pulsatile poiseuille flows in microfluidic channels with back-and...

© 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.


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.



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.



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.



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.



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.

References

Bialecki, B. and R.I. Fernandes, 2009, An alternating direction

implicit backward differentiation orthogonal spline collocation

method for linear variable coefficient parabolic equations,

SIAM J. Numer. Anal. 47, 3429-3450.

Bode, K., R.J. Hooper, W.R. Paterson, D.I. Wilson, W. Augustin,

and S. Scholl, 2007, Pulsed ow cleaning of whey protein foul-

ing layers, Heat Transfer Eng. 28, 202-209.

Chun, M.-S., T.S. Lee, and N.W. Choi, 2005, Microfluidic anal-

ysis of electrokinetic streaming potential induced by micro-

flows of monovalent electrolyte solution, J. Micromech.

Microeng. 15, 710-719.

Chun, M.-S., 2011, Electrokinetic secondary-flow behavior in a

curved microchannel under dissimilar surface conditions, Phys.

Rev. E 83, 036312/1-9.

Edwards, M.F. and M.A. Wilkinson, 1971, Review of potential

applications of pulsating flows in pipes, Trans. Inst. Chem.

Eng. 49, 85-94.

Fan, C. and B.-T. Chao, 1965, Unsteady, laminar, incompressible

flow through rectangular ducts, ZAMP 16, 351-360.

Finlayson, B.A., 1980, Nonlinear Analysis in Chemical Engi-

neering, McGraw-Hill, New York.

Fuerstman, M.J., A. Lai, M.E. Thurlow, S.S. Shevkoplyas, H.A.

Stone, and G.M. Whitesides, 2007, The pressure drop along

rectangular microchannels containing bubbles, Lab Chip 7,

1479-1489.

Gong, L., J. Wu, L. Wang, and K. Cao, 2008, Streaming potential

and electroviscous effects in periodical pressure-driven micro-

channel flow, Phys. Fluids 20, 063603/1-7.

Hodis, S. and M. Zamir, 2011, Coupled radial and longitudinal

displacements and stresses within the arterial wall in pulsatile

flow under tethered and free-wall conditions, Phys. Rev. E 83,

051923/1-8.

Hu, Y., A. Gopal, K. Lin, Y. Peng, E. Tasciotti, X. Zhang, and M.

Ferrari, 2011, Microfluidic enrichment of small proteins from

complex biological mixture on nanoporous silica chip, Biomi-

crofluidics 5, 013410/1-7.

Kim, K.S. and L. Simon, 2009, Optimal intravenous bolus-infu-

sion drug-dosage regimen based on two-compartment phar-

macokinetic models, Comput. Chem. Eng. 33, 1212-1219.

Kim, K.S., W. Ryoo, M.-S. Chun, G.-Y. Chung, and S.O. Lee,

2012, Transport analysis in reverse electrodialysis with pul-

satile flows for enhanced power generation, Korean J. Chem.

Eng. 29, 162-168.

Leal, L.G., 1992, Laminar flow and convective transport, But-

terworth-Heinemann, Boston.

Lejeunes, S., A. Boukamel, and S. Méo, 2011, Finite element

implementation of nearly-incompressible rheological models

based on multiplicative decompositions, Comput. Struct. 89,

411-421.

Leslie, D.C., C.J. Easley, E. Seker, J.M. Karlinsey, M. Utz, M.R.

Begley, and J.P. Landers, 2009, Frequency-specific flow con-

trol in microfluidic circuits with passive elastomeric features,

Nat. Phys. 5, 231-235.

Lide, D.R., 2006, CRC Handbook of Chemistry and Physics,

87th ed., CRC Press, Boca Raton.

Morris, C.J. and F.K. Forster, 2004, Oscillatory flow in micro-

channels-comparison of exact and approximate impedance

models with experiments, Exp. Fluids 36, 928-937.

Stone, H.A., A.D. Stroock, and A. Ajdari, 2004, Engineering

flows in small devices: microfluidics toward a lab-on-a-chip,

Annu. Rev. Fluid Mech. 36, 381-411.

Tikekar, M., S.G. Singh, and A. Agrawal, 2010, Measurement

and modeling of pulsatile flow in microchannel, Microfluid

Nanofluid 9, 1225-1240.

Vedel, S., L.H. Olesen, and H. Bruus, 2010, Pulsatile microf-

luidics as an analytical tool for determining the dynamic char-

acteristics of microfluidic systems, J. Micromech. Microeng.

20, 035026/1-11.

Womersley, R.J., 1955, Method for the calculation of velocity,

rate of flow and viscous drag in arteries when the pressure drop

is known, J. Physiol.-London 126, 553-563.

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