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Microfluidics Based Microsystems


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Microfluidics Based Microsystems
Fundamentals and Applications
edited by

S. Kakaç
TOBB University of Economics and Technology
Sögütözü, Ankara, Turkey

B. Kosoy
State Academy of Refrigeration
Odessa, Ukraine

D. Li
University of Waterloo
Waterloo, Ontario, Canada
and

A. Pramuanjaroenkij
Kasetsart University
Chalermphrakiat Sakonnakhon Province Campus
Sakonnakhon, Thailand

Published in cooperation with NATO Public Diplomacy Division


Proceedings of the NATO Advanced Study Institute on
Microfluidics Based Microsystems: Fundamentals
and Applications
Çeşme-Izmir, Turkey
August 23–September 4, 2009

Library of Congress Control Number: 2010930508

ISBN 978-90-481-9031-7 (PB)
ISBN 978-90-481-9028-7 (HB)
ISBN 978-90-481-9029-4 (e-book)

Published by Springer,
P.O. Box 17, 3300 AA Dordrecht, The Netherlands.
www.springer.com

Printed on acid-free paper

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© Springer Science + Business Media B.V. 2010
No part of this work may be reproduced, stored in a retrieval system, or transmitted
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CONTENTS
Preface

ix

Convective Heat Transfer Correlations in Some Common
Micro-Geometries
O. Aydin and M. Avci

1

Convective Heat Transfer in Microscale Slip Flow
A. Guvenc Yazicioglu and S. Kakaç
Direct and Inverse Problems Solutions in Micro-Scale
Forced Convection
C. P. Naveira-Cotta, R. M. Cotta, H. R. B. Orlande, and S. Kakaç

15

39

Conjugated Heat Transfer in Microchannels
J. S. Nunes, R. M. Cotta, M. R. Avelino, and S. Kakaç

61

Mechanisms of Boiling in Microchannels: Critical Assessment
J. R. Thome and L. Consolini

83

Prediction of Critical Heat Flux in Microchannels
J. R. Thome and L. Consolini

107

Transport Phenomena in Two-Phase Thermal Spreaders
H. Smirnov and B. Kosoy

121

An Investigation on Thermal Conductivity and Viscosity of Water
Based Nanofluids
I. Tavman and A. Turgut

139

Formation of Droplets and Bubbles in Microfluidic Systems
P. Garstecki

163

Transport of Droplets in Microfluidic Systems
P. Garstecki

183

The Front-Tracking Method for Multiphase Flows
in Microsystems: Fundamentals
M. Muradoglu

v

203


vi

CONTENTS

The Front-Tracking Method for Multiphase Flows
in Microsystems: Applications
M. Muradoglu

221

Gas Flows in the Transition and Free Molecular Flow Regimes
A. Beskok

243

Mixing in Microfluidic Systems
A. Beskok

257

AC Electrokinetic Flows
A. Beskok

273

Scaling Fundamentals and Applications of Digital Microfluidic
Microsystems
R. B. Fair

285

Microfluidic Lab-on-a-Chip Platforms: Requirements,
Characteristics and Applications
D. Mark, S. Haeberle, G. Roth, F. Von Stetten, and R. Zengerle

305

Microfluidic Lab-on-a-Chip Devices for Biomedical Applications
D. Li
Chip Based Electroanalytical Systems for Monitoring Cellular
Dynamics
A. Heiskanen, M. Dufva, and J. Emnéus

377

399

Perfusion Based Cell Culture Chips
A. Heiskanen, J. Emnéus, and M. Dufva

427

Applications of Magnetic Labs-on-a-Chip
M. A. M. Gijs

453

Magnetic Particle Handling in Microfluidic Systems
M. A. M. Gijs

467

AC Electrokinetic Particle Manipulation in Microsystems
H. Morgan and T. Sun

481

Microfluidic Impedance Cytometry: Measuring Single Cells
at High Speed
T. Sun and H. Morgan

507

CONTENTS

vii

Optofluidics
D. Erickson

529

Vivo-Fluidics and Programmable Matter
D. Erickson

553

Hydrophoretic Separation Method Applicable to Biological
Samples
S. Choi and J.-K. Park

