© 2012 The Korean Society of Rheology and Springer 89
Korea-Australia Rheology Journal, Vol.24, No.2, pp.89-95 (2012)DOI: 10.1007/s13367-012-0010-5
www.springer.com/13367
Pulsatile Poiseuille flows in microfluidic channels with back-and-forth mode
Kwang Seok Kim and Myung-Suk Chun*
Complex Fluids Laboratory, National Agenda Res. Div., Korea Institute of Science and Technology (KIST),Seongbuk-gu, Seoul 136-791, Republic of Korea
(Received February 22, 2012; final revision received April 1, 2012; accepted April 3, 2012)
Abstract
The numerical solver for the velocity field equation describing laminar pulsatile flows driven by a time-dependent pressure drop in the straight microfluidic channel of square cross-section is developed. In thecomputational algorithm, an orthogonal collocation on finite element scheme for spatial discretizations iscombined with an adaptive Runge-Kutta method for time integration. The algorithm with the 1,521 com-putational nodes and the accuracy up to O(10–5) is applied to the flow in the back-and-forth standing modewith the channel hydraulic diameter (Dh) in the range 10 – 500 µm and the oscillating frequency (f) of 1 to100 Hz. As a result, a periodic steady state is defined as the flow condition where there would be no netmovement after long time elapses. Following by the retardation phenomena in a cycle, reversal of the axialvelocity is observed at the channel center. Major attention is focused on the influences of the size of channelcross-section and the oscillating frequency. Increasing Dh and f results in the decrease in the amplitude ofmean velocity but the increase in the start-up time. Larger time delay occurs by low-frequency pulsation.
Keywords : pulsatile flow, microfluidics, Navier-Stokes equation, orthogonal collocation, adaptive integration
1. Introduction
The Poiseuille equation for steady-state and no-slip
boundary conditions can readily be applied into the pre-
diction of pressure drop by flow rate, or vice versa, in wide
range of scientific and engineering fields. Advancement of
cutting-edge technology allows that geometry and dimen-
sion of flow channel are being well-characterized as well
as more complicated. For example, lab-on-chips (LOC)
technologies put laboratory functions into single or mul-
tiple micrometer-sized chips so that mixing, particle char-
acterization (Hu et al., 2011), chemical reaction, and even
electric power generation can take place in miniaturized
portable devices. These LOC devices contain microfluidic
channels through which either simple or complex fluids
flow. Two or more channels of different sizes merge into a
single channel, and a stream bifurcates when needed. In
addition, there may be expansion or contraction of channel
cross-sections.
Unlike conventional pipes (circular cross-section), the
channel cross-section that is usually adopted in LOC is
rectangle since channel is fabricated through a lithographic
process where an image on the photomask is patterned
onto the photoresist-coated wafer using a ray of ultraviolet,
the details of which may be found elsewhere (Stone et al.,
2004). In the flow through a rectangular channel, the
velocity profile is a function of two independent variables,
i.e., lateral and transverse positions. Note that there are four
corners inside the channel where the shear stress by adja-
cent walls may cause significant resistance against flow
(Chun, 2011).
Time-dependent pulsatile flow in which the flow rate
regularly or irregularly fluctuates plays an indispensable
role in micro and biofluidic systems (Gong et al., 2008).
Such a flow can be often found in an artery (Hodis and
Zamir, 2011), since the seminal work of Womersley
(1955). The biological pulsatile flow was extended to the
industrial applications on the ground that the harmonic
motion with oscillatory frequency leads to the augmented
energy dissipation (Edwards and Wilkinson, 1971; Fan and
Chao, 1965). Later, this idea is employed to get rid of the
fouling layers in the duct. In LOC application, a periodic
oscillating pressure drop can be imposed to a channel flow
(Kim et al., 2012; Leslie et al., 2009; Vedel et al., 2010),
and it leads to a periodic form of an output after a certain
time (Morris and Forster, 2004). In laminar flows, a sinu-
soidal pressure drop brings about a sinusoidal flow velocity
with the same frequency accompanied by retardation (see
Fig. 1). The vertical lines in Fig. 1 denote mean velocities
of a constant pressure-drop flow (dashed curve) and a flow
with oscillating pressure drop (solid curve). Reduction in
mean velocity due to oscillation is of importance in terms
of amount of fluid that is transferred through a channel;
hence, a methodology of analysis on such oscillating fluid*Corresponding author: [email protected]
Kwang Seok Kim and Myung-Suk Chun
90 Korea-Australia Rheology J., Vol. 24, No. 2 (2012)
dynamics should be clearly addressed.
