Steady streaming: A key mixing mechanism in low-Reynolds-number acinar
Haribalan Kumar,1Merryn H. Tawhai,2Eric A. Hoffman,3and Ching-Long Lin1,a?
1Department of Mechanical and Industrial Engineering, The University of Iowa, Iowa City,
Iowa 52242, USA and IIHR-Hydroscience and Engineering, The University of Iowa, Iowa City,
Iowa 52242, USA
2Bioengineering Institute, The University of Auckland, Auckland 1010, New Zealand
3Department of Biomedical Engineering, The University of Iowa, Iowa City, Iowa 52242, USA;
Department of Internal Medicine, The University of Iowa, Iowa City, Iowa 52242, USA; and
Department of Radiology, The University of Iowa, Iowa City, Iowa 52242, USA
?Received 27 June 2010; accepted 14 February 2011; published online 18 April 2011?
Study of mixing is important in understanding transport of submicron sized particles in the acinar
region of the lung. In this article, we investigate transport in view of advective mixing utilizing
Lagrangian particle tracking techniques: tracer advection, stretch rate and dispersion analysis. The
phenomenon of steady streaming in an oscillatory flow is found to hold the key to the origin of
kinematic mixing in the alveolus, the alveolar mouth and the alveolated duct. This mechanism
provides the common route to folding of material lines and surfaces in any region of the acinar flow,
and has no bearing on whether the geometry is expanding or if flow separates within the cavity or
not. All analyses consistently indicate a significant decrease in mixing with decreasing Reynolds
number ?Re?. For a given Re, dispersion is found to increase with degree of alveolation, indicating
that geometry effects are important. These effects of Re and geometry can also be explained by the
streaming mechanism. Based on flow conditions and resultant convective mixing measures, we
conclude that significant convective mixing in the duct and within an alveolus could originate only
in the first few generations of the acinar tree as a result of nonzero inertia, flow asymmetry, and large
Keulegan–Carpenter ?KC? number. © 2011 American Institute of Physics. ?doi:10.1063/1.3567066?
Several studies have attempted to identify mechanisms
of mixing in low-Re flows. Some of the earlier works in this
regard are Refs. 1–7 and among others. The application areas
include transport of material in processing industries, mixers,
microfluidic applications, and physiological flows. In addi-
tion to these applications, the study of mixing is important in
understanding the transport of particles in the conducting and
respiratory regions of the lung. Understanding of transport to
and within the acinar region has practical applications in im-
proving delivery strategies of pharmaceutical aerosols or
other drugs, targeting deposition to specific locations and
henceforth reducing systemic absorption, and also for im-
proving estimates for retention of inhaled pollutants. Without
the assistance of turbulent mixing, how the low-Re acinar
flow achieves effective mixing is the topic of interest in this
Lung morphology and relevant terminologies are intro-
duced here. Alveoli are air pockets that occupy a part of, or
completely cover the walls of respiratory airways; on aver-
age beyond the 15th generation of the human airway tree. An
“acinus” consists of the entire region of alveoli and alveo-
lated ducts that are distal to the terminal bronchus. The al-
veoli can be visualized as an isolated or group of open cavi-
ties ventilated by the acinar airway. The typical Re ranges
approximately from 2 in the transitional region to 0.01 in the
terminal sacs. For further reading on acinar morphology and
morphometry, the reader is referred to Refs. 8 and 9.
During normal breathing, when the inspired volume is
larger than the anatomical deadspace, the inspired gas
“mixes” with the residual gas in the lung. In this process,
particles are transferred to the residual gas across the
inspired-residual interface front, which in case of aerosols is
referred to as aerosol mixing. Peclet number ?Pe=U0D/K,
where U0is the mean fluid velocity, D is the duct diameter,
and K is the diffusion coefficient? relates the magnitude of
convective to diffusive transport. For gas mixing in the aci-
nus, Pe?0.1–1, while for aerosol particles ?say, ?1 ?m
diameter?, Pe=3000–20 000. Consequently, convection and
Particles with diameter 0.5–1 ?m have very low depo-
sition efficiencies in the acinus and behave like nondiffusing
massless fluid particles.11Particles in this size range play a
very important role in various physiological processes.12
Heyder et al. performed mixing estimates with aerosol bolus
consisting of ?1 ?m particles.13The dispersion of the in-
haled bolus increased with increasing penetration volume.
The net transport of particles from the ?particle-laden? in-
spired air to the residual air was shown to occur as a result of
irreversible processes whose origins were unknown. This
motivated investigations of possible transport mechanisms
a?Author to whom correspondence should be addressed. Present address:
Department of Mechanical and Industrial Engineering, The University of
Iowa, Iowa City, Iowa 52242-1527, USA. Tel.: ?13193355673. FAX:
?13193355669. Electronic mail: email@example.com.
PHYSICS OF FLUIDS 23, 041902 ?2011?
1070-6631/2011/23?4?/041902/21/$30.00 © 2011 American Institute of Physics
and their origin. The accomplishment of mixing, particularly
in viscous flows deep in the lung are nontrivial. Advective
mixing was proposed as one such mechanism.
The early proponents of various acinar mixing mecha-
nisms within alveoli include Refs. 14–18. Tsuda et al. and
Henry et al. observed mixing in particle motion associated
with recirculation in an acinar model.14,18Tsuda et al. per-
formed experimental flow visualization in a rat lung using
blue and white colored dyes for inspired and resident
fluids.15Lateral images of acinar airways revealed delineated
interface patterns between the two dyes. After four breathing
cycles, an indistinguishable blue-white uniformity appeared
indicating a high degree of mixing. Recently, Henry et al.
demonstrated that alveolation is sufficient to produce con-
vective mixing in a rigid-wall oscillatory flow model with Re
pertaining to proximal generations of the acinus.19This ob-
servation completely shifts the onus from mixing originating
due to time-dependent wall motion and saddle point as
thought earlier. It also shifts the focus toward geometrical
features apart from revealing that even fundamental mixing
mechanisms are not completely understood. Sarangapani and
Wexler commented that the contribution from mixing toward
a dramatic increase in the interface area requires numerous
cycles and hence cannot completely explain the observed
dispersion in a single cycle.20Other works include those of
Lee and Lee, who used 30 identical toroidal alveolar cells
and modeled inspiration and expiration in isolated phases.21
The differences in dispersion between alveolated and nonal-
veolated tubes in the presence and absence of wall motion
were compared. From previous experimental and numerical
studies, although various possible causes have been sug-
gested, consistent and concrete evidence to origins of con-
vective mixing in the acinus is still missing. Darquenne and
Prisk compared dispersion of aerosol bolus between simula-
tion and experiments.22For particles of critical sizes 0.5 and
1 ?m in diameter, order-of-magnitude discrepancy was ob-
served in a zero-gravity environment. Flow-induced mixing
was suggested as one possible cause. Later, Darquenne and
Prisk designed a flow reversal ?FR? mechanism to study the
mixing in acinar flow.23Boli of particles were inhaled in
microgravity at two different penetration volumes. It was
then followed by a breathhold during which several flow
reversals were imposed. They found that “either the phenom-
enon of stretch and fold did not occur within the number of
FR… or that it had already occurred during the one breathing
cycle included in the basic maneuver.”
