Chapter 5 Thermofluid Engineering and Microsystems Microfluidic Dynamics Navier-Stokes equation 1. The momentum equation 2. Interpretation of the NSequation 3. Characteristics of flows in microfluidics (Re-number) 4. Examples of laminar flows 5. Summary 1
History of the Navier-Stokes Equation Navier-Stokes equation is the central relationship of fluid dynamics Basic assumptions continuous media continuum mechanics In case of liquids: Assumptions fulfilled in macrofluidics as well as microfluidics (down to 10-100 nm) for liquids How to describe the motion of a fluid? 2
Acceleration over Time 3
Acceleration along a Stream Line The Navier-Stokes (NS) Equation 4
Interpretation of N-S Equation Pressure gradient Body force (= volume force) 5
Different types of body forces (= volume forces) Example: static pressure under gravity 6
Friction 7
Simplifications in Microfluidics Characteristics of flows in microfluidics (Re-number) Behaviour of flow laminar = predictable turbulent = chaotic is dominated by the ratio between inertial effects (kinetic energy) and frictional effects (damping) Friction consumes kinetic energy and converts it into heat Motion is slowed/damped down Turbulences are possible only when friction is small compared to the kinetic energy 8
Reynolds Number Critical Reynolds Number 9
Flow Regimes Governing Equations 10
Newton s Law of Viscosity 11
Re = ρlv η Laminar flow, low Re High degree of laminarity implies that the streamlines are locally parallel. 12
Turbulent flows Velocity vectors unpredictably oscillating in time Couette Flow How does the flow look between two plates when One plate is at rest The other plate is moving at velocity v Situation is called Couette Flow Flow is driven by viscous drag force acting on the fluid Linear velocity gradient 13
Viscosity η internal friction of fluid transfer of momentum from one plane sliding parallel to another plane thickness is infinite 1. interlocked molecule layer 2. velocity gradient 14
Laminar pressure driven flow (PDF) through slit 15
Taylor Dispersion Laminar PDF through tube 16
Laminar PDF through tube: example Hagen-Poiseuille: analogy to electrical circuits 17
18
Hagen-Poiseuille: significant role of cross section 19
Summary At the boundaries usually no slip conditions are assumed Pressure driven flow (PDF) in micro channels typically shows a parabolic profile Zero velocity at the boundary Maximum velocity in the centre of the channel Shear force F ~ ηdv/dx Consider Taylor Dispersion when injecting fluid samples 20
v mean =v max /2 Example: capillary filling 21
Example: capillary filling 2πaσ cosθ 2σ cosθ R 2σ cosθ a 22
Self Priming of Microfluidic Chambers Self Priming of Microfluidic Blind-Channels 23
Bubble-free Priming Sequence Stokes Drag and Relative Velocity 24
Particle Motion in Fluid 1. Stokes force (drag) on particles 2. Buoyancy u g 3. DEP Spherical particles: γ = 6πηa Terminal velocity: F u = u + u particle fluid Time to reach equilibrium g τ m 2ρ pa γ 9η particle 2 a = = <10-6 sec for microparticles 3 = πa Δρmg / γ ~ 2Δρma g / 9η 3 γ 4 2 ε u DEP 0.03 η a 0.12 µm/s for 1 µm latex particles 2 V r 2 3 0.9 µm/s for 1 µm particles, 10 µm from electrode, @5Vrms Hydrodynamic Focussing Based on Sheath Flow 25
Hydrodynamic Focusing Sheath Flow Arrangement 26
Flow Cytometry Hydrodynamic Separation of Particles 27
Pinched-Flow Fractionation (PFF) Separation Principle successful separation of two particle sizes (diameter: 15 µm, 30 µm) 28
Working Principle: Fluid stream containing particles is focused to a wall until all particles are in contact to the wall ( pinched ) Small particles get closer to the wall than large particles All particles follow the stream lines of their center of gravity after the pinched section they leave towards different angles 29
Hydrodynamic Rectification Hydrodynamic Rectification: Performance 30