Microscale physics of fluid flows

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1 Microscale physics of flid flows By Nishanth Dongari Senior Undergradate Department of Mechanical Engineering Indian Institte of Technology, Bombay Spervised by Dr. Sman Chakraborty

2 Ot line What is microflidics Why microflidics? Potential applications Gas flow throgh microchannels 1. How gaseos flows are different from liqid flows Liqid flow throgh microchannels

3 What is microflidics? Microflidics refers to flid flow in microchannels as well as to microflidic devices (pmps, valves, mixers, etc.) and systems. Control of small volmes of flid (fl - μl, L) in microscopic channels ( μm)

4 Microflidics why stdy it? Reqired when the application demands handling of very small liqid volmes. (Ex Ink jet printers, drg administration, chemical microanalytical systems ) Redced liqid consmption Sample and reagent volmes, etc Faster response Shorter diffsion distances Efficient atomation Several fnctions on the same chip (Lab-On-A- Chip)

5 Microflidics why stdy it? (cont.) Redction of power consmption Parallel devices + faster processes = high throghpt Safety Reliability Integration + Mltifnctional Portable devices

6 Size characteristics (from Ngyen and Wereley, Fndamentals and Application of Microflidics, Artech Hose Pblishers, 00)

7 Potential applications Valves Channels Mixers

8 Potential applications (cont.) Pmps Ink jet printer

9 Gas flows throgh microchannels Difference between micro and macro domains with gases is presence of slip at the solid interfaces. The theory for incompressible, isothermal laminar flow in a macrochannel is well known de to the inherent simplicity of the problem. This is however not the case with microchannels, the difficlty stems from the fact that the flow is compressible and the slip at the walls has to be appropriately modeled. The flow of liqid in a microchannel is different from that of a gas in the same microchannel (see e.g. Gad-el-Hak [1999]). Whereas standard reslts sally apply with liqid flow, this is not the case with gases

10 IS THE CONTINUUM APPROXIMATION VALID? POSSIBLE NON-CONTINUUM EFFECTS: Slip at the bondary Stress and strain rate is nonlinear Continm approximation fails To a certain degree the sitation is similar to low pressre high-altitde aeronatical flows This can be seen by considering air flow nder standard conditions (λ = 70 nm) throgh a 10 μm channel which gives a Kndsen nmber, Kn of which is well within the slip regime (10-3 <Kn< 10-1 ) Flow is characterized by Kndsen nmber (Kn), defined as the ratio between moleclar mean free path and hydralic diameter of the test channel.

11 Flow Regimes Gad-el-hak (1999) Kndsen Nmber = λ/d h Kn= Continm Regime Transition Regime Slip Flow Regime Moleclar Regime

12 Review of slip models This is Maxwell's first-order slip bondary condition On a control srface, s, at a distance of λ/, half of the molecles come from one mean free path away from the srface with tangential velocity λ, and half of the molecles are reflected from the srface. On the assmption that a fraction σ of the molecles are reflected diffsively at the walls (i.e. their average tangential velocity corresponds to that of the wall, w ), and the remaining (1-σ) of the molecles are reflected speclarly (i.e. withot a change of their impinging velocity ) Maxwell obtained the following expression on expanding in terms of Taylor series and retaining terms p to second order :

13 Review of slip models (cont.) In 199 Prof. Harvey Lam at Princeton University has sggested the following slip bondary condition instead of Maxwell's first order slip conditions. In 1993, Beskok has determined the vale of b by a pertrbation expansion of the velocity field in terms of Kn and have fond that in the slip flow regime (Kn < 0.1), second order accracy is obtained if b is choosen as:

14 Review of slip models (cont.) Sreekanth [1969] sggested the following general form of secondorder slip model (1963) Vales of slip coefficients proposed in the literatre. The vales were obtained throgh theoretical considerations, DSMC simlations, or experiments. Ref: Nishanth Dongari and Amit Agrawal, Modeling of rarefied gas flow in long microchannels, Proceedings of 3 rd ICFMFP 006, Dec 7-9, IITBombay.

15 Governing Eqations Continity ρ t + ρ x + ρ v y = 0 Momentm ρ 1 ρ v ρ P μ v = t x y x x y 3 x xy Energy Eqation of State ρv 1 ρv v ρv P μ v v v = t x y y x y 3 y xy DT DP T T v v v ρcp k k μ = 0 Dt Dt x x y y x y x y 3 x y P = ρrt

16 Backgrond Arkilic et al. (1997) considered only first order slip model, both Arkilic et al. (1997) and Beskok et al. (1999, 00) neglected the inertial term.

17 Analytical Soltion The following assmptions are involved in this procedre: 1. The flow is two-dimensional and locally flly developed.. The flow conditions are isothermal. 3. The channel is long, and the entry and exit effects are negligible. 4. The viscos compressive stresses are negligible. 5. Profile of velocity assmed to be parabola. The momentm balance for the case of compressible flow on a finite elemental volme between two cross sections of a channel with axial length dz is given by [Sreekanth, 1969] 1

18 Soltion for pressre After sbstitting for and T w, Eq. 1 can be integrated to obtain an expression for pressre in terms of pressre (p o ) at some reference position z o and the corresponding Kndsen nmber (Kn 0 ). The following expression for pressre is obtained: (Ref: Nishanth Dongari et al., Analytical soltion of gaseos slip flow in long microchannels, IJHMT (In Press)) where

19 Redction to first order slip expression On differentiating soltion for pressre twice, with respect to z and sbstitting C = 0, we obtain: The extra term in Eq. appears becase the entire analysis is second order accrate; the magnitde of this term is dependent on Reynolds sqared.

