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Colloids and Surfaces A: Physicochem. Eng. Aspects 366 (2010) 1–11
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Colloids and Surfaces A: Physicochemical and
Engineering Aspects
journal homepage: www.elsevier.com/locate/colsurfa
Formation of vortices in a combined pressure-driven electro-osmotic flow
through the insulated sharp tips under finite Debye length effects
Zhi-Yuan Sun a,b , Yi-Tian Gao a,b,∗ , Xin Yu b , Ying Liu b
State Key Laboratory of Software Development Environment, Beijing University of Aeronautics and Astronautics, Beijing 100191, China
Ministry-of-Education Key Laboratory of Fluid Mechanics and National Laboratory for Computational Fluid Dynamics,
Beijing University of Aeronautics and Astronautics, Beijing 100191, China
a r t i c l e
i n f o
Article history:
Received 16 February 2010
Received in revised form 24 April 2010
Accepted 27 April 2010
Available online 4 May 2010
Pressure-driven electro-osmotic flow
Sharp tip
Finite Debye length
Poisson–Boltzmann model
Finite element method
a b s t r a c t
Formation of vortices in an electro-osmotic flow possesses engineering applications in enhancing and
controlling the microfluidic mixing. In this paper, we investigate the combined pressure-driven electroosmotic flow through the insulated sharp tips in a straight microchannel when a direct current electric
field is imposed. Maximum vorticity generated near the tip back is influenced by the local Reynolds
number and tip sharpness. Under a finite Debye length, the way to control the recirculation region for the
single tip is discussed. Such control is applied to a pair of sharp tips which are designed as the symmetrical
and asymmetrical ones in shape, or the ones staggered in position. Poisson–Boltzmann model is solved
to simulate the flow by the finite element method. Results are shown to support the assumption of finite
Debye length and expected to be helpful in controlling the vorticity and recirculation in the relevant
microfluidic devices.
© 2010 Elsevier B.V. All rights reserved.
1. Introduction
Electro-osmotic flow (EOF), attributed to the interfacial effects
in the micro- or nanometer scale, has become one of the nonmechanical techniques for manipulating fluids in microsystems,
such as the Lab-on-Chip devices and microfuel cells [1–5]. Generally
speaking, EOF is the bulk fluid motion driven by the electrokinetic
force acting on the net charged ions in the diffuse layer, the outer
part of an electrical double layer (EDL) [1,2]. Thickness of the EDL
can be characterized by the Debye length, which is in the order of
10 nm to 1 ␮m [3]. When such length is relatively small compared
with the geometry scale (i.e., under the assumption of infinitesimal Debye length), the linear EOF theory indicates the similitude
between the flow streamlines and electric field lines based on a
neutral fluid with uniform density, viscosity, conductivity and surface zeta potential at the fluid–solid interface [3–5]. One feature of
∗ Corresponding author at: State Key Laboratory of Software Development Environment, Beijing University of Aeronautics and Astronautics, Beijing 100191, China.
Tel.: +86 1082315701.
E-mail address: gaoyt@public.bta.net.cn (Y.-T. Gao).
0927-7757/$ – see front matter © 2010 Elsevier B.V. All rights reserved.
the linear EOF is the irrotationality which makes the existence of
vortices impossible [3–5]. Furthermore, due to the neglect of the
convection term, the linear EOF can be categorized as the Stokes
flow and the viscous term is balanced with the pressure term [3].
Therefore, in certain cases [3–6], the effect of EDL is represented by
the Helmholtz–Smoluchowski (HS) slip velocity at all the boundaries when the EOF is governed by the Navier–Stokes equations,
which admits the infinitesimal Debye length assumption [3–6].
In recent years, nonlinear electrokinetic phenomena in the
microfluids have risen as the practical aspects in some investigations [7–13]. For instance, one of those refers to the concept
induced-charge electro-osmosis (ICEO), which is directly related to
the polarizability of the wall surface [7–9], and the vortices have
been observed around the sharp tips with the symmetrical design
in Ref. [10] to support the ICEO effect. However, the ICEO is not the
only explanation for the nonlinear vortex dynamics. Another type
of nonlinear EOF, the one we care about in this paper, is caused by
the convection of the fluids and ionic species, in which the generation of vortices is also an attractive phenomenon [3,12,13]. In
the early work devoted to the combined pressure-driven EOFs in
the two-dimensional straight channels [14], the finite Debye length
effects have been discussed and the EDL vorticity thickness has
Z.-Y. Sun et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 366 (2010) 1–11
been defined. At the same time, utilization of the HS velocity as
the matching condition between the bulk flow and EDL has been
pointed to be incomplete for the mixed pressure-driven EOFs [14].
Further investigation reveals that the Reynolds number-dependent
EOF is caused by the finiteness of the Debye length [3]. In such a situation, the irrotationality is broken when the vorticity advects out
of the EDL and into the bulk flow [3,5]. Simulations show that the
convection of vorticity has close relationships with the Reynolds
numbers, Debye length, global and local geometry of the channels
[3]. Convective transport of vorticity is claimed by Ref. [5] to be
valuable for exploring further.
With those considerations, it would be necessary to study the
EOF, especially the combined pressure-driven one, under the finiteDebye-length assumption for the similar sharp-tip design as in Refs.
[9,10]. Appearance of vortices around the sharp tips is explained
by the ICEO mechanism and considered to be valuable for enhancing the mixing in the microfluidic devices under the direct current
(DC) electric field [7,8]. Meanwhile, the microchannels with those
tips are similar to the ones with different types of roughness, such
as the rectangular [15,16] and wave-like (or curve-like) [17,18]
roughness, but the generation of vortices for them has been found
to be scarce [15–18]. In this paper, we will mainly focus on the
combined pressure and electrokinetic effects on the generation of
vorticity around the tip back and recirculation region behind the tip.
Pressure-driven Poiseuille flow with the parabolic velocity profile
will be given at the inlet of the microchannel, and the zeta potential will be distributed uniformly on the surface of the insulated
sharp tips, while other sections of the channel will have the zero
potential. Convection term will be considered and may have to be
balanced with the pressure gradient, which brings complex cases
for the convective transport within the scale of the finite EDL. Additionally, one will be able to refer to Refs. [14,19,20] concerning the
combined pressure-driven EOFs, including the cases in the square
and wavy microchannels. Moreover, we will have to emphasize that
the thermal effects (Joule heating) will be neglected in our study to
avoid the additional influence on the dynamics of vortices [5].
