PHYSICS OF FLUIDS
VOLUME 15, NUMBER 2
FEBRUARY 2003
Dean vortices-induced enhancement of mass transfer through an interface
separating two immiscible liquids
A. Yu. Gelfgat
Department of Fluid Mechanics and Heat Transfer, Faculty of Engineering, Tel-Aviv University,
Ramat Aviv 69978, Israel
A. L. Yarin and P. Z. Bar-Yoseph
Faculty of Mechanical Engineering, Technion–Israel Institute of Technology, Haifa, 32000, Israel
共Received 9 May 2002; accepted 4 November 2002; published 7 January 2003兲
Two-fluid Dean vortex flow in a coiled pipe with vanishing torsion, and its effect on the mass
transfer through the liquid–liquid interface of two immiscible fluids are studied numerically. The
liquids are stratified by gravity, with the denser one occupying the lower part of the pipe. The
Navier–Stokes equations in both fluid layers are solved numerically by the finite volume method.
The results reveal a detailed structure of the transverse flow 共the Dean vortices兲 in coiled pipes with
the dimensionless curvature 0.1. Both cocurrent and countercurrent axial flows in the fluid layers are
considered. Using the flow fields predicted, the mass transfer equation is solved. It is shown that the
mass transfer of a passive scalar 共say, a protein with the Schmidt number of the order of 103 )
through the interface can be significantly enhanced by the Dean vortices, so that the mass transfer
rate can be increased by three to four times. This makes the Dean vortex flow an effective tool for
mass transfer enhancement at the liquid–liquid interface. It is shown that the Dean flow provides a
stronger mixing than the Taylor–Couette flow. It is also shown that there exists an optimal axial flow
rate in terms of this enhancement. The optimal flow corresponds to the value of the Dean number
of about 180. In the countercurrent flow case the Dean vortices can split, which has a negative effect
on the mass transfer enhancement. Both the cocurrent and countercurrent axial flows yield a similar
enhancement effect on the interfacial mass transfer rate. The problem is related to the search for
novel bioseparator devices. © 2003 American Institute of Physics. 关DOI: 10.1063/1.1532732兴
I. INTRODUCTION
vice of Refs. 1 and 2, and the results will be published in a
separate paper. Search for configurations in which the vortical flow can be generated without motion of the boundaries,
and which can be implemented in novel bioseparators/
bioreactors, led to study of stationary dc streaming generated
in emulsions,4 thin channels with capillary waves,5 and of
natural-convection flow in two-layer horizontal annuli.
The Dean vortices arising in coiled pipe single-liquid
flows are known as an effective means for heat and mass
transfer enhancement 共see Refs. 6 –11 and references
therein兲. In particular, Refs. 6 and 7 were motivated by the
need to prevent buildup of retained solute concentration near
a membrane in a reverse osmosis system. In these works it
was shown that the concentration buildup, known as concentration polarization, can be significantly reduced if solutions
are supplied through curved channels, where the Dean vortices arise. The Dean vortices effectively mix the retained solute and this results in depolarization and a significant increase in the transmembrane flux. In Refs. 6 and 7, as well as
in the previous works of that group, direct numerical simulation was used to calculate the velocity fields in curved
channels and pipes with single fluid flowing under different
conditions 共different Reynolds numbers, channel curvature,
etc.兲. Comparison with the experimental data obtained by
means of magnetic resonance flow imaging showed that the
numerical results are capable of reproducing the details of
the secondary flows 共the Dean vortices兲 rather accurately in-
In the present study, the effect of a two-fluid Dean vortical flow on mass transfer through an interface separating
two immiscible liquids is examined. The study is motivated
by the problem of extraction of admixtures having extremely
low diffusion coefficients, e.g., proteins. Since purely diffusional mass transfer of such admixtures is too slow and has
to be enhanced, mixing by vortical flows is called for.1 In
such flows the vortices inside each fluid transport the admixture toward the interface, which yields steeper concentration
gradients there. As a result, the mass fluxes through the interface increase and the mass transfer intensifies, even
though the diffusion coefficient remains small. The first experimental attempt in this direction was made in Ref. 2, with
two-fluid Taylor–Couette vortical flow used in a specially
designed bioseparator/bioreactor. This configuration was further studied theoretically and numerically in Ref. 3, where
the effect of the Taylor vortices and of the axial throughflow
on the mass transfer rates was investigated. Practical application of the two-fluid Taylor–Couette apparatus was found
to be problematic because the contact of the rapidly rotating
cylindrical boundaries 共e.g., in the experiments2 the rotation
rate exceeded 10 rev/s兲 with the stationary closed ends leads
to instability of the liquid–liquid interface. This instability
was also observed numerically by the present group. It was
compared to the experimental results obtained using the de1070-6631/2003/15(2)/330/18/$20.00
330
© 2003 American Institute of Physics
Downloaded 17 Jan 2003 to 132.68.1.29. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp
Phys. Fluids, Vol. 15, No. 2, February 2003
Dean vortices-induced enhancement
331
FIG. 1. Sketch of the problem.
cluding bifurcations of the additional vortices. Based on the
numerically generated velocity fields, the concentration field
was calculated numerically in Ref. 7, where it was shown
that concentration polarization in curved channels can be significantly inhibited by the Dean vortices as compared to that
in flat channels 共where the secondary vortices do not arise兲.
The present work deals in the same context, with mass
transfer enhancement by the Dean vortices developing in
coiled pipes of circular cross section filled by two liquid
layers formed by immiscible fluids. The flow is driven by the
pressure gradient, which acts along the pipe centerline and
can be the same or different in the fluid layers. The Dean
vortices, driven by the excessive centrifugal force, develop
in each layer. It is assumed that each fluid layer occupies half
of the pipe cross section. For a preliminary insight into the
possibility of the mass transfer enhancement, we consider a
helical pipe with vanishing torsion,12,13 so that the flow region reduces to a circular pipe coiled in a circle. The two
immiscible liquid layers, stratified by gravity, then occupy
the lower and upper parts of the torus so that the interface
coincides with the symmetry plane. To the best of our knowledge, such two-liquid flow in a coiled pipe has never been
considered before. The emergence of Dean vortices in twofluid Dean flows, which are the counterpart of the classical
Dean problem,14 was demonstrated in Ref. 15, where channel flow was considered. In the present work novel results on
two-fluid flow in coiled pipes 共the counterpart of the singlefluid flow of Ref. 16兲 are discussed. In particular, the patterns
of the Dean vortices, as well as the quantitative results on the
Dean-vortex-enhancement of the mass transfer through the
liquid–liquid interface are revealed.
Section II presents the problem formulation and briefly
discusses the numerical technique employed. The results are
presented and discussed in Sec. III. Conclusions are drawn in
Sec. IV.
II. FORMULATION OF PROBLEM
AND NUMERICAL TECHNIQUE
To estimate the effect of the Dean vortices on the mass
transfer we follow Ref. 13 and consider a helical pipe with
vanishing torsion. Thus, we consider the flow in a pipe with
circular cross section of radius a, coiled in a circle of radius
A. Following the classical Dean formulation,16 it can be interpreted as flow in a torus with centerline of radius A and
cross section of radius a. The flow region is shown schematically in Fig. 1. The upper and lower halves of the torus
are filled with two immiscible liquids, denoted 1 and 2. The
problem is considered in the coordinate system (r, ␪ ,s),
where (r, ␪ ) are the polar coordinates in the cross section of
the pipe, and s is the arclength along the centerline of the
torus.17 It is assumed that the flow is driven by constant
pressure gradients G 1 and G 2 in the s direction, which can
be different in each of the liquid layers. In the case of unequal pressures and pressure gradients in the layers, the interface is assumed to be slightly inclined and, as a result,
stabilized by gravity. It is also assumed that the liquid–liquid
interface is nondeformable and located approximately at the
plane ␪⫽⫾␲/2 and that the mass transfer does not affect the
flow. Denoting all the variables and parameters related to
liquids 1 and 2 by the corresponding subscripts, the flow in
each layer is described by the dimensionless momentum and
continuity equations
⳵ v1
1
⫹ 共 v1 "ⵜ 兲 v1 ⫽⫺es ⫺ⵜp 1 ⫹ ⌬v1 ,
⳵t
R
共1兲
ⵜ"v1 ⫽0,
共2兲
⳵ v2
␮ 21 1
G 21
1
⫹ 共 v2 "ⵜ 兲 v2 ⫽⫺
⌬v ,
es ⫺
ⵜp 2 ⫹
⳵t
␳ 21
␳ 21
␳ 21 R 2
共3兲
ⵜ"v2 ⫽0.