20 Validation of the theory This figre presents comparison of pressre from analytical soltion against the experimental data of Pong et al. The working gas was nitrogen and the otlet Kndsen nmber was The comparison between present theory and data is good. Comparison of pressre against experimental data of Pong et al. [3] for otlet Kndsen nmber of

21 Validation of the theory (cont.) This figre presents comparison of mass flx verss pressre ratio throgh the microchannel obtained from Eq. 0 against the experimental data of Arkilic et al. [1997]. The comparison between the theory and data is again good. Frther, the second order model is clearly an improvement over the earlier model. Comparison of mass flx against experimental data of Arkilic et al. [1997]

22 Kndson paradox It is emphasized that first-order slip models do not predict the Kndsen paradox" (appearance of a minima in normalized volme flx at Kndsen nmber approximately nity), or a change in crvatre of centerline pressre at high Kndsen nmbers. Comparison with Boltzmann's soltion (Cercignani et al. 004, Phys. Flids, 16, ) This reslt is significant from the perspective of nmerical simlations of rarefied gases. Comparison of normalized volme flx verss Kndsen nmber

23 Crvatre of pressre distribtion Figre shows a decrease in nonlinearity with an increase in Kn.At Kn = 0.16, the pressre distribtion becomes linear, and beyond it, the crvatre changes from convex with respect to the origin, to concave. At very large Kndsen nmbers, pressre again becomes almost linear, in agreement with the limiting case of free-moleclar flows.

24 Normalized Eqations Continity Momentm Energy Eqation of State 0 = + + y v x t ρ ρ ρ Re = y x v x y x x P y v x t μ ρ ρ ρ Re = y x x v y v x v y P y v v x v t v μ ρ ρ ρ = 3 Re Re.Pr y v x y x v y v x Ec y T x T k Dt DP Ec Dt DT Cp μ ρ = M T P γ ρ

25 Wall Conditions Wall Slip + = x T T y gas w V V wall gas ρ μ σ σ 4 3 l Maxwell (1879) + = Re 1) ( 3 x T Ec Kn y Kn w V V wall gas γ γ π σ σ w i r i V τ τ τ τ σ =

26 Thermal Bondary Condition Temperatre Jmp w T T wall gas y T T T + = Pr 1 l γ γ σ σ Von Smolchowski (1898) w T T y T Kn T T wall gas + = Pr 1 γ γ σ σ w i r i T de de de de = σ

27 Liqid flow throgh microchannels Pressre driven flows Electro-osmotic flow Diffsion Capillary flows etc. Srface effects tend to dominate, ths inertia forces are negligible to electro/viscos or capillary effects.

28 Pressre-driven flow zero flow at solid-flid interface no-slip condition adjacent horizontal layers flow at different speeds "slide over one another" P1 r layers exert resistive force on each other internal force viscosity viscosity, η (eta): flid s resistance to flow P Ref: I. Levine, Physical Chemistry, nd Ed.,1983, McGraw-Hill; Ref: G.T.A. Kovacs, Micromachined Transdcers Sorcebook, 1998, McGraw-Hill

29 Pressre-driven flow (cont.) Flow resistance Volmetric flow throgh a cylindrical tbe (laminar) Flow resistance for a cylindrical tbe Flow resistance for a rectanglar channel with the width w >> height h

30 Laminar vs. trblent flow laminar flow (Re (??)) trblent flow (Re (??)) well-defined streamlines nstable streamlines d = characteristic length, η = viscosity d = Diameter for Circlar tbes d = 4A / Lw for Other geometries: A - cross section area, Lw - wetted perimeter Disadvantages: Hydrodynamic dispersion Large pressre gradients are reqired Degradation of some flids

31 Electro-osmotic flow EOF generated at charged srfaces motion at srface propagates to adjacent soltion layers by viscos drag E applied, blk flow towards cathode typical veof: μm/s mm/s typical E: 100 s V/cm works well in φ < 100 μm

32 Electro-osmotic flow (cont.) pmping mechanism is plse less, and generates no back pressre valeless liqid handling liqids follow electric field lines no mechanical pmps or valves reqired very amenable to miniatrization Bt: Dependant on liqid composition (ph, ε, ζ, η)

EOF and P flow greater distortion in P flow 66 ms 165

33 Comparison of EOF and Poiseille flows P flow EOF 0 ms florescent dye visalizes distortion of a flid volme in EOF and P flow greater distortion in P flow 66 ms 165 ms Ref: P.H. Pal et al., Anal. Chem. 1998, 70,

34 NON-CONTINUUM EFFECTS - LIQUIDS FOR SUFFICIENTLY HIGH STRAIN RATE THE STRESS/RATE OF STRAIN AND HEAT FLUX/TEMPERATURE GRADIENTS RELATIONS BECOME NONLINEAR Slip occrs at the wall

35 Moleclar Dynamics Simlations Setp initial conditions and geometry Specification of interaction potential Integration of newtons eqations of motion

36 Smmary Gas flows Analytical soltions and their applicability Nmerical simlations Liqid flows

37 Acknowledgements I wold like to express my gratitde to Dr. Sman Chakraorty for his help in nmeros discssions and providing material. I am gratefl to Prof. A. W. Date and Dr. Amit Agrawal for recommending me for winter academy Thank yo very mch to all of my friends and Prof. Drst for making my stay intellectally stimlating and enjoyable.

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