Microfluidic Dynamics.pdf - Utk

Chapter 5 Thermofluid Engineering and Microsystems

Microfluidic Dynamics Navier-Stokes equation 1. 2. 3. 4. 5.

The momentum equation Interpretation of the NSequation Characteristics of flows in microfluidics (Re-number) Examples of laminar flows 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. 2.

interlocked molecule layer 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

vmean=vmax/2

Example: capillary filling

21

Example: capillary filling 2πaσ cos θ

2σ cos θ R

2σ cos θ a

22

Self Priming of Microfluidic Chambers

Self Priming of Microfluidic Blind-Channels

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Bubble-free Priming Sequence

Stokes Drag and Relative Velocity

24

Particle Motion in Fluid Terminal velocity:

Fparticle 1. Stokes force (drag) u particle = u fluid + γ on particles 2. Buoyancy ug Time to reach equilibrium 3. DEP 2ρ a 2 τa = mγ =

Spherical particles:

p

9η