Contents. Microfluidics  Jens Ducrée Physics: NavierStokes 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. InkJet Technology 9. Liquid Handling 10.Microarrays 11.Microreactors 12.Analytical Chips 13.ParticleLaden Fluids a. Measurement Techniques b. Fundamentals of Biotechnology c. HighThroughput Screening Microfluidics  Jens Ducrée Physics: NavierStokes 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 surfacetovolume ratio Shifted significance of physical effects in MF MFeffects Capillarity Electrokinetics Strictly laminar flow conditions Pros and cons of MFeffects New design principles Hazard for many applications Microfluidics  Jens Ducrée Physics: NavierStokes Equation 2
3 3. Physics of Microfluidic Systems 3.1. NavierStokes 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: NavierStokes Equation 3
4 3.1. NavierStokes Equation Central relationship of fluid dynamics Solutions for selected situations Assumptions Continuous media Viscous constant Microfluidics  Jens Ducrée Physics: NavierStokes Equation 4
5 3.1. NavierStokes Equations Lagrangian and Eulerian Description of Motion Derivation of the NSEquation Consequences from the Continuity Equation Interpretation of the Momentum Equation Common Boundary Conditions Simplifications Dynamic Similarity of Flows Numerical Solution of the NSEquations Example of an Analytical Solution Departure from Continuum Model Microfluidics  Jens Ducrée Physics: NavierStokes 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  Timederivative of {r i } Acceleration a i  Second timederivative 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: NavierStokes Equation 6
7 Lagrangian and Eulerian Description of Motion Euler Backbone of NSEquations 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: NavierStokes Equation 7
8 3.1. NavierStokes Equations Lagrangian and Eulerian Description of Motion Derivation of NSEquation Consequences from the Continuity Equation Interpretation of the Momentum Equation Common Boundary Conditions Simplifications Dynamic Similarity of Flows Numerical Solution of the NSEquations Example of an Analytical Solution Departure from Continuum Model Microfluidics  Jens Ducrée Physics: NavierStokes Equation 8
9 Derivation of the NSEquation 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: NavierStokes Equation 9
10 Transport Theorem Statement Time derivatives of integrals over timedependent region Differentiable, scalar function f (x, t ) Microfluidics  Jens Ducrée Physics: NavierStokes 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: NavierStokes Equation 11
12 Compressible fluids Equation of Continuity Incompressible Fluids Velocity vector for multiphase fluid Vector coordinates of each phase or substance i Microfluidics  Jens Ducrée Physics: NavierStokes Equation 12
13 Types of Forces Volume forces Gravity Coriolis Electromagnetic Overall volume force density summarized Surface forces Pressure Electrokinetic force Friction Stress tensor  Relation between mechanical shear stress and strain Microfluidics  Jens Ducrée Physics: NavierStokes Equation 13
14 Momentum Equation Insertion into Newtonian equation Volume forces Surface forces Integration and differentiation of vectors (componentwise) NS momentum equation Transport theorem Product rule Gaussian theorem Microfluidics  Jens Ducrée Physics: NavierStokes Equation 14
15 Structure of Stress Tensor Stress tensor Nonviscous fluid Neglecting inner friction Diagonal matrix with scalar pressure on diagonal Microfluidics  Jens Ducrée Physics: NavierStokes Equation 15
16 Euler Equation of Motion Diagonal matrix Decoupling of differential equations (to be read componentwise) Partial differential equations of first order Commonly used in gas dynamics I.e., for compressible, nonviscous fluids Microfluidics  Jens Ducrée Physics: NavierStokes Equation 16
17 Viscous Fluids Stokes postulates for viscous contribution to stress tensor Viscosity Characteristic constant With strain tensor Microfluidics  Jens Ducrée Physics: NavierStokes Equation 17
18 Viscous Fluids Nondiagonal 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: NavierStokes Equation 18
19 Incompressible Fluids (x,t) = = const. NavierStokes equation for incompressible fluids Microfluidics  Jens Ducrée Physics: NavierStokes Equation 19
20 3.1. NavierStokes Equations Lagrangian and Eulerian Description of Motion Derivation of the NSEquation Consequences from the Continuity Equation Interpretation of the Momentum Equation Common Boundary Conditions Simplifications Dynamic Similarity of Flows Numerical Solution of the NSEquations Example of an Analytical Solution Departure from Continuum Model Microfluidics  Jens Ducrée Physics: NavierStokes Equation 20
21 3.1. NavierStokes Equations Lagrangian and Eulerian Description of Motion Derivation of the NSEquation Consequences from the Continuity Equation Interpretation of the Momentum Equation Common Boundary Conditions Simplifications Dynamic Similarity of Flows Numerical Solution of the NSEquations Example of an Analytical Solution Departure from Continuum Model Microfluidics  Jens Ducrée Physics: NavierStokes Equation 21
22 Interpretation of the Momentum Equation Lefthand side Material derivative v times mass density Change in momentum (Newton) Righthand side Forces acting on fluid Microfluidics  Jens Ducrée Physics: NavierStokes Equation 22
