We present an analytical model predicting the experimentally observed gas mass flow rate in rectangular microchannels over slip and transition regimes without the use of any fitting parameter. Previously, Sone reported a class of pure continuum regime flows that requires terms of Burnett order in constitutive equations of shear stress to be predicted appropriately. The corrective terms to the conventional Navier-Stokes equation were named the ghost effect. We demonstrate in this paper similarity between Sone ghost effect model and newly so-called ‘volume diffusion hydrodynamic model’. A generic analytical solution to gas mass flow rate in a rectangular micro channel is then obtained. It is shown that the volume diffusion hydrodynamics allows to accurately predict the gas mass flow rate up to Knudsen number of 5. This can be achieved without necessitating the use of any adjustable parameters in boundary conditions or parametric scaling laws for constitutive relations. The present model predicts the non-linear variation of pressure profile along the axial direction and also captures the change in curvature with increase in rarefaction

Dadzie, Kokou S

Dongari, Nishanth

English

Glyndŵr University Research Online

Transition Regime Analytical Solution to Gas Mass FlowRate in a Rectangular Micro ChannelS Kokou Dadzie∗ and Nishanth Dongari †∗Mechanical and Aeronautical Engineering, Glyndwˆr University, Mold Road, Wrexham LL11 2AW, UK†Department of Mechanical and Aerospace Engineering, University of Strathclyde, Glasgow G1 1XJ, UKAbstract. We present an analytical model predicting the experimentally observed gas mass flow rate in rectangular microchannels over slip and transition regimes without the use of any fitting parameter. Previously, Sone reported a class ofpure continuum regime flows that requires terms of Burnett order in constitutive equations of shear stress to be predictedappropriately. The corrective terms to the conventional Navier-Stokes equation were named the ghost effect. We demonstratein this paper similarity between Sone ghost effect model and newly so-called ‘volume diffusion hydrodynamic model’. Ageneric analytical solution to gas mass flow rate in a rectangular micro channel is then obtained. It is shown that the volumediffusion hydrodynamics allows to accurately predict the gas mass flow rate up to Knudsen number of 5. This can be achievedwithout necessitating the use of adjustable parameters in boundary conditions or parametric scaling laws for constitutiverelations. The present model predicts the non-linear variation of pressure profile along the axial direction and also capturesthe change in curvature with increase in rarefaction.Keywords: mass diffusion; volume diffusion hydrodynamics; microchannel gas flow; Knudsen paradox; pressure distribution; slip flowsPACS: 47.61.-k, 47.61.Cb, 47.61.FgINTRODUCTIONGrowing demands in microdevice technology involving the motion of fluids at micro- and nano-length scales opena new branch in fluid mechanics requiring investigation of fluid flows occurring exclusively in ultra-small devices.Examples of these mechanical systems include micro-pumps, heat exchangers, jet polishing/cutting systems and,more importantly, the entire range of Micro-Electro-Mechanical Systems (MEMS) encompassing their various bio-engineering applications [1, 2, 3].Knudsen [4] experimental study, was among the first designed to report data pointing to anomalous behaviour ofrarefied gas flow through capillaries. Many experiments have since confirmed Knudsen’s initial observations [5, 6, 7].Meanwhile, it is now widely accepted that the standard set of fluid mechanical equations, namely those due to Navier-Stokes-Fourier, are inapplicable. Among recent developments in predicting these phenomena, Gallis and Torczynski[8] have presented a direct Monte Carlo simulation (DSMC). Gorji et al. [9] obtained the mass flowrate by solving,numerically, a velocity-space stochastic equation addressing molecular motions. Veltzke and Thaming [10] pointed outthat the mass flowrate in the slip flow regime can be accurately predicted by arguments of molecular spatial diffusioneffects without using any fitting parameter.Sone [11] suggested