Dispersion and mixing of aerosol boluses may be suffi-
ciently modeled by simple advection. The difference in fluid
particle location between the beginning and end of one os-
cillatory flow cycle may be recognized as a drift in a La-
grangian sense. This drift, large or small, is gained by a
particle purely due to flow topology, inertial and geometry
effects. Through-out this work, streaming and the associated
drift will be underlined as the fundamental mechanism. In
acinar flows, the flow conditions cause a large drift and
hence large increases in dye interface area from one cycle to
the next, which is commonly referred to as advective mixing.
Even though pure advection is considered, the term “kine-
matic irreversibility”18is more commonly used in this con-
text, which in essence is the Lagrangian drift at the end of
The objective of this work is to investigate the origin of
advective mixing and quantify mixing in physiologically mo-
tivated models of the acinus using three Lagrangian tech-
niques: tracer advection, stretching analysis, and axial dis-
persion. Unlike previous acinar mixing studies whose
attention was restricted to a specific region, this paper con-
siders mixing in three regions: the duct, the alveolar mouth,
and the alveolus. Of the three techniques, tracer advection
and stretching analysis provide a measure of the mixing rate
within the alveolus. The topological critical points, such as a
stagnation saddle point in the flow, are also identified for
assessing their roles in mixing from the viewpoint of chaotic
mixing.14,15The study of acinar mixing is not complete with-
out understanding axial transport in the alveolated duct.
Axial dispersion and effective diffusivity are estimated in the
alveolated airways. The remainder of the paper is organized
as follows. Section II describes the model and equations for
the different analysis techniques. Section III presents the re-
sults. For clarity, we first use a simple two-dimensional ?2D?
alveolated channel to illustrate some important, but not well
understood, dispersion and mixing mechanisms in a low-Re
multiple-open-cavity model before considering a three-
dimensional ?3D? geometry. Section IV summarizes conclu-
sions and discusses physiological implications.
A. Lagrangian methods
Geometrical structures that are representative of differ-
ent regions of the acinus, namely respiratory bronchioles and
alveolar ducts are shown in Fig. 1. Henceforth, we use “Case
I” ?Fig. 1?a?? to denote respiratory bronchioles which have
occasional alveoli. “Case II” ?Fig. 1?b?? is a model for alveo-
lar duct lined completely with 18 alveoli representing the
lung units in generations 18–22. The alveolar duct is well
defined in Case I. On the other hand, the walls of the duct in
Case II are not clearly defined and are formed from the space
between surrounding alveoli. The proximal wall of the alveo-
lus is conventionally the one closest to the alveolus mouth
during inspiration. Similarly the proximal generation is the
generation of airways, which has already been ventilated
along the path of the air. Analogously, a distal wall and a
distal generation are defined. To simulate the alveolar flow, a
sinusoidal flow rate is specified at the ductal entrance “E” in
Figs. 1?a? and 1?b?. The walls of alveoli and duct expand and
contract uniformly with a prescribed volumetric expansion of
25%. The Reynolds number ?Re=U0D/?? is measured at
peak inspiration, where U0is the average peak inspiratory
speed at the ductal entrance, D is the effective diameter of
the duct, and ? is the kinematic viscosity of air. The incom-
pressible Navier–Stokes equations are solved in an Arbitrary
Lagrangian–Eulerian framework.24,25The equations are nor-
=416 ?m and peak velocity of U0=0.032 m/s. A sinusoidal
flow rate is specified at one end of the duct while a Neumann
boundary condition is employed at the other. Homothetic
wall motion, where corresponding sides of the duct and al-
mouth dimension of
041902-2Kumar et al. Phys. Fluids 23, 041902 ?2011?
veolar wall remain parallel in a geometric expansion or con-
traction is prescribed. The domain is discretized using tetra-
hedral elements with a smallest element having a size of
?0.009L. A single alveolar unit consists of ?40 000 ele-
ments. For a detailed description of the computational
method and boundary conditions, the reader is referred to
Kumar et al.26The breathing period ?T? is chosen as 2.5 s ?at
24 breaths/min? to match the Womersley number ?Wo
=D??/?=0.2, ?=2?/T is the angular frequency? in previ-
ous studies.14,19The inverse of Strouhal number ?St? is
known as Keulegan–Carpenter number,27KC=1/St=U0T/L.
It determines the displacement or length of fluid particle ex-
cursion over a characteristic length, L. For analysis, Case I
?with Re=2, 1, and 0.52? and Case II ?with Re=1, 0.6, and
0.2? are investigated. For Case I, Re=2, 1, and 0.52 yield KC
of 386, 193, and 96.5, respectively. An additional case with
Re=0.06 of Case I is included to demonstrate that advection
with Re approaching zero, predicted by the current analysis,
exhibits essentially reversible behaviors. The flow phenom-
ena along the acinar pathway are commonly associated with
the fractional flow rate QA/QDintroduced by Tsuda et al.14
Here, QAis the alveolar flow rate and QDis the ductal en-
trance flow rate. Note that in a rigid-wall model, QA/QD=0.
The amount of alveolar expansion determines QAwhile the
volume change rate of the air volume in generations distal to
the current generation determines QD. This ratio, by defini-
tion increases down the acinar tree. For example, based on
the ductal flow rate in Case I, QA/QD=0.0024 for Re=2 and
FIG. 1. Representative geometrical models for regions of the acinus. ?a? “Case I” model for respiratory bronchiole, ?b? “Case II” model for alveolar duct,
where ?A, AD, and E? denote ?alveoli, alveolar duct, and ductal entrance?, ?c? and ?d? shows two presentations of an alveolar sac, denoted by ALV in ?a? to
be analyzed in Sec. III. ?c? ALV with solid and dashed edges for front and rear faces, respectively. ?d? ALV with three planes to show the orientation of the
unit. The y-direction is the axial direction. Some results are displayed in the y-z plane for clarity.
