basic principles of microfluidics -2

8/13/2019 Basic Principles of Microfluidics -2

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Basic Principles in Microfluidics


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Newtons Second Law for Fluidics

Newtons 2ndLaw (F= ma) :

Time rate of change of momentum of a system equal to net force

acting on system

Sum of forces acting on control volume =

Rate of momentum efflux from control volume

+

Rate of accumulation of momentum in control volume

!F = dPdt


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Navier - Stokes Equation

Navier-Stokesequation applies when:

(1) There are more than one million molecules

in smallest volume that a macroscopic change

takes place.

(2) The flow is not too far from thermodynamic

equilibrium.


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Navier - Stokes Equation

dUdt

= !"P#

+g + $#"2U

!dU

dt= "#P + !g +$#

2U

!iU = 0

!dU

dt= "#P + !g +$#

2U+

$

3#(#iU)

For noncompressible Fluid


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Navier - Stokes in Microfluidics

Terms become dominant based on physics of scale

In microfluidics inertial forces dominate due tosmall dimensions, even though velocity can behigh

dU

dt= !

"P

#+g +

$

#"

2U

dUdt

= ! 1"#P


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VISCOSITY


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Viscosity

Viscosityis a measure of resistance (friction)

of the fluid to the flow

This determines flow rate

Symbols: !and in some books

Units: Poise (gram/sec *Cm)


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Viscosity

Viscosityis a measure of resistance (friction)

of the fluid to the flow.

This determines flow rate.

Units: Poise (gram/sec Cm)


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Basic Properties - Viscosity

Fluids and gases are very different

Fluids become lessviscous as temperature

increases

Gases become moreviscous at temperature

increases


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Viscosity in Gases and Fluids

Gases

Fluids

!"!0 e#($#$

0

)

! = !0

(T0- constant)

(T0 - constant)

T

T0

"

#$%

&'

3

2


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Interfaces and Surface Tension


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Interfaces

Interface: Geometric Surface that delimits 2

fluids

Separation depends on molecular

interactions and Brownian diffusion


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Interfaces

Interface: Geometric Surface that delimits 2

fluids

Simplified view:

Interaction between

molecules

At interface:

different energies


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Interfaces

If U is the total cohesive energy per

molecule and d is a characteristic molecular

dimension, d2is its surface, then the energy

loss (surface tension) is given by:

! =U

2d

2


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Laplaces Law

Minimization of surface energy, create

curvature of fluids on other surfaces (fluids)

Curvature 1/R

Laplaces Law, the change in pressure is

related to the curvature of the surface.

For a sphere: !P = 2 (%/R)

For a cylinder: !P = %/R


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Droplet on a Surface of Two Properties

Simulations


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Coarsening

Two Droplets linked by a precursor film


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Coarsening

Two Droplets linked by a precursor film


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Contact Angle

Surface tension (force per length)

Angle is determined by the balance of

forces at the point of interface

Hydrophilic Hydrophobic


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Contact Angle

Surface tension (force per length)

Angle is determined by the balance of

forces at the point of interface

Oil on Water


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Hydrophilic - Hydrophobic


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Surface Tension

Droplet on a surface

Forces on cross section of drop

Surface tension along periphery

Pressure on section area

Pressure difference outside/inside drop

Force = !PA = "r2!P SurfaceTension=2!r"

! =r

2"P


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Forces - Capillary Effects

A wetting fluid will rise in a capillary tube

Equilibrium: pressure drop across meniscus

Surface tension

Viscosity

h =2!Cos(")

#gr


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Capillary Force


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Capillary Forces


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Small Channel (capillary) - Surface tension draws fluid of density&intothe channel of radius ( r)

'= contact angle

%= surface tension (N/m)

Height of Fluid in a tube in the presence of gravity

Capillary Forces

F =2!r"Cos(#)

h =2!Cos(")

#gr


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Forces - Capillary Effects


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Capillary Forces


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Droplet on Surfaces


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Droplet on Irregular Surfaces

r: roughness

f: ratio of contact angle to the total horizon surface

Youngs critical angle cos(') = (f-1) / (r-f)


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Wettability and Roughness


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Reynolds Number


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Fluids - Types of Flow

Laminar Flow (Steady)

Energy losses are dominated by viscosity effects

Fluid particles move along smooth paths in laminas or layers

Turbulent

Most flow in nature are turbulent!

