Lecture 2 Fluid dynamics in microfluidic systems

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1 Lecture 2 Fluid dynamics in microfluidic systems

2 1) The range of validity of the fluid mechanics equations

3 The hypothesis of the continuum in fluid mechanics (Batchelor, Introduction to Fluids Dynamics) Notion of «fluid particle»

4 What a fluid particle for «simple» liquids? L >> l fluidparticle >> l int ermolecular In practice, a few nanometers for simple fluids

5 Simple liquids Complex fluids Simple liquids (ex alcanes) Complex fluids (ex: polymers)

6 A few intermolecular scales are enough for continuum approach to apply in simple liquids J.Coll. Inter. Science, 110, 263 (1986)

7 How many intermolecular scales in a small microchannel? Intermolecular distance for an alcane ~ 0.3 nm 500 nm

8 Fluid particle and nanometric channels A fluid particle for water

9 Fluid particle and nanometric channels 5 nm A fluid particle for water L. Bocquet, P. Tabeling, LOC 2014

10 What a fluid particles in gases? Central notion: mean free path λ = 1 2πna 2 = kt 2πpa 2

11 How many mean free pathes in a small microchannel in normal conditons? A typical mean free path in normal conditions 500 nm

12 Nombre de Knudsen/ Régimes d écoulement dans les gaz Kn = λ L Nombre de Knudsen

13 Boltzmann equations

14 Burnett equations

15 Burnett equations

16 Knudsen number Flow regimes in gases Kn = λ L Knudsen number MICROFLUIDICS Hydrodynamic Regime Kn Slip regime Transition regime Rarefied gas regime

17 Conclusions - Simple liquid L >3 nm - Gas L > 1µm

18 2) The boundary conditions

19 Navier Boundary Condition u z 0 # u& u = L s % ( $ z '

20 Geometric interpretation of Navier Boundary Condition $ u' z = O u = L s & ) % z ( u(z) u(0) -L S 0 z

21 Slip length in gases GAS SOLID WALL Molecules scatter against the wall FLOW # u = L s % $ u z & ( ' Maxwell : L s λ (purely diffusive case)

22 Slip length in liquids over repulsive surfaces FLOW Bulk is dense Near-wall region Is slightly depleted JL Barrat, L.Bocquet, Phys. Rev. Lett. 82, 23 (1999)

23 Comparison numerics/experiment

24 3) Slippage over structured surfaces

25 Slippage on superhydrophobic surfaces

26 Prof Choi (S.Inst Techn. Hoboken, 2010)

28 Coating surfaces with a lubricating film

30 4) The Flow equations for microfluidics

31 The governing equations of fluid mechanics F=mγ

32 The flow equation for incompressible newtonian fluids + continuum F=mγ + incompressibility + newtonian Du Dt = u t + (u )u = 1 P +νδu ρ

33 Inertia forces are small in miniaturized systems 100 µm

34 Inertia forces are small in miniaturized systems 300 µm

35 Microhydrodynamics Stokes equation ρ Du 0 Dt = P + µδu Approximation valid for microfluidics Exceptions : microechangers spotters, inertial microfluidics

36 The Reynolds number is small in microfluidics Re = Ul/ν

37 Mathematical properties of the Stokes equations

38 Unicity of the solutions

39 Consequence of unicity : no hydrodynamic instability Mixing layer at high Re Kelvin Helmoltz instability

40 Instantaneity

41 τ ~ l2 ν

42 Philosophical inference: In microfluidic systems, time and space are miniaturized all at once

43 Linearity

44 Flow speeds are proportional to the intensity of the source of movement. P 1 U P 0

45 Reversibility

46 Reversibility of microfluidic Tesla Valves

47 Basic geometries, high Re behavior From H. Stone +PT Les Houches, 2010

48 Basic geometries, low Re behaviors From H. Stone Les Houches, 2010

49 Spiders too have chosen small Reynolds numbers

50 The job of the spider is to make a spiderweb

53 Trees too have chosen small Reynolds numbers

54 5) Resistance, capacitance, inductance

55 P 1 U P 0

56 Example of a hydrodynamic calculation of a microfluidic flow Poiseuilles flow u(z) = ΔP 2µL (z2 b2 4 ) Q = wb3 12µ ΔP L

