Contents. Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 1

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1 Contents 1. Introduction 2. Fluids 3. Physics of Microfluidic Systems 4. Microfabrication Technologies 5. Flow Control 6. Micropumps 7. Sensors 8. Ink-Jet Technology 9. Liquid Handling 10.Microarrays 11.Microreactors 12.Analytical Chips 13.Particle-Laden Fluids a. Measurement Techniques b. Fundamentals of Biotechnology c. High-Throughput Screening Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 1

2 3. Physics of Microfluidic Systems Behavior of fluids in microfluidic structures System approach Hydrostatics: fluids at rest Fluid dynamics: mechanics of fluids in motion Scaling laws Shift in surface-to-volume ratio Shifted significance of physical effects in MF MF-effects Capillarity Electrokinetics Strictly laminar flow conditions Pros and cons of MF-effects New design principles Hazard for many applications Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 2

3 3. Physics of Microfluidic Systems 3.1. Navier-Stokes Equation 3.2. Laminar and Turbulent Flow 3.3. Fluid Dynamics 3.4. Fluid Networks 3.5. Energy Transport 3.6. Interfacial Surface Tension 3.7. Electrokinetics Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 3

4 3.1. Navier-Stokes Equation Central relationship of fluid dynamics Solutions for selected situations Assumptions Continuous media Viscous constant Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 4

5 3.1. Navier-Stokes Equations Lagrangian and Eulerian Description of Motion Derivation of the NS-Equation Consequences from the Continuity Equation Interpretation of the Momentum Equation Common Boundary Conditions Simplifications Dynamic Similarity of Flows Numerical Solution of the NS-Equations Example of an Analytical Solution Departure from Continuum Model Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 5

6 Lagrangian and Eulerian Description of Motion Lagrange Systems with small numbers of particles of mass m i Description via set of position vectors {r i } Velocity v i - Time-derivative of {r i } Acceleration a i - Second time-derivative Relation to Newton s second law - Forces acting on each particle i Not suitable for fluid mechanics (n = 1 mol, N A = 6 x mol -1 ) Useful for treating special problems Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 6

7 Lagrangian and Eulerian Description of Motion Euler Backbone of NS-Equations Thermodynamic quantities (temperature, pressure) - Summarizing statistical details on molecular level Continuum Mechanics! Integral momentum of fluid in region t Material or substantial derivative Fundamental definition of acceleration a Particle mechanics: Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 7

8 3.1. Navier-Stokes Equations Lagrangian and Eulerian Description of Motion Derivation of NS-Equation Consequences from the Continuity Equation Interpretation of the Momentum Equation Common Boundary Conditions Simplifications Dynamic Similarity of Flows Numerical Solution of the NS-Equations Example of an Analytical Solution Departure from Continuum Model Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 8

9 Derivation of the NS-Equation Spatial region Vector function Representing change in particle position from t = 0 to given t Path of particle starting at r 0 at t = 0: t -> (r, t) Velocity of fluid observed at fixed position: r -> (r, t): ( / t) (r, t): Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 9

10 Transport Theorem Statement Time derivatives of integrals over time-dependent region Differentiable, scalar function f (x, t ) Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 10

11 Conservation of Mass Spatial integral of density over region Time derivative of mass integrals in transport theorem must vanish Integrands must vanish Equation holds for arbitrary regions Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 11

12 Compressible fluids Equation of Continuity Incompressible Fluids Velocity vector for multi-phase fluid Vector coordinates of each phase or substance i Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 12

13 Types of Forces Volume forces Gravity Coriolis Electro-magnetic Overall volume force density summarized Surface forces Pressure Electrokinetic force Friction Stress tensor - Relation between mechanical shear stress and strain Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 13

14 Momentum Equation Insertion into Newtonian equation Volume forces Surface forces Integration and differentiation of vectors (component-wise) NS momentum equation Transport theorem Product rule Gaussian theorem Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 14

15 Structure of Stress Tensor Stress tensor Non-viscous fluid Neglecting inner friction Diagonal matrix with scalar pressure on diagonal Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 15

16 Euler Equation of Motion Diagonal matrix Decoupling of differential equations (to be read component-wise) Partial differential equations of first order Commonly used in gas dynamics I.e., for compressible, non-viscous fluids Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 16

17 Viscous Fluids Stokes postulates for viscous contribution to stress tensor Viscosity Characteristic constant With strain tensor Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 17