The current study aims at building up a simulation
framework for analyzing an oscillating fluid flow formed
across a microfluidic channel for a LOC application per-
spective. The specific tasks conducted in this work include
model developments from the unsteady Navier-Stokes (N-
S) equation and a relevant simulation code, visualization
of start-up velocity profile, and quantitative analysis on a
mean velocity and a time delay (defined as phase dif-
ference between pressure drop and velocity) in terms of an
oscillating frequency and a channel size. The simulation
codes required to solve the problem have been made by
the authors in the Mathematica® (Wolfram Research, Inc.,
IL) environment for the purpose of the expanded appli-
cability of the fluid dynamic model considered in the
present study.
2. Problem Statements
2.1. Model formulationWe consider an incompressible, Newtonian fluid flowing
through a straight channel of a square cross-section with the
width W, the height H (or, referred to as depth), and the
length L (cf. W = H). The hydraulic diameter Dh represents
a size of the channel, defined as Dh = 2WH/(W + H). In Fig.
1, a rectangular coordinate is set up in a way that x-axis
goes along the channel width; y-axis is put parallel to the
channel height; and z-axis heads to the downstream. The
origin of the coordinate is given at the lower right corner of
the square channel from the viewpoint far behind the inlet.
Time-periodic pressure drop ∆p applied to both ends of
the channel with oscillating frequency f and time t is given
in the form of sine function:
. (1)
Here, pi and po stand for nominal values of pressure
imposed at the inlet and outlet of the channel, respectively,
and 2πf means the angular frequency. Other types of pres-
sure drop are also available. For example, if a rectangular
pressure drop is used, the equation describing the pressure
drop (e.g., Eq. (1)) needs additional factor such as “duty
cycle” which indicates during what portion of time in a
cycle a pressure is imposed (Tikekar et al., 2010). The
pressure drop is x or y independent, which is expected to
cause an oscillation of axial fluid velocity vz(t) with back-
and-forth standing mode. Since the fluid is initially at rest,
there will be a certain start-up behavior (Leal, 1992) that
becomes attenuated in the end, underlying the velocity pro-
file in a periodic mode. As a result, no net flux would be
expected within a cycle (or a period).
Ignoring an end effect, the N-S equation of motion for a
laminar flow assumption can be written as:
, (2)
where ρ and µ are the density and the viscosity of the fluid.
Equation (2) is subjected to a no-slip boundary condition
such that
. (3)
The use of the homogeneous condition implies that the
channel hydraulic diameter is sufficiently large (e.g., Dh >
10 µm) that complex interfacial phenomena (Chun et al.,
2005) can be ignored.
