3.2. Laminar and Turbulent Flow - DCU

Contents

1. Introduction

9. Liquid Handling

2. Fluids

10.Microarrays

3. Physics of Microfluidic Systems

11.Microreactors 12.Analytical Chips

4. Microfabrication Technologies Pr isbexa ispel: Ausar beitun gspha Au tsarbei ungde rStand ard-Ze el

13.Particle-Laden Fluids

5. Flow Control

a. Measurement Techniques

6. Micropumps 7. Sensors

b. Fundamentals of Biotechnology

8. Ink-Jet Technology

c. High-Throughput Screening

Microfluidics - Jens Ducrée

Physics: Laminar and Turbulent Flow

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3. Physics of Microfluidic Systems

1. Navier-Stokes Equations 2. Laminar and Turbulent Flow 3. Fluid Dynamics 4. Fluid Networks 5. Energy Transport

Pr isbexa ispel: Ausar beitun gspha Au tsarbei ungde rStand ard-Ze el

6. Interfacial Surface Tension 7. Electrokinetics



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3.2. Laminar and Turbulent Flow 1. Critical Reynolds Number 2. Shear-Driven Laminar Flow 3. Couette Flow 4. Laminar PDF through a Tube 5. Laminar PDF through a Gap Pr isbexa ispel: Ausar beitun gspha Au tsarbei ungde rStand ard-Ze el

6. Irrotational Flow 7. Centrifugal-Force Driven Flow 8. Effects in Laminar Flows 9. Turbulent Flows



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3.2.1. Critical Reynolds Number  Three types of flow conditions  Laminar  Low flow velocities  Smooth sliding of adjacent layers  Field of velocity vectors constant in time

 Turbulent     

Curling of field lines Mixing between adjacent layers „Unpredictable" development of field of velocity vectors Flow patterns increasingly turbulent towards high velocities Sometimes laminar flow preserved up to higher velocities Pr isbexa ispel: Ausar beitun gspha Au tsarbei ungde rStand ard-Ze el

 Periodic flow  3rd flow regime  Surface waves  Acoustic waves

 All three flow types solutions of NS-equation Microfluidics - Jens Ducrée


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3.2.1. Perturbation Analysis  Transition from laminar to turbulent flow regime  Mathematical perturbation analysis  Prediction whether velocity distribution belongs to distinct flow regime

 Ansatz  Known solution of NS-equation (guessed or measured)  Superimposing small perturbation Pr isbexa ispel: Ausar beitun gspha Au tsarbei ungde rStand ard-Ze el

 Product of - Amplitude - Oscillatory factor - Exponential term



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3.2.1. Perturbation Analysis

 Properties  Locally varying amplitude A   and  constant for given problem Pr isbexa ispel: Ausar beitun gspha Au tsarbei ungde rStand ard-Ze el

 Insertion of perturbed solution in NS as initial velocity field  Result: First order equations of  and   Sign of  indicates decay of perturbation into v0



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3.2.1. Critical Reynolds Number  Condition  = 0 defines critical Reynolds number Re*  Re < Re*  Perturbations damped in time

 Re > Re*  Exponential growth of perturbations in time  Perturbation theory not valid  „Unpredictable“ behavior of velocity field Pr isbexa ispel: Ausar beitun gspha Au tsarbei ungde rStand ard-Ze el

 Transition point Re = Re*  Flow oscillates between two flow regimes

 As Re increases further, turbulent character of flow increases



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3.2.1. Critical Reynolds Number  Re* ranges between 1 and 100,000  Re* depends on  Material properties (density, viscosity)  Boundary conditions  Critical velocity


 Microdevice  l = 100 µm  v* = 25 m s-1  Hardly reached in microdevices

 Re* geometries  Sphere: 2320  Flow parallel to plate: Re* = 500,000 Microfluidics - Jens Ducrée


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3.2. Laminar Flow




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3.2.1. Critical Reynolds Number  Transition point also depends on  Initial velocity field  Experimental environment

 Domain Re < Re*  No survival of initial turbulences

 Domain Re > Re*  Laminar flow still possible under certain conditions  Turbulences hampered by Pr isbexa ispel: Ausar beitun gspha Au tsarbei ungde rStand ard-Ze el

- Smooth walls - Smooth endings at orifices

 Laminar conditions up to Re = 100,000  Re > 100,000 - Thermal motion of molecules sufficient to trigger transition to turbulence



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3.2.1. Shift of Re* in MF Systems  In MF-systems    

Channel diameter 100 µm Flow velocity v = 10 mm s-1 Flow rate Av = 6 µl min-1 Re ~ 1 Re*  Turbulent regime

 Laminar regime restricted to Tiny layer  < d near moving body  "Prandtl layer"  Diffusion-limited mass and heat transfer  Decisive impact on mass and heat exchange in macrosystems Pr isbexa ispel: Ausar beitun gspha Au tsarbei ungde rStand ard-Ze el