a correction to the standard Navier-Stokes equation, where the additional terms with character-istics of Burnett equation terms were called ‘ghost effect terms’. This is because they affect flows of continuum regimedespite being of high order in terms of Knudsen number classification. Meanwhile, existence of different averagingmethods and the influence of molecular spatial stochasticity have been pointed out as means of obtaining a Burnettregime hydrodynamic model [12]. This model appears more consistent in terms of mechanical and thermodynamicproperties than a traditional full Burnett equation obtained by Chapman-Enskog series expansions applied to Boltz-mann kinetic equation [12]. In the present paper we derive an analytical solution to mass flow rate in microchannelsusing this new emerging hydrodynamics and compare results with experimental data.A CONSISTENT VOLUME DIFFUSION HYDRODYNAMIC MODEL: SONE GHOSTEFFECTAppearing in the following equations is the material derivative defined as D/Dt = ∂/∂ t +Um ·∇, where t is the timevariable and Um represents mass-based average velocity: that is a flow macroscopic velocity as seen in a conven-tional continuity equation. A volume diffusion hydrodynamic model consistent with mechanical and thermodynamicprinciples is written [12]:DρDt=−ρ∇ ·Um, (1)ρ DUmDt=−∇ ·Π, (2)ρ DDt[12U2m + ein]=−∇ · [Π ·Uv]−∇ · Ju, (3)that is closed withΠ = pI− 2µ ˚∇Uv, Uv =Um + Jv, (4)whereJv =κmρ ∇ρ , Ju =−κh∇T, (5)and˚∇Uv =12(∇Uv +∇Uv)− 13 ∇ ·UvI. (6)The single bar over the velocity gradient in this last equation denotes the transpose operator, with I the idemfactor. Inthe above set of equations (1) to (5): ρ is the mass density, p the pressure, T the temperature, and ein the fluid’s mass-specific internal energy density, the latter related to the temperature by ein = (3/2)RT, with R the specific gas constant.Velocity Uv appearing in expression of the shear stress is the volume velocity: that is a macroscopic velocity based onaveraging method that accounts for the fluid molecular spatial distributions and not only their masses. Therefore Jvis a volume diffusion flux that appears as we distinguish volume averaging from mass or gravimetric averaging [13].Presence of flux Jv in the shear stress makes the above hydrodynamic model of Burnett level. Indeed, using the idealgas law written in the form∇pp=∇ρρ +∇TT, (7)volume diffusion momentum equation (2) becomesρ DUmDt=−∇ ·[pI− 2µ ˚∇Um + 2µκm ˚∇∇ ln T], (8)if local relative pressure variations are neglected. In equation (8), we observe that the third term involving temperaturegradients on the right-hand-side, is a thermal stress term that is obtained traditionally at the second order in Chapman-Enskog expansion; so it is a Burnett shear stress term. This term is identical to corrective terms to the Navier-Stokesequation called “ghost effect terms”: conveying that this is a higher order Knudsen number term, which is found toinfluence flows in the pure hydrodynamics or pure continuum regime. Derivation of volume diffusion hydrodynamicset (1) to (5), does not involve any small parameter expansion procedure. Consequently, within this set of equations, thevolume diffusion presumably may have influence at any Knudsen number order; a situation that appears to corroboratewith Sone ghost effects.While a classical set of full Burnett hydrodynamic equations derived using Chapman-Enskog expansion seriesare known to have several mechanical and thermodynamic inconsistencies [14], the above set of equations (1) to(5) satisfies a series of mechanical principles and is compatible with linear irreversible thermodynamics [12, 15].Phenomenological transport coefficients involved consist of: µ the fluid dynamic viscosity, κm = k/(cpρ) the volumediffusivity coefficient, and κh = kcv/cp, with k the Fourier thermal conductivity, namely the conductivity appearing inthe Prandtl number, with cv and cp the specific heat conductivities [see 16, section5.4.1] and also [see 15, section 5.2].ANALYTICAL SOLUTIONWe consider a steady-state isothermal pressure-driven flow occurring in a rectangular microchannel. The streamwiseflow coordinate variable is denoted x and the wall normal coordinate is denoted y. The height