Poisson–Boltzmann (PB) model will be employed to describe the
EOF in our framework and the finite element method (FEM) will
be used to solve such model. PB model has its accuracy up to the
high ionic concentration or for the smaller EDL thickness compared
with the channel scale [21]. Simulations of such model have been
used to describe the complex electrokinetic cases, e.g., the electrokinetic transport in the microchannels with random roughness
[22] and the electrokinetic interactions in the microscale cross-slot
flow [11]. Refs. [23,24] have solved the PB-typed model with the
lattice Poisson–Boltzmann method (LPBM) to investigate the mixing enhancement in the microfluidics driven by both the pressure
gradient and electrokinetic forces. Further studies show that the
LPBM (including the coupled ones) is effective in simulating those
electrokinetic flows [25,26]. Actually, the linearized PB model for
the small zeta potential has been presented in Refs. [3,14] to discuss
the finite-Debye-length effects and convective transport in the EOF.
On the other hand, we will use the PB model with the Boltzmann
distribution in order to accord with the framework of Refs. [3,27],
which shows that the vorticity can be transported to the bulk region
by employing such distribution instead of solving the ion transport
Goals of this paper will be to clarify the following aspects: (1)
how to influence the vortices generated around the tip back by
the Reynolds number and tip sharpness in the combined pressuredriven EOF; (2) how to reduce the recirculation region generated
behind the sharp tip based on the infiniteness of Debye length;
(3) how to control the recirculation regions for a pair of sharp
tips designed in three typical manners. Question (1) will be mainly
focused on the small-scale vortices while Questions (2) and (3) will
be concerned about the large-scale eddies which can influence the
Fig. 1. Sketch of the microchannel with an isolated sharp tip.
flow structures. With those aims, the structure of this paper will
be arranged as follows: Geometry of the model and mathematical formulation will be presented in Section 2. Numerical results
and discussions will be provided in Section 3 to investigate the formation of vortices and control of recirculation under the effects
of finite Debye length. Finally, our summary will be addressed in
Section 4.
2. Formulation
Consider a straight finite-length channel with an isolated sharp
tip, which is located between two electrodes which produce an
electric field directed along the x-axis (see Fig. 1). The channel is
assumed to be sufficiently long upstream and downstream, while
the sharp tip is discussed in the regions far from the inlet and outlet (where the channel length L H). Such channel is filled with an
incompressible Newtonian electrolyte of uniform dielectric constant r , dynamic viscosity and density . When the externally
applied electric field is generated by the electrodes placed at the
channel inlet and outlet, the distribution of the external electric
potential is described by the Laplace equation [3],
∇ 2 = 0.
For the boundary conditions, at the inlet and outlet of the channel
is specified ( in = 0 , out = 0) and satisfies n · ∇ = 0 at the wall,
where n is the unit normal vector.
When the electrolyte contacts with the solid surface (tip surface), the surface charge causes rearrangement of both counter-ions
and co-ions in the liquid phase nearby, which leads to the formation of the EDL (we assume that the EDL is formed along the sharp
tips uniformly) [1–3]. The EDL thickness can be characterized by
the Debye length [1–4],
2z 2 e2 n0
0 r kB T
where z is the valance of ion, e is the charge of an electron, n0 is the
bulk ionic number concentration, 0 is the permittivity of vacuum,
kB is the Boltzmann constant and T is the absolute temperature.
For a 1:1 symmetric electrolyte, if we consider the very slow quasisteady-state-electrokinetic flow and neglect the ionic convection
within the EDL (the flow in the bulk region is considered to be electrically neutral), the conservation equations for the positive and
negative ionic species reduce
distribution in such
to the Boltzmann
a scale [3,21], n = n0 exp −ze /(kB T ) , where
is the inherent
electric potential in the EDL. Therefore the net charge density e in
the EDL can be expressed as [1,3,4]
e = −2zen0 sinh
kB T
According to the theory of electrostatics, within the EDL, the electric
potential obeys the Poisson equation, ∇ 2 = −e /(0 r ) [3,4,11].
Z.-Y. Sun et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 366 (2010) 1–11
Substituting Eq. (3) into the Poisson equation yields the nonlinear
PB equation [3,11,21,22,28] for the EOF,
0 r
kB T
Eq. (4) is a type of the nonlinear partial differential equations
(NPDEs). Such NPDEs have played an important role in various
fields of science and technology, and their solutions have been
investigated by symbolic computation [29]. Normally, the electric
field due to the EDL (−∇ ) is sufficiently larger than the externally
applied electric field (−∇ ), which indicates that the influence of
the applied electric field on the ionic distribution can be neglected
[3,11]. Eq. (4) is subjected to the following boundary conditions:
= at the sharp tip, where is the zeta potential (variation of
the zeta potential with the concentration can be referred to Ref.
[30]); = 0 at other sections of the channel.
For the quasi-steady-state EOF, all other electromagnetic forces
can be neglected compared with the static electric force [22], so
the body force f is induced by the electric filed −∇ on the net
charge density in the EDL region as f = −e ∇ . Incompressible fluid
flow in the steady state is described by the continuity equation and
Navier–Stokes equation as [3–6,11,22,28]
∇ · u = 0,
u · ∇ u = −∇ p + ∇ u + f,
where u is the velocity vector and p is the pressure. Eqs. (5a)
and (5b) are subjected to the following boundary conditions: the
parabolic velocity profile ux (y) = U0 (3/2 − 6y2 /H 2 ) at the inlet,
where U0 is the average velocity (while the influence of the spatial gradient in zeta potential on the assumed parabolic velocity
distribution upstream is neglected since the channel is sufficiently
long, and some revisions for such boundary conditions will be mentioned in the Remark of Section 3.1); no slip condition at the channel
walls; normal stress at the outlet, i.e., pout ≈ 0.
Introduce the following dimensionless parameters:
x∗ =
p∗ =
y∗ =
u∗ =
∗ =
kB T
Fig. 2. Sketch of the dimensionless boundary conditions.
solutions for the flow field. This solving process can save plenty of
computational resources since the variables are not coupled very
intensively [28]. The non-uniform grid applied to discretize the
domain in the bulk flow is extremely fine, while the finer case is
employed in the near-sharp-tip region, where the EDL locates. The
convergence criterion is chosen with the relative tolerance to be
1.0 × 10−6 and maximum number of iterations to be 100.