23 Interpretation of the Momentum Equation Pressure gradient Pressure force density Estimate for absolute value Microfluidics  Jens Ducrée Physics: NavierStokes 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: NavierStokes 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: NavierStokes 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: NavierStokes 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: NavierStokes 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: NavierStokes 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 Surfacetovolume ratio A / V l 1 Surfacerelated forces dominate in microworld Microfluidics  Jens Ducrée Physics: NavierStokes Equation 29
30 3.1. NavierStokes Equations Lagrangian and Eulerian Description of Motion Derivation of the NSEquation Consequences from the Continuity Equation Interpretation of the Momentum Equation Common Boundary Conditions Simplifications Dynamic Similarity of Flows Numerical Solution of the NSEquations Example of an Analytical Solution Departure from Continuum Model Microfluidics  Jens Ducrée Physics: NavierStokes 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: NavierStokes Equation 31
32 Common Boundary Conditions Full stiction of first fluid layer Impermeable walls Free slip Microfluidics  Jens Ducrée Physics: NavierStokes 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: NavierStokes Equation 33
34 3.1. NavierStokes Equations Lagrangian and Eulerian Description of Motion Derivation of the NSEquation Consequences from the Continuity Equation Interpretation of the Momentum Equation Common Boundary Conditions Simplifications Dynamic Similarity of Flows Numerical Solution of the NSEquations Example of an Analytical Solution Departure from Continuum Model Microfluidics  Jens Ducrée Physics: NavierStokes 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: NavierStokes Equation 35
36 Simplifications Incompressible fluids in MFsystems Neglecting Inertia term (v )v Volume forces g Stationary conditions v / t = 0 Simplified differential NSequation Microfluidics  Jens Ducrée Physics: NavierStokes Equation 36
37 Simplifications Assumptions Frictionless ( = 0) and stationary ( v / t = 0) flow Elimination of terms  Nonstationary  Proportional to viscosity Discarding gravity Bernoulli equation Vector analysis Important for dynamic pressure (later on) Microfluidics  Jens Ducrée Physics: NavierStokes Equation 37
38 3.1. NavierStokes Equations Lagrangian and Eulerian Description of Motion Derivation of the NSEquation Consequences from the Continuity Equation Interpretation of the Momentum Equation Common Boundary Conditions Simplifications Dynamic Similarity of Flows Numerical Solution of the NSEquations Example of an Analytical Solution Departure from Continuum Model Microfluidics  Jens Ducrée Physics: NavierStokes 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: NavierStokes 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: NavierStokes 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: NavierStokes Equation 41
42 3.1. NavierStokes Equations Lagrangian and Eulerian Description of Motion Derivation of the NSEquation Consequences from the Continuity Equation Interpretation of the Momentum Equation Common Boundary Conditions Simplifications Dynamic Similarity of Flows Numerical Solution of the NSEquations Example of an Analytical Solution Departure from Continuum Model Microfluidics  Jens Ducrée Physics: NavierStokes Equation 42
43 Numerical Solution of the NSEquations 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: NavierStokes Equation 43
44 3.1. NavierStokes Equations Lagrangian and Eulerian Description of Motion Derivation of the NSEquation Consequences from the Continuity Equation Interpretation of the Momentum Equation Common Boundary Conditions Simplifications Dynamic Similarity of Flows Numerical Solution of the NSEquations Example of an Analytical Solution Departure from Continuum Model Microfluidics  Jens Ducrée Physics: NavierStokes Equation 44
45 Example of an Analytical Solution NS (momentum) equation in cylindrical coordinates Discarding convective term yields Laplacetransformed profile of axial velocity for tube with circular cross section of radius r 0 Microfluidics  Jens Ducrée Physics: NavierStokes Equation 45
46 Example of an Analytical Solution With Laplacetransforms Arguments of Bessel function J 0 Integration of velocity profile yields Microfluidics  Jens Ducrée Physics: NavierStokes Equation 46
47 Example of an Analytical Solution Velocity profile for harmonic actuation Definition: Dynamic Reynolds number Reverse transform of Laplacetransformed profile of axial velocity Microfluidics  Jens Ducrée Physics: NavierStokes 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: NavierStokes Equation 48
49 3.1. NavierStokes Equations Lagrangian and Eulerian Description of Motion Derivation of the NSEquation Consequences from the Continuity Equation Interpretation of the Momentum Equation Common Boundary Conditions Simplifications Dynamic Similarity of Flows Numerical Solution of the NSEquations Example of an Analytical Solution Departure from Continuum Model Microfluidics  Jens Ducrée Physics: NavierStokes 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 < 109 Negligible fluctuations in concentration and composition Microfluidics  Jens Ducrée Physics: NavierStokes 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 MFsystems l mfp some 100 nm at STP l > 1 µm Kn < 0.1 even for smallest structures Continuum model applies to practically all MFsystems! Microfluidics  Jens Ducrée Physics: NavierStokes 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 Slipflow of multiphase liquids Molecular effects in thin films Particles and clogging Microfluidics  Jens Ducrée Physics: NavierStokes Equation 52
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