and width of the channelare denoted respectively by h and w, wherein w >> h such that the flow may be supposed two-dimensional. Therectangular channel height-to-length ratio h/L is assumed to be small to neglect the inlet and outlet effects. Velocitycomponent, Um =Um(x,y) is restricted to the streamwise direction and is a function of x and y. Disregarding energyequation, continuity and volume diffusion momentum equations (1) and (2) can be written as :∇ · [ρUm] = 0, (9)∇ · [ρUmUm] =−∇p−∇ ·Πv. (10)In terms of boundary conditions, impenetrability of mass at the channel walls requires normal component of massvelocity Um to vanish at y = ±h/2. In addition, we impose on the volume velocity a form of slip condition so that allboundary conditions are written:umy (x,±h/2) = 0, (11)anduvx (x,±h/2) = Jvx (x,±h/2) =∓λoρo1ρ[∂uvx∂y]y=h/2=∓λoρo1ρ[∂umx∂y]y=h/2, (12)In boundary conditions (11) and (12), subscript x and y refer to components in x and y directions, respectively. Subscript‘o’, refers to the channel outlet, which is simply used here as a convenient arbitrary reference. Furthermore, λo is thegas mean free path at the channel outlet. Note that equation (12) is not a standard slip condition, as the equation can beviewed as a constitutive equation for the volume flux Jv at the boundary when umx = 0 (i.e., when the no-slip boundarycondition is imposed on the mass velocity).A solution method starts with continuity and momentum equations (9) and (10), reformulated as:ρ ∂umx∂y +∂∂x [ρuvx ]−kcpd2 lnρdx2 = 0 , (13)µ ∂2uvx∂y2 =d pdx . (14)The solution of equation (14) satisfying boundary condition (12) isuvx =18µd pdx(4y2− h2− 8µE 1ρ), (15)in which we denote E = h2Kno/(2νo), wherein νo denotes the kinematic viscosity at the outlet, and Kno = λo/h is theoutlet Knudsen number. Substituting (15) into (13) and solving the resulting expression for umx , by subjecting to theboundary condition (11) results inumx =−18µ(43 y3− h2y)1ρddx[ρd pdx]+Ey1ρd2 pdx2 + ykcpρd2 lnρdx2 +C(x), (16)where the integration constant C(x) is a function only of x. However, application of boundary conditions (12) requiresthat C(x) = 0 as a result of symmetry. The preceding equations furnish the pressure distribution in accordance withthe following scheme. Evaluate equation (16) at y =±h/2, and use boundary condition (12) to obtain112h2 ddx[ρd pdx]+Ed2 pdx2 +kcpd2 lnρdx2 = 0. (17)Eliminate the density in equation (17) in favour of the pressure via use of the ideal gas law p = ρRT , and subsequentlyuse the identity pd p/dx = (d/dx)(p2/2) to obtainddx2[p2 +24µRTh2(E p+kcpln p)]= 0. (18)Integration of equation (18) followed by rearrangement yields(ppo)2+Fppo+G(ln(ppo)+ ln po)=CxL+D, (19)whereF =24µRTh2 poE and G = 24µRTh2 p2okcp. (20)Alternatively, in terms of the Knudsen and Prandtl numbers,Kno =kλhµpo√2RT , Pr = cpµ/k, F = 12Kno, G =24Prk2λKn2o, (21)where kλ =√pi/2 is a coefficient associated with the definition of the mean free path as related to the choice ofmolecular collision model. Equation (19) is the pressure distribution, in which constants C and D are determined fromknowledge of the channel inlet and outlet pressures, p(x = 0) = pi and p(x = L) = po. This furnishes expressionsC =−[(P2− 1)+F(P− 1)+G lnP] , D = P2 +FP+G(lnP+ ln po) , (22)where P= pi/po, denotes the inlet/outlet pressure ratio. The mass flowrate through the channel is given by the formula˙M = w∫ h/2−h/2ρUmdy = const. (23)Using Um =Uv− Jv, with equations (15) and (5) we have thatumx =18µd pdx(4y2− h2− 8µE 1ρ)− kcpρd lnρdx . (24)Multiply equation (24) by ρ and subsequently eliminate ρ in favor of p on the right-hand side. Introduction of theresulting expression into equation (23) yields˙M =− wh324µRTddx[p2 +24µRTh2(E p+kcpln p)]. (25)The bracketed term in equation (25) is seen to be identical to the bracketed term in equation (18). It follows that˙M =− wh3 p2o24LµRT C, (26)where C is the constant given by equation (22). Thus, the mass flow rate is given explicitly in terms of the parameterscharacterizing the problem as˙M =wh3 p2024LµRT[(P2− 1)+ 12Kno(P− 1)+24Prk2λKn2o lnP]. (27)Equation (27) has a structure frequently obtained when slip boundary conditions are used [17]. However, our goal inthis investigation is to