041902-3 Steady streaming: A key mixing mechanismPhys. Fluids 23, 041902 ?2011?
QA/QD=0.0047 for Re=1. For isolated alveolar representa-
tions, we use QA/QDand Re to identify flow conditions in
Cases I and II. For Case I with Re=1, due to nonzero QAthe
Re at the exit of the model is approximately reduced by 4%.
The transport model of small ?less than ?1 ?m? aerosol
particles across the interface of residual and inhaled air re-
duces to advection equation as sedimentational and deposi-
tional effects become negligible. Advection is governed by
the 3D kinematic vector equation:
dt= u?x,t?,x?0? = x0,
where x is the position vector of a passive particle ?some-
times referred as a marker?; u is the numerically generated
velocity field and x0is an initial condition. The advection is
passive involving no diffusion of particles and the particle
velocity exactly matches the fluid velocity. The advection is
carried out using a fourth-order Runge–Kutta scheme and a
time-step independent solution is ensured with a choice of
?t=T/500 000. For details on a typical procedure to solve
for particle transport, the reader is referred to Wang et al.28
All simulations are carried up to five cycles unless specified
The mixing measures used in this study are designed to
clarify the role of flow topology and geometry in the process
of tracer transport. Three independent techniques have been
used: tracer advection, stretching analysis, and axial disper-
sion. Each of these is chosen toward a specific objective and
has relative advantages. All three techniques are based on
massless particle tracking. The formulation details are given
below. Material advection ?i.e., tracer deformation? involves
passive tracking of tracers placed at strategic locations in a
flow. The tracer or dye is constructed from a uniform distri-
bution of particles. The second technique, stretch rate analy-
sis, is based on evolution of unit line elements computed
from velocity and its gradient. This technique is an extension
of the line element approach in Roberts and Mackley
?1995?29to 3D. Note that particle advection could also be
used to compute stretch rate by considering relative separa-
tion distance of adjacent particles, which however is a lower-
order approximation as it relies on only fluid velocity and
fails in high stretch regions.29The stretch rate computation
based on the line element approach relies on higher-order
approximation using fluid velocity and its gradient. In this
approach, each particle location is tagged with line elements,
= m . ?u-?D ? :mm?m,
= D ? :mm,
nT?D ln ?
where m=?dx,dy,dz?Tis the orientation vector of any line
element with ?mm?=1; u is the fluid velocity, D ? is the sym-
metric stretching tensor, and D?ln ??/Dt is the stretching
function. Time-averaged stretch rate, sl, is obtained from the
instantaneous stretching function ?of dimension s−1? aver-
aged from three initially orthogonal line elements.
The third measure of mixing employed here is axial dis-
persion. In this approach, the variance of particle displace-
ment is calculated based on axial location of particles.30The
initial condition is a bolus of particles released at the en-
trance of the alveolar duct. For a bolus consisting of N mark-
ers, the axial mean ?y? and variance ?y
breathing cycle are defined as
2at the end of the nth
?yi?nT? − yi?t0??/N,
??yi?nT? − yi?t0?? − ?y??2/?N − 1?,
where nT is the total time after “n” breathing periods and t0
may be the initial release time or some reference state, say
the end of first breathing period. Because the current acinar
model consists of alveoli attached to a straight duct, they
represent a section of the acinar tree. A periodic boundary
condition is employed so that particles exiting the domain
are allowed to reenter thus approximating multiple alveolar
units attached to the duct.
B. Eulerian streaming, Stokes drift, and Lagrangian
The mechanism of Lagrangian drift outlined here derives
its background from the well-known “steady streaming” in
oscillatory flows.27,31–37In an oscillatory flow setting, a non-
zero mean flow averaged over one time period may be ob-
served. This nonzero mean flow can result in significant drift
of particles at end cycle called steady streaming. An entire
treatise on steady streaming was given by Riley.33Suh and
Kang presented different instances of streaming and the im-
portance of Stokes drift.37Larrieu et al. presented an analyti-
cal treatment of drift in a simple setting of Couette flow
weakly perturbed by a wavy bottom.34For any given passive
particle under the assumption of small displacement,37Eq.
?1? with x?t=0?=x0may be expanded in a Taylor series.
dt= u?x,t? ? u?x0,t? + ??x · ??u?x0,t? + H.O.T.,
where the small displacement ?x=?0
indicates higher-order terms of the series. Let the time aver-
age of a variable ? over one period of T be denoted as ???.
Then, to the first-order approximation,
tu?x0,t?dt and “H.O.T.”
x?T? − x?0?
= uL?x0? = uE?x0? + uS?x0?,
where uE?x0?=?u?x0,t?? and uS?x0?=??x·?u?x0,t??. uEis
known as the “Eulerian mean” ?or “Eulerian streaming”? ve-
locity and uSis the “Stokes drift” velocity. uL, which is the
sum of uEand uS, is called the “Lagrangian mean” ?or “La-
grangian streaming” or “Lagrangian drift”? velocity. La-
grangian streaming is often referred to as steady streaming.
+??x·??u?x0,t? or uL=uE+uSresembles the material deriva-
041902-4Kumar et al.Phys. Fluids 23, 041902 ?2011?
tive D?/Dt=??/?t+u·?? in a Lagrangian framework that
consists of a local derivative ?the former? and a convective
derivative ?the latter?. More precisely, the acceleration of a
fluid parcel is obtained by taking the time derivative of Eq.
?4?, which yields the material derivative of the parcel’s ve-
locity. Although steady streaming is a nonlinear phenom-
enon, the Eulerian streaming is caused by the fluid dynami-
cal interaction ?between fluid parcels and with the geometry
of the fluid system under consideration? and the Stokes drift
arises from a kinematic viewpoint, depending on the pathline
of the tracked particle.37The decomposition Eq. ?5? allows
distinction between the two effects. Nonetheless, Eq. ?5?
holds true only when the displacement of a particle is small
such that u?x,t? can be expanded in series with respect to the
initial reference location xo. Typically, acinar flows are char-
acterized by a large KCnumber. Hence the above differential
form of Eq. ?5? must be modified in discrete form for large
particle displacements. Consider a particle at a location xjat
tj=j?t with ?t=T/N. To the first-order approximation of the
Taylor series expansion of Eq. ?1?, the generalized stencil for
the particle location is as follows:
P = 1,
= u?x0,tP−1? + uI
P = 2,N,
where the “instantaneous” Stokes drift velocity at time tP−1
uI= ??x1− x0? · ??u?x0,tP−1? + ??x2− x1? · ??u?x1,tP−1?
