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.
Quantities proportional L2
Drag, surface charge, etc.
Quantities proportional L1
Surface tension
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Who Rules
!