共4兲
Here vk ⫽(u k , v k ,w k ) is the flow velocity vector, and p k the
pressure (k⫽1,2); es is the axial unit vector in the pipe.
Furthermore, ␳ 21⫽ ␳ 2 / ␳ 1 , ␮ 21⫽ ␮ 2 / ␮ 1 and G 21⫽G 2 /G 1
are the ratios of the densities, dynamic viscosities, and pressure gradients, respectively. The length, time, velocity and
pressure in 共1兲–共4兲 are rendered dimensionless by a,
( ␳ 1 a/G 1 ) 1/2, (G 1 a/ ␳ 1 ) 1/2, and G 1 a, respectively, so that the
value of the dimensionless external pressure gradient in layer
1 is reduced to unity. The nondimensional parameter R
⫽ (a/ ␮ 1 ) 冑G 1 a ␳ 1 is thus the analog of the Reynolds number. Details on the momentum and continuity equations in
Downloaded 17 Jan 2003 to 132.68.1.29. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp
332
Phys. Fluids, Vol. 15, No. 2, February 2003
Gelfgat, Yarin, and Bar-Yoseph
the chosen coordinate system are given in Appendix A. Note
that an additional governing parameter is the dimensionless
curvature of the pipe, ␧⫽a/A. Note also that gravity is assumed to be fully offset by the steady-state component of the
vertical pressure gradient, whereby both factors are eliminated from Eqs. 共1兲 and 共3兲.
It is emphasized that the above-introduced nondimensional parameter R is a formal ratio of the inertial to viscous
forces. On the other hand, it is desirable to introduce also the
Reynolds number Re based on the mean flow velocity W̄ 1
corresponding to the single-layer flow 共liquid 1 only兲 in a
straight pipe of radius a. According to the Poiseuille formula
W̄ 1 ⫽a 2 G 1 /(8 ␮ 1 ). Hence, Re⫽␳1aW̄1 /␮1⫽␳1a3G1 /(8␮21).
The latter shows that Re⫽R2/8. Introducing the Dean number
as13 D⫽ 冑2␧ ␳ 1 a 3 G 1 / ␮ 21 ⫽8(2␧) 1/2 Re, we have D
⫽(2␧) 1/2R 2 . The dimensionless group K of Ref. 12 is K
⫽D/25/2.
The flow is assumed to be independent of the coordinate
s. Thus, the problem is treated in the plane (r, ␪ ), in the
domain 0⭐r⭐1, ⫺ ␲ /2⭐ ␪ ⭐3 ␲ /2. Following the commonly used notation12,18 –20 we call the velocity component
w ‘‘axial velocity,’’ and the flow in the (r, ␪ ) plane ‘‘transverse flow.’’
The no-slip boundary conditions are imposed on the pipe
wall
r⫽1,
⫺ ␲ /2⭐ ␪ ⭐3 ␲ /2:
v⫽0,
共5兲
also
r⫽0:
兩 v兩 ⬍⬁,
共6兲
and continuity of the velocities and tangent stresses is required on the liquid–liquid interface
␪ ⫽⫾ ␲ /2,
0⭐r⭐1:
⳵u1
⳵u2
⫽ ␮ 21
,
⳵␪
⳵␪
u 1 ⫽u 2 ,
⳵w1
⳵w2
⫽ ␮ 21
.
⳵␪
⳵␪
v 1 ⫽ v 2 ⫽0,
w 1 ⫽w 2 ,
共7兲
共8兲
With the steady-state flow calculated, the mass transfer
problem is considered. Here we assume that the time needed
to reach the steady state of the flow is much shorter than the
characteristic diffusion time, so that the mass transfer during
the transient stage is negligible. The mass transfer equations
are
1
⳵c1
⫹ 共 v1 "ⵜ 兲 c 1 ⫽ ⌬c 1 ,
⳵t
Pe
共9兲
⳵c2
D 21
⫹ 共 v2 "ⵜ 兲 c 2 ⫽
⌬c 2 ,
⳵t
Pe
共10兲
where c k , k⫽1,2, are the concentrations of a passive scalar
in the liquid layers 1 and 2, D 21⫽D 2 /D 1 ⫽O(1) is the ratio
of the diffusion coefficients, Pe⫽R Sc⫽ (a/D 1 ) 冑G 1 a/ ␳ 1 the
Peclet number and Sc⫽ ␮ 1 /D 1 ␳ 1 the Schmidt number. The
excess dimensionless concentration of solute is c⫽(c
⫺c min)/(cmax⫺cmin), where c max and c min are the maximal and
minimal concentration values at t⫽0. Initially, liquid 1 is
assumed to contain a uniformly distributed admixture at a
concentration c max , c⫽1, which begins to diffuse into the
admixture-free liquid 2 (c min⫽0, c⫽0). The torus is assumed to be impermeable for the admixture, while the concentrations and mass fluxes are continuous at the interface.
Thus the boundary and initial conditions for the mass transfer problem read
r⫽0:
c 1 ⬍⬁,
r⫽1,
⫺ ␲ /2⭐ ␪ ⭐3 ␲ /2:
␪ ⫽⫾ ␲ /2,
c 2 ⬍⬁,
⳵c
⫽0,
⳵r
c 1 ⫽c 2 ,
0⭐r⭐1:
c 1 ⫽1,
t⫽0:
共11兲
共12兲
⳵c1
⳵c2
⫽D 21
, 共13兲
⳵␪
⳵␪
c 2 ⫽0.
共14兲
For characterization of the mass transfer we introduce the
local Sherwood number
Sh共 x 兲 ⫽
1 ⳵c2
r ⳵␪
冏
,
x⫽⫺r sin共 ␪ 兲 ,
共15兲
␪ ⫽⫾ ␲ /2
and the mass fraction that has diffused into the liquid layer 2
mf⫽
2
␲
冕 冕
3 ␲ /2
␲ /2
1
0
rc 2 共 r, ␪ 兲 dr d ␪ ,
共16兲
where ␲/2 is the area of half the dimensionless pipe cross
section of unit radius. The local Sherwood number 共15兲 is
unobtainable near r⫽0 and should be calculated separately
for x⬍0 and x⬎0 or, respectively, ␪⫽⫹␲/2 and ␪⫽⫺␲/2. It
is emphasized that in our discussion of the mass transfer we
use the flow pattern and x axis corresponding to cross section
A of Fig. 1.
The problems 共1兲–共8兲 and 共9兲–共14兲 were solved numerically. The finite volume method was used with the SIMPLE
velocity-pressure decoupling algorithm and a semi-implicit
three-level time integration scheme. It was used for swirling
flows in a cylindrical container21 and for study of masstransfer enhancement due to Taylor3 and natural convection
vortices. The code was validated additionally for single-fluid
flow against the analytical solution for small R, ␧, Re, and D
共see Appendix B兲 and against the results of Refs. 13, 22, and
23 for relatively large Dean numbers. The steady-state fluid
flow could be calculated with a dimensionless time step ␶
⫽0.1 for D⭐200, and with ␶ ⫽0.01 for 200⭐D⭐6000 共usually, several thousands of time steps are needed to reach a
converged solution at large D兲.
To ensure numerical stability of the mass transfer calculations at the large Peclet numbers characteristic of proteins
(Pe⬃O(105 )), the calculations were performed with a
smaller time step ␶ ⫽10⫺3 . The reported flow and masstransfer calculations were done on the same uniform staggered grid. Jumps of the physical properties over the liquid–
liquid interface were smoothed as in Ref. 3. To ensure meshand time-step independence of the results, the calculations
for the largest value of the Peclet number (Pe⫽105 ) were
repeated on 100⫻200 and 150⫻300 grids in the r and ␪
directions, respectively, with time steps ␶ ⫽10⫺3 and 10⫺4 .