18 Viscous Fluids Non-diagonal elements Transformation to system of partial differential equations Second order Additional constant of integration Additional information for solution required, e.g. - Equation of state - Caloric equation of state Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 18

19 Incompressible Fluids (x,t) = = const. Navier-Stokes equation for incompressible fluids Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 19

20 3.1. Navier-Stokes Equations Lagrangian and Eulerian Description of Motion Derivation of the NS-Equation Consequences from the Continuity Equation Interpretation of the Momentum Equation Common Boundary Conditions Simplifications Dynamic Similarity of Flows Numerical Solution of the NS-Equations Example of an Analytical Solution Departure from Continuum Model Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 20

21 3.1. Navier-Stokes Equations Lagrangian and Eulerian Description of Motion Derivation of the NS-Equation Consequences from the Continuity Equation Interpretation of the Momentum Equation Common Boundary Conditions Simplifications Dynamic Similarity of Flows Numerical Solution of the NS-Equations Example of an Analytical Solution Departure from Continuum Model Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 21

22 Interpretation of the Momentum Equation Left-hand side Material derivative v times mass density Change in momentum (Newton) Right-hand side Forces acting on fluid Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 22

23 Interpretation of the Momentum Equation Pressure gradient Pressure force density Estimate for absolute value Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 23

24 Interpretation of the Momentum Equation Viscosity term Force density f Inner friction of fluid Force density - Viscosity - Flow profile Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 24

25 Interpretation of the Momentum Equation Approximation Tube of diameter d Scales with - Maximum flow velocity v max - Inverse square of diameter d -2 Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 25

26 Interpretation of the Momentum Equation Gravity Force density term On earth - Fluids experience pressure associated with own weight (gravity) = const. Barometric formula - Thermalized compressible fluids Gravitational effects negligible for microfluidic devices! Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 26

27 Interpretation of the Momentum Equation Buoyancy Principle of Archimedes Body plunges in fluid Different pressures p 1 < p 2 on top and bottom surface Pressure - Weight of liquid column p i ~ g h Buoyancy force F buoy = F 2 - F 1 - Propelling body towards surface Body stops when F buoy matched by force of gravity F g h 2 < 0: swimming Force density Relevance to MF Large 10 4 N m -3 for = water Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 27

28 Interpretation of the Momentum Equation Centrifugal forces Present in MF systems like CDs Possible pumping mechanism - Depending on angular frequency = 2 Electrostatic forces Discarded so far (for simplicity) - Product of charge density and electrical field strength Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 28

29 Interpretation of the Momentum Equation Scaling of volume and surface forces Surface forces proportional to A l 2 Volume forces V l 3 Surface-to-volume ratio A / V l -1 Surface-related forces dominate in microworld Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 29

30 3.1. Navier-Stokes Equations Lagrangian and Eulerian Description of Motion Derivation of the NS-Equation Consequences from the Continuity Equation Interpretation of the Momentum Equation Common Boundary Conditions Simplifications Dynamic Similarity of Flows Numerical Solution of the NS-Equations Example of an Analytical Solution Departure from Continuum Model Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 30

31 Common Boundary Conditions Required for complete definition of problem Determine evolution in time Initial field Initial values for entire vector field v Boundary regions Behavior at system boundaries Spatial boundary types Vector field components on boundary surface Derivatives in direction normal to surface Combinations Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 31

32 Common Boundary Conditions Full stiction of first fluid layer Impermeable walls Free- slip Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 32

33 Common Boundary Conditions Inflow boundary conditions Velocity components kept constant over time Outflow conditions Constant gradient of velocity field components in normal direction Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 33

34 3.1. Navier-Stokes Equations Lagrangian and Eulerian Description of Motion Derivation of the NS-Equation Consequences from the Continuity Equation Interpretation of the Momentum Equation Common Boundary Conditions Simplifications Dynamic Similarity of Flows Numerical Solution of the NS-Equations Example of an Analytical Solution Departure from Continuum Model Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 34

35 Simplifications Equations of motion very complex System of differential equations Coupled Second order Analytical solutions Only for special situations - High symmetry - Neglect of coupling Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 35

36 Simplifications Incompressible fluids in MF-systems Neglecting Inertia term (v )v Volume forces g Stationary conditions v / t = 0 Simplified differential NS-equation Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 36