+ ¯ + ??xP−1− xP−2? · ??u?xP−2,tP−1?.
Summation of the above discrete equations for P=1 to N
N???xj− xj−1? · ??
where uL=?xN−x0?/T is the Lagrangian streaming velocity,
Stokes drift velocity.
Furthermore uS=?uI/N. For validation of formula ?7?,
please refer to Appendix A.
Nu?x0,ti−1? is the Eulerian streaming velocity,
u?xj−1,ti−1??? is the
C. Code validation
Henry et al. recently investigated mixing in a stationary
wall axisymmetric model to study the effect of unsteadiness
induced by the oscillatory flow and nonzero inertia.19The
flow conditions pertained to proximal generations in the aci-
nus. Their model has been chosen here for validation. The
problem details and some of the results are given below. The
geometry is a central channel surrounded by three cavities as
shown in Fig. 2?a?. The solution is obtained for Re=2
?QA/QD=0? and a time period of 3 s. The flow streamlines in
the central cavity extracted onto the y-z plane are shown in
FIG. 2. ?a? An axisymmetric alveolated duct model used for validation. The model geometry and dimensions are chosen from Henry et al. ?Ref. 19?. ?b?
Streamlines in the cross-section of an axisymmetric alveolus near peak inspiration.
041902-5Steady streaming: A key mixing mechanismPhys. Fluids 23, 041902 ?2011?
Fig. 2?b?. Typical of open cavity flows, a single recirculatory
flow region is observed in the cavity. A pair of fluid-particles
initially separated by an infinitesimal distance of d0?10−11
are advected for 50 cycles. The rate of separation of this
particle pair is estimated by finding the Liapunov exponent
given as: ?n=?1/t?ln?dt/d0?. Here dtis the final distance af-
ter time t. ?nconverges to a positive value of ?0.02 in good
agreement with Henry et al.19
A. Flow in a 2D channel
Before embarking on the analysis of flow structure and
mixing in the 3D alveolar geometry, a representative 2D case
is used to demonstrate various important, but not well-
understood, mixing patterns, some of which had been re-
ported before.10,18We will first discuss the steady streaming
phenomenon and its characteristics, and then relate it to the
origin of various interface stretching and folding patterns
observed in the duct, in the duct mouth, and within the cav-
ity. Consider an oscillating flow in a 2D long, straight chan-
nel with multiple rectangular grooves located periodically on
the lower part of the channel as shown in Fig. 3?a?. All di-
mensions and flow conditions are chosen to match closely
with Case I. Unlike the 3D cases to be discussed later, the
channel and cavity walls remain rigid. Flow is simulated
only in the midsection of length ?L+2dEL?. Due to low-Re,
the flow becomes fully developed within distances much less
than one channel height. Hence a parabolic profile with a
sinusoidal waveform is imposed at the ductal entrance E,
while a Neumann outflow condition is applied at “N.” The
Re in a 2D setting is defined using the channel height ?H? as
Re=U0H/?. The simulation is carried out for Re=1 and
KC=193. This combination of low-Re and high-KCis unique
for acinar flows. For the given flow conditions, the flow
separates and forms one single recirculation eddy in the cav-
ity. The flow structure is not symmetric with respect to the
vertical centerline in the cavity as shown in Fig. 3?b? and the
deviation from symmetry is resulted from nonzero-Re inertia
effects. The flow in the channel is separated from flow in the
cavity by a separation line, which attaches itself to the side-
walls of the cavity. This separation line or “separatrix”7pen-
etrates roughly to 25% of the cavity depth.
In the following presentation, the advection and ob-
served drift are understood in relation to the Eulerian mean
FIG. 3. ?a? Schematic of a 2D channel with multiple rectangular cavities.
The flow streamlines shown correspond to t/T=0.24, close to peak inspira-
tion. dEis the entrance length parameter. E, model entrance where a para-
bolic velocity profile is imposed; N, model exit where a Neumann outflow
boundary condition is imposed. ?b? Streamlines at near peak inspiration
?t/T=0.24? and expiration ?t/T=0.76? in the cavity of a channel flow with
Re=1. Arrows indicate the axial flow direction in the channel.
FIG. 4. ?Color? Contours of Eulerian mean velocities computed using Eq. ?7?: ?a? axial velocity ?y-component? and ?b? transverse velocity ?x-component?.
041902-6Kumar et al.Phys. Fluids 23, 041902 ?2011?
TABLE I. Lagrangian streaming, Eulerian mean and Stokes drift velocities for points 1, 2, 3, 4, and 5 inside the cavity marked in Fig. 5?a?.
PointLagrangian streaming ?uL,vL?
Eulerian mean ?uE,vE?
Stokes drift ?uS,vS?
FIG. 5. ?a? Drifts ?solid line? of two initially vertical line dyes ?dotted line? at end cycle t/T=1. ?b? Location of line dye Ao-Boat end inspiration t/T=0.5,
A?-1-2-3-4-5-B?. Points 1-2-3-4-5 correspond to those in ?a?. The pathlines for points 1 and 5 on inspiration and expiration ?marked by I and E, respectively?
are also plotted. ?c? Location of line dye Co-Doat end inspiration t/T=0.5, C?-D?. ?d? The distribution of y-component instantaneous Stokes drift velocity vI
for point 1 over one period. The instants at the three locations along the pathlines in the upper left insert are marked by the same symbols in the main plot.
In the lower right insert, the inspiratory curve with an inverted sign ?solid line? and the expiratory curve with the reversed time t?/T=1−t/T ?dashed line? are
overlapped to compare their magnitudes.
041902-7Steady streaming: A key mixing mechanismPhys. Fluids 23, 041902 ?2011?
and Stokes drift velocities in Eq. ?7?. The axial and trans-
verse components of Eulerian mean flow is shown in Fig. 4.