577

Programmable Cell Manipulation Using Lab-on-a-Display
H. Hwang and J.-K. Park

595

Index

615

PREFACE
This volume contains an archival record of the NATO Advanced Study
Institute on Microfluidics Based Microsystems – Fundamentals and Applications held in Çeşme-Izmir, Turkey, August 23–September 4, 2009. ASIs
are intended to be high-level teaching activity in scientific and technical
areas of current concern. In this volume, the reader may find interesting
chapters and various microsystems fundamentals and applications.
As the world becomes increasingly concerned with terrorism, early onspot detection of terrorist’s weapons, particularly bio-weapons agents such
as bacteria and viruses are extremely important. NATO Public Diplomacy
division, Science for Peace and Security section support research, Advanced
Study Institutes and workshops related to security. Keeping this policy of
NATO in mind, we made such a proposal on Microsystems for security. We
are very happy that leading experts agreed to come and lecture in this
important NATO ASI.
We will see many examples that will show us Microfluidics usefulness
for rapid diagnostics following a bioterrorism attack. For the applications in
national security and anti-terrorism, microfluidic system technology must
meet the challenges. To develop microsystems for security and to provide a
comprehensive state-of-the-art assessment of the existing research and
applications by treating the subject in considerable depth through lectures
from eminent professionals in the field, through discussions and panel
sessions are very beneficial for young scientists in the field.
Microfluidics are great tools for security and anti-terrorism with many
applications. New and better diagnostic technology must be developed in
order to be prepared for an act of bio-terrorism. The subject will be treated
through lectures by experts on biosensors, microsystems, bio micro-electromechanical devices, and nanofluidics. To establish the objectives of this
Institute, important lectures by prominent expert on the field are presented
and are included in this volume of the Institute.
Basics of Electrokinetic Microfluidics, Lab-on-a-Chip Devices for Biomedical Applications, Microfluidic Biological Application Specific Integrated

Circuits, Integrated Optofluidics and Nanofluidics, Cell Culture Revolution
via Dynamical Microfluidic Controls, Fundamentals of droplet flow in
microfluidics, Implementation of fluidic functions in digital microfluidics,
Chip architecture and applications for digital microfluidics, Mixing in
microfluidic systems are presented and discussed in detail. In addition more
presentations such as Optofluidics – Fusing Nanofluidics and Nanophotonics,
Programmable Matter – Micro and milliscale fluid dynamics of reconfigurable
assembly for control of living systems, An Overview on Microfluidic

ix


x

PREFACE

Platforms for Lab-on-a-Chip Applications, Centrifugal Microfluidics for
Lab-on-a-Chip Applications are also given.
Transport of droplets and bubbles in microfluidic systems – from flow
through a simple pipe to logic gates and automated chips for chemical
processing, Analytical, Synthesis and Bio-Medical Applications of Microchip Technology, Hydrophoretic separation method for blood sample
analysis, Magnetophoretic multiplexed immunoassays in a microchannel,
programmable particle manipulation using lab-on-a-display are discussed in
details with fundamentals and applications.
During the 10 working days of the Institute, the invited lecturers covered
fundamentals and applications of Microsystems in various fields including
the security.
The sponsorship of the NATO Science for Peace and Security Section
(SPS) is gratefully acknowledged; in person, we are very thankful to Dr.
Fausta Pedrazzini director of the ASI programs and the executive secretary,

Ms Alison Trapp who continuously supported and encouraged us at every
phase of our organization of this Institute. We are appreciative to TOBB
University of Economics and Technology and International Centre of Heat
and Mass Transfer for their sponsorships. We are also very grateful to
Annelies Kersbergen, publishing editor of Springer Science; our special
gratitude goes to Drs. Nilüfer Eğrican, Şepnem Tavman, Almıla Yazıcıoğlu,
Ahmet Yozgatlıgil, Derek Baker, Selin Aradağ, Nilay S. Uzol for coordinating
sessions and we are very thankful to Büryan Apaçoğlu, Gizem Gülben,
Sezer Özerince, and Cahit C. Köksal for smooth running of the Institute.
S. Kakaç
B. Kosoy
D. Li
A. Pramuanjaroenkij


CONVECTIVE HEAT TRANSFER CORRELATIONS IN SOME
COMMON MICRO-GEOMETRIES
ORHAN AYDIN AND METE AVCI
Department of Mechanical Engineering
Karadeniz Technical University, 61080 Trabzon,
Turkey,

