Open Journal of Applied Sciences, 2012, 2, 28-34
doi:10.4236/ojapps.2012.21003 Published Online March 2012 (http://www.SciRP.org/journal/ojapps)
On Laminar Flow in Microfabricated Channels with
Partial Semi-Circular Profiles
William J. Federspiel,2,3,4, Isabella Valenti1,2
1Medical Devices Laboratory, McGowan Institute for Regenerative Medicine, University of Pittsburgh, Pittsburgh, USA
2Department of Bioengineering, University of Pittsburgh, Pittsburgh, USA
3Department of Chemical Engineering, University of Pittsburgh, Pittsburgh, USA
4Department of Critical Care Medicine, University of Pittsburgh, Pittsburgh, USA
Received January 18, 2012; revised February 17, 2012; accepted April February 27, 2012
ABSTRACT
Soft and hard micromachining techniques used to develop microfluidic devices can yield microchannels of many dif-
ferent cross-sectional profiles. For semi-circular microchannels, these techniques often produce only partial semicircu-
lar (PSC) cross-sections. This study investigated fully developed laminar flow in PSC microchannels as a function of a
circularity index, , defined as the ratio of the radiuses along the curved and flat surfaces of the PSC profile. A correc-
tion factor, K, to the Hagen-Poiseuille relation was determined and was well-fitted by the power-law relationship

2.56
5.299K  . Actual correction factors were compared to estimates based on several hydraulic models for flow in mi-
crochannels of arbitrary cross-section, as well as the half-ellipsoid cross-section. The level of wall shear stress, when nor-
malized by the pressure drop per unit length, increased approximately linearly with increase in the circularity index,

.
Keywords: Hagen-Poiseuille Flow; Poiseuille Flow; Flow Resistance; Wall Shear Stress
1. Introduction
The trend toward miniaturization that has been driven by
advances in fabrication processes derived from micro-
electromechanical systems (MEMs) and other microsys-
tem technologies has led to microfluidic devices for use in
numerous chemical, biological and medical applications.
Microfluidics refers to devices for accommodating and
controlling flow through microchannels with cross-sec-
tional scales of 10 - 100 microns. For many traditional
applications, microchannels are fabricated on silicon or
quartz substrates using photolithographic methods derived
directly from MEMs processes. Whitesides and coworkers
extended these “hard” microfabrication techniques into
the novel realm of soft lithography to create microchan-
nels and other microfluidic devices in elastomeric materi-
als such as polydimethylsiloxane (PDMS), principally
with advantages in biological and biomedical applications
[1]. The processes underlying hard and soft microfabrica-
tion allow for different microchannel shapes tailored to
specific applications. While rectangular microchannels
are ubiquitous, the second most common microchannel
geometry may be the semi-circular profile, which occurs
naturally in hard micromachining with isotropic etching
in the photolithographic steps [2]. In soft lithography,
plasma etching can also be tailored to provide rounded
corners to silicon negative masters, that then yield elas-
tomeric microchannels with semi-circular profiles after
spin coating [3,4]. The rounded edges of semicircular
profiles help promote cell seeding and proliferation in
microfluidic devices developed for tissue engineering [3].
Recent studies that have developed microchannels with
semi-circular profiles include an analysis of shear stress
effects on endothelial cells in curved microvessels [5], the
evaluation of micromachined flow cytometers with inte-
grated optics [6,7], the creation of a novel magnetohy-
drodynamic micropump [8], and studies of microfluidic
devices for capillary electrophoresis [9]. Our own group
recently developed biohybrid artificial lung modules with
semi-circular endothelialized blood microchannels subja-
cent to rectangular gas microchannels [10].
The semi-circular microchannels that are fabricated
using hard or soft micromachining can best be charac-
terized as partial semi-circular (PSC) microchannels, in
which the “radius” to the curved-side is appreciably less
than the “radius” along the flat-side. Figure 1(a) displays
an SEM image of the typical PSC cross-section obtained
using soft-lithography from our work on biohybrid artifi-
cial lung modules [10], while Figure 1(b) shows an SEM
image of a PSC microchannel developed using hard-
micromachining for a microfluidic capillary electropho-
resis device [9]. A circularity index, , can be defined
as the ratio of the radius to the curved-side relative to that