Downloaded 17 Jan 2003 to 132.68.1.29. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp
Phys. Fluids, Vol. 15, No. 2, February 2003
Dean vortices-induced enhancement
333
FIG. 2. Isolines of the axial velocity w
共upper frames兲 and streamlines of the
transverse flow 共lower frames兲 for
cocurrent flow G 21⫽1. The left-hand
side of each frame corresponds to the
inner part of the toroidal boundary, the
right-hand side to the outer part. The
cross-section B of Fig. 1 is shown. 共a兲
R⫽20 (D⫽179); 共b兲 R⫽40 (D
⫽716); 共c兲 R⫽60 (D⫽1610); 共d兲 R
⫽80 (D⫽2862); 共e兲 R⫽100 (D
⫽4472). In all cases ␧⫽0.1, ␮ 12
⫽0.96.
No significant changes were found. Most of the masstransfer results reported in the following refer to the 100
⫻200 grid and ␶ ⫽10⫺3 .
Following Ref. 24 we define the characteristic turnover,
averaging, and diffusion time as T e ⫽( ␳ 1 a/G 1 ) 1/2, T a
⫽T d Pe⫺2/3, T d ⫽a 2 /D 1 , respectively. Using, as before, the
time scale ( ␳ 1 a/G 1 ) 1/2, we obtain the following dimensionless values: T e ⫽1, T a ⫽Pe1/3, T d ⫽Pe. In the calculations
reported in the following 103 ⭐Pe⭐105 . The range of integration in time for the mass-transfer problem was 200 for
Pe⫽103 , 1000 for Pe⫽104 , and 10 000 for Pe⫽105 , i.e., the
integration time was always longer than the averaging time
and shorter than the diffusion time.
the maximal Schmidt number Sc⫽Pe/R was 103 for R
⫽100 and even larger for smaller R. The dimensionless curvature of the tube was typically taken as ␧⫽0.1. The variation of R in the range 1⭐R⭐120 corresponds to that of the
Dean number 0.45⭐D⭐6440. The other governing parameters were fixed. We considered the cases of cocurrent and
countercurrent axial flow in the two layers, corresponding to
G 21⫽1 and ⫺1, respectively. In most of the calculations the
fixed property ratios of the liquids were chosen as ␳ 21
⫽1.4, ␮ 21⫽0.96, and D 21⫽1.1, and the values of the density
and viscosity ratios were taken from the experiment.1 Since
the physical properties of the two protein solutions should be
close, the value of D 21 was chosen close to unity.
III. RESULTS AND DISCUSSION
A. Flow patterns
In the calculations the parameter R was varied from 1 to
100 or 120 and the Peclet number Pe from 103 to 105 . Thus,
The calculated flow patterns in the cocurrent and countercurrent flows are plotted in Figs. 2 and 3. All the isolines
FIG. 3. Isolines of the axial velocity w
共upper frames兲 and streamlines of the
transverse flow 共lower frames兲 for
countercurrent flow G 21⫽⫺1. The
left-hand side of each frame corresponds to the inner part of the toroidal
boundary, the right-hand side to the
outer part. The cross-section B of Fig.
1 is shown. 共a兲 R⫽20 (D⫽179); 共b兲
R⫽40 (D⫽716); 共c兲 R⫽60 (D
⫽1610); 共d兲 R⫽80 (D⫽2862); 共e兲
R⫽100 (D⫽4472). In all cases ␧
⫽0.1, ␮ 12⫽0.96. The sections of the
streamline ␺ ⫽0, which do not coincide with the interface, are highlighted
by numeral zero.
Downloaded 17 Jan 2003 to 132.68.1.29. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp
334
Phys. Fluids, Vol. 15, No. 2, February 2003
Gelfgat, Yarin, and Bar-Yoseph
FIG. 4. Maximal values of the axial
velocity vs the R number. Symbols
correspond to the calculated points.
The range 0⭐R⭐100 corresponds to
that of the Dean number 0⭐D⭐4472;
␧⫽0.1. 共a兲 Cocurrent flow, G 21⫽1,
共b兲 countercurrent flow, G 21⫽⫺1,
␮ 12⫽0.96.
are equally spaced, except for the streamlines in Fig. 3共a兲
where two additional levels are added to illustrate the appearance of additional vortices. The positive and negative
contours of the stream function correspond to the clockwise
and counterclockwise motion, respectively.
In the case of the cocurrent flow (G 21⫽1, Fig. 2兲 the
maximal value of the axial velocity is located close to 共but
not at兲 the center plane 共note that in the single liquid case
this maximum is always exactly at the center plane3,13,18 –20兲.
In the two-fluid case considered here the advection of the
axial velocity takes place in each of the layers separately.
Therefore, one can expect appearance of one maximum of
the axial velocity in each layer. However, with the param-
eters used in the present calculations, development of two
local maxima was not observed; it could probably happen if
the density and viscosity differentials between the liquids
were larger. The transverse flows consist of a pair of the
Dean vortices with one vortex in each layer as in the classical Dean case of a single-liquid flow. With increase of the
parameter R 共i.e., of the Reynolds or the Dean numbers, Re
and D兲, the maximum of the axial velocity is advected toward the outer part of the solid boundary, and a boundary
layer of the axial velocity forms there. Also, with the increase of the parameter R 共or Re and D兲 the ‘‘eyes’’ of the
vortices are pushed backwards and outwards, as shown in
Fig. 2. A similar observation was done in Ref. 7 regarding a
Downloaded 17 Jan 2003 to 132.68.1.29. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp
Phys. Fluids, Vol. 15, No. 2, February 2003
Dean vortices-induced enhancement
335
FIG. 5. Maximal values of the stream
function and the quantity ⌽ vs the R
number. The range 0⭐R⭐100 corresponds to that of the Dean number
0⭐D⭐4472; ␧⫽0.1. 共a兲 Cocurrent
flow, G 21⫽1, 共b兲 countercurrent flow,
G 21⫽⫺1, ␮ 12⫽0.96.
single-fluid flow in a curved pipe, based on their numerical
and experimental results. The maximum and minimum of the
stream functions shift in the opposite direction—toward the
inner boundary and the liquid–liquid interface. Development
of the boundary layer is a common observation for the
single-liquid case.18 –20 Note that due to the action of the
centrifugal force, which is directed toward the outer part of
the toroidal boundary and is maximal near the interface, the
motion along the latter is also directed from the inner 共lefthand兲 to the outer 共right-hand兲 part of the solid 共toroidal兲
boundary.
In the case of countercurrent flow (G 21⫽⫺1, Fig. 3兲
there are two maxima of the axial velocity, one in each layer.
Near these maxima the centrifugal force drives the transverse
flow toward the outer part of the boundary, so that the flow
returns to the inner part along the interface. Therefore, the
direction of motion along the interface is opposite to that in
the cocurrent case 共cf. Figs. 2 and 3兲. Besides this, already at
rather low values of R or of the Dean number, R⬇10 or D
⬇45, each Dean vortex of the transverse flow splits in two. It
will be shown in the following that this split reduces the
mixing of a passive scalar. Note also that in the case of
countercurrent flow, the boundary layers develop not only
near the outer part of the boundary, but also near the interface.
The dependences of the maximal absolute values of the
axial velocity and the stream function on the Reynolds number are shown in Figs. 4 and 5. The axial velocity, as ex-
Downloaded 17 Jan 2003 to 132.68.1.29. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp
336
Phys. Fluids, Vol. 15, No. 2, February 2003
Gelfgat, Yarin, and Bar-Yoseph
FIG. 6. Local Sherwood number calculated via the analytical boundary
layer model for R⫽1 and ␧⫽0.01. The Reynolds and Dean numbers are
Re⫽0.125 and D⫽0.141, respectively. The value of D 21 was chosen as
D 21⫽1. The plot corresponds to the cross-section A of Fig. 1, x⫽⫺1 corresponds to the outer part of the toroidal surface, x⫽1—to the inner one.
The Schmidt number was Sc⫽105 and thus the Peclet number Pe⫽105 .
pected, increases monotonically 共Fig. 4兲; the increase is linear at small values of R⬍20 (D⬍179), where viscous
dissipation is significant, and slows down at larger R 共or
larger D兲.
As mentioned, in the case of cocurrent flow there is a
single Dean vortex in each layer 共Fig. 2兲. The maximal value
of the stream function yields a coarse estimate of the vortex
intensity, and consequently an estimate of the mixing speed
of a passive scalar inside the vortex. It was unexpectedly
found that in the cocurrent flow case the dependencies
␺ max(R) are nonmonotonic and reach a maximum at a certain
value of the parameter R 关Fig. 5共a兲兴. Therefore the dependence of the mass transfer rate on the Reynolds number
should also be nonmonotonic, and indeed, we shall see in the
following that it also has a maximum at a certain moderate
value of R 共or D兲.