Note that the local maximum positive and negative axial
Eulerian streaming velocities occur near the two corners of
the cavity and are asymmetric in sign with respect to the
vertical centerline of the cavity. The nonzero Eulerian
streaming components are a result of asymmetries from
nonzero-Re inertia effects, which do not cancel out between
inspiration and expiration cycles. The distribution of nonzero
axial mean in Fig. 4 agrees with the asymmetry of the
streamlines shown in Fig. 3?b?. Hence, the presence of a
nonzero Eulerian mean flow is due to asymmetry in the fluid
flow between the two half-cycles and the effect decreases
with decreasing Re. On inspiration the fluid flows from right
to left, and on expiration the flow is reversed ?see Fig. 3?b??.
The flow fields on inspiration and expiration become more
asymmetric near the upper corners of the cavity than near the
bottom because fluid in the channel experiences immediate
expansion and contraction when flowing over the cavity. The
Eulerian mean magnitude is about three orders smaller
than the mean velocity of the fluid. Such nonzero mean flow
effects are typically observed in other low-Re settings such
as flow over a wavy bottom. For example, Larrieu et al.
showed the formation of positive and negative peaks of Eu-
lerian mean velocities observed between two crests of the
wavy wall.34Because the magnitude of the Eulerian mean
velocity uEis small, to assess the sensitivity of uEon mesh
size we present in Appendix B the comparison of the ob-
served maximum Eulerian mean velocities for three different
mesh sizes. The results show that a mesh-independent solu-
tion is ensured. The “fine mesh” of Table IV is henceforth
FIG. 6. ?Color? Tracer advection in the cavity of a 2D channel flow with Re=1. Distributions of tracer particles at t/T=: ?a? 0, ?b? 0.5, and ?c? 1. ?d? Stretch
rate map calculated in the same region as ?a? within the cavity.
041902-8Kumar et al. Phys. Fluids 23, 041902 ?2011?
We now consider advection in three regions: the cavity,
the cavity-channel mouth, and the outer channel. First, let us
consider advection of particles inside the cavity. Let A0-B0
and C0-D0denote the respective right and left vertical mate-
rial lines ?comprising a number of Lagrangian particles? at
t/T=0 in Fig. 5?a?. The drifts of two initial vertical lines of
particles at the end of one cycle are also displayed. At end
cycle, the right line dye A0-B0forms a fold denoted by 1-2-
3-4-5 in Fig. 5?a?. On the other hand, the left line dye C0-D0
is almost reversed back to its initial location. The shapes of
the two dyes at half-cycle t/T=0.5, A?-B?and C?-D?, are
displayed in Figs. 5?b? and 5?c?, respectively. The locations
of points ?Lagrangian particles? 1, 2, 3, 4, and 5 at t/T=1 in
Fig. 5?a? are also marked along A?-B?at t/T=0.5 in Fig.
5?b?. At t/T=0 these points are aligned vertically along
A0-B0. During inspiration, these points are advected upward
toward the mouth region where nonzero streaming is domi-
nant ?see Fig. 4?. Point 1 is advected to the left wall of the
cavity on inspiration, passing through positive and negative
streaming zones and resting away from the nonzero stream-
ing zone. The ?uppermost? thin line marked by “I” in Fig.
5?b? delineates the pathline of point 1 during forward excur-
sion. On expiration, the pathline E almost follows the in-
spiratory pathline I, which is expected in very low-Re flow.
Points 1 and 5 experience almost zero drift. Points 2, 3, and
4 spread along from negative to positive streaming zones.
The resulting Lagrangian drift of the material line at end
cycle in Fig. 5?a? takes the shape of a fold with the under-
shoot of point 2 to the left and the overshoot of point 4 to the
right. For the left line C?-D?at end inspiration shown in Fig.
5?c?, only the tip of the dye reaches the peripheral lower
portion of the positive Eulerian streaming zone, producing
To better understand the roles played by Eulerian mean
and Stokes drift on the total ?Lagrangian? drift, Table I com-
pares the three terms of Eq. ?7?: Lagrangian streaming veloc-
ity ?uL?, Eulerian mean velocity ?uE?, and Stokes drift veloc-
ity ?uS? for the five points along the right line A0-B0. For all
the points, the independently calculated uL, uE, and uSsatisfy
uL?uE+uS, ?with average error of ?0.15%? again validat-
ing Eq. ?7?. The undershoot of point 2 to the left and the
overshoot of point 4 to the right that forms a fold as illus-
trated in Fig. 5?a? are reflected by the negative and positive
y-component steady streaming velocities vL.
Table I further shows the dominant contribution of
Stokes drift to the total drift, suggesting the kinematic ?or
pathline-dependent? nature of the folding pattern. To further
understand this kinematic nature, we shall examine the con-
tribution of the instantaneous Stokes drift velocity uIin Eq.
?6? from various locations along the inspiratory and expira-
tory pathlines to the total Stokes drift velocity uS?=?uI/N?.
Figure 5?d? shows the distribution of y-component instanta-
neous Stokes drift velocity vIfor point 1 ?see the insert in the
upper left corner where the inspiratory and expiratory path-
lines are almost overlapped?. On inspiration ?solid line? vIis
negative, whereas on expiration ?dashed line? vIbecomes
positive. The time instants at the three locations along the
pathlines in the insert are marked by the same symbols in the
main plot. The distributions of vIon inspiration and expira-
tion are of nearly the same shape, but opposite sign. vI
reaches a local minimum or maximum when the particle
reaches around the midway between the left and right walls
of the cavity. By inverting the sign of vIfor t/T=0−0.5 and
reversing the time axis ?t?/T=1−t/T? for t/T=0.5–1 as
shown in the lower right insert, we can compare vIof the
same particle at approximately the same location, but oppo-
site phase. On the left side of the peak value ?in the lower
FIG. 7. ?Color online? Tracer advection in a 2D channel with dE=1.2. ?a? Advection of an initial line dye ?gray? into multiple cavities downstream is shown
in blue. ?b? Appearance of layer structure at end inspiration t/T=0.5. The dye in multiple cavities shown in ?a? is overlaid on to a single cavity due to
periodicity. ?c? Fold structure is formed after one cycle at t/T=1 and covers the cavity-channel mouth region.