Fluid particles move in irregular paths,somewhat similar to the molecularmomentum transfer but on a muchlarger scale

Reynolds Number

Reis a measure of turbulence


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Reynolds Number

Reynolds number (Re) = inertial forces / viscous forces

Re = Kinetic energy / energy dissipated by shear

Implies inertia relatively important

VD= Drag velocity, L = characteristic length,!=viscosity, & = density

Re < 2100 : laminar (Stokes) flow regime slow fluid flow, no inertial effects

laminar flow in microfluidics

slow time constants, heavy damping

Re > 4000: unstable laminar flow - turbulent flow regime

Re = !VD

L

"Re =

1

2mV

D

2

1

2!V

DA Re

=

(!AL)VD

"A


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High and Low Reynolds number fluidics

When the Reynolds number is low, viscous

interaction between the wall and the fluid is

strong, and there is no turbulences or vortices


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Is this Flow Turbulent?

Channel Geometry - Use a characteristic length : Dh

Dhis a geometric constant

Re =!

"VD

h


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Is this Flow Turbulent?


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Mixing

Re = 12 and Re = 70

Cycle 1

Cycle 2

Cycle 3


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Microchannels Cross Sections


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Re and Size


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Re

Re - Some examples

Friction factor ~ 1/ Re


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Human Circulatory System


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Knudsen Number

Knudsen number assumes that we can treat the material as acontinuum

Continuum hypothesis holds better for liquids than gases

also,

(mfp= mean free path of molecules, Dh= hydraulic diameter

Kn measures deviation of the state of the material continuum

Kn< 0.01 continuum

0.01 < Kn

< 0.1 slip flow

0.1 < Kn< 10 transition region

10 < Kn molecular flow

Kn =!mfp

DhK

n =

!"

2

(M

Re

)


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The Smallest Length Scale of a Continuum

Low ReHigh Re

Kn =

M

Re

!"

2


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Stokes - Einstein Diffusion

Stokes - Einstein Equation

Diffusion of a particle

(gas, fluid)

Translational Diffusivity

Rotational Diffusivity

!

Dt =

KBT

6!"a

Dr =

KBT

8!"a3


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Diffusion in Fluids

Very short diffusion times

D = diffusion constant

X = diffusion length

)= diffusion rate

Laminar flow limits benefits for fluid mixing.

Highly predictable diffusion has enabled a new class ofmicrofluidic diffusion mixers

x = 2D! ! =1

2

x2

D


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Squeezed film damping

Squeeze a film by pushing on the plates (one is not moving) Viscous drag is opposing the motion of thefluid

Beam displacement

Flow of fluid (Reynolds equation)

Knudsen number, K,is the ratio of the mean free path to gap

Squeeze number: relative importance of viscous to spring forces

!"2U

"t2 + EI

"4U

"u4 = P +

F

L

12!d(Ph)

dt

= "{(1+6k)h3P"P}

P =bdU

dtb =

96!W3

"4h

3 L


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Concluding Remarks


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Summary

Re= turbulent / viscous stresses

Re < 2100 : laminar (Stokes) flow regime,

slow fluid flow, no inertial effects

laminar flow in microfluidics

slow time constants, heavy damping

Re > 4000 : turbulent flow regime


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Fluid Behavioral

What happens when the fluid is on the micro -nano scale?

We discussed scaling- this is a review

Quantities proportional L3

Inertia, buoyancy, etc.


Drag, surface charge, etc.


Surface tension


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Who Rules

!

basic principles of microfluidics -2

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