In the case of countercurrent flow the maximal absolute
values of the stream function increase monotonically with
the Reynolds number 关Fig. 5共b兲兴. However, in this case there
can be two vortices in each of the layers subdivided by sections of the streamline ␺ ⫽0, which do not coincide with the
interface 共Fig. 3兲, and part of the passive scalar should be
transported through the boundary separating two steady vortices. Therefore, there is no direct relation between the intensity of the strongest vortices and the mixing rate of a passive
scalar at the interface. To characterize the mixing rate inside
a vortex 共Sec. III兲 we introduce the following measure:
⌽⫽
2
S
兩␺兩 ,
␲ vortex max
共17兲
where S vortex is the dimensionless area occupied by the vortex and ␲/2 is the total area of the layer 共the dimensionless
radius of the pipe cross section is equal to 1兲. The area S vortex
corresponds to the area occupied by the vortices bounded by
the interface and the streamline ␺ ⫽0 marked by numeral
zero in each of the layers in the lower frames in Fig. 3. The
value of ⌽ is sensitive to both the vortex intensity 兩 ␺ max兩 and
the fraction of the layer occupied by it. Variation of the quantity ⌽ with R is shown in Fig. 5共b兲. It is seen that for R
⬎60, ⌽ does not change strongly, and has a smooth maximum at R⬇100. Therefore in the countercurrent case, the
mixing rate is not expected to change strongly for 60⭐R
⭐120 (1610⭐D⭐6440).
For aqueous solutions ( ␳ 1 ⫽1 g/cm3 and ␮ 1
⫽10⫺2 g/(cm s)) in a pipe of a⫽1 cm the pressure gradient
G 1 can be estimated as 0.08 and 8 Pa/m for R⫽10 and 100,
respectively. This corresponds to the respective maximal
transverse velocities of approximately 0.11 and 2.8 cm/s for
the cocurrent, and to 0.045 and 1.9 cm/s for the countercurrent flow. The smaller maximal transverse velocity in the
latter case reflects the splitting of the vortices in each fluid
layer in the former. For comparison, the velocity of the secondary meridional flow in a two-layer Taylor–Couette
apparatus3 was found to be of the order of 0.6 cm/s under
realistic conditions, and the velocity of a two-layer natural
convection flow was estimated as 0.17 cm/s. The results
show that the Dean vortices in two-layer systems are stronger than the corresponding Taylor–Couette or naturalconvection counterparts. Thus novel bioseparators based on
the Dean vortices can be expected to be more effective than
those based on the Taylor–Couette apparatus. This fact is a
consequence of the nature of the Dean and Taylor–Couette
vortices: the former appear at any flow rate as an intrinsic
part of the basic flow, and the latter—as a result of bifurcation of the secondary flow from the basic one beyond a certain threshold. Thus the Dean vortices take up a significantly
larger amount of the flow energy than the Taylor–Couette
ones, and are accordingly stronger than the latter.
B. Mass transfer
A general analytical solution for quasistationary mass
transfer through a liquid–liquid interface between a pair of
counterrotating vortices is given in Ref. 4 for large Peclet
numbers. In the case of PeⰇ1 the mass transfer takes place
in a thin boundary layer adjoining the interface, whereas far
from this layer the concentration can be assumed to be constant. The local Sherwood number, corresponding to the quasistationary concentration distribution, can then be expressed
as a function of the interfacial velocity. In the present case
the only analytical flow field available is the one corresponding to a single-liquid case at small values of the Dean number 共cf. 共B5兲–共B9兲 in Appendix B兲. It is appliable to the
two-layer cocurrent (G 21⫽1) case only at ␳ 21⫽ ␮ 21⫽1 and
D→0, for which we undertake to derive an analytical solution for the mass transfer rate. Then, employing the general
solution4 for the concentration field and Eq. 共C19兲 共see Appendix C兲, we arrive in the present case at the following
result:
Downloaded 17 Jan 2003 to 132.68.1.29. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp
Phys. Fluids, Vol. 15, No. 2, February 2003
Dean vortices-induced enhancement
337
FIG. 7. The instantaneous local Sherwood number calculated for R⫽1, ␧⫽0.01, G 21⫽1, D 21⫽1, and Pe⫽105 . The Reynolds and Dean numbers are Re
⫽0.125 and D⫽0.141, respectively. The x axis represents the liquid–liquid interface, x⫽⫺1 corresponding to the outer part of the toroidal pipe, and x
⫽1—to the inner part. The cross-section A of Fig. 1 is shown. 共a兲 10⭐t⭐80, 共b兲 80⭐t⭐4000, 共c兲 t⫽4000.
ShBL共 x 兲 ⫽
1
48冑␲
1/2
D Sc
D 1 /D 2
1⫹ 冑D 1 /D 2
f 共 x 兲,
⫺1⭐x⭐1,
共18兲
where Sc⫽ ␮ 1 /( ␳ 1 D 1 ). The function f (x) is given by
f 共 x 兲⫽
兩 x 兩 共 1⫺x 2 /4兲共 1⫺x 2 兲 2
.
关 22/315⫺x 3 /3⫹9x 5 /20⫺3x 7 /14⫹x 9 /36兴 1/2
共19兲
It is readily seen that ShBL(⫾1)⫽0.
Figure 6 shows the local Sherwood number ShBL calculated as per Eqs. 共18兲 and 共19兲 for R⫽1 and ␧⫽0.01. The
profile of ShBL(x) is characterized by a zero value in the
middle of the interface and two local maxima, with the larger
maximum corresponding to the inner part of the flow along
the interface. Cross-section A of Fig. 1 corresponds to the
results in Fig. 6.
The instantaneous local Sherwood numbers obtained via
time-dependent numerical modeling for the above-mentioned
case (R⫽1, ␧⫽0.01, G 21⫽1, ␩ 21⫽ ␳ 21⫽D 21⫽1) using the
model 共9兲–共14兲—are shown in Figs. 7 and 8 for Pe⫽105 and
106 , respectively. Cross-section A of Fig. 1 corresponds to
these results as well.
At Pe⫽105 and relatively short integration times 共Fig.
7共a兲兲 the local Sherwood number is distributed almost symmetrically about the central point x⫽0 and has two almost
equal maxima located to the right and left of the central
point. With increase of the integration time, the mass transfer
in the central part of the interface intensifies, so that these
two maxima disappear 共Fig. 7共b兲兲. At long integration times
(t⬎500), the local Sherwood number slowly decreases with
time and its profile approaches a certain asymptotic shape,
illustrated in Fig. 7共c兲. At a long integration time the profile
is characterized by slow variation in the central part of the
boundary and a steep decrease near the boundaries.
For Pe⫽106 and short integration times, t⬍500, the distribution of the local Sherwood number is likewise characterized by two almost equal and symmetric maxima 共Figs.
8共a兲 and 8共b兲兲. With increase of the integration time, these
Downloaded 17 Jan 2003 to 132.68.1.29. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp
338
Phys. Fluids, Vol. 15, No. 2, February 2003
Gelfgat, Yarin, and Bar-Yoseph
FIG. 8. The instantaneous local Sherwood number calculated for R⫽1, ␧⫽0.01, G 21⫽1, D 21⫽1, and Pe⫽106 . The Reynolds and Dean numbers are
Re⫽0.125 and D⫽0.141, respectively. The x axis represents the liquid–liquid interface, x⫽⫺1 corresponding to the outer part of the toroidal pipe, and x
⫽1—to the inner part. The cross-section A of Fig. 1 is shown. 共a兲 10⭐t⭐150, 共b兲 200⭐t⭐2000, 共c兲 2500⭐t⭐5000, 共d兲 t⫽5000 and 7000.
maxima are shifted from the central part of the interface
toward the boundaries. At larger times, t⬎2000, the profile
likewise acquires an asymptotic shape 共Fig. 8共c兲兲, qualitatively different from that for Pe⫽105 . The profiles at t
⫽5000 and 7000 are zoomed in Fig. 8共d兲 and are seen to
have two non-symmetric and nonequal maxima, the larger
one near the inner part of the torus boundary (x⫽1) and the
smaller one—near the outer part (x⫽⫺1). This agrees
qualitatively with the analytical result 共18兲 and 共19兲 as per
Fig. 6. In agreement with the analytical boundary layer
model, the profiles have a rather steep minimum between the
two maxima, but unlike the former the minimum is located
near x⫽⫺0.5 instead of the analytically predicted x⫽0. The
magnitudes of the local Sherwood number at the maxima in
Fig. 8共d兲 are also significantly larger than those in Fig. 6.