57 Example of a hydrodynamic calculation of a microfluidic flow Slippy Poiseuille flow u(z) = L SΔPb 2µL ΔP 2µL (z2 b2 4 ) Q = wb2 12µ (6L S + b) ΔP L

58 The notion of hydrodynamic resistance ΔP = RQ m Case Hele Shaw R = 12ν b 2 L S

59 The hydrodynamic capacitance Q m = C dp dt Example : Deformable tube: dp=κ -1 dv/v Q m =ρdv/dt Q m =mκdp/dt et donc C=κm Volume V Pressure P

60 Inductance are zero in microfluidics ΔP = L dq m dt L = 0

61 Microfluidic networks From H. Stone Les Houches, 2010

62 Resistance network of a flow focusing Ps R. Austin et al (1999)

63 Resistance network of a flow focusing R. Austin et al (1999)

64 Electrical scheme of a microvalve PDMS PDMS GLASS

65 The bottleneck effect Q C=m/E =πd 2 Lρ/4E R=12νl/b 3 w Calcul : C=m/E=πD 2 Lρ/4E, R=12νl/b 3 w et τ=rc τ = 3π µd2 L Eb 3 w

66 Bottleneck effect is suppressed if we use pressure sources ΔP C R

67 Be scared by the dead volumes This increases the capacitances embedded in the system. PS: No bubbles, please. Orders of magnitude?

68 The particular case of Hele-Shaw flows

69 Darcy law governs flows in Hele Shaw geometries V = 1 b b / 2 b / 2 V = b2 12µ p U(x, y,z)dz In a Hele-Shaw cell, the flow is potential

70 Recirculation zones can exist in non Hele Shaw cell geometries

71 7) Microfluidics exploiting inertial effects

72 Reminder on boundary layer detachment Boundary layer U Favorable case

73 Boundary layer detachment: reminder Unfavorable case U Detachment

74 Application to pumping Stemme et al (1993)

75 Microfluidics exploiting centrifugal forces

76 Centrifugal microfluidics

77 Centrifugal microfluidics/developing countries

78 Reminder on Dean Flows

79 Inertial microfluidics

80 8) Introduction to nanofluidics

81 Flow in nanometric channels A typical fluid particle for simple fluids L. Bocquet, P. Tabeling, LOC 2014

84 The flow equations for nanofluidic flows below one nanometer A new physics must be built

85 The flow equation for nanofluidic flows above a few nanometers F=mγ + F nano

86 Forces at the nanoscale

87 Van der Waals attraction between surfaces D P = F S A 6π D 3 Adhesion: P = A 6π D 3 0 Surface energy: γ = E 2S = 1 2 F dr = S D 0 A 24πD 0 2

88 Measurement of Van der Waals forces between a cylinder and a plane in vacuum

89 Tabor-Israelachvilii force machine

93 Manifestation of Van der Waals forces: particle deposition and clogging

94 EXPERIMENTS MADE IN 20 µm HIGH MICROCHANNELS parallel channels PDMS-coated slide outlet inlet reservoir 400µm 8mm Clogging at entry W/D = 2, real time ~5s 40 µm 94

95 THE PHYSICS OF CLOGGING Lateral displacement during the travel ~ (DL/U) 1/2 ~ 30 nm 95

96 Oscillating force Oscillating force in organic liquid films Static force in confined organic liquid films (alkanes, OMCTS ). Oscillations reveal liquid structure in layers parallel to the surfaces The Horn & Israelachvili, J. Chem Phys 1981

97 Electrostatic forces λ D = εd σ l D l D

98 Measurement of electrostatic forces

99 Consequences of electrostatic forces: EOF

100 Overlap of Debye layers: nanofluidic diode w >> λ D w < λ D w Pénètre Ne pénètre pas CAS 1 CAS 2

101 Slippage in nanofluidic systems

102 Slippage in nanofluidic systems

103 Nanofluidic optimization of aquaporines Quick tour in Microfluidics 103

104 What must be known: - Conditions of validity of the Navier Stokes equations - Elementary properties of gas flows in small systems - Properties of Stokes equations - Notion of slippage, orders of magnitudes in gas/liquid - Calculation of simple situations (ex: slippy Poiseuille) - Notion of resistance, capacitance and inductance - Bottleneck effect - Basic phenomena of inertial bases microfluidics - Forces at play in nanofluidic systems

105

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