Abstract. This work summarizes some of our recent theoretical studies on
convective heat transfer in micro-geometries. Only pure analytical solutions
are presented here. At first, forced convection is studied for the following
three geometries: microtube, microchannel between two parallel plates and
microannulus between two concentric cylinders. Constant heat flux is assumed
to be applied at walls. Then mixed convection in a vertical parallel-plate
microchannel with symmetric wall heat fluxes is investigated. Steady and
laminar internal flow of a Newtonian is analyzed. In the analysis, the usual

continuum approach is coupled with the two main characteristics of the
microscale phenomena, the velocity slip and the temperature jump. In the
forced convection problems, viscous dissipation is also included, while it is
neglected for the mixed convection problem. Internal velocity and temperature
distributions are obtained for varying values of governing parameters. Finally,
fully analytical Nusselt number correlations are developed for the cases
investigated.

1. Introduction
Microelectromechanical systems (MEMS) have gained a great deal of
interest in recent years. Such small devices typically have characteristic size
ranging from 1 mm to 1 μm, and may include sensors, actuators, motors,
pumps, turbines, gears, ducts and valves. Microdevices often involve mass,
momentum and energy transport. Modeling gas and liquid flows through
MEMS may necessitate including slip, rarefaction, compressibility, intermolecular forces and other unconventional effects [1].
The interest in the area of microchannel flow and heat transfer has
increased substantially during the last decade due to developments in the
electronic industry, microfabrication technologies, biomedical engineering,
etc. In general, there also seems to be shift in the focus of published articles,
S. Kakaç et al. (eds.), Microfluidics Based Microsystems: Fundamentals and Applications,
DOI 10.1007/978-90-481-9029-4_1, © Springer Science + Business Media B.V. 2010

1


2

O. AYDIN AND M. AVCI

from descriptions of the manufacturing technology to discussions of the

physical mechanisms of flow and heat transfer [2].
Readers are referred to see the following recent excellent reviews related
to transport phenomena in microchannels. Ho and Tai [3] summarized
discrepancies between microchannel flow behavior and macroscale Stokes
flow theory. Palm [2], Sobhan and Garimella [4] and Obot [5] reviewed the
experimental results in the existing literature for the convective heat transfer
in microchannels. Rostami et al. [6, 7] presented reviews for flow and heat
transfer of liquids and gases in microchannels. Gad-el-Hak [1] broadly
surveyed available methodologies to model and compute transport phenomena
within microdevices. Guo and Li [8, 9] reviewed and discussed the size
effects on microscale single-phase fluid flow and heat transfer. In a recent
study, Morini [10] presents an excellent review of the experimental data for
the convective heat transfer in microchannels in the existing literature. He
critically analyzed and compared the results in terms of the friction factor,
laminar-to-turbulent transition and the Nusselt number.
It is shown that fluid flow and heat transfer at microscale differ greatly
from those at macroscale. At macroscale, classical conservation equations
are successfully coupled with the corresponding wall boundary conditions,
usual no-slip for the hydrodynamic boundary condition and no-temperaturejump for the thermal boundary condition. These two boundary conditions
are valid only if the fluid flow adjacent to the surface is in thermal equilibrium.
However, they are not valid for gas flow at microscale. For this case, the
gas no longer reaches the velocity or the temperature of the surface and
therefore a slip condition for the velocity and a jump condition for the
temperature should be adopted.
The Knudsen number, Kn is the ratio of the gas mean free path, λ, to the
characteristic dimension in the flow field, D, and, it determines the degree
of rarefaction and the degree of the validity of the continuum approach. As
Kn increases, rarefaction become more important, and eventually the
continuum approach breaks down. The following regimes are defined based
on the value of Kn [11]:

(i)
(ii)
(iii)
(iv)

Continuum flow (ordinary density levels)
Kn ≤ 0.001
Slip-flow regime (slightly rarefied)
0.001 ≤ Kn ≤ 0.1
Transition regime (moderately rarefied) 0.1 ≤ Kn ≤ 10
Free-molecule flow (highly rarefied)
10 ≤ Kn ≤ ∞

Viscous dissipation is another parameter that should be taken into
consideration at microscale. It changes temperature distributions by playing
a role like an energy source induced by the shear stress, which, in the
following, affects heat transfer rates. The merit of the effect of the viscous
dissipation depends on whether the pipe is being cooled or heated.