Consider the microchannel with the height H = 100 ␮m, the
height and width of the sharp tip for the standard model shown
in Fig. 1 being h = 30 ␮m and d = 40 ␮m, respectively. The two
sides of the tip are arranged symmetrically. In the simulation, KCl solution (valence |z| = 1) is used as the electrolyte,
the physical properties of which contain = 1.0 × 103 kg m−3 ,
= 0.9 × 10−3 kg m−1 s−1 and r = 80. The physical constants
include 0 = 8.854 × 10−12 C2 J−1 m−1 , kB = 1.38 × 10−23 J K−1 and
e = 1.6 × 10−19 C. The temperature is chosen as T = 298 K. The bulk
ionic number concentration is considered as n0 = 1.0 × 10−6 M
with the zeta potential = 0.2 V [30] for the initial simulation. With
those parameters, the characteristic length of the Debye length is
estimated to be 307 nm [the dimensionless parameter K∼325.3
correspondingly, and other dimensionless parameters can be confirmed via Eqs. (6a) and (6b)], and the size of the initial layer
thickness (thickness of the first element layer, that is, the layer
adjacent to the corresponding boundary) within the EDL is chosen as 5 nm. The numbers of the nodes and meshes are 165,505
and 18,649, respectively.
3.1. Development of vortices with Reynolds number
Re =
K = −1 H,
2zen0 U02
where ∗ represents the dimensionless parameters, K is the electrokinetic parameter and Re is the average reference Reynolds number. Then Eqs. (1), (4), (5a) and (5b) can be non-dimensionalized by
those parameters as
∇ 2 ∗ = 0,
= K sinh(
∇ · u = 0,
1 2 ∗
∇ u + G sinh(
u∗ · ∇ u∗ = −∇ p∗ +
)∇ ∗ .
The corresponding dimensionless boundary conditions for Eqs.
(7)–(10) are illustrated in Fig. 2.
3. Results and discussions
Simulation of the pressure-driven EOFs in our model is carried
out by the 2D code of the COMSOL Multiphysics 3.4 (COMSOL Inc.),
which is based on the FEM. The code solves Eqs. (7) and (8) for the
steady external electric field and inherent electric field within the
EDL. Then the solutions are applied to Eqs. (9) and (10) to derive the
In the corner region of the microchannel, there exists transport
of the vorticity from the EDL region to the bulk region, which is
considered to be caused by the convective fluid motion [3]. Such
transport becomes remarkable with the Reynolds number increasing [3]. In the present work, first we will investigate how the
Reynolds number influences the vortices induced by the combined
effects of the pressure and electroosmosis. We fix 0 to be 2.3 V (the
dimensionless 0∗ ∼11.5) in order to provide an external electric
field approximately 1 × 104 V/m, and control the Reynolds number to be less than 20 (Ref. [3] has discussed the vorticity transport
near the channel corners up to Re = 100, which is considered to
be difficult to achieve experimentally [5]). Fig. 3 shows the velocity streamlines and vorticity contours for the maximum Re = 0.02,
0.05, 0.66 and 0.98 (appearing around the tip), respectively. It is
evident that with the low Reynolds number increasing, the vorticity contour around the tip is pushed directly to the downstream
region, and a recirculation region (the eddy) is formed behind the
tip. As seen in Fig. 3(a), when Re is relatively small, the flow exhibits
the EOF feature around the tip; as the eddy being generated, there
exists a flow layer between the recirculation region and channel
wall [see Fig. 3(c) and (d)], which is due to the finite EDL and different from the case for the pure pressure-driven flow (details of
which will be discussed in Part 3.3).
Z.-Y. Sun et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 366 (2010) 1–11
Fig. 3. Numerical results for the vorticity contours (left column) and velocity streamlines (right column) for four Reynolds numbers (flow direction: from the left to the
right): (a) Re = 0.02; (b) Re = 0.05; (c) Re = 0.66; (d) Re = 0.98.
With respect to the high Reynolds number (less than 20), the
recirculation region expands (the scale of the downstream region
is selected in order to avoid the eddy crossing the outlet boundary),
and the maximum positive value of vorticity max is calculated to
occur at the tip back. Fig. 4 illustrates the variation of max with the
Reynolds number for the pressure-driven EOF and pure pressuredriven flow as comparison. As the Reynolds number increasing,
max for the whole comparison is experiencing three phases: when
Re ≤ 2, max for the pressure-driven EOF keeps constant while
max for the pure pressure-driven flow increases nearly linearly;
when 2 ≤ Re ≤ 8, max for both cases are almost equivalent; when
Re ≥ 8, the deviation between the two increases gradually. The variation could be understood as the following: in the stage with the
relatively low Reynolds number, the EOF dominates the vorticity
generation, i.e., the vorticity contributed by the pressure gradient is finite; as the Reynolds number increasing, especially up to
the high value, the absolute value of vorticity is mainly influenced
by the pressure-driven flow. However, the deviation at the relative high Reynolds number indicates the more remarkable vorticity
transport within the EDL.
Furthermore, Fig. 5 shows the vorticity contours for three groups
of Reynolds numbers Re = 1.7, 5.7 and 12.1 around the wall of tip
back with the EDL scale. In the view of such microscale, the vorticity contours are pushed to an expanded region with the Reynolds
number increasing, and the tails of vorticity approach to the bulk
flow region. Ref. [3] attributes the Reynolds-number-related vorticity transport to the effect of convective term in taking the curl of
the Navier–Stokes equation.
In addition, efforts have been made to capture the Moffatt
eddies, a sequence of weak eddies generated near the corners in
the pure viscous flow as the Reynolds number approaches to zero
[31,32]. The angle of the corner in our model, as seen in Fig. 1, is
123.7◦ , which does not exceed the critical angle 2critical ≈ 146.3◦
[31], and our angle admits the Moffatt eddies in the pure pressure-
Z.-Y. Sun et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 366 (2010) 1–11
Fig. 4. Variation of max with the Reynolds number for the pressure-driven EOF
(square plot) and pure pressure-driven flow (round plot).
driven flows. We use the same numerical approach as the one
employed above and apply a refined local grid refinement to simulate the Moffatt eddies at the tip corners only considering the
pressure-driven flow. Two illustrative results are shown in Fig. 6(a)
and (b), and the corresponding pressure-driven-EOF cases have
been provided in Fig. 3(a) and (d) previously. Sizes of Moffatt eddies
in those two figures are ∼1.63 × 0.98 ␮m2 and ∼0.67 × 0.44 ␮m2 ,
respectively, which are much smaller than the scale of the channel
tip { Note that the Moffatt eddy can also exist at Re ≈ 1, as seen in
Fig. 6(b), which accords with the results in Ref. [32]}. Moffatt eddies
at the back corner of the tip for Re = 0.005 are similar to the ones at
the front corner in Fig. 6(a), and a recirculation region has replaced
the back corner vortex for Re = 0.90. As the corner is approached,
successive eddies are of the quickly decreasing size for our angle
[31], and it is hard to capture the smaller neighboring eddies with
reasonable resources. Furthermore, for our case of pressure-driven
EOFs, the weak Moffatt eddies can be greatly influenced by the electrokinetic effects. Numerical simulation shows that in the region of
Moffatt eddies, the velocity filed is of the order 10−5 to10−4 m s−1
for the pressure-driven EOFs, while 10−9 to10−8 m s−1 (Re = 0.005)
or 10−7 to10−6 m s−1 (Re = 0.90) for the pure pressure-driven flow.
