Basic Principles in Microfluidics

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1 Basic Principles in Microfluidics 1

2 Newton s Second Law for Fluidics Newton s 2 nd Law (F= ma) : Time rate of change of momentum of a system equal to net force acting on system!f = dp dt Sum of forces acting on control volume = Rate of momentum efflux from control volume + Rate of accumulation of momentum in control volume 2

3 Navier - Stokes Equation Navier-Stokes equation applies when: (1) There are more than one million molecules in smallest volume that a macroscopic change takes place. (2) The flow is not too far from thermodynamic equilibrium. 3

4 Navier - Stokes Equation! du dt = "#P +!g + $# 2 U + $ 3 #(#iu) For noncompressible Fluid!iU = 0! du dt = "#P +!g + $# 2 U du dt =! "P # + g + $ # "2 U 4

5 Navier - Stokes in Microfluidics Terms become dominant based on physics of scale In microfluidics inertial forces dominate due to small dimensions, even though velocity can be high du dt du dt =! "P # + g + $ # "2 U =! 1 " #P 5

6 VISCOSITY 6

7 Viscosity Viscosity is a measure of resistance (friction) of the fluid to the flow This determines flow rate Symbols: η and in some books µ Units: Poise (gram/sec * Cm) 7

8 Viscosity Viscosity is a measure of resistance (friction) of the fluid to the flow. This determines flow rate. Units: Poise (gram/sec Cm) 8

9 Basic Properties - Viscosity Fluids and gases are very different Fluids become less viscous as temperature increases Gases become more viscous at temperature increases 9

10 Viscosity in Gases and Fluids Gases! =! 0 (T 0 - constant) (T 0 - constant) " # $ T T 0 % & ' 3 2 Fluids η η 0 e (Τ Τ 0 ) 10

11 Interfaces and Surface Tension 11

12 Interfaces Interface: Geometric Surface that delimits 2 fluids Separation depends on molecular interactions and Brownian diffusion 12

13 Interfaces Interface: Geometric Surface that delimits 2 fluids Simplified view: At interface: different energies Interaction between molecules 13

14 Interfaces If U is the total cohesive energy per molecule and d is a characteristic molecular dimension, d 2 is its surface, then the energy loss (surface tension) is given by:! = U 2d 2 14

15 Laplace s Law Minimization of surface energy, create curvature of fluids on other surfaces (fluids) Curvature 1/R Laplace s Law, the change in pressure is related to the curvature of the surface. For a sphere: For a cylinder: P = 2 (γ/r) P = γ/r 15

16 Droplet on a Surface of Two Properties Simulations 16

17 Coarsening Two Droplets linked by a precursor film 17

18 Coarsening Two Droplets linked by a precursor film 18

19 Contact Angle Surface tension (force per length) Angle is determined by the balance of forces at the point of interface Hydrophilic Hydrophobic 19

20 Contact Angle Surface tension (force per length) Angle is determined by the balance of forces at the point of interface Oil on Water 20

21 Hydrophilic - Hydrophobic 21

22 Surface Tension Droplet on a surface Forces on cross section of drop Surface tension along periphery Pressure on section area Pressure difference outside/inside drop Force =!PA = "r2!p Surface Tension=2!r"! = r 2 "P 22

23 Forces - Capillary Effects A wetting fluid will rise in a capillary tube Equilibrium: pressure drop across meniscus Surface tension Viscosity h = 2! Cos(") #gr 23

24 Capillary Force 24

25 Capillary Forces 25

26 Capillary Forces Small Channel (capillary) - Surface tension draws fluid of density ρ into the channel of radius ( r) F = 2!r" Cos(#) θ = contact angle γ = surface tension (N/m) Height of Fluid in a tube in the presence of gravity h = 2! Cos(") #gr 26

27 Forces - Capillary Effects 27

28 28

29 Capillary Forces 29

30 Droplet on Surfaces 30

31 Droplet on Irregular Surfaces r: roughness f: ratio of contact angle to the total horizon surface Young s critical angle cos(θ) = (f-1) / (r-f) 31

