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Journal of Applied Fluid Mechanics, Vol. 13, No. 6, pp. 1847-1858, 2020.
Available online at www.jafmonline.net, ISSN 1735-3572, EISSN 1735-3645.
DOI: 10.47176/jafm.13.06.31238
Numerical Analysis of Flow Geometry in I-Shaped
Viscous Micropumps using LB-IBM
A. Alimoradi and S. Ali Mirbozorgi†
Department of Mechanical Engineering, University of Birjand, Birjand, Iran
†Corresponding Author Email: samirbozorgi@birjand.ac.ir
(Received December 10, 2019; accepted May 7, 2020)
ABSTRACT
In viscous micropumps one of the main reasons for a flow rate reduction is vortices which are located at the top
of the rotating rotor. In this paper, we have tried to add proper additional walls in the micropump channel, to
eliminate or decrease the size of these vortices. Among the all investigated new models, only one, the I-Shaped
micropump with an extra step above the rotor, could reduce the size of the vortices and also increase the outlet
flow rate. In this paper, the numerical simulations were conducted by using the Lattice Boltzmann Method and
by exploiting the Immersed Boundary method and the Blocking technique in order to overcome the LBM
drawbacks. The results show that at the channel height ∗3.7, this new model can produce a flow rate of
150% more than the normal I-Shaped micropumps. Also, one can tune the maximum produced pressure by
adjusting the height of this step and micropump with higher channel height can be much more efficient and
usable. In addition, by using this new structure for micropump, the designers can also use bigger channel heights
which were not efficient in the original design.
Keywords: Lattice Boltzmann method; Immersed boundary method; Viscous micropump.
1. INTRODUCTION
The scientific advancement in the miniaturization of
electromechanical devices necessitates
miniaturization of all systems associated with these
devices. One of these systems is the fluid transfer
system, and it is the most important part, which is a
pump.
Micropumps perform the fluid transfer in the micro
dimensions. This fluid transfer may be exploited to
cool a microchip, a fuel cell, or a micro-reactor, or to
transfer, combine, react, or analyze the reactants in a
micro total analysis system. Practically, it has been
proven that the application of some micropumps,
compared to their macro-sized peers, is along with
some advantages including reduction in the use of
samples and reactants, automating processes, and
improving the quality of the experiments.
The surface-to-volume ratio in micro dimensions is
much larger than that on the macro scale, hence
surface effects will be prominent than the volumetric
ones. When the Reynolds number has a small value
due to the small flow cross-section or velocity, the
viscous forces dominate the inertial ones. Therefore,
micropumps using inertial forces as the pumping
mechanism have low efficiency. A lot of
investigations have been carried out to find new
mechanisms that no longer have this problem,
leading to the creation of pumping mechanisms
including positive displacement using a piezoelectric
actuator, phase change, pumping using viscous
forces, etc. (Iverson and Garimella 2008).
Among micropumps, positive-displacement
micropumps are of higher popularity and are used in
various devices, particularly those needing to
transfer a specific amount of fluid (for instance,
automatic insulin-injection devices). Despite the
high popularity, these micropumps require precise
fasteners and seals which will make their design
complex and therefore, this will make the process of
fabrication more difficult.
Electro-hydrodynamic (EHD) pumps are another
class of micropumps. In this model, dielectric fluid
is exposed to an electric field. This electric field is
created by several electrodes located inside the
channel, and the fluid is displaced due to induced
charges and creates a flow rate inside the channel.
Electro-chemical pumps and bubble pumps transfer
fluids by utilizing the volume changes from phase
transition to displace fluid. Bubble pumps consist of
a micro-channel with several independent heaters on
its surface. These micro-pumps operate as follows:
A. Alimoradi and S. Ali Mirbozorgi / JAFM, Vol. 13, No. 6, pp. 1847-1858, 2020.
1848
the first heater works enough to form a bubble of
fluid’s vapor on the surface with the same size as the
width of the channel. When the second heater is
activated, the bubble expands. Then, by turning on
the third heater and turning off the first heater, the
vapor bubble starts moving due to the pressure
difference in its sides and pushes the fluid forward,
toward the micropump outlet. This sequential
process continues until the bubble exits the channel,
and by that, the process will restart from the
beginning. The electrochemical pumps consist of a
pair of electrodes located inside a water tank which
is connected to a channel filled with fluid. By turning
on the pump, the electrodes split the water and create
bubbles of oxygen and hydrogen. These bubbles are
then transferred to the channel and the fluid inside
the channel is driven forwards by these bubbles,
creating a flow of fluid.