2.2. Nondimensionalization Once the velocity profile is obtained from Eqs. (2) and
(3), the mean velocity vzm can be estimated, given by
. (4)
Equations (1), (2), and (3) can be rewritten in dimen-
sionless forms by rearrangements with employing the non-
dimensionalized variables: dimensionless time τ = (2π f )t,
dimensionless lateral coordinate X = x/Dh, dimensionless
transverse coordinate Y = y/Dh, and P = p/(pi − po). The
velocity Vz = vz/vzm,st is normalized by the mean velocity
obtained when a constant pressure drop (i.e., pi − po) is
imposed to the channel as vzm,st = (pi − po)W
2/aµL, because
the mean velocity of pulsatile flow vzm has a tendency to
become zero. The geometric constant a equals 28.264 for
the square cross-section channel (W = H = Dh), while the
full expression can be found in the work by Fuerstman et
al. (2007). A reciprocal to an angular frequency corre-
sponds to the characteristic time (= 1/(2π f )). Then, we
obtain the followings
p∆ po pi–( ) 2πft( )sin=
ρ∂vz
∂t-------
p∆L
------– µ∂2
vz
∂x2
---------∂2
vz
∂y2
---------++=
vz x 0=vz x W=
vz y 0=yz y H=
0= = = =
vz
mt( )
vz t( ) AdArea
∫A
---------------------------≡vz t x y, ,( ) xd yd
0
W
∫0
H
∫WH
-------------------------------------------=
Fig. 1. Top view of axial velocity profiles formed across a chan-
nel and their mean velocities: oscillating (solid) and constant
(dashed) pressure drop. The y-coordinate comes out from the
paper.
Pulsatile Poiseuille flows in microfluidic channels with back-and-forth mode
Korea-Australia Rheology J., Vol. 24, No. 2 (2012) 91
, (5)
(6)
. (7)
Note that Eq. (6) is written in terms of the Womersley number
Wo, which means the ratio of oscillation to a viscous effect
(Womersley, 1955) and defined as . It is
already known that a flow with Wo < 1 shows no or neg-
ligible retardation in a mean velocity and time delay, while
a flow with Wo > 10 shows significant retardation and time
delay.
3. Numerical Solver
In the solution of Eq. (6), the parabolic type partial dif-
ferential equation is spatially discretized to a set of ordi-
nary differential equations with respect to time. It follows
a two-dimensional orthogonal collocation on finite ele-
ment (OCFE) strategy (Finlayson, 1980; Bialecki and
Fernandes, 2009). Then, the resulting initial value prob-
lems (IVPs) have been integrated up by using an adaptive
Runge-Kutta (R-K) method. The solution algorithm is
illustrated in Fig. 2.
3.1. Spatial discretizationIn the OCFE strategy, the square computational domain
(i.e., 0 < X < 1, 0 < Y < 1) is first divided evenly into 3×3
square subdomains. Note that the number of subdomains is
adjustable according to a specific accuracy required. For
simple application of an OCFE, coordinates in all sub-
domains are redefined to 0 < Xm,n < 1 and 0 < Ym,n < 1
where the subscripts m and n indicate the lateral and the
transverse sequences of a subdomain, respectively. In one
subdomain, the collocation points are located at the zeros
of the eleventh-order Legendre polynomial that is shifted
and fitted to 0 < Xm,n < 1 and 0 < Ym,n < 1, respectively.
Hence, the collocation points are not evenly spaced, the
distribution is symmetrical to 0.5 and more populated near
0 and 1.
The computational nodes where Eq. (6) will be evaluated
are the elements of the Cartesian set of {Xm,n} and {Ym,n}.
