MICRO FLOWS: AN
INTRODUCTION
Michael Shusser
1
SIZE RANGES OF MACRO,
MICRO, AND NANO DEVICES
2
FLUID FLOW AND HEAT
TRANSFER IN SINGLE-PHASE
FLOW OF A NEWTONIAN
FLUID IN A MICRO-CHANNEL
•
•
•
•
NO MULTIPHASE FLOW
NO POLYMERS OR BIO-FLUIDS
NO COMPLEX GEOMETRIES
NO ELECTRO-KINETIC FLOWS
3
IS EVERYTHING
DIFFERENT OR JUST
SMALLER?
4
IS THE CONTINUUM
APPROXIMATION VALID?
POSSIBLE NON-CONTINUUM
EFFECTS:
• SLIP AT THE BOUNDARY
• STRESS/RATE OF STRAIN
RELATION IS NONLINEAR
• CONTINUUM APPROXIMATION
FAILS
5
FOR THE TIME BEING WE
ASSUME THAT THE
CONTINUUM THEORY IS
VALID
• LIQUIDS
• GASES FOR L > 5 μM
6
MANY OF STUDIES OF BASIC HEAT
AND FLUID FLOW PROBLEMS IN
BASIC GEOMETRIES FOUND LARGE
DEVIATIONS FROM EXPECTED
RESULTS
• FRICTION FACTOR f
• NUSSELT NUMBER Nu
0.5 
f MICRO
 3.5
f MACRO
Nu MICRO
0.2 
 16
Nu MACRO
• CRITICAL REYNOLDS NUMBER ReC
0.13 
Re C,MICRO
Re C,MACRO
 0.43
7
LAMINAR FLOW OF AN
INCOMPRESSIBLE FLUID WITH
CONSTANT PROPERTIES IN A
CIRCULAR PIPE
1   u  dp
r  
r r  r  dx
 dp 

D
dx 

FACTOR
f 
u 2m 2

• FRICTION
• REYNOLDS NUMBER
• POISEUILLE NUMBER
Re D 
u m D

Po  f  Re D
8
64
f 
Re D
Po  64
9
SCALING EFFECTS
• THE EFFECTS THAT CAN BE
NEGLECTED IN MACRO SCALES BUT
ARE IMPORTANT IN MICRO SCALES
ARE CALLED SCALING EFFECTS
• PROVIDED THE CONTINUUM
APPROXIMATION REMAINS VALID, ALL
THE DISCREPANCIES BETWEEN
MICRO AND MACRO FLOWS CAN BE
EXPLAINED AS SCALING EFFECTS
10
• ENTRANCE EFFECTS
• VISCOUS HEATING
• TEMPERATURE- AND PRESSURE
DEPENDENT PROPERTIES
• WALL ROUGHNESS
• COMPRESSIBILITY
• CONJUGATE HEAT TRANSFER
• AXIAL HEAT CONDUCTION
11
ENTRANCE EFFECTS
FOR LAMINAR FLOW IN A CIRCULAR PIPE
X fd ,hyd
D
 0.05Re D
X fd ,therm
D
 0.05 Re D Pr
12
WATER FLOW IN A 2D CHANNEL –
CFD/EXPERIMENT
x
x 
D h Re

13
• ENTRANCE EFFECTS ARE NOT
ALWAYS NEGLIGIBLE IN MICRO
FLOWS
• DEVELOPING FLOW IS STRONGLY
INFLUENCED BY THE INLET
VELOCITY PROFILE
• THERE IS NOT ENOUGH DATA ON
ENTRANCE EFFECTS FOR
VARIOUS CROSS-SECTIONS
14
VISCOUS HEATING
ENERGY EQUATION FOR FLOW IN A PIPE
T
1   T 1   T   Br  du 
 2 
u 
 r   
 
x Re Pr  x r r  r   Re Pr  dr 
2
2
VISCOUS HEATING
(VISCOUS DISSIPATION)
u 2m
Br 
kT
15
BRINKMAN NUMBER
• THE IMPORTANCE OF THE VISCOUS
HEATING TERM IS DETERMINED BY
THE BRINKMAN NUMBER
• FOR EXAMPLE, FOR CONSTANT HEAT
48
1
FLUX
Nu 
11  48Br