In viscous micropumps, as the name suggests, the
viscosity is utilized as the pumping mechanism. The
geometric structure of this class of micropumps
includes a cylindrical or disk rotor (a low-height
cylinder) placed in a micro-channel in different
shapes. In the case of using a disk rotor, the transfer
of momentum to the fluid is performed through the
base surface of the rotating disk. This category of
micropumps can be divided into disk and spiral
micropumps. However, when the cylindrical rotor is
used, the momentum transfer to the fluid is
conducted through the lateral surface of the rotating
cylinder. This category can include various types,
such as I-shaped, L-shaped, and U-shaped
micropumps. All of these micropumps consist of a
rotating cylinder asymmetrically located inside the
channel. Due to the asymmetry in the rotor position
inside the channel, unequal shear forces are created
on the top and bottom sides of the rotor during its
rotation which causes the fluid to displace. The very
simple structure and the lack of need for seals and
fasteners are of the advantages of viscous
micropumps over other types of micropumps.
Moreover, this category of micropumps can be used
for all fluids, and energy consumption, compared to
other types, is rather low.
I-shaped micropumps were first introduced (Sen et
al. 1996). In their experimental study, they observed
that the average velocity of the fluid in the
micropump outlet was about 10% of the rotor linear
velocity. Besides, they showed that the average
velocity and flow rate first increased and then
decreased by increasing the channel width at a fixed
linear velocity of the rotor. (Sharatchandra et al.
1997) analyzed this micropump numerically and
examined the impact of parameters such as the
distance of the rotor from the wall, Reynolds
number, and pressure difference at the two ends of
the micropump on the output flow rate. Moreover,
they pointed out in their investigation that two
vortices formed above the rotor blocking the flow
path and as the channel width increases, vortices will
become larger and they start to merge into a single
vortex, and as a result, the width of the channel,
which flow can pass, will also decrease. In 2004,
(Abdelgawad et al. 2004) numerically analyzed the
I-shaped micropump using the Fluent software. Their
analysis was carried out unsteadily from t=0, i.e.
when the rotor was off until the flow reached the
steady-state. In this study, they assessed the effect of
geometrical parameters on the micropump
efficiency, as well as the impact of the Reynolds
number, channel width, and the rotor-wall distance
on the stability time of the flow and found that the
Reynolds number had a higher impact on the stability
time that the other parameters. In the same year,
(Phutthavong and Hassan 2004)selected different
shapes of rotors for an I-shaped micropump and
analyzed the flow through the micropump using the
Fluent software. One of the findings of their work
indicated that the circular rotor was capable of
producing more flow rate in comparison to the
polygon models. While introducing two novels L-
shaped and U-shaped models (Da Silva et al. 2007)
obtained optimal values of geometric dimensions for
all three models of micropumps. They performed
their simulation with the help of the Comsol
software.
The aforementioned studies revealed that although in
all cases, the effect of geometric parameters, rotor
speed, pressure differences, and rotor shape have
been investigated, the effect of the channel geometric
shape and the vortex formed above the rotor on the
flow rate and maximum pressure difference has been
neglected. It seems that the position of the vortex or
vortices, and eventually the pump flow rate and head
can be influenced by changing the flow geometry
using additional walls.
In this study, the impact of the additional walls inside
the channel on the size of the vortex above the rotor
was investigated numerically. From all the
investigated cases, only one case could reduce the
size of the vortex above the rotor, so that the passage
width of the main flow increased and ultimately
increased the outlet flow rate. Besides, these new
walls enabled us to control the output pressure which
micropump produces. In the following, the
configuration of these micropumps and the problem
geometry are introduced first. The Lattice Boltzmann
Method (LBM) and the immersed boundary method
(IBM) were used to numerically analyze the flow and
define the rotor, respectively. Moreover, the
blocking method and the Zou-He model were used to
define the additional walls. Finally, the findings will
be compared with those reported in previous studies.
2. PROBLEM GEOMETRY
The configuration of a simple I-shaped micropump
has been depicted in Fig. 1. As shown in this figure,
the flow paths are aligned in the inlet and outlet and
indicate the letter I (straight line). It should be noted
that the naming of the L-shaped and U-shaped
micropumps, in which the flow path in the outlet
relative to the inlet, changes 90 and 180 degrees,
respectively, has been performed in this way. Figure
2 demonstrates the model of an I-shaped micropump
with an additional wall (step). The first goal of the
selection of this additional wall is to interfere with
the flow pattern around the rotor. Furthermore, the
A. Alimoradi and S. Ali Mirbozorgi / JAFM, Vol. 13, No. 6, pp. 1847-1858, 2020.
1849
reason why this configuration has been chosen, in
addition to the simplicity of the final geometry, is:
based on the definition of the I-shaped micropumps,
the flow direction in the inlet and the outlet of the
micropump remains the same and fixed. In these
figures, D, Lu and Ld, H, Hs, and Ls respectively
designate the rotor diameter, the inlet and outlet
distance to the center of the rotor, the channel height,
the step height, and the step length which is used only
in the model presented in Fig. 2.