37 Simplifications Assumptions Frictionless ( = 0) and stationary ( v / t = 0) flow Elimination of terms - Non-stationary - Proportional to viscosity Discarding gravity Bernoulli equation Vector analysis Important for dynamic pressure (later on) Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 37

38 3.1. Navier-Stokes Equations Lagrangian and Eulerian Description of Motion Derivation of the NS-Equation Consequences from the Continuity Equation Interpretation of the Momentum Equation Common Boundary Conditions Simplifications Dynamic Similarity of Flows Numerical Solution of the NS-Equations Example of an Analytical Solution Departure from Continuum Model Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 38

39 Dynamic Similarity of Flows Scalability of fluidic experiments Mathematical point of view Transformation to dimensionless variables Substitutions Dimensionless NS equation Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 39

40 Dynamic Similarity of Flows Identical results for similar geometries and C Scaled by scalar constant C Coincidence of - Reynolds number - Froude number MF: discarding gravity Re completely determines dynamics of momentum equation Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 40

41 Dynamic Similarity of Flows Reynolds number Re Measure for ratio - Work spent on acceleration - Energy dissipated by friction Approximating frictional energy Ratio Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 41

42 3.1. Navier-Stokes Equations Lagrangian and Eulerian Description of Motion Derivation of the NS-Equation Consequences from the Continuity Equation Interpretation of the Momentum Equation Common Boundary Conditions Simplifications Dynamic Similarity of Flows Numerical Solution of the NS-Equations Example of an Analytical Solution Departure from Continuum Model Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 42

43 Numerical Solution of the NS-Equations Modeling the system Reduction of complexity Making system as simple as possible, but not any simpler. (A. Einstein) Discretization of continuous space Grid Number of grid points sets computational requirements Adaptive mesh refinement Free boundary problems CFD packages Special lecture Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 43

44 3.1. Navier-Stokes Equations Lagrangian and Eulerian Description of Motion Derivation of the NS-Equation Consequences from the Continuity Equation Interpretation of the Momentum Equation Common Boundary Conditions Simplifications Dynamic Similarity of Flows Numerical Solution of the NS-Equations Example of an Analytical Solution Departure from Continuum Model Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 44

45 Example of an Analytical Solution NS (momentum) equation in cylindrical coordinates Discarding convective term yields Laplace-transformed profile of axial velocity for tube with circular cross section of radius r 0 Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 45

46 Example of an Analytical Solution With Laplace-transforms Arguments of Bessel function J 0 Integration of velocity profile yields Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 46

47 Example of an Analytical Solution Velocity profile for harmonic actuation Definition: Dynamic Reynolds number Reverse transform of Laplace-transformed profile of axial velocity Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 47

48 Example of an Analytical Solution Solution within "Microfluidic Limit" For small arguments r o *, i.e., Re dyn 1 Expansion of pressure flow relation Which is of the form Later on we will see that - Corresponds to hydrodynamic resistance R hd - Corresponds to hydrodynamic inertance L hd Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 48

49 3.1. Navier-Stokes Equations Lagrangian and Eulerian Description of Motion Derivation of the NS-Equation Consequences from the Continuity Equation Interpretation of the Momentum Equation Common Boundary Conditions Simplifications Dynamic Similarity of Flows Numerical Solution of the NS-Equations Example of an Analytical Solution Departure from Continuum Model Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 49

50 Departure from Continuum Model Typically averaging over ensemble of N = 6 x particles 1 l of water corresponds to about 55 mol 1 µl thus contains about 3 x molecules State quantities of thermodynamics - Statistical average Microscopic picture Discrete molecules Large absolute fluctuations N = N 0.5 Small relative fluctuations N / N =1 / N 0.5 < 10-9 Negligible fluctuations in concentration and composition Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 50

51 Knudsen Number: Rarefaction Applicability of continuum model for fluidic system Ratio between Mean free path Characteristic dimension Three regimes Kn < 0.1: continuum approximation Kn > 10: free particle motion Intermediate regime: handled by allowing slip at walls Kn for gases in MF-systems l mfp some 100 nm at STP l > 1 µm Kn < 0.1 even for smallest structures Continuum model applies to practically all MF-systems! Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 51

52 Departure from Continuum Model Molecular structure Many degrees of freedom per molecule For instance, rotation about molecular axis Deviations from conventional theory Surface viscosity Slip-flow of multiphase liquids Molecular effects in thin films Particles and clogging Microfluidics - Jens Ducrée Physics: Navier-Stokes Equation 52

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