CONVECTIVE HEAT TRANSFER CORRELATIONS

3

In this work, heat and fluid flow in some common micro geometries is
analyzed analytically. At first, forced convection is examined for three
different geometries: microtube, microchannel between two parallel plates
and microannulus between two concentric cylinders. Constant wall heat
flux boundary condition is assumed. Then mixed convection in a vertical
parallel-plate microchannel with symmetric wall heat fluxes is investigated.

Steady and laminar internal flow of a Newtonian is analyzed. Steady, laminar
flow having constant properties (i.e. the thermal conductivity and the thermal
diffusivity of the fluid are considered to be independent of temperature) is
considered. The axial heat conduction in the fluid and in the wall is assumed to
be negligible. In this study, the usual continuum approach is coupled with
the two main characteristics of the microscale phenomena, the velocity slip
and the temperature jump.
Effects of the main governing dimensionless parameters on the momentum
and heat flow transfer will be analyzed. Pure analytical correlations for Nusselt
number as a function of the Brinkman number and the Knudsen number are
developed for both hyrodynamically and thermally fully developed flow. In
fact, this work will be a summary view of our recent studies [12–15].
2. Nu Correlations
In this part, three different geometries are considered and corresponding
results for the Nusselt number these geometries are given respectively in the
following.
2.1. FORCED CONVECTION IN A MICROPIPE

For this geometry, the fully developed velocity profile taking the slip flow
condition at the wall is
2(1 − (r / r0 ) 2 + 4 Kn)
u
=
um
(1 + 8Kn)

(1)

where um is the mean velocity and Kn is the Knudsen number, Kn = λ / Dh .
The Nusselt number correlation for this geometry is obtained as follows

[12]:
Nu =

Brq

2
Brq

1 + 16 Brq
1 ⎛ 16γ Kn ⎞
1
1+
+
+
+
+
⎜
⎟
4
3
4 ⎝ γ + 1 Pr ⎠ 3(1 + 8 Kn) (1 + 8 Kn) 24(1 + 8 Kn)2 6(1 + 8 Kn)

(2)


O. AYDIN AND M. AVCI

4

where Brq, represents the modified Brinkman number, whose value is

determined from
μ um2
(3)
Brq =
D qw′′

2.2. FORCED CONVECTION IN A MICROCHANNEL BETWEEN
TWO PARALLEL PLATES

The fully developed velocity profile for this microchannel is:
u
3 ⎡1 − ( y / w) 2 + 4 Kn ⎤
= ⎢
⎥
um 2 ⎣
1 + 6 Kn
⎦

(4)

where Kn is the Knudsen number, Kn = λ / 2w .
In terms of the modified Brinkman number, Brq, the Nusselt number
receives the following form [13]:
Nu =

2
2Brq
11Brq
2(1 + 21Brq )
1 ⎛ 12γ Kn ⎞

2
+
+
+
+
1+
⎜
⎟
4
3
2
3 ⎝ γ + 1 Pr ⎠ 35(1 + 6Kn) 35(1+ 6Kn) 105(1 + 6Kn) 15(1+ 6Kn)

(5)

where
Brq =

μ um2

w qw

(6)

2.3. FORCED CONVECTION IN A MICROANNULUS BETWEEN
TWO CONCENTRIC CYLINDERS

The dimensionless velocity distributions is obtained as [14]:
u
= 2 (1 − R 2 + 2rm*2 ln ( R ) + A ) / B

um

where A and B are, respectively:

(7)


CONVECTIVE HEAT TRANSFER CORRELATIONS

5

A = 4 Kn (1 − r * )(1 − rm*2 )

(8)

⎛
⎞
⎛1
⎞
r *2
ln ( r * ) ⎟ + 8Kn (1 − r * )(1 − rm*2 ) ⎟⎟
B = ⎜⎜ 1 − r *2 − 4rm*2 ⎜ +
*2
⎝ 2 1− r
⎠
⎝
⎠

(9)

Here Kn is Knudsen number ( Kn = λ Dh ) and rm* designates the
dimensionless radius where the maximum velocity occurs (∂u / ∂r = 0) . It is
given by [14]
1/ 2