This shows that steady-state mass transfer has not yet been
achieved in Fig. 8共d兲. Closer agreement with the analytical
model may perhaps be obtained for larger Peclet numbers,
e.g., Pe⫽107 or 108 . However, in this case much finer grids
Downloaded 17 Jan 2003 to 132.68.1.29. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp
Phys. Fluids, Vol. 15, No. 2, February 2003
FIG. 9. The instantaneous local Sherwood number calculated for R⫽10,
␧⫽0.01, G 21⫽1, D 21⫽1, and Pe⫽105 . The Reynolds and Dean numbers
are Re⫽12.5 and D⫽14.1, respectively. The x axis represents the liquid–
liquid interface, x⫽⫺1 corresponding to the outer part of the toroidal pipe,
and x⫽1—to the inner part. The cross-section A of Fig. 1 is shown.
and calculation over much longer times will be required,
which makes the process unaffordable.
Profiles qualitatively similar to the analytical one can
also be observed at larger R numbers at very short integra-
Dean vortices-induced enhancement
339
tion times. This is illustrated in Fig. 9. The calculations were
performed for R⫽10, ␧⫽0.01, G 21⫽1, and ␩ 21⫽ ␳ 21⫽D 21
⫽1. The profiles have two maxima with the larger maximum
near the inner part of the torus boundary, as predicted by the
analytical model, and the minimum closer to the center of the
interface x⫽0 共cf. Fig. 8共d兲兲 and shifting toward x⫽⫺0.5 as
time increases. This deviation from the analytical prediction
can be explained by mixing of the admixture inside the bulk
of the Dean vortices, which affects the boundary conditions
for the interfacial mass transfer already at a very early stage
of the process.
The instantaneous concentration fields at large Reynolds
and Peclet numbers, Re⫽100 and Pe⫽105 , are shown in
Figs. 10 and 11. It is seen that the concentration isolines
reproduce the streamlines of the transverse flow in the bulk
of both liquids 共cf. Figs. 2 and 3兲. This means that at the
large Peclet number, the mass transfer is almost completely
dominated by the convective transport due to the transverse
flow. However, diffusion represents the only mechanism of
mass transfer through the interface. As was shown in Ref. 3,
enhancement of the mass transfer is due to the steeper concentration gradients at the interface created by the enhanced
convective transport of the admixture. This leads to a higher
mass flux even though the diffusion coefficients continue to
be low. The corresponding distributions of the local Sher-
FIG. 10. The instantaneous concentration contours at different time moments. The case of cocurrent flow,
G 21⫽1, ␧⫽0.1, R⫽100, and Pe⫽105
共D⫽4472兲. 25 concentration contours
are equally distributed between c⫽0
and c⫽1. The cross-section B of Fig.
1 is shown.
Downloaded 17 Jan 2003 to 132.68.1.29. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp
340
Phys. Fluids, Vol. 15, No. 2, February 2003
Gelfgat, Yarin, and Bar-Yoseph
FIG. 11. The instantaneous concentration contours at different time moments. The case of countercurrent
flow, G 21⫽⫺1, ␧⫽0.1, R⫽100, and
Pe⫽105 共D⫽4472兲. 25 concentration
contours are equally distributed between c⫽0 and c⫽1. The crosssection B of Fig. 1 is shown.
FIG. 12. The instantaneous local Sherwood number at different time moments. The case of cocurrent flow, G 21⫽1, ␧⫽0.1, R⫽100 共D⫽4472兲, and
Pe⫽105 . The cross-section A of Fig. 1 is shown.
FIG. 13. The instantaneous local Sherwood number at different time moments. The case of countercurrent flow, G 21⫽⫺1, ␧⫽0.1, R⫽100 共D
⫽4472兲, and Pe⫽105 . The cross-section A of Fig. 1 is shown.
Downloaded 17 Jan 2003 to 132.68.1.29. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp
Phys. Fluids, Vol. 15, No. 2, February 2003
Dean vortices-induced enhancement
341
FIG. 14. Time histories of the mass
fraction m f in layer 2. R⫽100 共D
⫽4472兲. 共a兲 Cocurrent flow, G 21⫽1,
共b兲 countercurrent flow, G 21⫽⫺1,
␮ 12⫽0.96.
wood number over the interface are shown in Figs. 12 and
13. In the case of cocurrent flow 共Fig. 2兲 the local Sherwood
number increases starting from the inner part of the flow
region, x⫽⫺1 共where the streamlines in the two layers become closer near the interface兲 and then decays toward the
outer part, x⫽1. In the case of countercurrent flow 共Fig. 3兲
the direction of the interfacial flow is opposite, and the local
Sherwood number increases from the outer boundary and
then decreases as the inner one is approached. In both cases
the Sherwood number is larger in the areas where the liquid
motion in the vortices is directed toward the interface. The
rapid growth of Sh(x) near x⫽0 should not be overestimated, since a significant numerical error may be involved
there 共see Eq. 共15兲兲, but still the distributions corresponding
to higher values of the Dean number 共or the higher values of
R) are radically different from those for D→0. Indeed, com-
Downloaded 17 Jan 2003 to 132.68.1.29. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp
342
Phys. Fluids, Vol. 15, No. 2, February 2003
Gelfgat, Yarin, and Bar-Yoseph
FIG. 15. 共a兲 Time histories of the mass
fraction m f in layer 2 for different viscosity ratios ␮ 12 . R⫽50 共D⫽1118兲,
Pe⫽105 , ␧⫽0.1. Cocurrent flow, G 21
⫽1. 共b兲 Isolines of the axial velocity
共1兲 and streamlines of the transverse
flow 共2兲. R⫽50 共D⫽1118兲, Pe⫽105 ,
␧⫽0.1. Cocurrent flow, G 21⫽1, ␮ 12
⫽0.5.
parison of Figs. 6, 7, and 12 shows that the minimum of Sh
at the pipe center, characteristic of D→0, disappears for
higher values of D.
Time histories of the mass fraction m f 共Eq. 共16兲兲 in layer
2 are shown in Fig. 14 for R⫽100 共D⬇4500兲 and the Peclet
number varying from 103 to 105 . With increase of the Peclet
number, the mass transfer slows down, since the limiting
purely diffusional transport through the interface becomes
weaker. The dimensionless time necessary to reach about
90% of complete equilibration of the admixture concentration in both layers can be estimated as Pe/10.
The effect of the viscosity ratio ␮ 12 is illustrated in Figs.
15 and 16 for the cocurrent and countercurrent flows, respectively. It is seen 共Fig. 15共a兲兲 that for the cocurrent flow mass
transfer increases significantly for ␮ 12⫽0.5. This results
from the fact that a single-vortex system in each fluid layer
characteristic of the values ␮ 12⫽0.96– 1.5 共cf. Fig. 2兲 is replaced in the case of ␮ 12⫽0.5 by a double-vortex system in
FIG. 16. Time histories of the mass fraction m f in layer 2 for different
viscosity ratios ␮ 12 . R⫽50 共D⫽1118兲, Pe⫽105 , ␧⫽0.1. Countercurrent
flow, G 21⫽⫺1.
Downloaded 17 Jan 2003 to 132.68.1.29. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp
Phys. Fluids, Vol. 15, No. 2, February 2003
Dean vortices-induced enhancement
343
FIG. 17. Volume fraction m f in layer 2 vs the parameter R 共or the Dean number at three different times兲. Cocurrent flow, 0⭐R⭐100, 0⭐D⭐4472, G 21
⫽1, ␮ 12⫽0.96. 共a兲 Pe⫽103 , 共b兲 Pe⫽104 , 共c兲 Pe⫽105 .
each fluid layer 共Fig. 15共b兲兲. On the other hand, the effect of
viscosity ratio ␮ 12 on the mass transfer rate in countercurrent
flows is relatively small 共Fig. 16兲.