Such difference indicates that the Moffatt eddies illustrated in
Fig. 6(a) and (b) can vanish and be neglected in our discussions
Fig. 5. Vorticity contours around the wall of tip back with the EDL scale for (a) Re = 1.7; (b) Re = 5.7; (c) Re = 12.1.
Fig. 6. Moffatt eddy at the front corner of the tip in the pure pressure-driven flow with the Reynolds number (a) Re = 0.005; (b) Re = 0.90 (the streamlines are arranged
Z.-Y. Sun et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 366 (2010) 1–11
Fig. 7. Dimensionless velocity profiles at different channel cross-sections. The separation distances between those cross-sections and the central line of sharp tip vary
from 100 ␮ m (the first parabolic curve from the left to the right) to 30 ␮ m (the last
distorted parabolic curve from the left to the right) with the step about 11.7 ␮m.
Fig. 8. Comparison of Cases A (40 ␮m, 30 ␮m) and B (20 ␮m, 30 ␮m) on the variation
of max with the Reynolds number for the pressure-driven EOF.
based on the pressure-driven EOFs although the Reynolds number
is very small.
Remark on the distribution of inlet velocity
As mentioned above, we have employed the parabolic velocity
distribution at the inlet in our numerical simulations. In fact, the
spatial gradient in zeta potential can influence the assumed velocity distribution in the finite region near the sharp tip upstream.
For instance, Fig. 7 shows the velocity profiles for the upstream
cross-sections adjacent to the sharp tip with the same physical
conditions of the flow in Fig. 3(d). We notice that the velocity
profiles keep the parabolic shape when the separation distance is
beyond about 65 ␮m, i.e., the assumption that the channel is sufficiently long can be approximated in such a situation. Further,
if we have not chosen a channel long enough upstream, the distorted parabolic velocity profiles in Fig. 7 (e.g., the last two ones
from the left to the right) can be selected to revise the boundary
conditions for Eqs. (5a) and (5b) in the actual simulations. Additionally, as the Reynolds number and zeta potential on the tip increase,
the channel with longer separation distance between the inlet and
sharp tip should be used to satisfy our assumption. More works
based on the long-channel-length assumption can be found in Refs.
[6,12,13,32,33], in which the influence induced by the non-uniform
zeta-potential gradient on the inlet velocity is negligible under such
increasing, max , which is mainly affected by the pressure gradient,
increases faster for the sharper tip.
Fig. 9 provides us with the more detailed images at the location
where max is induced. We notice that the vorticity around the tip
back for Case B expands to wider regions along both tangential and
normal directions to the tip wall. At the same time, the normalized
velocity arrows indicate the larger convective transport region for
Case B, where the fluid moves to the bulk flow along the normal
direction to the tip wall.
Through above analysis, it can be inferred that keeping appropriate Reynolds number and tip sharpness is necessary for controlling
the maximum vorticity around the tip back. If max is required to
be finite and not to be very sensitive to the Reynolds number, relatively small Reynolds number and selected sharp tip (less sharper)
might be the good choices for such pressure-driven EOFs.
3.2. Effect of tip sharpness on vorticity formation
3.3.1. Effect of externally applied electric field
For the outer boundary of the EDL without irrotationality, the
HS formula
Global geometry (large-scale configuration, e.g., channel crosssection) and local geometry (small-scale configuration, e.g.,
roughness) have considerable influence on the EOFs including the
pressure-driven ones [3,10,20,22,28]. Numerical simulations in Ref.
[3] reveal that the vorticity leakage (convective transport) can be
reduced remarkably by increasing the corner radius of the channel at even high Reynolds number. This part will be devoted to the
comparison of the vorticity generation for two different sharpness
of the channel tips. The shape parameters (d, h) are chosen as A:
(40 ␮m, 30 ␮m) and B: (20 ␮m, 30 ␮m), respectively. Fig. 8 gives
two curves to compare the variation of max with the Reynolds
number between the two cases. The flow condition for the sharper
tip (Case B with apex angle 36.9◦ while A with 67.4◦ ) is more
likely to be dominated by the pressure-driven flow effect. The first
stage for Case B, i.e., max being kept as almost constant, lasts relatively shorter than that for Case A, and such constant approaches
to a smaller value for the sharper tip. As the Reynolds number
3.3. The way to reduce recirculation region under finite Debye
The vorticity generated around the wall of tip back has been
investigated in above two parts. In fact, the finite Debye length is
able to influence the recirculation region behind the tip when the
Reynolds number is relatively high, which will be discussed in this
u =−
0 r E ,
gives the effective slip velocity u at the electrolyte–solid interface
induced by the tangential electric field E [7]. Furthermore, the flux
of vorticity Jω parallel to the channel wall within the finite EDL can
be calculated as Jω = 0.5u2 [3]. If large electric field is applied, the
corresponding u can be derived to pull the adjacent fluid in the
bulk flow to counteract the opposite velocity induced by the recirculation flow. We have carried out numerical simulation to study
such phenomenon and some results have been sketched in Fig. 10.
With 0 increasing (the externally applied electric field increases),
the recirculation region for the same Reynolds number is shrinking and the front stagnation point (where the flow reattaches to
the upper horizontal wall) is moving along the wall to the lower
position. We define the distance between the front and back stagna-
Z.-Y. Sun et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 366 (2010) 1–11
Fig. 9. Vorticity contours and velocity arrows (normalized) around the wall of tip back with the Reynolds number Re = 5.7 for (a) Case A; (b) Case B.
Fig. 10. Streamlines of the recirculation region behind the sharp tip (flow direction: from the left to the right) at Re = 1.1 for (a) 0 = 2.3 V (0∗ ∼11.5); (b) 0 = 17.3 V
(0∗ ∼86.5); (c) 0 = 34.5 V (0∗ ∼172.5).