⎛
⎞
⎜
⎟
2
rm ⎜
⎟
(1 − r * )(1 + 4 Kn)
*
rm = = ⎜
⎟
2
*
ro ⎜
⎛ r −1 ⎞
2 ln(1 / r * ) − 4 Kn ⎜ * ⎟ ⎟
⎜ r ⎟⎟
⎜
⎝
⎠⎠
⎝

(10)

For this geometry, two different forms of the thermal boundary conditions

are applied, which are shown in Fig. 1. In the following, these two different
cases are treated separately [14]:
insulated

qw

ro

Flow

ri

r
z

qw

insulated

(a) Case A

(b) Case B

Figure 1. Schematic of the problem [14].

For the Case A, the dimensionless temperature distribution is obtained
as follows [14]:


O. AYDIN AND M. AVCI

6

θ (R) =

T − Ts
qw′′ ro / k

*2
2
*2
2
a ⎛⎜ −3 − A + 2rm + R (1 + A − 2rm + R / 2) ⎞⎟
=
⎟
2B ⎜ − ln R (1 + 2 A − 2rm*2 (1 + ln R ) )
⎝
⎠
Br
2
+ 2 (1 − R2 )(1 + R2 − 8rm*2 ) + 4ln R (1 − 4rm*2 ) − 8rm*4 ( ln R ) + ln R
B

(

where
a=

(11)

)

−2 B 2 r * + 8Br ( r *2 − 1)(1 − 4rm*2 + r *2 ) + 32 Br rm*4 ln ( r * )

(

B (1 + 2 A − 2rm*2 − r *2 )( r *2 − 1) + 4 rm*2 r *2 ln ( r * )

)

(12)

and
Br =

μ um2

(13)

qw′′ ro

Similarly, the dimensionless temperature distribution is obtained for the
Case B as in the following [14]:
θ ( R) =

T − Ts
qw′′ ro / k

2
*2

*2
2
*2
*
*2
a ⎛⎜ ( R − r )(1 + A − 2rm ) − ( R − r ) − ( ln R − ln r )(1 + 2 A − 2rm ) ⎞⎟
=
⎟
2 B ⎜ +2rm*2 ( R 2 ln R − r *2 ln r * )
⎝
⎠
⎛ ( R 2 − r *2 ) 8rm*2 − ( R 2 + r *2 ) + 4 ( ln R − ln r * )(1 − 4rm*2 ) ⎞
Br ⎜
⎟
+ 2⎜
⎟
B ⎜ −8r *4 ( ln R )2 − ( ln r * )2
⎟
⎝ m
⎠

(

(

)

(14)

)

where
a=

−2 B 2 r * + 8Br ( r *2 − 1)(1 − 4rm*2 + r *2 ) + 32 Br rm*4 ln ( r * )

(

B (1 + 2 A − 2rm*2 − r *2 )( r *2 − 1) + 4 rm*2 r *2 ln ( r * )

)

(15)

After performing necessary substitutions, the Nusselt number is obtained
as follows [14]:


CONVECTIVE HEAT TRANSFER CORRELATIONS
Nu =

7

qw Dh
2
= − * (1 − r * )
(Tw − Tm ) k θ m

(16)

2.4. MIXED CONVECTION IN A VERTICAL PARALLEL-PLATE
MICROCHANNEL WITH SYMMETRIC WALL HEAT FLUXES

For this problem under the above mentioned assumptions, therefore, the
dimensionless velocity profile is obtained as [15]:

U = C1eξ Y cos(ξ Y ) + C2 e−ξ Y cos(ξ Y ) + C3 eξ Y sin(ξ Y ) + C4 e−ξ Y sin(ξ Y )

(17)

where
1/ 4

⎡ Gr ⎤
ξ =⎢ qU⎥
⎣ Re ⎦

(18)

By applying the boundary conditions given in Eq. (10), the four unknown
constants C1, C2, C3 and C4 can be obtained. Some typical values of these
constants for different values of Grq/Re and Kn are tabulated in Table 1.
TABLE 1. Typical values of constants C1, C2, C3, and C4 [15].