demonstrate that, due to the use of volume diffusive flux, equation (27) can reproduced faithfullythe experimental data without the use of any fitting parameter.Indeed, in the course of gas-kinetic molecular description of the hydrodynamic set of equations (1)-(5), volumediffusion is a manifestation of the molecular-level spatial diffusion process that used to be ignored [18]. This, in turn,generates a second level of non-equilibrium scaling beyond, or parallel to, traditional Knudsen number scaling [12].Veltzke and Thaming [10] demonstrated in this context that volume diffusivity coefficient depends not only on theproperties of the gas but also on the channel geometry. This leads us to identify a volume diffusivity coefficient forthe rectangular channel corrected by the channel dimensions as, κ∗m = κmL/w. As thermodynamic expression of thevolume diffusivity coefficient is κm = k/(cpρ), this geometrical correction only affects the Prandtl number coefficientTABLE 1. Summary of fluid properties and physical coefficients used in figure 1w L h Pr4.92×10−4 [m] 9.39×10−3 [m] 9.38×10−6 [m] 0.67µ kλ Sc P1.97513×10−5 [Pa · s ]√pi/2 0.67 5 1 2 3 4 5 0.1  1  10dimless mass flow rate, GmMean Knudsen number, KnNavier-StokesVolume diffusion ExperimentsFIGURE 1. Mass flow rate as predicted by volume diffusion hydrodynamics compared with experimental databy a factor L/w in the final pressure distribution equation (19), and the mass flow rate equation (27). Full expressionof the micro gas mass flow rate in rectangular channel using the volume diffusion momentum equation is thereforefinally given by˙M =wh3 p2024LµRT[(P2− 1)+ 12Kno(P− 1)+24wPrLk2λKn2o lnP], (28)in which, there appear only physical properties with no coefficient playing a fitting role such as a slip coefficient.These physical properties in the case of Ewart et al. [7] experimental data for gaseous helium are summarized in table1. A full comparison between predictions by equation (28) and experiments are seen in figure 1, plotted in the form ofdimensionless flowrate, Gm, vs the mean Knudsen number that are defined byKn[mean] =kλhµpi+p02√2RT , Gm = ˙M ∗(wh2L√2RT(pin− pout))−1. (29)From the figure, the volume diffusion model, i.e. equation (28), agrees with experimental data up to a Knudsen numberof about 5 with all parameters possessing clear physical meanings as given in table 1 and a viscosity coefficient havingits appropriate experimental value. This agreement can be said an excellent achievement considering the fact thatconventional Navier-Stokes modified by slip boundary corrections, gives agreement with the data within the slip butnot deep into the transition regime as the present volume diffusion model does.Figure 2 presents a comparison of normalized streamwise pressure distributions against the experimental data ofPong et al. [19] at various pressure ratios (P). Here, pressure is normalized with the outlet pressure of the channel, andthe pressure ratio is defined as the ratio of inlet to outlet pressures. The measurements by Pong et al. [19] were madeby embedding measurement ports in a microchannel in which pressure transducers were mounted. The working gaswas nitrogen and the outlet Knudsen number was 0.044. So theoretical graphs are plotted using pressure distributionequation (19) including nitrogen Prandtl number of Pr = 0.72 and the geometrical correction factor L/w from thevolume diffusivity coefficient. The comparison between the present volume diffusion model and the data is good.0 0.2 0.4 0.6 0.8 1123x/Lp(x)/p out  P = 2.70P = 2.36P = 2.02Experimental DataFIGURE 2. Pressure distribution as predicted by volume diffusion model compared with experimental data of Pong et al. [19]for an outlet Knudsen number of 0.044 and various pressure ratios (P).