Fig. 11. Variation of the dimensionless number with the applied voltage 0∗ for (a) three Reynolds numbers Re = 1.1, 2.1 and 3.2 when the zeta potential is fixed as ∗ = 7.8;
(b) three zeta potentials ∗ = 0.97, 1.95 and 2.92 with Re = 1.08.
Z.-Y. Sun et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 366 (2010) 1–11
Fig. 12. Variation of
with (a) the dimensionless zeta potential ∗ = ze/(kB T ); (b) the bulk concentration n0 for Re = 2.2 and 0 = 92 V (0∗ ∼460).
tion points (where the flow reattaches to the lower horizontal wall)
as the reattachment length LR . A dimensionless number quantifying the recirculation region can be introduced as the reattachment
length divided by the tip height, i.e., = LR /h. Fig. 11(a) illustrates
the numerical results on the variation of with the applied electric field (represented by 0∗ ) for three different Reynolds numbers.
For the same Reynolds number, decreases monotonously with the
applied electric field increasing (when 0 increases to certain value,
the recirculation region becomes not visible, and is not recorded
beyond such value in our investigation). For larger Reynolds number, controlling the recirculation region to the same level requires
the higher applied voltage to provide the driven flow near the
In order to further study the variation of with 0∗ , Fig. 11(b)
illustrates the similar conditions to the square plots in Fig. 11(a)
for three relative small zeta potentials when the Reynolds number is fixed as ∼1.08. Results reveal that in the range of ∗ =
0.97 − 7.8 ( = 25–200 mV), the ∼0∗ variation holds the similar
features describing the reduction of recirculation. For the smaller
zeta potential [e.g., the circle plots in Fig. 11(b)], larger applied
electric field is necessary to reach the same decrement of . Meanwhile, those discussions indicate the consistency of our numerical
simulation within the zeta potentials of order 101 to102 mV.
3.3.2. Effects of zeta potential and bulk concentration
Due to the Boltzmann distribution within the finite EDL, the
zeta potential and bulk concentration can influence the net charge
density, which could act the similar effect on reducing the recirculation region behind the tip. Fig. 12(a) shows the variation of
with the dimensionless zeta potential ∗ for Re = 2.2 and 0 = 92 V
(the dimensionless 0∗ ∼460). With increasing, decreases almost
as the linear manner. For small value of ∗ , Eqs. (3) and (4) can
be linearized, which indicates that the net charge density is proportional to the internal electric potential distribution within the
EDL. And the increasing net charge density leads to the corresponding increase of the body force under the given electric field, which
results the growth of flow velocity for the finite EDL, if the convective term in Eq. (5b) is considerably weak. Such growth induces
larger velocity field near the EDL to counteract the recirculation
region behind the tip. This process provides a simple explanation
for the variation shown in Fig. 12(a).
With regard to the effect of the bulk concentration, the variation of with n0 is illustrated in Fig. 12(b). When n0 is relatively
small (about 10−8 to10−6 M), decreases rapidly with n0 increasing,
and in such range, the recirculation region can be effectively controlled by selecting appropriate concentration. When n0 is higher
than approximately 0.5 × 10−5 M, keeps almost constant with n0
increasing, and it is limited to control the recirculation region by
changing the concentration. We will give a simplified understanding of such variation by virtue of the analytic solution for the EOF
near the parallel plate [14,34]. For the two-dimensional parallel
plates, if the zeta potential is assumed to be constant along the
channel surface, Eq. (4) can be simplified and its analytic solution
has been obtained as [14,34]
tanh−1 tanh
) ,
where ∗ is the normalized distance from the wall, the parameters
˛ = ze/(kB T ) and ˇ = 8 n0 z 2 e2 H 2 /(0 r kB T˛). From Formulae (3)
and (12), we can derive the following relation,
e ∼(2ze)n0 sinh 4 tanh−1 tanh
where = [8 z 2 e2 H 2 /(0 r kB T )] . Further, if we assume that
u∼e is applied to the velocity field near the finite EDL to counteract the recirculation flow, the variation trend of u with n0 can
be estimated at the same positions near the EDL when other physical conditions are fixed. Numerical calculations of Expression (13)
reveal that close to the wall, the net charge density increases
quickly to a balanced value with the bulk concentration increasing, which is sketched in Fig. 13 (in fact, if n0 rises to enough high
level, e will present the decreasing trend, which is not considered
in our study). Correspondingly, the simplified parallel-plate case
might provide a local description on the variation of flow condition
with n0 (similar to that for e ) within the finite-EDL framework,
Fig. 13. Estimation of e varying with n0 at = 0.2 V for
curve) and 4 × 10−4 (the lower curve).
= 3 × 10−4 (the upper
Z.-Y. Sun et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 366 (2010) 1–11
Fig. 14. (a) Variation of with 0∗ for the single tip and symmetrical tip pairs at the inlet U0 = 0.2 m/s; (b) reduction of the recirculation for the symmetrical tip pairs with
Re = 1.6 and 0 = 46 V (0∗ ∼230).
and the change of recirculation region shown in Fig. 12(b) is experiencing the opposite process. In other words, the fast decrease for
is induced by the corresponding increase for the fluid velocity of
flow layer between the recirculation and tip wall. When such velocity steps into the balanced region (varying little with n0 ), keeps
balanced as illustrated in Fig. 12(b).
Therefore, appropriate control of the externally applied electric
field and zeta potential can bring us the similar effects on reducing
the recirculation region behind the tip, while changing the bulk
concentration in the balanced region is not a very effective choice.
3.4. Control of recirculation for a pair of sharp tips
In above parts, we have discussed the vorticity generation with
different Reynolds numbers and tip sharpness for a single tip. Based
on the finite EDL, the way to control the recirculation through varying the externally applied electric field, zeta potential and bulk
concentration has also been investigated. In the applications of
microfluidic systems, a pair of symmetrical sharp tips in a straight
channel are designed with the intention to enhance the nonlinear
effects [9], and the micromixer fabricated incorporating a mixing
section comprised of four symmetrically opposed corner pairs can
effectively improve the mixing in the EOF [10]. Thereby, in this
part we will give a brief study on controlling the recirculation for
a pair of sharp tips, which are designed as the symmetrical and
asymmetrical ones in shape, or the ones staggered in position.