Grq/Re
1

50

100

Kn
0.00
0.02
0.06
0.10
0.00
0.02
0.06
0.10
0.00
0.02
0.06
0.10

C1
2.87634
2.39220
1.82928
1.51201
0.82091
0.69563
0.55026
0.46847
0.59553
0.50312
0.39610
0.33599

C2

−2.87634
−2.19857
−1.41051
−0.96634
−0.82091
−0.49796
−0.12322
0.08763
−0.59553
−0.30136
0.03930
0.23064

C3
−8.90402
−7.16036
−5.13293
−3.99024
−0.42563
−0.30682
−0.16895
−0.09137
−0.11637
−0.05680
0.01218
0.05093

C4
15.15670
12.25113

8.87272
6.96858
3.39702
2.82999
2.17201
1.80179
2.88858
2.44242
1.92576
1.63556

After several steps of derivations, the Nusselt numbers is obtained as [15]:
Nu1 = −

1

θ m*

(19)


O. AYDIN AND M. AVCI

8

where
0.5

T −T
θ = m 1 =

( q Dh / k )
*
m

∫ Uθ

*

dY

0
0.5

(20)

∫ UdY
0

and
θ* =
=

(T − Ts ,1 ) (Ts ,1 − T1 )
T − T1
=
+
q2 Dh / k q2 Dh / k
q2 Dh / k

(

Re
2ξ 2 e −ξ Y ( (C4 − C3 )eξ Y + (C3e 2ξ Y − C4 ) cos(ξ Y ) + (C2 − C1e 2ξ Y )sin(ξ Y ) )
Grq

)

(21)

− β t Kn(q1 / q2 )

3. Results and Discussion

Here, only summary results are given for three different geometries considered
separately.
3.1. FORCED CONVECTION IN A MICROPIPE

Figure 1 shows the variation of the Nusselt number with the Knudsen number
for different values of the modified Brinkman number. For Brq = 0, an
increase at Kn decreases Nu due to the temperature jump at the wall.
Viscous dissipation, as an energy source, severely distorts the temperature
profile. Positive values of Brq correspond to wall heating (heat is being
supplied across the walls into the fluid) case (qw > 0), while the opposite is
true for negative values of Brq. In the absence of viscous dissipation the
solution is independent of whether there is wall heating or cooling.
However, viscous dissipation always contributes to internal heating of the
fluid, hence the solution will differ according to the process taking place.
Nu decreases with increasing Brq for the hot wall (i.e. the wall heating
case). As expected, increasing dissipation increases the bulk temperature of
the fluid due to internal heating of the fluid. For the wall heating case, this

increase in the fluid temperature decreases the temperature difference
between the wall and the bulk fluid, which is followed with a decrease in
heat transfer. When wall cooling is applied, due to the internal heating effect
of the viscous dissipation on the fluid temperature profile, temperature
difference is increased with the increasing Brq (Fig. 2). For more details,
readers are referred to Ref. [12].


CONVECTIVE HEAT TRANSFER CORRELATIONS

9

8

7

Br q
-----------0.1
-0.01
0.0
0.01
0.1

Pr=0.7

Nu

6

5

4

3

2
0,00

0,02

0,04

0,06

0,08

0,10

Kn

Figure 2. The variation of Nu with Kn at different values of Brq [12].

3.2. FORCED CONVECTION IN A MICROCHANNEL BETWEEN
TWO PARALLEL PLATES

For this geometry, Fig. 3 illustrates the variation of the Nusselt number with
the Knudsen number for different values Brinkman numbers. As seen, an
increase at Kn decreases Nu due to the temperature jump at the wall. The
effect of the viscous dissipation is discussed above. For more details,
readers are referred to Ref. [13].

5,6
5,2

Br q
-------------

Pr=0.7
4,8

-0.1
-0.01
0.0
0.01
0.1

4,4

Nu

4,0
3,6
3,2
2,8
2,4
2,0
0,00

0,02

0,04

0,06

0,08

0,10

Kn

Figure 3. The variation of Nu with Kn at different values of Brq [13].