(a)0 0.2 0.4 0.6 0.8 111.522.53x/Lp(x)/p out  Kn = 0.01Kn = 0.1Kn = 0.5Kn = 2(b)0 0.2 0.4 0.6 0.8 1−0.1−0.0500.050.10.15x/L[ p − p lin ]/[ p in − p out ]  Kn = 0.01Kn = 0.1Kn = 0.5Kn = 2FIGURE 3. (a) Normalized pressure distribution along the streamwise direction for various Knudsen numbers, (b) dimensionlessdeviation of the pressure distribution from the linear pressure distribution as a function of the dimensionless distance.The volume diffusion effect on the streamwise pressure distribution can be assessed by varying the Knudsen number.Interesting results are observed as shown in the Fig. 3(a) for a pressure ratio of 3. The curvature of the pressurevariation decreases with an increase in Knudsen number, and the pressure distribution becomes linear in the earlytransition regime. With further increases in Knudsen number the curvature changes from convex (in the continuumand early transition regimes) to concave (in the late transition regime). Change in curvature of pressure profile canalso be captured by the second-order slip model [20], however the classical first-order slip model fails to predict thisphenomena. Fig. 3(b) shows the dimensionless deviation of the streamwise pressure distribution from a linear pressureprofile, as a function of dimensionless distance along the channel. With an increase in Knudsen number, the rarefactioneffect tends to dominate compressibility effects, and the dimensionless deviation becomes zero at a Knudsen number ofaround 0.5. With further increases in Knudsen number, the dimensionless deviation changes from positive to negativevalue. In the slip flow regime, where compressibility effects are still dominant, the pressure profile is unsymmetricalwith respect to the streamwise distance, but turns out to be quite symmetric in the transition regime at Kn = 2, whererarefaction/volume diffusion effects are dominant.CONCLUSIONIn this paper we derived an analytical solution to Knudsen enhanced mass flowrate phenomena in micro gas channelsusing a volume diffusion hydrodynamic model. The solution equation is similar in form to that obtained using first andsecond order slip boundary conditions. However, with the volume diffusion approach and volume diffusivity coefficientthat depends on the geometry of the channel, we obtain agreement with the experimental data up to Knudsen numberof 5. It also allows the direct investigation of the non-linear pressure distribution, and predicts the change in curvatureof the streamwise pressure profile with an increase in rarefaction (when the diffusive effects dominate the convectivefluxes).ACKNOWLEDGMENTSND would like to thank the European Community’s Seventh Framework Programme FP7/2007-2013 under grantagreement ITN GASMEMS no 215504.REFERENCES1. G. Karniadakis, A. Beskok, and N. R. Aluru, Microflows and Manoflows: Fundamentals and simulations, Springer, 2005.2. S. Colin, Microfluidics, ISTE, 2010.3. G. Kandlikar, S. Garimella, D. Li, S. Colin, and M. R. King, Heat transfer and fluid flow in minichannels and microchannels,Elsevier, 2006.4. M. Knudsen, Ann. Phys-Leipzig 28, 75 – 130 (1909).5. E. B. Arkilic, K. S. Breuer, and M. A. Schmidt, J. Fluid Mech. 437, 29–43 (2001).6. J. Maurer, P. Tabeling, P. Joseph, and H. Willaime, Phys. Fluids 15, 2613 (2003).7. T. Ewart, P. Perrier, I. Graur, and J. M. Meolans, J. Fluid Mech. 584, 337 – 356 (2007).8. M. A. Gallis, and J. R. Torczynski, Phys. Fluids 24, 012005 (2012).9. M. H. Gorji, M. Torrilhon, and P. Jenny, J. Fluid Mech. 680, 574–601 (2011).10. T. Veltzke, and J. Thaming, J. Fluid Mech. 698, 406–422 (2012).11. Y. Sone, Annu. Rev. Fluid Mech. 32, 779–811 (2000).12. S. K. Dadzie, and J. M. Reese, Phys. Rev. E 85, 041202 (2012).13. S. K. Dadzie, and J. M. Reese, Phys. Fluids 22, 016103 (2010).14. L. García-Colin, R. Velasco, F. Uribe, Phys. Rep. 465, 149–189 (2008).15. H. Brenner, Int. J. Eng. Sci. 47, 930–958 (2009).16. H. Brenner, Physica A 349, 11–59 (2005).17. I. A. Graur, J. G. Meolans, and D. E. Zeitoun, Microfluid Nanofluid 2, 64–77 (2006).18. S. K. Dadzie, J. M. Reese, and C. R. McInnes, Physica A 387, 6079–6094 (2008).19. K. Pong, C. M. Ho, J. Liu, Y. C. Tai, Proceedings of Application of Microfabrication to Fluid Mechanics, ASME WinterAnnual Meeting, Chicago, 51-56 (1994).20. N. Dongari, A. Agrawal and A. Agrawal, Int. J. Heat. Mass. 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Transition regime analytical solution to gas mass flow rate in a rectangular micro channel

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Transition regime analytical solution to gas mass flow rate in a rectangular micro channel

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