3.4.1. Effect of applied electric field and control of recirculation
for a pair of symmetrical tips
Similar to the discussion in Part 3.3.1, first we will carry out brief
numerical simulation aiming to investigate how the applied electric
field influences the recirculation region for a pair of symmetrical
tips. Fig. 14(a) compares the variation of with 0∗ between the
single-tip and tip-pair cases when the inlet U0 is set to be the same
value. It can be inferred that the maximum cell reference Reynolds
number (appears around the sharp tip) for the tip-pair case is larger
than that for the single one, and such condition is due to the shorter
distance between the two tip apexes, which results in the higher
velocity for the fluid passing the tips. The variation of with 0∗
for higher Reynolds number accords with that shown in Fig. 11(a)
for the single tip, which indicates that a nonlinear increase in 0∗
is needed to reduce the recirculation region to the same level as
the Reynolds number increasing (the absolute value of slope for the
curve with higher Reynolds number is smaller). Fig. 14(b) illustrates
the symmetrical reduction of the recirculation regions for the tip
In fact, for the symmetrical sharp tips, the asymmetrical recirculation can be generated if each tip has the unequal zeta potential
distribution on its surface, which can be viewed as the combined
effects discussed in Part 3.3.2 for the single tip. Fig. 15(a) provides
such an example, in which the stronger zeta potential for the up
tip reduces the recirculation to a smaller region than that for the
down tip. Furthermore, for those two recirculation regions, one can
be eliminated completely while another one still exists if the zeta
potential for each tip is well selected.
3.4.2. Control of recirculation for a pair of asymmetrical tips in
shape and staggered tips in position
Regarding to a pair of sharp tips asymmetrically in shape, we
adopt the two tips (Cases A and B) introduced in Part 3.2 and arrange
them in apex-opposed position. The numerical results showing the
recirculation control are provided in Fig. 15(b). If the two tips have
the same zeta potential (e.g., both at 0.1 V), the recirculation region
for the shaper tip (the up one) is larger. Enhancing up to 0.25 V, the
recirculation can be reduced to almost the same level as the one for
the down tip by virtue of the zeta-potential effect investigated in
Part 3.3.2.
Fig. 15(c) and (d) illustrates the application of recirculation control for a pair of sharp tips staggered in position (the distance
between symmetrical axes of both tips is 10 ␮m). From Fig. 15(c), it
can be found that the recirculation region for the back tip is visibly
larger than that for the front one since the jet-like flow passing the
two tips is directed downwards and the local Reynolds number for
the back tip increases (the situation for the pure pressure-driven
flow has the similar feature). The difference between those two
recirculation regions can be reduced if we increase up and decrease
down , which is shown in Fig. 15(d). Through such control, the flow
layer between the recirculation region and tip wall is increasing for
the up tip and decreasing for the down tip. In this part, we have
briefly demonstrated some examples of the recirculation control.
Further applications are under investigation and will be presented
in a future publication.
3.5. Effectiveness of Eqs. (3) and (4)
In this part, the assumption that the ionic convection is negligible within the EDL will be briefly examined with the scale analysis
[35,36] on the following Nernst-Planck equations [12,21,22,26]:
J = −D0 ∇ n −
zeD0 n
kB T
+ ∇ · J = 0,
+ nu,
Z.-Y. Sun et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 366 (2010) 1–11
Fig. 15. Sketches of controlling the recirculation regions for (a) a pair of the symmetrical tips with U0 = 0.2 m/s, 0 = 34.5 V, up = 0.25 V and down = 0.1 V; (b) a pair of the
asymmetrical tips with U0 = 0.2 m/s, 0 = 23 V, up = 0.25 V and down = 0.1 V; (c) a pair of the staggered tips with U0 = 0.2 m/s, 0 = 23 V and up = down = 0.2 V; (d) a pair
of the staggered tips with U0 = 0.2 m/s, 0 = 23 V, up = 0.26 V and down = 0.06 V.
where J is the ionic flux, n is the number density of the ionic
species in the EDL and D0 is the ion diffusivity. Such equations are
employed to describe the ionic transport in the electrokinetic flows
[12,21,22,26]. Using Dimensionless Parameters (6) and n∗ = n/n0 ,
we can non-dimensionalize Eqs. (14) and (15) for the quasi-steadystate case as
∇ 2 n∗ + ∇ · (n∗ ∇
∇ · (n∗ u∗ ) = 0.
If we consider the direction along the Debye length within the EDL,
we have x∼y∼ and ∼, the scale of the diffusion term in Eq.
(16) is given by ∇ 2 n∗ ∼nH 2 /(n0 2 ), where n is the ionic density
difference. By the similar approach, the electrochemical migration
term appears to have ∇ · (n∗ ∇ ∗ )∼nH 2 ze/(n0 2 kB T ) and the
ionic convection term presents ∇ · (n∗ u∗ )∼nHu/(n0 U0 ), where u
is the characteristic flow speed along the normal direction against
the tip wall within the EDL. Thereby in a scale sense, Eq. (16) leads
to the scale comparison as
ze u
kB T D0
In our discussions, varies in the range of approximately
0.05–0.2 V, and such values lead to ze/(kB T )∼(1.9–7.8) in Formula (17). Additionally, the Debye length is of the order ∼102 nm
(about 70–300 nm), and u is estimated to be ∼(10−4 to10−3 ) m s−1
from the numerical results within the EDL. Using those approximate values and D0 = 2 × 10−9 m2 s−1 for KCl, we can derive
u/D0 ∼(10−2 to10−1 ), which indicates that the convective transport of the ions in the EDL can be neglected (with errors not beyond
about 10% compared with the diffusion term) if the Reynolds number, applied electrical field and zeta potential are appropriate in
most of our calculations. In fact, Eqs. (3) and (4) are employed to
describe the ion distribution in Refs. [3,11,13,27] when the vortices
are existing. Other NPDEs can be found, e,g, in Refs. [29,37,38].
4. Summary
Motivated by the finite-Debye-length assumption, we have
studied the combined pressure-driven EOF through the insulated
sharp tips in a straight microchannel by numerically solving Eqs.
(7)–(10). The maximum vorticity generated around the tip back
has been investigated and the way to control the recirculation
region (eddy) behind the tips (single one and a pair) has been
provided for the relatively high-Reynolds-number flow, as seen
in Figs. 10 and 15. Our simulation and analysis have revealed the
following results:
• Maximum vorticity generated around the tip back experiences
three stages with the Reynolds number increasing, as seen in
Fig. 4. In the first two stages, features of the EOF and pressuredriven flow dominate the vorticity generation, respectively. In the
third stage, the vorticity transport within the finite EDL becomes
remarkable for the high-Reynolds-number flow. For the sharper
tip, effects of the pressure-driven flow are more obvious and the
vorticity gets stronger around the tip back, as seen in Fig. 8.