To illustrate how the Dean vortices in two-layer systems
enhance the mass transfer, the mass fraction m f at several
fixed time moments is plotted in Figs. 17 and 18 versus the
parameter R 共or the Dean number兲 for several Peclet numbers. It is seen that increase of R from 0 to approximately 10
or 15 共increase of D from 0 to about 45 or 100兲 leads to steep
increase of the mass transfer rates in both the cocurrent and
countercurrent flows. Further increase of R 共or of the Dean
number兲 does not yield any significant increase of the mass
transfer and sometimes even a slight reduction 共Fig. 17兲.
Note that the trend of variation of the mass transfer rate with
R 共or with the Dean number兲 corresponds to the trend of
intensity of the vortical motion illustrated in Fig. 5. In the
case of cocurrent flow 共Figs. 2 and 5共a兲兲 the stream function
reaches its largest value at R⬇25 共D⬇280兲 and then virtually saturates versus R 共or D兲. Figure 17 shows that the mass
transfer affected by this flow, steeply intensifies up to R
⬇15 共D⬇100兲 and then likewise practically saturates. The
position of the maximum of m f in Figs. 17共a兲–17共c兲 is almost independent of the Peclet number. Thus, the dependence of the mass transfer rate on the parameter R 共or on the
Dean number兲 correlates with the corresponding dependence
of the vortex intensity 共cf. Figs. 5共a兲 and 17兲. In the case of
countercurrent flow 共Figs. 3 and 5共b兲兲 the maximum of the
stream function of the transverse flow, as well as the quantity
⌽, steeply increase up to R⬇40 共D⬇715兲, and at larger values of R the quantity ⌽ almost saturates. The increase of the
mass transfer rate in this case 共cf. Fig. 18兲 corresponds to the
range R⬍50 共D⬍1118兲, which also correlates with the estimated intensity of the vortices adjoining to the liquid–liquid
interface. The steep growth of the mass fraction 共and, therefore, of the mass transfer rate兲 in the countercurrent case of
Fig. 18 corresponds to Re⬍25 共D⬍280兲 at Pe⫽103 , to R
⬍20 共D⬍180兲 at Pe⫽104 , and to R⬇15 共D⬍100兲 at Pe
⫽105 . This means that there exists an optimal value of the
parameter R 共an optimal axial flow rate兲, which yields the
most effective mass transfer through the liquid–liquid interface. This optimal value depends on the Peclet number, on
the pipe curvature, and on the mode of axial flow 共cocurrent
or countercurrent兲. In any case, it is not very large and can be
roughly estimated as R⬇20, which corresponds to Re⬇50 or
D⬇180 for ␧⫽0.1. From the practical point of view this
means that optimal mass transfer enhancement is reached at
a moderate pressure drop, and there is no need to increase the
axial flow rate beyond a certain point. This conclusion is
valid for both the cocurrent and countercurrent cases, which
do not differ too much.
It should be noted that we did not find any well-defined
power scaling in the dependence of the intensity of the transverse flow on the parameter R 共or on the Reynolds and Dean
numbers; cf. Fig. 5兲. As a result, there is no power-law scaling of the mass transfer rate versus the Peclet number either.
Finally, it is emphasized that the effect of variation of
Downloaded 17 Jan 2003 to 132.68.1.29. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp
344
Phys. Fluids, Vol. 15, No. 2, February 2003
Gelfgat, Yarin, and Bar-Yoseph
FIG. 18. Volume fraction m f in layer 2 vs the parameter R 共or the Dean number at three different times兲. Countercurrent flow, 0⭐R⭐100, 0⭐D⭐4472,
G 21⫽⫺1, ␮ 12⫽0.96. 共a兲 Pe⫽103 , 共b兲 Pe⫽104 , 共c兲 Pe⫽105 .
dimensionless curvature of the pipe ␧ on the flow field and
the mass transfer rate was studied in the present work for
two-layer flows in the range of ␧⫽0.01–0.1. Some additional
numerical and experimental results on the effect of ␧ on
single-layer fluid flows at the values of ␧ as high as 0.317
can be found in Ref. 7. Both works, however, fully dispense
with the possible torsion of the pipe axis, which might limit
application of the present results to helical pipes. A recently
published Ref. 25 and references therein shed some light on
the effect of torsion on the transverse flow field in singlefluid flows. The numerical and experimental results of the
latter work show that as pipe torsion increases at a constant
Dean number, the structure of the transverse flow changes
from two counterrotating vortices to a single-vortex type.
Similar results for two-fluid flows are unavailable, and the
present group has undertaken such calculations, which will
be reported separately. It is clear that a transition from a
two-vortex secondary flow to a single-vortex type will reduce efficiency of the method of enhancement of the interfacial mass transfer discussed in the present work. Therefore
there should exist a limitation on the value of the dimensionless pipe torsion, which guarantees a two-vortex structure of
the secondary flow and the enhanced interfacial mass transfer.
IV. CONCLUSIONS
Two-liquid flows in a coiled circular pipe with vanishing
torsion and their effect on the mass transfer through the
liquid–liquid interface are studied. Patterns of the Dean vortices appearing in the transverse flow are reported. In particular, it is shown that the flow along the immiscible liquid–
liquid interface is in opposite directions in the cases of
cocurrent and countercurrent axial flow. In the countercurrent
case the Dean vortices are found to split into two vortices in
each liquid layer, which leads to a certain decrease in the
velocity of the transverse flow. It is also shown that the intensity of these vortices does not increase monotonically
with increase of the Reynolds 共or Dean兲 number, albeit it
virtually saturates at higher values of Re 共or D兲. This results
in a similar behavior of the mass transfer rate versus the
Reynolds 共or Dean兲 number. Both the cocurrent and countercurrent cases studied lead eventually to similar enhancement
of the mass transfer.
It is shown that the Dean vortex flow considered can be
an effective tool for enhancement of the mass transfer
through liquid–liquid interfaces, especially in cases when the
Schmidt number is large 共e.g., Sc⬃103 , as for proteins兲. It is
emphasized that the Dean vortices provide a stronger mixing
mechanism than the Taylor–Couette counterparts. It is also
emphasized that such a coiled-pipe-based bioseparator/
bioreactor should not contain any moving boundaries and the
flow inside it can be sustained for a long time. There exists
an optimal value of the Reynolds 共or Dean兲 number 共optimal
axial flow rate兲 which yields the most effective mass transfer
enhancement. The optimal value was found to be about R
⫽20 共or Re⫽50 and D⬇180兲, which is not very high and
Downloaded 17 Jan 2003 to 132.68.1.29. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp
Phys. Fluids, Vol. 15, No. 2, February 2003
Dean vortices-induced enhancement
h r ⫽1,
can be easily reached in practice. No definite scaling law for
the diffused mass fraction versus the Reynolds, Dean, or Peclet numbers was found.
h ␪ ⫽r,
h s ⫽1⫹␧r sin ␪ .
共A1兲
The corresponding equations can be derived from their counterparts in general orthogonal curvilinear coordinates 共see,
for example Ref. 17兲. Assuming ⳵ / ⳵ s⬅0, the continuity
equation reads
ACKNOWLEDGMENTS
This research was supported by BSF 共US–Israel Binational Science Foundation兲 Grant No. 97-118, the Israel Ministry of Immigrant Absorption 共to A.Yu.G.兲, and by the Israel
High Performance Computer Unit.