10

O. AYDIN AND M. AVCI

3.3. FORCED CONVECTION IN A MICROANNULUS BETWEEN
TWO CONCENTRIC CYLINDERS

Figure 4 illustrates the variation of the Nusselt number with the aspect ratio
of the annulus, r* for different values of the Knudsen number at Cases A
and B without viscous dissipation (Br = 0), respectively. For the both cases,
the influence of the increasing Kn is to decrease the heat transfer rates. As
expected, for the Case A, an increase in r* increases Nu, while it decreases
Nu for the Case B. However, this Nu-dependence on r* becomes negligible
with increasing Kn.
The variation of the Nusselt number with the Knudsen number for
different values of the Brinkman number at r* = 0.2 for Cases A and B,
respectively, is shown in Fig. 5. An increase at Kn decreases the Nu due to
the temperature jump at the wall. Nu decreases with increasing Br for the

hot wall (i.e. the wall heating case). For this case, the wall temperature is
greater than that of the bulk fluid. Viscous dissipation increases the bulk
fluid temperature especially near the wall since the highest shear rate occurs
in this region. Hence, it decreases the temperature difference between the
wall and the bulk fluid, which is the main driving mechanism for the heat
transfer from wall to fluid. However, for the cold wall (i.e. the wall cooling
case), the viscous dissipation increases the temperature differences between
the wall and the bulk fluid by increasing the fluid temperature more.
Therefore, increasing Br in the negative direction increases Nu. As seen
from the figure, the behavior of Nu versus Kn for lower values of the
Brinkman number, either in the case of wall heating (Br = 0.01) or in the
case of the wall cooling (Br = −0.01) is very similar to that of Br = 0. In
addition, as observed from the figure, Br is more effective on Nu for lower
values of Kn than for higher values of Kn. For more details, readers are
referred to Ref. [14].
3.4. MIXED CONVECTION IN A VERTICAL PARALLEL-PLATE
MICROCHANNEL WITH SYMMETRIC WALL HEAT FLUXES

For this problem, the variation of Nu with Grq/Re is plotted for different
values of Kn in Fig. 6. As expected, increasing Grq/Re increases Nu while
increasing Kn decreases Nu. Because of the lower values of the Grq/Re
present at microscale, the aiding effect of the buoyancy forces on the inertia
forces are not much. Therefore, increasing Grq/Re in this limited range will
not have a profound effect on Nu. For example, at Kn = 0.02, increasing
Grq/Re from 1 to 200 will lead to an increase of about 2% in Nu. For more
details, readers are referred to Ref. [15].


CONVECTIVE HEAT TRANSFER CORRELATIONS

11

6
Br = 0.00
Pr = 0.71

5

Kn = 0.00

Nu

Kn = 0.02

4

Kn = 0.04
Kn = 0.06
Kn = 0.08

3

2
0.2

Kn = 0.10

0.3

0.4

0.5

0.6

0.7

0.8

r*

(a)

10
Br = 0.00
Pr = 0.71

8

Nu

Kn = 0.00

6
Kn = 0.02
Kn = 0.04

4

Kn = 0.06

Kn = 0.08
Kn = 0.10

2
0.2

0.3

0.4

0.5
r

0.6

0.7

0.8

*

(b)
Figure 4. The variation of Nu with r* at different values of Kn for Br = 0.0, (a) Case A,
(b) Case B.


O. AYDIN AND M. AVCI

12

7
Br = -0.10
Br = -0.01
Br = 0.00
Br = 0.01
Br = 0.10

6

5
Nu

r* = 0.2
Pr = 0.71

4

3

2
0.00

0.02

0.04

0.06

0.08

0.10

Kn

(a)
16
Br = -0.10
Br = -0.01
Br = 0.00
Br = 0.01
Br = 0.10

14
12

r* = 0.2
Pr = 0.71

Nu

10
8
6
4
2
0.00

0.02

0.04

0.06

0.08

0.10

Kn

(b)
Figure 5. The variation of Nu with Kn at different values of Br for r* = 0.2, 0.5 and 0.8,
(a) Case A, (b) Case B.


CONVECTIVE HEAT TRANSFER CORRELATIONS

13

9
Kn = 0.00

8

Nu (Nu1=Nu2)

7

Kn = 0.02

6

Kn = 0.04

5

Kn = 0.06
Kn = 0.08

4

Kn = 0.10

Pr = 0.7
rq = 1.0

3
0

50

100

150

200

Grq / Re

Figure 6. The variation of the Nu with the Grq/Re at different values of Kn.

Acknowledgment

The first author of this work is indebted to the Turkish Academy of Sciences
(TUBA) for the financial support provided under the Programme to Reward
Success Young Scientists (TUBA-GEBIT).

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