• At the high Reynolds number, recirculation region behind the
tip can be reduced through the change of the externally applied
electric field, zeta potential and bulk concentration, as seen in
Figs. 11 and 12. Control of the recirculation by the bulk concentration is not so effective if the concentration varies in the balanced
range [with n0 > 0.5 × 10−5 M in Fig. 12(b)].
• For a pair of the sharp tips designed in three manners (symmetrical and asymmetrical ones in shape, staggered ones in position),
recirculation behind each tip can be controlled if the selected
zeta potentials are distributed, as seen in Fig. 15. Correspondingly, flow layer between the recirculation region and tip wall
can be visible during such control.
Under the finite-Debye-length assumption, numerical results
and discussions could be expected to be helpful in understanding
Z.-Y. Sun et al. / Colloids and Surfaces A: Physicochem. Eng. Aspects 366 (2010) 1–11
the nonlinear convective EOFs, and further applied to the microfluidic devices for the purpose of controlling the mixing.
We express our sincere thanks to Editor N. Furlong and the
Referees for their valuable suggestions. We are also very grateful to all the members of our discussion group for their beneficial
comments. This work has been supported by the National Natural Science Foundation of China under Grant No. 60772023, by
the Open Fund (No. BUAA-SKLSDE-09KF-04) and Supported Project
(No. SKLSDE-2010ZX-07) of the State Key Laboratory of Software
Development Environment, Beijing University of Aeronautics and
Astronautics, by the National Basic Research Program of China (973
Program) under Grant No. 2005CB321901, and by the Specialized
Research Fund for the Doctoral Program of Higher Education (Nos.
20060006024 and 200800130006), Chinese Ministry of Education.
[1] D.Q. Li, Electrokinetics in Microfluidics, Academic, Oxford, 2004.
[2] H.A. Stone, A.D. Stroock, A. Ajdari, Engineering flows in small devices microfluidics toward a lab-on-a-chip, Annu. Rev. Fluid Mech. 36 (2004) 381–411.
[3] J.M. Oh, K.H. Kang, Conditions for similitude and the effect of finite Debye length
in electroosmotic flows, J. Colloid Interface Sci. 310 (2007) 607–616.
[4] E.B. Cummings, S.K. Griffiths, R.H. Nilson, P.H. Paul, Conditions for similitude
between the fluid velocity and electric field in electroosmotic flow, Anal. Chem.
72 (2000) 2526–2532.
[5] J.G. Santiago, Comments on the conditions for similitude in electroosmotic
flows, J. Colloid Interface Sci. 310 (2007) 675–677.
[6] J. Zhang, G. He, F. Liu, Electro-osmotic flow and mixing in heterogeneous
microchannels, Phys. Rev. E 73 (2006) 056305.
[7] M.Z. Bazant, T.M. Squires, Induced-charge electrokinetic phenomena: theory
and microfluidic applications, Phys. Rev. Lett. 92 (2004) 066101.
[8] T.M. Squires, M.Z. Bazant, Induced-charge electro-osmosis, J. Fluid Mech. 509
(2004) 217–252.
[9] Y. Eckstein, G. Yossifon, A. Seifert, T. Miloh, Nonlinear electrokinetic phenomena
around nearly insulated sharp tips in microflows, J. Colloid Interface Sci. 338
(2009) 243–249.
[10] J.K. Chen, R.J. Yang, Vortex generation in electroosmotic flow passing through
sharp corners, Microfluid Nanofluid 5 (2008) 719–725.
[11] Y.J. Juang, X. Hu, S. Wang, L.J. Lee, C. Lu, J. Guan, Electrokinetic interactions in
microscale cross-slot flow, Appl. Phys. Lett. 87 (2005) 244105.
[12] S. Bhattacharyya, A.K. Nayak, Electroosmotic flow in micro/nanochannels with
surface potential heterogeneity: An analysis through the Nernst-Planck model
with convection effect, Colloid Surf. A 339 (2009) 167–177.
[13] J.C. Ramirez, A.T. Conlisk, Formation of vortices near abrupt nano-channel
height changes in electro-osmotic flow of aqueous solutions, Biomed. Microdevices 8 (2006) 325–330.
[14] P. Dutta, A. Beskok, Analytical solution of combined electroosmotic/pressure
driven flows in two-dimensional straight channels: finite Debye layer effects,
Anal. Chem. 73 (2001) 1979–1986.
[15] M. Wang, J. Wang, S. Chen, Roughness and cavitations effects on electro-osmotic
flows in rough microchannels using the lattice Poisson–Boltzmann methods, J.
Comput. Phys. 226 (2007) 836–851.
[16] S. Kang, Y.K. Suh, Numerical analysis on electroosmotic flows in a microchannel with rectangle-waved surface roughness using the Poisson–Nernst–Planck
model, Microfluid Nanofluid 6 (2009) 461–477.
[17] P.K. Das, S. Bhattacharjee, Electrostatic double-layer interaction between
spherical particles inside a rough capillary, J. Colloid Interface Sci. 273 (2004)
[18] M. Jain, A. Yueng, K. Nandakumar, Induced charge electro osmotic mixer: Obstacle shape optimization, Biomicrofluidics 3 (2009) 022413.
[19] R. Monazami, M.T. Manzari, Analysis of combined pressure-driven electroosmotic flow through square microchannels, Microfluid Nanofluid 3 (2007)
[20] Z. Xia, R.W. Mei, M. Sheplak, Z.H. Fan, Electroosmotically driven creeping flows
in a wavy microchannel, Microfluid Nanofluid 6 (2009) 37–52.
[21] M. Wang, S. Chen, On applicability of Poisson–Boltzmann equation for
micro- and nanoscale electroosmotic flows, Commun. Comput. Phys. 3 (2008)
[22] M. Wang, Q. Kang, Electrokinetic transport in microchannels with random
roughness, Anal. Chem. 81 (2009) 2953–2961.
[23] J.K. Wang, M.R. Wang, Z.X. Li, Lattice Boltzmann simulations of mixing enhancement by the electro-osmotic flow in microchannels, Mod. Phys. Lett. B 19 (2005)
[24] J.K. Wang, M.R. Wang, Z.X. Li, Lattice Poisson–Boltzmann simulations of electroosmotic flows in microchannels, J. Colloid Interface Sci. 296 (2006) 729–736.