⳵
⳵
关 r 共 1⫹␧r sin ␪ 兲 u 兴 ⫹
关共 1⫹␧r sin ␪ 兲v兴 ⫽0;
⳵r
⳵␪
APPENDIX A: CONTINUITY, MOMENTUM, AND MASS
TRANSFER EQUATIONS IN HELICAL
COORDINATES „ r , ␪ , s …
The Lamé coefficients of the coordinate system shown in
Fig. 1 are
再 冋
冋
共A2兲
the r, ␪, and s projections of the momentum equations are
册
⳵u
⳵u v ⳵u v2
⳵p 1 ⳵
1
⳵
⳵
1
␧ sin ␪
⫺ ⫺
w 2 ⫽⫺
⫹u ⫹
⫹
关 r 共 1⫹␧r sin ␪ 兲 u 兴 ⫹ 2
⳵t
⳵r r ⳵␪
r
1⫹␧r sin ␪
⳵ r R ⳵ r r 共 1⫹␧r sin ␪ 兲 ⳵ r
r 共 1⫹␧r sin ␪ 兲 ⳵ ␪
⫻ 共 1⫹␧r sin ␪ 兲
345
册
冎
⳵u
cos ␪
2 ⳵v
cos ␪ ⫹␧r sin共 2 ␪ 兲
2
⫺ 2
⫺␧
w ,
2 v ⫺␧
⳵␪
r ⳵␪
r 共 1⫹␧r sin ␪ 兲
共 1⫹␧r sin ␪ 兲 3
共A3兲
⳵v
⳵v v ⳵v uv
␧ cos ␪
⫹u ⫹
⫺
⫹
w2
⳵t
⳵r r ⳵␪
r
1⫹␧r sin ␪
⫽⫺
⫹
冋
再
1 ⳵p 1
1
⳵ 1
⳵共 rv 兲
⫹
共 1⫹␧r sin ␪ 兲
r ⳵ ␪ R 1⫹␧r sin ␪ ⳵ r r
⳵r
冋
册
册
⳵w
⳵w v ⳵w
␧ sin ␪
␧ cos ␪
⫹
uw⫹
⫹u
⫹
vw
⳵t
⳵ r r ⳵ ␪ 1⫹␧r sin ␪
1⫹␧r sin ␪
再 冋
r
⳵
1 1 ⳵
⫽⫺G i ⫹
关共 1⫹␧r sin ␪ 兲 w 兴
R r ⳵ r 1⫹␧r sin ␪ ⳵ r
⫹
冋
1
⳵
1 ⳵
关共 1⫹␧r sin ␪ 兲 w 兴
2
r ⳵ ␪ 1⫹␧r sin ␪ ⳵ ␪
⫹␧ 2
册
冎
册
cos ␪
␧r⫹sin ␪
2
v , 共A5兲
3 u⫺␧
1⫹␧r
sin
␪
1⫹␧r
sin ␪ 兲 3
兲
共
共
⳵c
⳵c v ⳵c
⫹u ⫹
⳵t
⳵r r ⳵␪
再
冎
␧ cos ␪ 1 ⳵ c
.
1⫹␧r sin ␪ r ⳵ ␪
u⫽
⳵␺
1
,
r 共 1⫹␧r sin ␪ 兲 ⳵ ␪
v ⫽⫺
⳵␺
1
,
1⫹␧r sin ␪ ⳵ r
APPENDIX B: ANALYTICAL SOLUTION
FOR THE FLOW FIELD IN A SINGLE-LIQUID CASE
FOR SMALL CURVATURE
To derive the analytical solution for small curvature we
assume that the axial velocity is of order 1 and the transverse
flow of order ␧. Thus, separating all the terms of order 1 in
Eq. 共A5兲 and all those of order ␧ in Eqs. 共A2兲–共A4兲, we
arrive at the following equations.
The continuity equation
and the momentum equations
共A6兲
共A7兲
and is rendered dimensionless by (G 1 a 3 / ␳ 1 ) 1/2.
1 ⳵ 共 ru 兲 1 ⳵ v
⫹
⫽0,
r ⳵r
r ⳵␪
1 1 ⳵ 共 rc 兲 1 ⳵ 2 c
␧ sin ␪ ⳵ c
⫹ 2 2⫹
Pe r ⳵ r
r ⳵␪
1⫹␧r sin ␪ ⳵ r
⫹
共A4兲
The stream function ␺ of the transverse flow is given by
where G i ⫽1 or G 21 for liquids 1 and 2, respectively.
Note that Re⫽R2/8, and D⫽8(2␧) 1/2 Re⫽(2␧)1/2R 2 .
The mass transfer equation is
⫽
冎
1
⳵
␧r⫹sin ␪
1 ⳵
2 ⳵u
cos ␪
⫹␧
u⫹␧ 2
w ,
关共 1⫹␧r sin ␪ 兲v兴 ⫹ 2
r 2 ⳵ ␪ 1⫹␧r sin ␪ ⳵ ␪
r ⳵␪
r 共 1⫹␧r sin ␪ 兲 2
共 1⫹␧r sin ␪ 兲 3
⫺
再 冋
册
共B1兲
冎
⳵ p 1 ⳵ 1 ⳵ 共 ru 兲
1 ⳵ 2u 2 ⳵ v
⫹ 2 2⫺ 2
⫽⫺␧w 2 sin ␪ ,
⫹
⳵r R ⳵r r ⳵r
r ⳵␪
r ⳵␪
共B2兲
Downloaded 17 Jan 2003 to 132.68.1.29. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp
346
⫺
Phys. Fluids, Vol. 15, No. 2, February 2003
再 冋
册
1 ⳵ p 1 ⳵ 1 ⳵共 rv 兲
1 ⳵ 2v 2 ⳵ u
⫹
⫹ 2 2⫹ 2
r ⳵␪ R ⳵r r ⳵r
r ⳵␪
r ⳵␪
冉
冎
⫽⫺␧w 2 cos ␪ ,
⫺1⫹
Gelfgat, Yarin, and Bar-Yoseph
冋 册
1 1 d
dw
r
⫽0.
R r dr dr
␧Pek ⫺
共B3兲
共B4兲
Equations 共B1兲–共B4兲 must be solved with the no-slip boundary condition at r⫽1. The solution is13,16
w⫽
R
共 1⫺r 2 兲 ,
4
冉 冊
r2
R3
1⫺
u⫽␧
共 1⫺r 2 兲 2 sin ␪ ,
1152
4
v ⫽⫺␧
R3
共 r 2 ⫺1 兲共 7r 4 ⫺23r 2 ⫹4 兲 cos ␪ ,
4608
共B5兲
共B6兲
共B7兲
R
r 共 2r 4 ⫺6r 2 ⫹9 兲 sin ␪ .
192
共B8兲
This solution corresponds to small values of the Dean number D⫽8(2␧) 1/2 Re⫽(2␧)1/2R 2 . The dimensionless stream
function of the flow ␺ is given by 共A7兲, 共B6兲, and 共B7兲 in
this limit 共␧→0兲 as per
␺ ⫽⫺␧
冉 冊
R3
r2
r 1⫺
共 1⫺r 2 兲 2 cos ␪ ,
1152
4
共B9兲
where it is rendered dimensionless by (G 1 a 3 / ␳ 1 ) 1/2.
APPENDIX C: CALCULATION OF THE SHERWOOD
NUMBER WITHIN THE FRAMEWORK OF THE
BOUNDARY LAYER MODEL
We assume that the admixture concentration varies only
in a thin boundary layer near the interface in both liquids 1
and 2, which is a plausible approximation based on the fact
that ScⰇ1. In the present case c k ⫽c k (r, ␪ ) and the dimensionless Eqs. 共9兲 and 共10兲 in the boundary layer approximation take the form
␧U k
⳵ck
1 1 ⳵ 2c k
Vk ⳵ck
⫽
,
⫹␧
⳵r
r ⳵ ␪ Pek r 2 ⳵ ␪ 2
共C1兲
where ␧U k ⫽u k and ␧V k ⫽ v k are the u and v velocity components in the kth layer, respectively, and Pek are the Peclet
numbers defined as Pek ⫽W̄ 1 a 1 /D k . In the present case we
assume ␳ 21⫽ ␮ 21⫽1, as well as G 21⫽1. As a result, U 1
⫽U 2 ⫽U and V 1 ⫽V 2 ⫽V. According to Eqs. 共B6兲, 共B7兲,
and 共B9兲,
U⫽
V⫽
A共 r 兲
sin ␪ ,
r
共C2兲
dA
cos ␪ ,
dr
共C3兲
冉 冊
Re
r2
A 共 r 兲 ⫽ r 1⫺
共 1⫺r 2 兲 2 .
18
4
At the interface ␪⫽⫿␲/2, Eq. 共C1兲 takes the form
⳵ 2c k
⳵ck
⳵ck
⫽Q k 共 x 兲
,
2 ⫹ P k共 x 兲 y
⳵y
⳵y
⳵x
where
冉
P k ⫽␧Pek ⫿
dA/dr
r
冊
A
r
冎
at
共C6兲
␪ ⫽⫿ ␲ /2.