0.229  Br
• IN MACRO FLOWS VISCOUS HEATING
IS IMPORTANT ONLY FOR VERY
VISCOUS FLUIDS OR VERY HIGH
VELOCITIES
16
• IN MICRO FLOWS BRINKMAN NUMBER IS
USUALLY VERY SMALL
u 2m
Br 
kT
• WATER: μ = 8.55·10-4 kg(m·s) k = 0.613 W/(m·K)
ΔT = 1 ºC um = 0.1 m/s
Br ≈ 1.4·10-5
• AIR: μ = 1.846·10-5 kg(m·s) k = 0.0263 W/(m·K)
ΔT = 1 ºC um = 1 m/s
Br ≈ 7·10-4
• THE INFLUENCE OF VISCOUS HEATING ON
HEAT TRANSFER IN MICRO FLOWS IS
USUALLY NEGLIGIBLE
17
VISCOUS HEATING CAN BE IMPORTANT
DUE TO VERY STRONG DEPENDENCE OF
LIQUID VISCOSITY ON TEMPERATURE
WATER T = 300 K ν = 8.576·10-7 m2/s
T = 310 K ν = 6.999·10-7 m2/s
TEMPERATURE RISE OF 10 K CAUSES
18% DECREASE IN KINEMATIC VISCOSITY
WHICH RESULTS IN CORRESPONDING
INCREASE OF THE LOCAL Re NUMBER
AFFECTING THE FRICTION FACTOR
18
THERMAL EXPLOSION
THE MOMENTUM AND ENERGY
EQUATIONS FOR FULLY DEVELOPED
FLOW IN A CIRCULAR PIPE ARE
dp 1 d  du 

 r 
dx r dr  dr 
1 d  dT    du 
r      0
r dr  dr  k  dr 
2
FOR EXPONENTIAL DEPENDENCE OF
LIQUID VISCOSITY ON THE
TEMPERATURE
 E   ET  T0  
 E 


 
   0 exp
exp 
   0 exp

 RT 
 RT0  
RT02


19
INTRODUCING NEW VARIABLES
r2
 2
r0
ET  T0 

RT02
THE ENERGY EQUATION REDUCES TO
d 2  1 d




e
0
2
d  d
1  0
  const
d
0
d   0
IT HAS NO SOLUTION FOR   2
NO FULLY DEVELOPED FLOW!
20
ISOPROPANOL FLOW IN A SQUARE
MICRO CHANNEL
• L = 11.4 cm; D = 74.1 μm; (L/D = 1543)
• FOR Re ≈ 300
Tin - Tout =6.2 oC
21
EXAMPLE OF A CFD RESULT
• INLET CONDITIONS
D= 20 μm; T = 300 K
ν = 8.576·10-7 m2/s
Re = 2000
V = 85.76 m/s !
22
• VISCOUS HEATING HAS USUALLY
NO INFLUENCE ON HEAT
TRANSFER IN MICRO FLOWS
• ITS INFLUENCE ON FRICTION
FACTOR CAN BE IMPORTANT DUE
TO VERY STRONG DEPENDENCE
OF LIQUID VISCOSITY ON
TEMPERATURE, ESPECIALLY FOR
LONG CHANNELS
23
VARIABLE PROPERTIES
• DUE TO LARGE GRADIENTS IN MICRO
FLOWS THE DEPENDENCE OF
PROPERTIES ON PRESSURE AND
TEMPERATURE IS IMPORTANT
• LIQUIDS SHOULD BE MODELED AS
INCOMPRESSIBLE WITH
TEMPERATURE-DEPENDENT
VISCOSITY
• SOMETIMES PRESSUREDEPENDENCE OF VISCOSITY SHOULD
ALSO BE TAKEN INTO ACCOUNT
24
LIQUID FLOW AT 30 MPa
25
COMPRESSIBILITY EFFECTS
• THE FRICTION-INDUCED PRESSURE
DROP PER TUBE LENGTH COULD BE
LARGE IN FLOW THROUGH A NARROW
CHANNEL
• COMPRESSIBILITY EFFECTS CAN BE
IMPORTANT IN GAS FLOWS EVEN FOR
LOW MACH NUMBERS
26
PRESSURE AND DENSITY VARIATIONS
ALONG THE TUBE AT DIFFERENT INLET
MACH NUMBERS
27
WALL ROUGHNESS
• ROUGHNESS LEADS TO INCREASING
FRICTION FACTOR AT THE SAME Re
NUMBER AND DECREASING VALUE OF
THE CRITICAL Re NUMBER (EARLIER
TRANSITION FROM LAMINAR TO
TURBULENT FLOW)
• THE INFLUENCE OF THE ROUGHNESS
IS DETERMINED BY ITS GRAIN SIZE ks
AND FRICTION VELOCITY v* (OR WALL
SHEAR STRESS τw)
w
v* 