The same approach is also applied to all boundaries, e.g.,
(0, Ym,n), (1, Ym,n), (Xm,n, 0), and (Xm,n, 1). The number of
total computational nodes is 1,521 (= 13×13×9). The
detailed formulation and application of OCFE can be
found in the previous work (Kim and Simon, 2009). There
are total 24 boundaries: 12 (= 3/wall for 4 walls) in contact
with the channel wall plus 12 in contact with adjacent sub-
domain. For the former boundaries, predescribed Eq. (7) is
applied. For the latter boundaries, the continuities of veloc-
ity and shear stress conditions like below hold for each
direction:
, (8)
, (9)
, (10)
. (11)
3.2. Initial value problem with R-K methodBy using the OCFE scheme described above, Eq. (6) is
discretized into a set of the ordinary differential equation
with respect to time. These IVPs are solved by using an
adaptive R-K method, in which the step size (∆t) is auto-
matically adjusted during iterations. From the initial veloc-
ity profile and the initial stepsize, the truncation error
difference between the R-K fourth and the fifth order meth-
ods is computed. If the error is less than the preset lower
limit (εmin), the stepsize will be doubled. On the contrary, if
the error is greater than the preset upper limit (εmax), the
stepsize will be cut in half (see Fig. 2). Then, using a new
stepsize, the truncation error is estimated again. This loop
is broken when the truncation error falls between εmin and
εmax, and the results from the R-K fifth order method is
P∆ τ( )sin–=
∂Vz
∂τ--------
a P∆
Wo2
----------–1
Wo2
----------∂2
Vz
∂X2
----------∂2
Vz
∂Y2
----------+ ,+=
Vz X 0=Vz X 1=
Vz Y 0=Vz Y 1=
0= = = =
Wo Dh 2πf µ ρ⁄( )⁄=
Vz
m n,
Xm n,
0=Vz
m 1 n,–
Xm 1 n,–
1= for m 2 3,=( )=
∂Vz
m n,
∂X------------
Xm n,
0=
∂Vz
m 1– n,
∂X-----------------
Xm 1 n,–
1=
for m 2 3,=( )=
Vz
m n,
Ym n,
0=Vz
m n 1–,
Ym n 1–,
1= for n 2 3,=( )=
∂Vz
m n,
∂Y------------
Ym n,
0=
∂Vz
m n 1–,
∂Y-----------------
Ym n 1–,
1=
for n 2 3,=( )=
Fig. 2. Framework of numerical algorithm for unsteady gov-
erning equation in the present study.
Kwang Seok Kim and Myung-Suk Chun
92 Korea-Australia Rheology J., Vol. 24, No. 2 (2012)
recorded and taken as the initial values for the next iter-
ation.
In this work, εmin = 10-6 and εmax = 10-5. For fast com-
putation, the R-K fourth and fifth order integrations follow
the Fehlberg’s suggestion such that
, (12)
, (13)
where the coefficients ki’s are given in Lejeunes et al.
(2011).
4. Results and Discussion
In the simulation, results are presented for hydraulic
diameters in the range 10 µm≤Dh≤500 µm, and for chan-
nel length of 3 cm. The effects of electrokinetic and hydro-
dynamic interactions (cf. electroviscous effect, fluid slip,
surface conductance, and etc.) become pivotal to the char-
acterization of the velocity profile as the Dh is reduced less
than 10 µm. However, we do not take them into consid-
eration because, according to our experiences (not shown
here), acceleration by pulsation makes negligible differ-
ence in a velocity profile from a steady-state value (with a
constant ∆p) in the channels with the order of micrometers.
The oscillating frequency is varied from 1 to 100 Hz, and
the fluid properties (ρ = 0.993 g/cm3, µ = 0.68×10-2 g/m·s)
are taken from water at 310 K (Lide, 2006). The maximum
pressure drop is set to 3 mbar, providing the mean velocity
to be in the range from 5.3×10-2 mm/s (for Dh = 10 µm) to
133 mm/s (for Dh = 500 µm). In this condition, the Wo
ranges from 3.1×10-2 (for Dh = 10 µm and f = 1 Hz) to
15.5 (for Dh = 500 µm and f = 100 Hz). The Reynolds
number Re (=ρvzm,stDh/µ) is calculated to 8.2×10-4 (for Dh
= 10 µm) and 103 (for Dh = 500 µm), verifying the laminar
flow assumption used in setting up Eq. (2).
4.1. Periodic steady-state velocity profileThe periodic steady state refers to the condition that a
quantity is a periodic function of time. Due to an oscil-
lating pressure drop, the axial fluid velocity, initially at rest,
should undergo start-up behavior, which is gradually
diminished. In the end, the velocity will be put into a peri-
odic steady state with the same frequency as the pressure
does. We will deal with the periodic steady state first, and
the start-up behavior later.
V t t∆+( ) V t( ) 25
216---------k1
1408
2565------------k3
2197
4104------------k4
1
5---k5–+ +⎝ ⎠
⎛ ⎞ t∆+=
V t t∆+( ) V t( )=
16
135---------k1
6656
12825---------------k3
28561
56430---------------k4
9
50------k5–
2
55------k6+ + +⎝ ⎠
⎛ ⎞ t∆+
Fig. 3. (Color online) Periodic steady-state velocity profiles formed in a square cross-section with Dh = 500 µm at f = 50 Hz. Sequences
are (a), (b), (c), (d), and (a) with time interval of 5 ms equivalent to a quarter period.