W. J. FEDERSPIEL ET AL. 29
along the flat-side, with representing a complete
semi-circular profile. The values of are approximately
0.84 and 0.80 for the PSC microchannels shown in Fig-
ures 1(a) and (b), respectively.
1  

In this study, we analyze fully-developed flow in par-
tial semi-circular (PSC) microchannels using an analytic-
cal solution () and numerical simulations (
1  0.5


) using Comsol Multiphysics®. The analysis and
results presented focus primarily on determining a cor-
rection factor to the Hagen-Poiseuille pressure drop-
flowrate relationship for fully-developed flow in circular
channels and on characterizing the maximum and aver-
age wall shear stress. While the equation of motion for
fully-developed flow in a complete semi-circular micro-
channel () can be analytically solved, the analytical
solution was not found in our literature search and our
solution is presented in the Appendix. Several recent
1.0
1  
(a)
(b)
Figure 1. (a) Example 1 of the partial semi-circular (PSC)
cross-sectional profiles for microchannels fabricated using
soft techniques. Adapted from (Burgess et al. 2009); (b)
Example 2 of the partial semi-circular (PSC) cross-sectional
profiles for microchannels fabricated using hard micromach-
ing techniques. Adapted from (Peeni et al. 2005).
studies have addressed fully-developed flow in micro-
channels of arbitrary and/or specific cross-sectional sha-
pes. Oosterbroek [11] used analytical and approxima-
tion techniques to determine or estimate the velocity pro-
files in different microchannel geometries. The geometry
which models closest the PSC microchannel was the
half-ellipsoid. Variational principles based on minimize-
tion of work were adapted from analogous structural
mechanics problems related to beam torsion to estimate
the velocity profile and flow resistance for the half-el-
lipsoid microchannel. For microchannels of arbitrary
cross-section, Mortensen et al. [12] showed that the hy-
draulic resistance can be approximated as a linear func-
tion of a compactness factor, which is defined as the ratio
of the perimeter squared to the cross-sectional area of the
microchannel. The linear relationship was determined
from the analytical velocity profiles for flow in micro-
channels of full-ellipsoid, rectangular, and triangular shape.
Bahrami et al. [13] developed a novel approximate solu-
tion for estimating the flow resistance of microchannels of
any arbitrary cross-section based solely on a single geo-
metrical feature, the dimensionless polar moment of iner-
tia of the cross-section. The approximate solution com-
pared well with analytical and numerical results for flow
in microchannels of several different geometries. The
results of our study on fully-developed flow in PSC mi-
crochannels are compared and evaluated against the theo-
ries described above in these recent studies.
2. Theoretical Model and Simulations
The geometry and geometrical parameters for the partial
semi-circular (PSC) microchannel are shown in Figure
2(a) in the x, y plane, where z is the direction of flow.
The flat, bottom-side of the PSC is of length, D, and the
height of the PSC is given as 2D
 , so that 1


represents a complete semi-circular channel. This study
considers the flow resistance of PSC microchannels in
the range of

from 0.5 to 1.0.
Fully developed laminar flow in the z direction for a
Newtonian fluid is governed by
21P
VL


 (1)
where


,VVxy
2
is the magnitude of the velocity in
the z direction,

is the Laplacian operator in ,
x
y
space,

is the fluid viscosity, and PL is the pres-
sure drop per unit length in the direction of flow. The
flow is also governed by the no-slip condition on the wall
(w) surfaces of the microchannel:
0 for w VS

R (2)
The governing equation can be solved numerically
most conveniently by introducing dimensionless de-
pendent and independent variables. The spatial coordi-

W. J. FEDERSPIEL ET AL.
30
nates (,
x
y) are scaled by the flat surface width, D, while
the flow velocity is scaled according to:
2
DP
UV L