32 Wettability and Roughness 32

33 Reynolds Number 33

34 Fluids - Types of Flow Laminar Flow (Steady) Energy losses are dominated by viscosity effects Fluid particles move along smooth paths in laminas or layers Turbulent Most flow in nature are turbulent! Fluid particles move in irregular paths, somewhat similar to the molecular momentum transfer but on a much larger scale Reynolds Number Re is a measure of turbulence 34

35 Reynolds Number Reynolds number (Re) = inertial forces / viscous forces Re = Kinetic energy / energy dissipated by shear Implies inertia relatively important Re = 1 2 mv 2 D 1 Re = 2!V A D (!AL)V D "A Re =!V D L " V D = Drag velocity, L = characteristic length, η= viscosity, ρ = density Re < 2100 : laminar (Stokes) flow regime slow fluid flow, no inertial effects laminar flow in microfluidics slow time constants, heavy damping Re > 4000 : unstable laminar flow - turbulent flow regime 35

36 High and Low Reynolds number fluidics When the Reynolds number is low, viscous interaction between the wall and the fluid is strong, and there is no turbulences or vortices 36

37 37

38 Is this Flow Turbulent? Channel Geometry - Use a characteristic length : D h Re =! " VD h D h is a geometric constant 38

39 Is this Flow Turbulent? 39

40 Mixing Re = 12 and Re = 70 Cycle 1 Cycle 2 Cycle 3 40

41 Microchannels Cross Sections 41

42 Re and Size 42

43 Re - Some examples Re Friction factor ~ 1/ Re 43

44 Human Circulatory System 44

45 Flow associated with Skin 45

46 Knudsen Number Knudsen number assumes that we can treat the material as a continuum Continuum hypothesis holds better for liquids than gases also, K n =! mfp D h K n =!" 2 ( M Re ) λ mfp = mean free path of molecules, D h = hydraulic diameter K n measures deviation of the state of the material continuum K n < 0.01 continuum 0.01 < K n < 0.1 slip flow 0.1 < K n < 10 transition region 10 < K n molecular flow 46

47 The Smallest Length Scale of a Continuum High Re Low Re K n = M Re!" 2 47

48 Stokes - Einstein Diffusion Stokes - Einstein Equation Diffusion of a particle (gas, fluid) η Translational Diffusivity Rotational Diffusivity D t = K BT 6!"a D r = K BT 8!"a 3 48

49 Diffusion in Fluids Very short diffusion times x = 2D!! = 1 2 D = diffusion constant X = diffusion length τ = diffusion rate x 2 D Laminar flow limits benefits for fluid mixing. Highly predictable diffusion has enabled a new class of microfluidic diffusion mixers 49

50 Fluid Squeeze 50

51 Squeezed film damping Squeeze a film by pushing on the plates (one is not moving) Viscous drag is opposing the motion of the fluid Beam displacement Flow of fluid (Reynolds equation) Knudsen number, K, is the ratio of the mean free path to gap Squeeze number: relative importance of viscous to spring forces! "2 U "t 2 + EI "4 U "u 4 = P + F L 12! d(ph) dt P = b du dt = "{(1+ 6k)h 3 P"P} b = 96!W 3 " 4 h 3 L 51

52 Concluding Remarks 52

53 Summary Re = turbulent / viscous stresses Re < 2100 : laminar (Stokes) flow regime, slow fluid flow, no inertial effects laminar flow in microfluidics slow time constants, heavy damping Re > 4000 : turbulent flow regime 53

54 Fluid Behavioral What happens when the fluid is on the micro - nano scale? We discussed scaling - this is a review Quantities proportional L 3 Inertia, buoyancy, etc. Quantities proportional L 2 Drag, surface charge, etc. Quantities proportional L 1 Surface tension 54

55 Who Rules η 55

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