041902-9Steady streaming: A key mixing mechanismPhys. Fluids 23, 041902 ?2011?
insert?, the dashed line is slightly higher than the solid line,
signifying the net vIthat contributes to vSwhen the particle
is near the right upper corner ?e.g., the triangle in the upper
insert? is positive. On the other hand, on the right side of the
peak value, the dashed line is slightly lower than the solid
line, resulting in the negative net vIthat the particle experi-
ences near the left upper corner of the cavity. The positive
and negative local Stokes drift velocities cancel out at end
cycle, resulting in nearly zero displacement in Fig. 5?a? for
point 1. This analysis suggests that if a particle is advected
through positive and negative Eulerian mean regions and
rests at a zero Eulerian mean region at end inspiration t/T
=0.5, the total drift at end cycle may be zero due to the
Next, particle advection and stretch rate maps are dem-
onstrated inside the cavity for two purposes. First, it high-
lights the connection between particle advection and fluid
kinematics reflecting through deformation of line elements
expressed by Eq. ?2?. Second, it helps elucidate the behavior
of stretch rate in association with Lagrangian drifts. Figures
6?a?–6?c? show a time sequence of the advection of Lagrang-
ian particles at t/T=0, 0.5, and 1, respectively. Here, the
particles initially fill a rectangle, expanding from previous
two vertical line dyes to a surface, to map out the drift in the
core region of the cavity. At end inspiration ?Fig. 6?b?? par-
ticles exhibit a spiral shape, equivalent to a combination of
Figs. 5?b? and 5?c?. At end expiration ?i.e., end cycle t/T
=1? Fig. 6?c? shows the majority of particles are reversed
back to their original locations, except that a series of folds
are formed and distributed along a curved strip from the right
wall to the bottom wall, and then to the left wall. Figure 6?d?
shows the stretch rate map of the same region within the
cavity. A tongue of higher stretch rate region surrounds an
almost zero mixing core. The higher stretch rate region co-
incides with the advection map shown in Fig. 6?c?, indicating
that the stretch rate can capture and distinguish the regions of
small and large drifts. A higher stretch rate yields greater
separation and fold of dyes.
TABLE II. Lagrangian streaming, Eulerian mean and Stokes drift velocities for points 1, 2, 3, 4, and 5 in the outer channel marked in Fig. 8?b?.
Point Lagrangian streaming ?uL,vL?
Eulerian mean ?uE,vE?
Stokes drift ?uS,vS?
FIG. 8. Drifts of two line dyes in the channel near the two cavity corners for dE=: ?a? 1.2, ?b? 5. ?c? Locations of points 1-2-3-4-5 at end inspiration t/T
=0.5. Their locations at end expiration t/T=1 are marked in ?b?.
041902-10Kumar et al. Phys. Fluids 23, 041902 ?2011?
Having demonstrated typical advection patterns in the
cavity, we present results of advection and drift in the mouth
region. The visualization of advection of a line dye for which
periodic boundary condition has been applied is shown in
Fig. 7. The initial line is stretched into multiple cavities at
t/T=0.5 as shown in Fig. 7?a?. A large increase in length of
the tracer during the positive half-cycle is noted. The dye is
overlaid back on to a single cavity, resulting in an appear-
ance of “layer structure” in the mouth as shown in Fig. 7?b?.
This is equivalent to the pattern of advection that can be
expected when a dye is placed periodically in the alveolar
mouth of all cavities in a physical scenario. Figure 7?c?
shows the final shape of the dye after one complete advec-
tion cycle, exhibiting multiple folds ?as opposed to a single
fold of the right line dye inside the cavity in Fig. 5?a??. The
origin of folding is also attributable to steady streaming and
will be discussed next in conjunction with the drift charac-
teristics in the outer channel.
Advection in particle motion in the outer channel is stud-
ied by releasing two vertical lines of particles in the channel
adjacent to the two corners of the cavity. The final shape of
the tracer after one cycle shows the Lagrangian drift from its
initial condition. Two values of dE=1.2 and 5 are considered.
As seen in Fig. 3, the upstream channel length is dEL. By this
specification, dE→? is a model with a single cavity in an
infinitely long channel, and dE=0 is one where the channel is
completely lined with grooves with no spacing in between.
The drifts of these line tracers at end cycle calculated from
Eq. ?1? are shown in Figs. 8?a? and 8?b?. Like Fig. 7?c?,
multiple folding patterns, hereafter referred to as “fold struc-
ture,” are observed. The number of folds is higher for dE
=1.2 than for dE=5. Multiple folds in the tracer clearly indi-
cate that two particles initially located close to each other
might overshoot or undershoot its initial location as observed
inside the cavity ?Fig. 5?a??, in the mouth region ?Fig. 7?c??,
and in the outer channel ?Figs. 8?a? and 8?b??.
The advection characteristics of selected points ?La-
grangian particles? 1, 2, 3, 4, and 5 marked in the dye for
dE=5 in Figs. 8?b? and 8?c? for t/T=1 and 0.5, respectively,
are studied. The initial vertical line tracer on the right hand
side is stretched over multiple cavities at end inspiration
t/T=0.5. Only a section of the stretched tracer is shown in
Fig. 8?c?. Points 1 and 4 that consistently fall short of their
initial positions ?undershoot? after one cycle are always lo-
cated near the left corner of the cavity at end inspiration. On
the other hand, points 2 and 5 that overshoot their initial
positions after one cycle are located at the right corner of the
cavity at end inspiration. Point 3 which is almost traced back
to its initial location is located over the channel at t/T=0.5.
Table II shows that points 1 and 2 ?or points 4 and 5? have
nonzero y-component streaming velocities vLof opposite
signs, and the calculated uL, uE, and uSsatisfies uL?uE
+uSwith an average error of 0.7%. Hence, it can be con-
cluded that nonzero steady streaming velocities combined
with pathlines of particles that traverse over multiple cavities
?due to large KC? result in the observed advection drift and
folding patterns. The number of folds increases with increas-
ing KC?or decreasing cavity spacing?. If a horizontal line dye
is released in the mouth region as shown in Fig. 7, the re-
sulting fold structure covers the mouth region as shown in
Fig. 7?c?. Unlike the fold structure formed at end cycle, the
layer structure in Fig. 7?b? is irrelevant to mixing and is
observed at end inspiration when a number of stretched trac-
ers appear periodically over multiple cavities.
Axial dispersion in the channel caused by steady stream-
ing shown in Fig. 8 is quantified by Eq. ?3?. The data at the
end of first cycle are used as the reference initial condition to
remove the effect of initial transient drift in the first cycle,
which is usually greater than subsequent cycles. This is be-
cause of larger increase of dye interface from a line to folds
in the first cycle while the fold pattern is retained in subse-
quent cycles. The variance shows an exponentially increas-
ing trend with cycle number as seen in Fig. 9 for both cases
dE=1.2 and 5. Additional experiments were carried out and
two important variables that alter the observed fold structure
significantly were realized. These are the Keulegan–
Carpenter number, KCand geometrical ratio L/H ?ratio of
cavity length to channel height?. Nevertheless, an elaborate
parametrical study and identification of critical threshold val-
ues of these parameters is beyond the scope of this paper.