 (3)
The resulting dimensionless equation of motion is
given by:
21U 
 (4)
where is the dimensionless Laplacian operator on
the domain shown in Figure 2(b).
2

The no-slip boundary condition requires 0U

on
the surfaces in Figure 2(b) .
Comsol Multiphysics® Version 3.4 was used to nu-
merically solve Equation (4), subject to its boundary con-
dition, on the dimensionless domain (Figure 2(b)) using
the Poisson equation solver in the basic multiphysics
module of Comsol. The solution for the dimensionless
flow velocity was integrated over the dimensionless do-
main (subdomain integration in Comsol) to obtain a
“correction factor” to the Hagen-Poiseuille relation for
laminar flow in a circular conduit. The correction factor,
K, is defined from the integrated dimensionless velocity
for convenience as
(a)
(b)
Figure 2. (a) Schematic illustrating the cross-sectional ge-
ometry of the PSC microchannel in dimensional forms. The
circularity index denotes the departure from a full semi-
circular channel (); (b) Schematic illustrating the
cross-sectional geometry of the PSC microchannel in di-
mensionless forms. The circularity index denotes the de-
parture from a full semi-circular channel ().
1  
1  
1
π
d128
A
UA K




 (5)
where d
A

and
A

indicate integration over the di-
mensionless domain (Figure 2(b)). With this definition,
transforming Equation (5) back into dimensional form
yields:
4
128
π
L
PK Q
D

 (6)
where Q is the volumetric flowrate through the channel.
Thus, the correction factor K represents the proportion by
which the flow resistance (PQ

) is increased in a PSC
microchannel compared to a circular duct of diameter, D.
Also of interest is the shear stress exerted on the wall
by the flowing fluid. The wall shear stress is the z com-
ponent of the fluid stress vector exerted on the wall:



 n, where

is the stress tensor, given for a
Newtonian fluid by

t






 VV, and

n is a
unit normal vector at the wall directed into the domain.
Applying these relationships and noting thV at

Vk
by
,
the wall shear stress in the PSC microchannel is given
wV



n (7)
which can be determined in a normalized form directly
from the numerical solution by computing the normal
derivative of the dimensionless velocity at the wall:

wU
DPL



n (8)
3. Results and Discussion
3.1. Hagen-Poiseuille Correction Factor
Fully developed laminar flow in partial semi-circular
(PSC) microchannels was numerically simulated using
Comsol Multiphysics®. The “partial” nature of the PSC
microchannel was specified by the dimensionless pa-
rameter,

wherein the height of the PSC microchannel
was 2D

, with D being the diameter or width of the
flat-side of the PSC microchannel. Laminar flow was
characterized over the range . Simulations
were performed using dimensionless variables with the
0.5 1.0  
velocity (V) normalized according to
2
DP
VL





,
where

is fluid viscosity and PL is the pressure
drop per unit length in the channel. Figure 3 shows the
contours of the normalized velocity field in a PSC mi-
crochannel with 0.5


.
The pressure drop and flowrate relationship for PSC
microchannels was characterized by introducing a cor-
rection factor, K, to the Hagen-Poiseuille relation (see
Equation (6)), in which K represents the proportion by

W. J. FEDERSPIEL ET AL. 31
Figure 3. Normalized velocity contours for fully-developed
laminar flow in a PSC microchannel for . .05  
which the flow resistance is increased in a PSC micro-
channel compared to a circular duct of the same diameter.
The flow correction factor increased by 6-fold as

decreased from (complete semi-circular micro-
channel) to (
Figure 4(a)).
1.0  
0.5  
In this range the correction factor was well fit by the
power-law relation: 2.56
5.299K  . For the complete
semi-circular microchannel (1.0

), the correction
factor computed from the simulations (5.279) was in
agreement with that from the analytical solution (5.279)
for a semi-circular microchannel (see Appendix). Figure
4(b) displays the correction factor plotted against the
microchannel compactness factor, C. Mortensen et al.
[12] developed a theoretical analysis showing that for
microchannels of arbitrary cross-section, the hydraulic
resistance varies approximately linearly with a compact-
ness factor defined by:
2
P
C
A
 (9)
in which P and A are the microchannel perimeter and
cross-sectional area, respectively. In the range of the cir-
cularity index () studied here, the correction factor did
follow the linear relationship: K = 3.517C – 55.35 with
an value of 0.98.