u
w  
r
r  r0
28
FLOW REGIMES FOR ROUGH PIPES
HYDRAULICALLY
SMOOTH
TRANSITION
COMPLETELY
ROUGH
k S v*
0
5

f  f Re
LAMINAR
TURBULENT
 ks

k S v*
5
 70 f  f  , Re 

 Re 
TURBULENT
 ks 
f  f 
 Re 
TURBULENT
k S v*
 70

29
• FOR LOW Re (D < 100 μm) SOME
EXPERIMENTS OBSERVED
DEVIATIONS FROM THE CLASSICAL
THEORY INCLUDING THE INFLUENCE
OF ROUGHNESS IN LAMINAR FLOW
• ONE POSSIBLE REASON FOR THE
DISCREPANCY IS NON-UNIFORMITY
OF THE ROUGHNESS
• THERE IS NOT ENOUGH DATA ON
INFLUENCE OF ROUGHNESS ON HEAT
TRANSFER
30
CONJUGATE HEAT TRANSFER
• IN MICRO FLOWS THE RELATIVE
THICKNESS OF THE CHANNEL WALL
s/Dh IS USUALLY MUCH LARGER THAN
IN MACRO FLOWS
• THEREFORE CONVECTIVE HEAT
TRANSFER IN THE FLUID AND HEAT
CONDUCTION IN THE WALL MUST BE
ACCOUNTED FOR SIMULTANEOUSLY
• THIS CONJUGATED HEAT TRANSFER
IS USUALLY NEGLIGIBLE FOR MACRO
FLOWS
31
EXPERIMENT
• LAMINAR FLOW Re ≈ 50 L/D ≈ 160
• CONSTANT WALL HEAT FLUX
32
THEORETICAL SOLUTION
• WALL TEMPERATURE
dTw
qw

 const
 cp
dx
m
• BULK TEMPERATURE
dTm q w

 const
 cp
dx m
• NUSSELT NUMBER
q w D 48
Nu 
  4.36
Tw  Tm k 11
33
EXPERIMENT - RESULTS
34
CFD – CONJUGATE HEAT
TRANSFER INCLUDED
35
AXIAL CONDUCTION NUMBER
• THE IMPORTANCE OF THE CONJUGATE
HEAT TRANSFER IS GIVEN BY THE AXIAL
CONDUCTION NUMBER M
 cond //
M
 conv
es
ks
L

cef V
36
• THE NUMBER M IS USUALLY VERY
LOW FOR MACRO CHANNELS (HIGH V,
SMALL eS/ef, LARGE L) BUT CAN BE
LARGE FOR MICRO CHANNELS (LOW
V, eS/ef IS NOT SMALL, SMALL L)
• FOR LARGE M THE WALL HEAT FLUX
BECOMES STRONGLY NON-UNIFORM:
MOST OF THE HEAT IS TRANSFERRED
TO THE FLUID NEAR THE ENTRANCE
TO THE CHANNEL
37
AXIAL HEAT CONDUCTION
ENERGY EQUATION FOR FLOW IN A PIPE
T 1   T 1   T  
 2 
u