In summary, the relation between advection, stretch rate,
steady streaming, and Lagrangian drift has been established
in a simple 2D channel flow for Re=1 and KC=193. It is
shown that a mechanism that leads to large increase in dye
interface length exists in a low-Re flow that arises due to
streaming. With the physical insights gained from this case,
we proceed to report the results in 3D acinar models. Unlike
the 2D channel flow, all cases, except one, presented below
consider uniform expansion and contraction of the 3D duct
and alveolar walls.
B. Alveolar flow structure
The characteristics of the flows in Cases I and II are
briefly described here. In the proximal generations, the main
cavity flow typically consists of a recirculation near the
FIG. 9. Axial dispersion quantified by axial variance as a function of num-
ber of cycle. Subscript “L” is used for variance to indicate that the compu-
tation is carried out on the right vertical line of tracer shown in Figs. 8?a?
041902-11Steady streaming: A key mixing mechanismPhys. Fluids 23, 041902 ?2011?
proximal wall. The only nondiffusive interaction between the
duct and the alveoli occurs through an entrainment region.
The presence of such an entraining flow is a consequence of
expansion and contraction of alveolar walls. Hence the fluid
exchange to and from the alveoli occurs through a region
located near the proximal wall corner.18,26An instantaneous
portrait of stream-traces above and within the alveolus for
Case I is given in Figs. 10?a?–10?c? for Re=2, 1, and 0.52,
respectively. For flows with Re=2 and 1, a recirculation is
present within the cavity near the proximal wall.26Figure
10?c? corresponds to a lower-Re flow where recirculation is
absent. Also, for all Re?0.52 including Re=0.06 case which
will be considered later, no recirculation is observed within
the alveolus. The flow structure within an alveolus in Case II,
in general is similar to Case I. However, due to the asym-
metric arrangement of alveoli around the duct, alveolar units
surrounding the duct are ventilated nonuniformly.
Since steady streaming plays an important role in mixing
as demonstrated before with a 2D alveolated channel flow,
Fig. 10?d? shows the contours of the steady streaming Eule-
rian mean axial ?y-component? velocity in the y-z plane of an
alveolus for Case I with Re=1 ?refer to Fig. 1?d? for the
location of the y-z plane?. The positive and negative stream-
ing velocities are observed in the proximal ?right? and distal
?left? corners of the alveolus, exhibiting the same feature as
the 2D rigid-channel case in spite of the 3D moving wall.
For a given Re, the major difference from the rigid-wall case
is that the magnitude of the streaming velocity in the
FIG. 10. ?Color? 3D view of instantaneous stream-trace portrait near peak inspiration at t/T=0.24 within and over the alveolar cavity ALV for Case I ?see Fig.
1? with ?a? Re=2; QA/QD=0.0024, ?b? Re=1; QA/QD=0.0047, ?c? Re=0.52; QA/QD=0.0095. P, proximal wall; D, distal wall. ?d? Contours of Eulerian mean
axial ?y-component? velocity in the y-z plane of the cavity ALV for Case I with Re=1.
041902-12Kumar et al. Phys. Fluids 23, 041902 ?2011?
moving-wall case is slightly lower than that of the rigid-wall
case and is more asymmetric, having greater positive stream-
ing velocity than negative one. With increasing ?decreasing?
Re to 2 ?0.52?, the maximum Eulerian axial velocity in-
creases ?decreases? approximately by twofold.
C. Flow topology classification
Figure 11 illustrates the flow topology and associated
critical points within the alveolus by showing the instanta-
neous stream-traces in the y-z plane of the alveolus “ALV”
displayed in Fig. 1?d?. The reasons for identifying critical
points are twofold. First, a stagnation saddle point had been
attributed before for chaotic mixing in the alveoli.10,14,15,18
Second, the observed flow topology helps identify advection
regions of interest for analysis in subsequent sections.
The flow topology can be classified by the first-order
critical points of the flow in the y-z midplane. The type of
critical points depends on the eigenvalues of the velocity
gradient in the vicinity of a critical point.38,39The magnitude
of fluid velocity vanishes at a critical point. The eigenvalues
are computed from the characteristic equation ??A?−??I??
=0, where ?A?=Aij=?ui/?xjis the Jacobian matrix based on
the velocity gradient in the midplane and ?s are eigenvalues,
?1and ?2. The eigenvalues and the type of critical points for
Case I, Re=2 and 1, are listed in Table III at time instants
that roughly correspond to the maximum flow rate during
inspiration and expiration. Two critical points are observed in
Fig. 11. By definition, if both eigenvalues are real with at
least one of them being negative, the critical point is a
“saddle” point ?see Fig. 11?b??. If both eigenvalues are com-
plex conjugates, the critical point is called a “center” ?see
Fig. 11?c??. The center point is found at the center of the
recirculation, whereas the saddle point is near the proximal
wall. The presence of the saddle point is consistent with
Refs. 14, 17, and 18 who also recognized its presence within
the cavity. Tsuda et al. explained that such a saddle point
arises from superposition of a main recirculating cavity flow
and the radial flow generated by wall motion.14
As the critical point is in essence a stagnation point, the
magnitude of velocity near the saddle point is relatively
small. During expansion, the saddle point and recirculation
are displaced deeper into the cavity. As shown in Fig. 11?b?,
a saddle point diverts streamlines that pass into it to different
regions, thus leading to uncertainty. The nonzero real part of
the eigenvalue of the center-like point depicts a spiral behav-
ior. The change in sign between inspiration and expiration
indicates the direction of this spiral. That is, the negative
?positive? real part of the center-like spiral point implies that
particles in its vicinity tend to move toward ?away from? that
point in a spiral fashion. Although the particles could move
toward the spiral point from any direction, there is no guar-
antee that they would follow the same path in a reversible
manner when moving away from it, resulting in uncertainty.
Hence it was postulated that the presence of critical points
might give rise to mixing in its neighborhood.