 Estimate for thickness  by energy analysis  Viscous work

 Spent when body traveling at v0 covers distance of its own length l Microfluidics - Jens Ducrée


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3.2.2. Prandtl Boundary Layer  Setting adjacent fluid into motion requires kinetic energy

 Assuming linear flow profile within Prandtl layer  Setting equal kinetic energy and viscous work yields Pr isbexa ispel: Ausar beitun gspha Au tsarbei ungde rStand ard-Ze el

   

Typical MF-values: l = 1cm, d = 100 µm and Re = 1   1 cm >> d Fully developed Prandtl layer therefore not found in MF systems Attention - Re increases with speed of flow



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3.2. Laminar and Turbulent Flow 1. Critical Reynolds Number 2. Shear-Driven Laminar Flow 3. Couette Flow 4. Laminar PDF through Tube 5. Laminar PDF through Gap Pr isbexa ispel: Ausar beitun gspha Au tsarbei ungde rStand ard-Ze el




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3.2.3. Couette Flow  Azimuthal symmetry     

Purely azimuthal fluid motion Cylindrical coordinates (r, , z) Velocity field v(r) Pressure distribution p Symmetry reduces NS-equations and continuity equation to Pr isbexa ispel: Ausar beitun gspha Au tsarbei ungde rStand ard-Ze el



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3.2.3. Couette Flow  Azimuthal symmetry  Purely azimuthal fluid motion  Cylindrical coordinates (r, , z)  Velocity field v(r)

 Ansatz


 Solution

small ~r



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3.2.4. Laminar PDF through Tube

 Pressure-driven flow  Important phenomenon in nature  E.g., transport of nutrients in plants and animals by heart Pr isbexa ispel: Ausar beitun gspha Au tsarbei ungde rStand ard-Ze el

 Law of Hagen-Poisseuille  Pressure drop  Throughput

 Symmetry  Parabolic flow profile  Cylindrical symmetry Microfluidics - Jens Ducrée


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 Pressure forces Pr isbexa ispel: Ausar beitun gspha Au tsarbei ungde rStand ard-Ze el

 Viscous forces

 Relationship for stationary flow (dvz/dt = 0) Fp = F Microfluidics - Jens Ducrée


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 Integration  Extension of auxiliary cylinder of radius r to tube radius r0 Pr isbexa ispel: Ausar beitun gspha Au tsarbei ungde rStand ard-Ze el

 Flow velocity profile



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 Maximum velocity (in center at r = 0) Pr isbexa ispel: Ausar beitun gspha Au tsarbei ungde rStand ard-Ze el

 Average velocity



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3.2.4. Flow Rate  Volumetric flow IV determined by integration of vz(r) dA over r0


 Law of Hagen-Poiseuille  IV scales with r 4

 Average velocity  Alternative expression for Reynolds number Microfluidics - Jens Ducrée


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3.2.4. Throughput

IV ~ r 4 = A2

Hagen-Poiseuille

A0/4 A0

I0,V ~ A02


IV ~ 4 (A0 / 4)2 = ¼ I0V IV ~ ´N (A0 / N)2 = (1/N) I0V



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3.2.4. Hydraulic Diameter  Based on law of Hagen-Poiseuille for cylindrical geometry  PDF through duct with non-circular cross-section  Equivalent hydraulic diameter




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3.2.4. Hydraulic Diameter  Round tube

 Square tube  Edge length Pr isbexa ispel: Ausar beitun gspha Au tsarbei ungde rStand ard-Ze el

 Annular geometry



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3.2.5. Laminar PDF through Gap


 Pressure-driven flow    

No (external) shear or volume forces Parallel plates Laminar regime Pressure gradient antiparallel to direction of flow

 No-slip conditions



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3.2.5. Laminar PDF through Gap


 Rectangular element     

Width 2x Length l Depth b Cross section Ax = b l Fore-part Az = 2 x b

 Total velocity gradient across element 2 dv / dx |+/-x Microfluidics - Jens Ducrée


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3.2.5. Laminar PDF through Gap  Differential relationship

 Parabolic flow profile


 Peak velocity

 Overall volume flow rate IV per channel width y



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3.2.6. Irrotational Flows  Vorticity

 Vanishes in irrotational flows

 Vector identity Pr isbexa ispel: Ausar beitun gspha Au tsarbei ungde rStand ard-Ze el

 Vanishing divergence of vorticity

 For vanishing vorticity, i.e. irrotational flow, v can be written



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3.2.6. Potential Flow Theory  Basic building blocks  Set of special flow schemes  Analogous to multipole concept in electrodynamics

 Mathematical point of view  Special instances of Green’s function Pr isbexa ispel: Ausar beitun gspha Au tsarbei ungde rStand ard-Ze el



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3.2.6. Velocity Potentials (2-dim.)  Simplification  2-dim. velocity field v = (vx, vy)

