u t ) and along the wall part corresponding to the second N u-plateau (N u t,plateau#2 ), with respect to the average Knudsen number, for the weakly compressible micro-ows of Table 5, at D h = 6 µm, ∆p ref = 0.1 bar and a decreasing average pressure.

In the micro tube experiments by Demsis et al. [START_REF] Demsis | Experimental determination of heat transfer coecient in the slip regime and its anomalously low value[END_REF][START_REF] Demsis | Heat transfer coecient of gas owing in a circular tube under rareed condition[END_REF], a linearly decreasing slope in Log-Log scale of the relation N u t = f (Kn av ) is also observed. However, in these experiments, the values of the average Nusselt number, N u t , along the whole wall vary between 3 × 10 -4 and First, in the momentum and energy conservation equations ( 39)-( 41), the advection and transport terms are of the order of O( ). Furthermore, it can be checked from Tables 6a and6b that all the other terms of these equations are at least of the order of O( 0.5 ) when Re w O(1)

and M a w < O(1). Therefore the advection and transport terms terms are negligible with respect to the other terms in this parameter range. Since the analytical solutions are established by neglecting these terms, they are thus valid for Re w O(1) and M a w < O [START_REF] Kandlikar | Heat transfer in microchannels -2012 status and research needs[END_REF]. In this parameter range, Tables 6d and6e O(1).

The pressure work (P W ) and the viscous dissipation (V D) in the energy equation ( 41) are of the order of 6b). As a consequence P W and V D cannot be neglected in the energy equation whatever the values of M a w and Kn w . They are indeed at least of the order of the transverse diusion term (∼

) or of the transport term (∼ ). Similarly, the power of the viscous stress in the expression of the wall heat ux (second R.H.S term in Eq. ( 47)) is of the order of the diusive term (rst R.H.S term), independently of Kn w and M a w . Consequently, N u V SP in Eqs. ( 64) and ( 93) is never negligible.

The thermal creep (T C) term (second R.H.S term in Eq. ( 44)) is of the order of P r w Kn 2 w Re w u ref ūref

with a quite small factor ahead: σ θ 2(γ-1) πγ ∼ 0.1 to 0.2. As for the velocity slip term (rst R.H.S term of ( 44)), it is of the order of Kn w . Therefore, from Table 6c, the thermal creep term appears to be always negligible, whatever M a w and Kn w , for the developed micro-ows considered here. This conclusion is obviously not valid when the entrance eects are important: in this case, the scale for the temperature dierence should be taken as

Then it can be shown that the T C term must be taken into account at small Mach numbers (M a w

2 ) for ∆T ref T in ∼ 1, which corresponds to small Reynolds numbers (Re w 1).

Appendix B: asymptotic solutions for long micro channels with isothermal walls

Following the asymptotic analyzes by Arkilic et al. [START_REF] Arkilic | Gaseous slip ow in long microchannels[END_REF] and Cai et al. [START_REF] Cai | Gas ows in microchannels and microtubes[END_REF], considering weakly rareed ows with small temperature variations in long micro-channels ( = D h /L << 1), the dimensionless quantities can be written as asymptotic expansions in :

Note that V 0 = 0 because the continuity equation [START_REF] Loyalka | Some numerical results for the BGK model: thermal creep and viscous slip problems with arbitrary accommodation of the surface[END_REF] gives

= 0 at the zero order with and V 0 = 0 at the wall. At the zero order with , the dimensionless temperature is also supposed constant, equal to θ 0 = 0, i.e. T = T w . The temperature expansion is thus compatible with walls at constant or weakly varying temperature.

Then, the dimensionless conservation equations ( 38)-( 41) can be written at the zero or rst order with accounting for the ranges of variations of the dimensionless parameters expressed as a function of in


Adrien Malaisé
email: adrien.malaise@inria.fr

Pauline Maurice

Francis Colas

Serena Ivaldi

Online Human Activity Recognition for Ergonomics Assessment

Keywords: Human Activity Recognition, Ergonomics, Wearable Sensors, Hidden Markov Model

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