Pulsatile Poiseuille flows in microfluidic channels with back-and-forth mode
Korea-Australia Rheology J., Vol. 24, No. 2 (2012) 93
In Fig. 3, the representative velocity profiles in a periodic
steady state (= vzm/vz
m,st for sufficiently large t) are shown
with back-and-forth mode at Dh = 500 µm and f = 50 Hz.
The time interval between consecutive images is 5 msec
which is equivalent to a quarter period. The Wo of this con-
dition is estimated to 11.0. As expected from the high Wo,
the retardations are observed in the center region of Figs.
3(a) and 3(c). The pressure drop in Fig. 3(a) is imposed
forward. Accordingly, the fluid near the channel walls
(especially near four corners) moves forward. However,
the fluid around the center still shows a backward flow
(blue colored). This is because the fluid near the channel
walls is subject to strong friction by the stationary wall
attenuating the backward inertia as well as oscillatory
motion in a short time. On the other hand, the friction
becomes weak around the center so that relatively larger
inertia causes the retardation phenomena.
4.2. Mean velocity behaviorAs mentioned before, an introduction of the pulsatile
flow to micro and biofluidics brings about two issues: the
reduction in flow rate (or mean velocity) and the time
delay (cf. phase difference). A ratio of the mean velocity of
pulsatile flow to that of steady flow with constant pressure
drop is estimated as
. (14)
By using Eq. (14), an example of a time course of a dimen-
sionless mean velocity Vzm is presented in Fig. 4. The solid
curve is the dimensionless mean velocity (the values are
read from the left axis), and the dashed curve is the pres-
sure drop (the values are read from the right axis). The
shaded area is bounded by two lines that pass through all
local maxima and minima of each peak as the time elapse,
which is named the range of the mean velocity, corre-
sponding to twice the amplitude. The Vzm range is decreas-
ing at initial stage, but a periodic steady state occurs and
the range does not change anymore after a certain time
(i.e., indicated as a start-up time tS in Fig. 4). We can deter-
mine the start-up time tS, as described in the later section.
Fig. 5 demonstrates the half range (or amplitude) of
dimensionless mean velocity at its periodic steady state
(i.e., t > tS) for various channel hydraulic diameters and
oscillating frequencies. Since the mean velocity is nor-
malized with respect to vzm,st, a possible value of the max-
imum in a half range is one. This state can be obtained by
applying either very low frequency (i.e., f < 1 Hz with solid
curve) or small channel cross-section with Dh < 100 µm.
Oscillation with higher frequency and/or in a wider chan-
nel leads to a smaller range, implying the decrease in the
fluid displacements around its initial position (i.e., ampli-
tude). In the practical view point, provided that 50 Hz is
imposed to the LOC, the channel hydraulic diameter
should be less than about 50 µm in order to obtain a half
range close to one. On the other hand, if channel hydraulic
diameter larger than 250 µm and the frequency was greater
than 50 Hz, it is expected that the flow rate through the
channel will be only about 60% or less than that of the con-
stant-pressure-drop flow.
Fig. 6 shows the variations of time delay tD, which is a
phase difference between the pressure drop (∆p) and the
mean velocity (vzm). This parameter plays an important role
in appropriate and durable operations of the miniaturized
devices by means of optimum timing (e.g., opening or
Vz
mτ( )
vz
mτ( )
vz
m st,
------------ Vz τ X Y, ,( ) Xd Yd0
1
∫0
1
∫= =
Fig. 4. Time evolution of pulsatile dimensionless mean velocity
(solid) and pressure drop (dashed) at Dh = 500 µm and f = 50 Hz.
Fig. 5. Half range of periodic steady-state velocity for various
values of Dh and f.