 Velocity potential   Scalar

 Stream function   Scalar Pr isbexa ispel: Ausar beitun gspha Au tsarbei ungde rStand ard-Ze el

monopole dipole



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3.2.6. Hele-Shaw Table  Visualization of basic 2-dim. flows  Uniform stream over floor to drain  Bottles  Raised or lowered to adjust gravitational pressure  Connected to through holes

 2-dim. flow (top view)  Sources and drains (monopoles)


 Holes

 Doublets (dipoles)  Source and sink very close to each other  Bottles spaced by same distance above and below floor

 Sometimes transparent cover to ensure uniform depth Microfluidics - Jens Ducrée


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3.2.6. Bernoulli Equation  Continuity equation  Irrotational flow  Navier-Stokes  Rewritten  Using vector analysis Pr isbexa ispel: Ausar beitun gspha Au tsarbei ungde rStand ard-Ze el

 General form of Bernoulli

vanishing vorticity

 Integration in space

 Bernoulli  Stationary conditions  Integration in space



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3.2.7. Centrifugal-Force Driven Flow  Flow Profile     

Laminar flow only driven by centrifugal forces Fluidic duct with radius r0, angular frequency  Stationary conditions, incompressible fluids No-slip boundary conditions Neglecting inertia and pressure effects

 Solution: z-dependent flow profile Pr isbexa ispel: Ausar beitun gspha Au tsarbei ungde rStand ard-Ze el

 At center r

z

 Velocity profile typically more flat than in PDF Microfluidics - Jens Ducrée


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3.2.7. Comparison to PDF CD

PDF

 Parabolic velocity profiles  Relation between steepness of velocity profiles Pr isbexa ispel: Ausar beitun gspha Au tsarbei ungde rStand ard-Ze el

 Example  -

z = l = 1 cm  = 1000 kg m-3 (water) p = 1000 hPa  = 500 rpm (single speed CD player)


v^  2.7  103 PDF profile much steeper


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3.2.8. Taylor Dispersion  Axial dispersion of solute in laminar flow  Dispersion of drugs in blood flow

 Situation  Steady state flow  Round tube

 Hypothetical absence of diffusion  Solute follows flow profile

 Molecular diffusion


 Counteracts dispersion  Axial spreading at Deff t  Radial diffusion exchanges solute molecules between layers

 MF example  v = 1 mm s-1 , r0 = 100 µm , D = 3 x 10-9 m2 s-1  Second term prevails over unity  Effective constant for axial diffusion ~ D(1 + c D –2)



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3.2.8. Hydrodynamic Focusing  Situation  Microscopic tip at end of capillary  Sucking in liquid from larger vessel  Laminar regime Pr isbexa ispel: Ausar beitun gspha Au tsarbei ungde rStand ard-Ze el

 Full solid angle projected onto tiny orifice cross section



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3.2.8. Hydrodynamic Focusing  Example:  Ink dispenser near orifice of capillary  Vertical position within capillary adjusts to transversal shift of dispenser




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3.2.8. Reversed Experiment  Fluid plug expelled from orifice of capillary into larger tank  Small velocity  Laminar

 High velocity  Turbulent




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3.2.8. Application to Cytometry and Mixing




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3.2.9. Turbulent Flows


 Turbulent flow in tube for Re > Recrit  Turbulent profile    

Velocity vectors unpredictably oscillating in time Time-averaged profile Much flatter profile than laminar flow Tendency for flattening grows with Re



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3.2.9. Turbulent Flows  Throughput according to Blasius (1883-1970)

Laminar

 Approximations well above 3Re*

 p     l 

0.57

r0

2.71


 Mean velocity

 p    l  



0.57

r0

0.71

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3.2.9. Scaling of Mean Velocity Turbulent

Laminar Pressure gradient Radius Density Pr isbexa ispel: Ausar beitun gspha Au tsarbei ungde rStand ard-Ze el

Viscosity

 Same pressure gradient applied to tube  Smaller turbulent flow velocity

 Turbulent velocity varies with density   Flow energy dissipated by turbulent mixing

 Laminar flow  Viscous forces between smoothly sliding layers

 Turbulent regime  Enhanced flow resistance Microfluidics - Jens Ducrée


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3.2.9. Entrance Effects

 Laminar


 Turbulent  Microfluidic systems  Re ~1 and r0 = 100 µm


zdevel = 10 µm (laminar)


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3.2.9. Friction Losses  Surface roughness  Local eddy formation  Darcy-Weissbach relation  Pressure loss  Flow velocity  Friction factor Pr isbexa ispel: Ausar beitun gspha Au tsarbei ungde rStand ard-Ze el

f = const. for smooth tube and laminar conditions



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3.2.9. Roughness-Viscosity Model  Surface roughness induces turbulence in boundary layer  Surface roughness height   Roughness viscosity  Adding to bulk viscosity 

 Surface roughness Reynolds number Pr isbexa ispel: Ausar beitun gspha Au tsarbei ungde rStand ard-Ze el

 Empirical factor



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