共C7兲
Solution of Eq. 共C6兲 is subject to the following boundary
conditions:
y⫽⫺⬁,
c 2 ⫽c ⬁2 ,
共C8兲
y⫽⫹⬁,
c 1 ⫽c ⬁1 ,
共C9兲
⳵c2
⳵c1
⫽⫺D 1
,
⳵y
⳵y
y⫽0,
⫺D 2
y⫽0,
c 1 ⫽c 2 ,
共C10兲
共C11兲
and solution of the problem 共C6兲–共C11兲 yields4
冉冊
冉冊
c 2⫽
Z
c ⬁2 ⫹ ␬ c ⬁1 ␬ 共 c ⬁2 ⫺c ⬁1 兲
⫺
erf
,
1⫹ ␬
1⫹ ␬
2
共C12兲
c 1⫽
Z
c ⬁2 ⫹ ␬ c ⬁1 共 c ⬁2 ⫺c ⬁1 兲
⫺
erf
,
1⫹ ␬
1⫹ ␬
2
共C13兲
where
⫺1/2
␬ ⫽D 21
,
Z⫽y
冋冕
x
1
W 共 x 兲 ⫽2
e W( ␰ )⫺W(x)
d␰
Q k共 ␰ 兲
冕
册
共C14兲
1/2
共C15兲
,
P k共 ␰ 兲
d␰.
const Q k 共 ␰ 兲
x
Given Eqs. 共C4兲, 共C7兲, and 共C16兲, we find
冋
W 共 ␰ 兲 ⫺W 共 x 兲 ⫽2 ln
␰ 共 1⫺ ␰ 2 /4兲共 1⫺ ␰ 2 兲 2
x 共 1⫺x 2 /4兲共 1⫺x 2 兲 2
共C16兲
册
共C17兲
for ␪⫽⫿␲/2.
Therefore the mass flux j 2 ⫽⫺(D 2 /a) ⳵ c 2 / ⳵ y 兩 y⫽0 is
found from 共C12兲, 共C15兲, and 共C17兲 as
j 2⫽
D 2 ␬ 共 c ⬁2 ⫺c ⬁1 兲
1⫹ ␬
a 冑␲
⫻
共C4兲
共C5兲
where Eqs. 共C2兲 and 共C3兲 were used and the coordinate y is
normal to x 共normal to the interface; the coordinates are
dimensionless兲. Since at the interface ␪⫽⫿␲/2 we have x
⫽⫾r, and Eq. 共C5兲 can be rearranged to the following form:
Q k ⫽⫺␧Pek
2
p⫽␧
冊
⳵ 2c k
A ⳵ c k dA/dr ⳵ c k
⫾
y
⫽ 2,
r ⳵x
r
⳵y
⳵y
冋
␧Pe2 共 Re/18兲 x 2 共 1⫺x 2 /4兲 2 共 1⫺x 2 兲 4
22/315⫺x 3 /3⫹9x 5 /20⫺3x 7 /14⫹x 9 /36
册
1/2
.
共C18兲
Obviously, j 1 ⫽ j 2 . The Sherwood number is defined as
Downloaded 17 Jan 2003 to 132.68.1.29. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp
Phys. Fluids, Vol. 15, No. 2, February 2003
ShBL共 x 兲 ⫽
j 2a
,
共 c ⬁2 ⫺c ⬁1 兲 D 2
Dean vortices-induced enhancement
11
共C19兲
which yields, with the aid of Eq. 共C18兲, the expressions 共18兲
and 共19兲.
1
G. Baier, T. M. Grateful, M. D. Graham, and E. N. Lightfoot, ‘‘Prediction
of mass transfer rates in spatially periodic flows,’’ Chem. Eng. Sci. 54, 343
共1999兲.
2
G. Baier, M. D. Graham, and E. N. Lightfoot, ‘‘Mass transport in a novel
two-fluid vortex extractor,’’ AIChE J. 46, 2395 共2000兲.
3
A. L. Yarin, A. Yu. Gelfgat, and P. Z. Bar-Yoseph, ‘‘Enhancement of mass
transfer in a two-layer Taylor–Couette apparatus with axial flow,’’ Int. J.
Heat Mass Transf. 45, 555 共2002兲.
4
A. L. Yarin, ‘‘Stationary d.c. streaming due to shape oscillations of a
droplet and its effect on mass transfer in liquid–liquid interfaces,’’ J. Fluid
Mech. 444, 321 共2001兲.
5
A. L. Yarin, ‘‘Steady streaming and mass transfer due to capillary waves in
a two-layer system,’’ Fluid Dyn. Res. 31, 79 共2002兲.
6
K. Y. Chung, M. E. Brewster, and G. Belfort, ‘‘Dean vortices with wall
flux in a curved channel membrane system. 3. Concentration polarization
in a spiral reverse osmosis slit,’’ J. Chem. Eng. Jpn. 31, 683 共1998兲.
7
H. Mullubhotla, G. Belfort, W. A. Edelstein, and T. A. Early, ‘‘Dean vortex
stability using magnetic resonance flow imaging and numerical analysis,’’
AIChE J. 47, 1126 共2001兲.
8
J. J. Sillekens, C. C. M. Rindt, and A. A. van Steenhoven, ‘‘Developing
mixed convection in a coiled heat exchanger,’’ Int. J. Heat Mass Transf.
41, 61 共1998兲.
9
N. Acharya, M. Sen, and H.-C. Chang, ‘‘Analysis of heat transfer enhancement in coiled-tube heat exchangers,’’ Int. J. Heat Mass Transf. 44, 3189
共2001兲.
10
S. Lugue, H. Mallubhotla, G. Gehlert, R. Kuriel, S. Dzengeleski, S. Pearl,
and G. Belfort, ‘‘A new coiled hollow-fiber module for enhanced microfiltration performance in biotechnology,’’ Biotechnol. Bioeng. 65, 247
共1999兲.
347
T. Kluge, C. Rezende, D. Wood, and G. Belfort, ‘‘Protein transmission
during Dean vortex microfiltration of yeast suspensions,’’ Biotechnol.
Bioeng. 65, 649 共1999兲.
12
S. A. Berger and L. Talbot, ‘‘Flow in curved pipes,’’ Annu. Rev. Fluid
Mech. 15, 461 共1983兲.
13
T. J. Pedley, Fluid Mechanics of Large Blood Vessels 共Cambridge University Press, Cambridge, 1980兲.
14
W. R. Dean, ‘‘Fluid in a curved channel,’’ Proc. R. Soc. London, Ser. A
121, 402 共1928兲.
15
A. Yu. Gelfgat, A. L. Yarin, and P. Z. Bar-Yoseph, ‘‘Three-dimensional
instability of a two-layer Dean flow,’’ Phys. Fluids 13, 3185 共2001兲.
16
W. R. Dean, ‘‘The stream-line motion of fluid in a curved pipe,’’ Philos.
Mag. 5, 673 共1928兲.
17
M. Germano, ‘‘On the effect of torsion on a helical pipe flow,’’ J. Fluid
Mech. 125, 1 共1982兲.
18
L. Zabielski and A. J. Mestel, ‘‘Steady flow in a helically symmetric
pipe,’’ J. Fluid Mech. 370, 297 共1998兲.
19
T. J. Hüttl, C. Wagner, and R. Friedrich, ‘‘Navier-Stokes solutions of laminar flows based on orthogonal helical coordinates,’’ Int. J. Numer. Methods Fluids 29, 749 共1999兲.
20
Y. Fan, R. I. Tanner, and N. Phan-Thien, ‘‘Fully developed viscous and
viscoelastic flows in curved pipes,’’ J. Fluid Mech. 44, 327 共2001兲.
21
A. Yu. Gelfgat, P. Z. Bar-Yoseph, and A. Solan, ‘‘Vortex breakdown and
instability of swirling flow in a cylinder with rotating top and bottom,’’
Phys. Fluids 8, 2614 共1996兲.
22
W. M. Collins and S. C. R. Dennis, ‘‘The steady motion of viscous fluid in
a curved tube,’’ Q. J. Mech. Appl. Math. 28, 133 共1975兲.
23
D. J. McConaloque and R. S. Srivastava, ‘‘Motion of fluid in a curved
tube,’’ Proc. R. Soc. London, Ser. A 307, 37 共1968兲.
24
P. B. Rhines and W. R. Young, ‘‘How rapidly is a passive scalar mixed
within closed streamlines?’’ J. Fluid Mech. 133, 133 共1983兲.
25
K. Yamamoto, A. Aribowo, Y. Hayamizu, T. Hirose, and K. Kawahara,
‘‘Visualization of the flow in a helical pipe,’’ Fluid Dyn. Res. 30, 251
共2002兲.
Downloaded 17 Jan 2003 to 132.68.1.29. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/phf/phfcr.jsp