Kwang Seok Kim and Myung-Suk Chun
94 Korea-Australia Rheology J., Vol. 24, No. 2 (2012)
closing in the actuator). The time delay can be determined
in Fig. 4, by subtracting the n-th zero of pressure drop (i.e.,
(n+1/2)/f in Eq. (1)) from that of the mean velocity for suf-
ficiently large n. In Fig. 6, the time delay decreases with
increasing frequency for the whole range of Dh, but this
trend becomes weak for f > 10 Hz.
4.3. Start-up timeNow, we are turning our attention to the transient state
that the flow should undergo before the periodic steady
state. To this end, the start-up time tS is defined as the time
that elapses until the periodic steady state is achieved. One
can determine it as the earliest time point at which a dif-
ference between the maximum value of mean velocity (i.e.,
vzm,max/vz
m,st) and the half range at a large time (i.e., vzm/2 for
large t) should be less than 1%.
Fig. 7 represents the start-up time tS for various channel
hydraulic diameters and oscillating frequencies. It is evi-
dent that the start-up time shows an increasing tendency
along the channel hydraulic diameter at a given frequency
except f = 1 Hz. We would point out that the flow system
reaches the periodic steady state immediately after the pul-
satile pressure is applied if Dh is less than 100 µm regard-
less of the frequency. The start-up time increases with
increasing f up to 50 Hz, but f dependency is negligible
between 50 and 100 Hz.
5. Conclusions
In the present study, characteristics of a time-dependent
flow through microfluidic channels of the square cross-sec-
tion have been explored by introducing an oscillating pres-
sure drop. The N-S equation was solved by the orthogonal
collocation on finite element scheme combined with the
adaptive R-K integration. The ranges chosen as 10 µm ≤
Dh ≤ 500 µm and 1 Hz ≤ f ≤ 100 Hz are frequently encoun-
tered in the micro and biofluidic systems.
Defining a periodic steady state of pulsatile flow with
back-and-forth mode, we observed the retardation phe-
nomena caused by a relatively large inertia. In a period of
an oscillating flow, the axial velocity profile at the channel
center can have the opposite direction to the velocity near
the walls. The amplitude of mean velocity decreases with
increasing channel hydraulic diameter, and both amplitude
of mean velocity and time delay decrease with increasing
oscillating frequency. The start-up time increases with
increasing channel hydraulic diameter and frequency up to
50 Hz. Our findings are expected to provide helpful infor-
mation when relevant flow systems through microfluidic
channels are designed and fabricated.
Nomenclatures
a : geometric constant, 28.264 [-]
Dh : channel hydraulic diameter [µm]
f : oscillating frequency [s-1]
H : channel height [µm]
L : channel length [mm]
P : dimensionless pressure [-]
p : pressure [bar]
Re : Reynolds number [-]
tD : time delay [s]
tS : start-up time [s]
Vz : dimensionless axial velocity [-]
Vzm : dimensionless mean velocity [-]
Fig. 6. Time delay for various values of Dh and f.
Fig. 7. Start-up time for various values of Dh and f.
Pulsatile Poiseuille flows in microfluidic channels with back-and-forth mode
Korea-Australia Rheology J., Vol. 24, No. 2 (2012) 95
vz : axial fluid velocity [µm·s-1]
vzm : mean velocity of pulsatile flow [µm·s-1]
vzm,st : mean velocity of steady flow [µm·s-1]
W : channel width [µm]
Wo : Womersley number [-]
x : lateral coordinate [µm]
X : dimensionless lateral coordinate [-]
y : transverse coordinate [µm]
Y : dimensionless transverse coordinate [-]
z : streamwise axial coordinate [µm]
Greek Letters
ρ : density [g·cm-3]
µ : fluid viscosity [g·m-1s-1]
τ : dimensionless time [-]
Acknowledgments
This work was supported by the Basic Research Program
(No. 20100021979) from the National Research Founda-
tion (NRF) of Korea, and the authors would like to grate-
fully acknowledge it.
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