2
R
The Hagen-Poiseuille correction factor can also be
compared with predictions based on several hydraulic
models for flow in arbitrary microchannels. The simplest
hydraulic model would be based on using the hydraulic
diameter, 4
h
DAP, in the Hagen-Poiseuille relation.
The correction factor is then given by 44 D
h. Table 1
indicates that the hydraulic diameter model significantly
over-predicts the correction factor by 36% to 75% in the
range of studied.
D

Bahrami et al. [13] (ref: doi:10.1016/j.ijheatmasstrans
fer.2006.12.019) developed a novel approach for esti-
mating the flow resistance of microchannels of any arbi-
trary cross-section. Their analysis introduces the dimen-
Table 1. Comparison of actual and predicted Hagen-
Poiseuille correction factors.

act
K
1
hd
K
2
pmi
K
3
s
e
K
4
0.5 32.21 56.37 31.86 34.66
0.6 19.46 31.19 19.19 20.74
0.7 12.87 19.37 12.64 13.56
0.8 9.107 13.09 8.905 9.479
0.9 6.787 9.447 6.610 6.969
1 5.279 7.174 5.117 5.333
(a)
(b)
Figure 4. (a) Correction factor, K, for the Hagen-Poiseuille
relation as a function of circularity index ; (b) Correc-
tion factor, K, for the Hagen-Poiseuille relation as a func-
tion of the compactness factor C.

sionless polar moment of inertia, *
p
I
, computed for the
microchannel geometry:

22
2
1d
pcc
A
I
xxy A
A
y







(10)
in which the dimensionless coordinates and cross-sec-
tional area (denoted with asterisks) are scaled by the
same dimension. (Note: Comsol automatically calculates

W. J. FEDERSPIEL ET AL.
32
the polar moment of inertia and all other pertinent geo-
metric parameters for any defined domain.) Applying
their analysis to determine the Hagen-Poiseuille correc-
tion factor yields:
3
2
1π
8
p
I
K
A


 (11)
in which the dimension used to normalize for length is
that used in the Hagen-Poiseuille relation itself. Table 1
indicates that Equation (11) provides a very good esti-
mate for the correction factor of PSC microchannels,
underestimating the correction factor by only 1.1% at
to 3.1% at .
0.5  1.0  
An approximate model for the cross-sectional profile a
PSC microchannel is a half-ellipsoid. While an analytical
solution to the equation of fluid motion does not exist for
the half-ellipsoid, Oosterbroek used a variational approach
based on minimization of work, adapted from the analo-
gous structural mechanics problem of beam torsion, to
approximate the velocity profile and flow resistance for
half-ellipsoid microchannels [11]. The approximate re-
sistance, R, of a half-ellipsoid microchannel was deter-
mined as
22
33
32 3
π3
La b
Rab
 
 (12)
where a and b are the major and minor axes of the
half-ellipsoid. Since ba approximates the circularity
index, , Equation (12) yields the Hagen-Poiseuille
correction factor:

2
3
3
43 K



 (13)
Table 1 indicates that the correction factor based on the
half-ellipsoid is also a good estimate for the correction
factor for PSC microchannels, in this case over-estimating
the correction factor most at small : 7.6% at
 0.5


,
and improving to a 1.0 % over-estimate at . 1.0  
3.2. Wall Shear Stress Distribution
The wall shear stress associated with fully developed
laminar flow in PSC microchannels would also be of
interest in many applications, especially biomedical ap-
plications involving cells. The wall shear stress, w

,
computed directly from the dimensionless flow field is
normalized according to wP DL 