r
 
x Pe  x
r r  r  
2
AXIAL HEAT CONDUCTION
• AXIAL HEAT CONDUCTION CAN
USUALLY BE NEGLECTED UNLESS
PECLET NUMBER IS VERY LOW
Pe  Re Pr  50
38
• OILS: Pr >>1 LIQUIDS: Pr ~ 5
GASES: Pr ~ 0.7 LIQUID METALS: Pr << 1
• IN MACRO FLOWS THE AXIAL HEAT
CONDUCTION IS NEGLIGIBLE EXCEPT
LIQUID METAL FLOWS
• IN MICRO FLOWS THE AXIAL HEAT
CONDUCTION SOMETIMES MUST BE
TAKEN INTO ACCOUNT
39
TURBULENCE IN MICRO FLOWS
• MICRO FLOWS ARE USUALLY LAMINAR
(Re < 2000)
• MOST EXAMPLES OF TURBULENT
FLOW ARE USUALLY FOR RELATIVELY
LARGE DIAMETERS (D > 300 μm)
• FOR LARGE PRESSURE DIFFERENCE,
GAS FLOWS CAN BE TURBULENT
EVEN FOR SMALL DIAMETERS
40
CFD: PIPE FLOW
• D = 50 μm; PIN ≈ 20 atm; POUT ≈ 2 atm
• VISCOUS COMPRESSIBLE TURBULENT FLOW
• INLET: VX ≈ 125 m/s Re ≈ 25,000
• DO STANDARD TURBULENCE MODELS (LIKE
41
k-ε) WORK IN THIS CASE?
NON-CONTINUUM EFFECTS GASES
• THE FLOW IS RAREFIED FOR GASES
AND THE WALLS “MOVE”
• TO A CERTAIN DEGREE THE
SITUATION IS SIMILAR TO LOWPRESSURE HIGH-ALTITUDE
AERONAUTICAL FLOWS
• HOWEVER, REYNOLDS AND MACH
NUMBERS ARE MUCH LOWER
42
MOLECULAR MAGNITUDES
• NUMBER DENSITY OF MOLECULES n
p
n
k BT
n  2.691025m3
• MEAN MOLECULAR SPACING δ
n
1/ 3
  3.3  109 m
• MOLECULAR DIAMETER d
10
d

3
.
7

10
m
DILUTE GAS: δ/d > 7
AIR:
THE DATA FOR p = 1 atm; T = 0 ºC
43
MEAN FREE PATH
• THE DISTANCE TRAVELED BY THE
MOLECULES BETWEEN COLLISIONS
IS KNOWN AS MEAN FREE PATH λ

AT p = 1 atm; T = 25 ºC
GAS
AIR
λ, nm 61.1
N2
CO2
60.4
40.2
O2
1
d 2 n 2
He
Ar
65.0 176.5 64.4
44
KNUDSEN NUMBER
• THE KEY DIMENSIONLESS PARAMETER
IS THE KNUDSEN NUMBER Kn

Kn  
L
 M
2 Re
Kn < 0.01
CONTINUUM
0.01 < Kn <0.1
SLIP FLOW
0.1 < Kn < 10
TRANSITIONAL FLOW
Kn > 10
FREE-MOLECULAR FLOW
45
LIMITS OF APPROXIMATIONS
46
NON-CONTINUUM EFFECTS LIQUIDS
• FOR SUFFICIENTLY HIGH STRAIN RATE
THE STRESS/RATE
OF STRAIN AND

HEAT FLUX/TEMPERATURE GRADIENTS
RELATIONS BECOME NONLINEAR
u 2
 

y 
HERE τ IS THE MOLECULAR TIME-SCALE
• THE CRITICAL VALUE IS VERY HIGH FOR
ORDINARY LIQUIDS BUT NOT SO FOR
COMPLEX FLUIDS
47
FUTURE DIRECTIONS OF
RESEARCH
48
CONCLUSIONS
• PROVIDED THE CONTINUUM
APPROXIMATION REMAINS VALID, ALL THE
DISCREPANCIES BETWEEN MICRO AND
MACRO FLOWS CAN BE EXPLAINED AS
SCALING EFFECTS
• THE MAIN SCALING EFFECTS ARE VARIABLE
PROPERTIES, COMPRESSIBILITY,
CONJUGATE HEAT TRANSFER
• SOME INFLUENCE OF ENTRY LENGTH,
VISCOUS HEATING, AXIAL HEAT
CONDUCTION AND ROUGHNESS IS ALSO
49
POSSIBLE
REFERENCES
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
Bayraktar & Pidugu, Int J Heat Mass Trans, 2006
Cui et al, Phys Fluids, 2004
Gad-el-Hak, Int J Heat Mass Trans, 2003
Gamrat et al, Int J Heat Mass Trans, 2005
Guo & Li, Int J Heat Mass Trans, 2003
Herwig & Hausner, Int J Heat Mass Trans, 2003
Herwig, ZAMM, 2002
Hetsroni et al, Int J Heat Mass Trans, 2005, p. 1982
Hetsroni et al, Int J Heat Mass Trans, 2005, p. 5580
Judy et al, Int J Heat Mass Trans, 2002
Karniadakis & Beskok, Micro Flows, 2002
Koo & Kleinstreuer, Int J Heat Mass Trans, 2004
Maranzana et al, Int J Heat Mass Trans, 2004
50
THANKS!
51