The entrainment region, a thin layer attached to the duc-
tal wall, is also marked in Fig. 11. The stream-trace, which
represents the upper bound of this layer, is overlaid with
solid circles. This stream-trace is open to the channel on the
proximal wall side, but is closed and attached to the distal
wall, allowing advection of fluid into or out of the cavity.
The presence of the entrainment region restricts the recircu-
lation eddy to the right hand side of the cavity close to the
proximal wall, and forms a saddle point where recirculating
flow interacts with entrained flow and radial flow induced by
wall motion. In contrast, for the rigid-wall case, like the 2D
channel flow shown in Fig. 3?b?, a separation line is formed
at the mouth region to separate ductal flow from cavity flow
and the recirculation is located approximately at the center of
D. Tracer advection and deformation
In this section, we investigate the Lagrangian drift that
arises in tracer transport in an alveolus. For the tracer advec-
tion experiments below we shall consider the alveolus ALV
marked in Fig. 1?a?. For clarity, different 3D views of the
alveolus are provided in Figs. 1?c? and 1?d?. A cross-shaped
dye is introduced within the cavity and its deformation is
monitored for ten cycles to assess the effect of recirculation
on mixing. A visualization of the final shape of the dye for
three different flow conditions is shown in Fig. 12. The four
arms of the dye are numbered, aiming to observe advection
TABLE III. Critical points in the flow.
Re Time EigenvaluesType
FIG. 11. ?a? Stream-traces in the y-z plane of the alveolus ALV ?see
Fig. 1?d?? for Case I, Re=1 at t/T=0.24. The entrainment layer is denoted
by a double-sided arrow, and its upper bound is delineated by the stream-
trace marked with circles. P, proximal wall; D, distal wall. Enlarged view of:
?b? a saddle point, ?c? a center point ?associated with recirculation?. For a
saddle point, Im??1?=Im??2?=0, Real??1??Real??2??0; for a center point,
041902-13Steady streaming: A key mixing mechanismPhys. Fluids 23, 041902 ?2011?
in four regions of the cavity. The dye experiences an increase
in the interface area due to the alveolar flow: ?45% increase
=0.0047? and ?3.5% for Re=0.52?QA/QD=0.0095?. Similar
to the observed folding phenomena in the 2D case of Fig. 6,
a stretch of material into the proximal corner is observed due
to steady streaming. Again indicative of the Re effects, the
drift is more pronounced for Re=2 than Re=1 and for Re
=1 than Re=0.52. In particular, at Re=2, blue arm #1 is
folded and stretched toward the upper proximal wall, and
arm #3 near the mouth region is also highly stretched. At
Re=1, the recirculation zone is reduced ?see Figs. 10 and
11?, and thus only stretch of arm #2 is evident. Arm #3 still
experiences stretching because of its proximity to the mouth
region. At Re=0.52, the recirculation is absent ?see Fig. 10?
and only the stretch of arm #3 seems significant. The in-
crease in the red-blue interface area ratio is correlated as
roughly quadratic with increasing Re ?see Fig. 12?d??. It is
also useful to plot the ratio as a function of the flow ratio
QA/QDdefined earlier. The interface area increases almost
exponentially with decreasing values of QA/QD.
Previously the advection and deformation of tracers in-
side the cavity are examined by a cross-shaped dye. Next, a
planar tracer is advected for one full cycle. The tracer is
FIG. 12. ?Color? Advection patterns for an initial cross-shaped dye within the cavity ALV of Case I for ?a? Re=2; QA/QD=0.0024, ?b? Re=1; QA/QD
=0.0047, and ?c? Re=0.52; QA/QD=0.0095 after ten periods of breathing. The blue dye is the initial shape of the dye and the red dye is the deformed shape
after advection. ?d? Correlation of red-blue interface area ratio with QA/QDand Re.
041902-14Kumar et al.Phys. Fluids 23, 041902 ?2011?
constructed using ?20 000 particles and placed initially in Download full-text
the y-z plane shown in Fig. 1?d?. The advection is carried out
in a complete 3D sense utilizing all the three velocity com-
ponents. Figure 13 shows advection characteristics in the
midplane for Case I, which are of particular interest for rea-
sons discussed in Sec. III A in association with steady
streaming. For studying the characteristics of deformation
within the alveolus, the advection pattern in strategic regions
within the planar tracer are separately followed. These re-
gions are depicted by different colors and illustrated in Fig.
13. The blue tracer, on average, covers the region inside the
cavity where recirculation ?in the neighborhood of the center
point? is observed. Mixing in the neighborhood of the proxi-
mal region ?associated with the saddle point? within the al-
veolus is identified using the green tracer. As observed in
Fig. 11, the midplane contains a saddle point near the proxi-
mal cavity wall. The rest of the cavity is colored in gray.
The layered appearance of the gray tracer at end inspi-
ration, as seen in the middle panel of Figs. 13?a?–13?c?, is a
visualization of traverse of the tracer in multiple alveolar
units and has already been discussed in conjunction with Fig.
7?b?. Because the axial distance traveled is greater for Re
=2 than Re=1, and for Re=1 than Re=0.52, the number of
layers observed is higher in Fig. 13?a? than Fig. 13?b?, and in
Fig. 13?b? than Fig. 13?c?.
Mixing within the alveolus is visualized by the deforma-
tion of the blue and green regions. For Re=2 the blue region
at end inspiration t/T=0.5 is stretched in a manner similar to
the 2D channel case shown in Fig. 6?b?. Recall that in the
presence of alveolar and ductal wall expansion, the recircu-
lation eddy is shifted toward the proximal wall to accommo-
date the entrained flow. Therefore, the blue dye can hardly
reach the distal wall of the cavity. At end expiration ?i.e., end
cycle, t/T=1?, the fold structure similar to the 2D case in
Fig. 6?c? is observed. The fold structure extends further up to
the green tracer region. At the mouth region, the fold struc-
FIG. 13. ?Color? Material advection in the y-z plane of the cavity ALV of Case I ?see Fig. 1?d?? with ?a? Re=2; QA/QD=0.0024 ?b? Re=1; QA/QD
=0.0047, ?c? Re=0.52; QA/QD=0.0095 at t/T=0 ?top panel?, 0.5 ?middle panel?, and 1.0 ?bottom panel?. P, proximal wall; D, distal wall. The blue ?green? dye
in ?a? and ?b? covers a center-like spiral point ?a stagnation saddle point?. There is no critical point in ?c? due to very low-Re.
041902-15 Steady streaming: A key mixing mechanismPhys. Fluids 23, 041902 ?2011?