. Figure 5
shows the variation of the normalized wall shear stress
on the flat and curved surfaces of a PSC microchannel
for . As would be anticipated, the maximum
shear stress occurs in the middle of the flat surface for
PSC microchannels.
0.5  
The maximum and average values of the normalized
Figure 5. Distribution of the normalized wall shear stress,
along the flat and curved surfaces of a PSC microchannel
for .05


.
Figure 6. The linear variation in the maximum and average
normalized wall shear stress with the circularity index

.
wall shear stress varied approximately linearly with

,
as shown in Figure 6. The linear correlations for the
maximum and average normalized wall shear stresses
are: m and (re-
spectively). The shear stress decreased by a factor of 0.57
as
0.1820.032 
0.122 0.031
a



decreased from unity to . 0.5  
This decrease occurs because the shear stress was nor-
malized by the pressure drop per unit length in the mi-
crochannels. For a given flowrate, the pressure drop per
unit length increases about 6-fold as decreases from

1.0


to 0.5


(see Figure 4(a)). Accordingly, for
a given flowrate the shear stress would increase by ap-
proximately 3.4-fold as

decreases from 1.0


to
0.5


. For 1.0


the normalized maximum shear
stress from the simulations (0.212) was in agreement with
that from the analytical solution (0.212) for a semi-circu-
lar channel (Appendix).

W. J. FEDERSPIEL ET AL.
33
4. Conclusions
1) For fully developed laminar flow in partial semi-cir-
cular (PSC) microchannels, the correction factor, K, to
the Hagen-Poiseuille relationship is well fitted by the
power law relation: 2.56
5.299K   in the range of
, where is an index of circularity relat-
0.5 1.0   
ing the radius of the curved surface, 1
2D
 , to that of the
flat surface, 2D.
2) The correction factor can be predicted within 3% by
a novel relationship developed for microchannels of ar-
bitrary cross-section, which accounts for the shape of the
cross-sectional profile using only the dimensionless polar
moment of inertia for the profile.
3) The normalized wall shear stress, wP
DL





increases approximately linearly with the circularity index
. The maximum shear stress occurs at the center of the
flat surface. The maximum (m) and average (a) normal-
ized wall shear stresses are given
and .

0.1820.032
m


0.122 0.031
a


5. Acknowledgements
The work presented in this publication was made possi-
ble by grants HL70051 and HL080926 from the National
Heart, Lung, and Blood Institute at the National Institutes
of Health. The contents are solely the responsibility of
the authors and do not necessarily represent the official
views of the National Heart, Lung, and Blood Institute or
National Institutes of Health. We would like to recognize
the University of Pittsburgh’s McGowan Institute for
Regenerative Medicine for support of this study.
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W. J. FEDERSPIEL ET AL.
34
Appendix
The normalized equation of motion can be analytically
solved readily for the special case of a semi-circular
channel (). Using polar coordinates (
1.0  ,r

) cen-
tered on the flat side of the normalized geometry, the
dimensionless equation of motion is:
2
2
11
1
UU
r
rrrr

 



 
 
(A1)
with the no-slip boundary conditions:



12, ,0UU  r
. A separation of variables approach can be
undertaken after the substitution:

,π0Ur
 
2
1
,, cos2
4
Ur Wrr   

1  (A2)
which leads to Laplace’s equation for

,Wr

:
2
2
11
0
WW
r
rrr r

 



 

with the same no-slip conditions as for U with the excep-
tion

11
,cos2
216
W 
 1



 .
A straight-forward separation of variables solution to
Equation (A3) can be performed resulting in a series so-
lution for


,Wr

. Substituting that solution into Equa-
tion (A2) results in the normalized velocity profile for a
semi-circular channel:
 


2
21
2
1
1
,cos21
4
sin 21
1
2
2π212123
n
n
n
Ur r
rn
nnn










(A4)
The integrals and derivatives leading to the Hagen-
Poiseuille correction factor and normalized wall shear
stress are straightforward and hence are not provided here.
 (A3)