[25] M. Wang, N. Pan, J. Wang, S. Chen, Lattice Poisson–Boltzmann simulations of
electroosmotic flows in charged anisotropic porous media, Commun. Comput.
Phys. 2 (2007) 1055–1070.
[26] M. Wang, Q. Kang, Modeling electrokinetic flows in microchannels using coupled lattice Boltzmann methods, J. Comput. Phys. 229 (2010) 728–744.
[27] N.A. Patankar, H.H. Hu, Numerical simulation of electroosmotic flow, Anal.
Chem. 70 (1998) 1870–1881.
[28] D. Yang, Y. Liu, Numerical simulation of electroosmotic flow in microchannels
with sinusoidal roughness, Colloid Surf. A 328 (2008) 28–33.
[29] M.P. Barnett, J.F. Capitani, J. Von Zur Gathen, J. Gerhard, Symbolic calculation
in chemistry: selected examples, Int. J. Quant. Chem. 100 (2004) 80–104;
G.C. Das, J. Sarma, Response to Comment on ‘A new mathematical approach for
finding the solitary waves in dusty plasma’, Phys. Plasmas 6 (1999) 4394–4397;
B. Tian, Y.T. Gao, Cylindrical nebulons, symbolic computation and Bäcklund
transformation for the cosmic dust acoustic waves, Phys. Plasmas (Lett.) 12
(2005) 070703;
B. Tian, Y.T. Gao, Comment on ‘Exact solutions of cylindrical and spherical dust
ion acoustic waves’, Phys. Plasmas 12 (2005) 054701;
B. Tian, Y.T. Gao, On the solitonic structures of the cylindrical dust-acoustic and
dust-ion-acoustic waves with symbolic computation, Phys. Lett. A 340 (2005)
B. Tian, Y.T. Gao, Symbolic-computation study of the perturbed nonlinear
Schrödinger model in inhomogeneous optical fibers, Phys. Lett. A 342 (2005)
B. Tian, Y.T. Gao, Variable-coefficient higher-order nonlinear Schrödinger
model in optical fibers: New transformation with burstons, brightons and symbolic computation, Phys. Lett. A 359 (2006) 241–248;
B. Tian, Y.T. Gao, Symbolic computation on cylindrical-modified dust-ionacoustic nebulons in dusty plasmas, Phys. Lett. A 362 (2007) 283–288;
B. Tian, Y.T. Gao, Spherical nebulons and Bäcklund transformation for a space
or laboratory un-magnetized dusty plasma with symbolic computation, Eur.
Phys. J. D 33 (2005) 59–66.
[30] G.M. Mala, D. Li, C. Werner, H.J. Jacobasch, Y.B. Ning, Flow characteristics of
water through a microchannel between two parallel plates with electrokinetic
effects, Int. J. Heat Fluid Flow 18 (1997) 489–496;
S. Arulanandam, D. Li, Liquid transport in rectangular microchannels by electroosmotic pumping, Colloid Surf. A 161 (2000) 89–102.
[31] H.K. Moffatt, Viscous and resistive eddies near a sharp corner, J. Fluid Mech. 18
(1964) 1–18;
P.N. Shankar, Moffatt eddies in the cone, J. Fluid Mech. 539 (2005) 113–135;
F.E. Laine-Pearson, P.E. Hydon, Particle transport in a moving corner, J. Fluid
Mech. 559 (2006) 379–390;
C.J. Heaton, On the appearance of Moffatt eddies in viscous cavity flow as the
aspect ratio varies, Phys. Fluids 20 (2008) 103102.
[32] G. Biswas, M. Breuer, F. Durst, Backward-facing step flows for various expansion
ratios at low and moderate Reynolds numbers, J. Fluid Eng. 126 (2004) 362–374.
[33] S.A. Mirbozorgi, H. Niazmand, M. Renksizbulut, Electro-osmotic flow in
reservoir-connected flat microchannels with non-uniform zeta potential, J.
Fluid Eng. 128 (2006) 1133–1143.
[34] R.J. Hunter, Zeta Potential in Colloid Science: Principles and Applications, Academic, New York, 1981.
[35] D.B. Ingham, A. Bejan, E. Mamut, I. Pop, Emerging Technologies and Techniques
in Porous Media, Kluwer Academic Publishers, Netherlands, 2004.
[36] V.A.F. Costa, A time scale-based analysis of the laminar convective phenomena,
Int. J. Therm. Sci. 41 (2002) 1131–1140.
[37] W.J. Liu, B. Tian, H.Q. Zhang, L.L. Li, Y.S. Xue, Soliton interaction in the
higher-order nonlinear Schrödinger equation investigated with Hirota’s bilinear method, Phys. Rev. E 77 (2008) 066605;
W.J. Liu, B. Tian, H.Q. Zhang, Types of solutions of the variable-coefficient nonlinear Schrödinger equation with symbolic computation, Phys. Rev. E 78 (2008)
W.J. Liu, B. Tian, H.Q. Zhang, T. Xu, H. Li, Solitary wave pulses in optical
fibers with normal dispersion and higher-order effects, Phys. Rev. A 79 (2009)
W.J. Liu, B. Tian, T. Xu, K. Sun, Y. Jiang, Solitary wave pulses in optical
fibers with normal dispersion and higher-order effects, Ann. Phys. (2010),
[38] Z.Y. Sun, Y.T. Gao, X. Yu, W.J. Liu, Y. Liu, Bound vector solitons and soliton complexes for the coupled nonlinear Schrödinger equations, Phys. Rev. E 80 (2009)
Z.Y. Sun, Y.T. Gao, X. Yu, X.H. Meng, Y. Liu, Inelastic interactions of the
multiple-front waves for the modified Kadomtsev–Petviashvili equation in
fluid dynamics, plasma physics and electrodynamics, Wave Motion 46 (2009)
Y. Liu, Y.T. Gao, T. Xu, X. Lü, Z.Y. Sun, X.H. Meng, X. Yu, X.L. Gai, Soliton solution,
Bäcklund transformation, and conservation laws for the Sasa-Satsuma equation in the optical fiber communications, Z. Naturforsch. 65a (2010) 291–300;
X. Yu, Y.T. Gao, Z.Y. Sun, Y. Liu, N-soliton solutions, Bäcklund transformation and
Lax pair for a generalized variable-coefficient fifth-order Korteweg-de Vries
equation, Phys. Scr. 81 (2010) 045402.