Fig. 1. Simple I-Shaped Viscous Micropump.
Fig. 2. I-Shaped Viscous Micropump with a step
above the rotor.
is the rotor distance from the bottom wall, with the
negative values meaning that part of the rotor is
placed within the wall. ω is the rotor rotational speed
in the clockwise direction and PL and PH are the
micropump inlet and outlet pressures, respectively. It
is worth noting that the distance of the rotor from the
upper wall is not equal to in Fig. 2, and the vertical
position of the upper wall depends only on the value
of Hs measured from the upper wall.
By choosing D as the reference length, the geometric
dimensions can be nondimensionalized as relations
(1).
*
*
*
*
*
*
u
u
d
d
x
y
xy
D
D
LH
LH
D
D
Ld
Ld
D
D
(1)
Assuming a Newtonian fluid and a laminar, steady,
and incompressible flow, the velocity and pressure
can be nondimensionalized using (2) and (3),
respectively.
**
,
2
uvD
Uu v
UU
(2)
**
22
**
/HL
P
P
PP
DP
(3)
In the current research, the Reynolds number is
written as (4) based on the rotational speed of the
rotor.
2
Re 2
D
(4)
Where ν is the fluid kinematic viscosity. The
boundary conditions of the walls in the channel and
rotor areas are as the non-slip boundary condition
and the inlet boundary condition as
∗
∗
0
,
moreover, the outlet boundary condition is as
∗
∗
, in which
∗
can vary from zero to the
maximum pressure of that model of the
micropump. In addition to these conditions, the
derivative of the velocity perpendicular to the
boundary at the inlet and outlet boundaries is
considered to be zero (
0
). The linear velocity
of the rotor is considered to be fixed as
/2
. The Lu and Ld values are assumed to be
equal to 8D so that a fully developed flow can be
obtained in the inlet and the outlet and the selected
boundary conditions don’t harm the simulation.
Also, the length of the L
S
has chosen to be 3D.
The power consumption per rotor length can be
specified based on the (5).
WFR
(5)
In this relation, the rotor force per unit of length ′
can be obtained using the following relation.
2
Wall
0dFR
(6)
The shear stress on the rotor surface is calculated as
(7).
Wall
RotorFluid
rR
U
r
r
UU
(7)
If the rotor starts to rotate at the zero moment, its
power consumption will be the maximum, and the
power consumption will reduce as the fluid
accelerates and will ultimately reach a fixed value.
To nondimensionalize the power consumption, the
maximum power consumed by the rotor can be
selected as the reference power. Assuming
0, the maximum rotor power per unit length can be
obtained by simplifying the above relations, as (8).
32
Max
WDRe
r
(8)
Therefore, by dividing the power consumption by the
maximum power consumption, the dimensionless
power consumption will be obtained as the following
relation.
A. Alimoradi and S. Ali Mirbozorgi / JAFM, Vol. 13, No. 6, pp. 1847-1858, 2020.
1850
*
Max
W
WW
(9)
3. NUMERICAL SOLUTION METHOD
3.1 LBM
The LBM is one of computational fluid dynamics
methods in which, instead of solving Navier-Stokes
equations, the discrete Boltzmann equation
governing the particle distribution function, f, is
solved. The values of the distribution functions f, in
addition to dependence on the discrete locations x
and time t, they are also dependent on the directions,
as each direction is defined by its different particle
velocity i
c. Due to this dependence, the distribution
functions f are represented as
. Selection of the
number of directions and the velocity of the particles
in each direction leads to different models, with the
D2Q9 model often used in the two-dimensional
problems; there are 9 particle velocities in this two-
dimensional model. In the case of using the
Bhatnagar–Gross–Krook (BGK) operator, the LBM
can be written as (10). The numerical solution of this
relation consists of two stages of the collision and
propagation of particles.
,
,,
,
ii i
i
i
eq
i
f
xc fxttt t
f
xt f xt
F
t
t
(10)
In this relation
, i
f
, and eq
i
f
are the relaxation
time constant, discrete force term, and equilibrium
distribution functions, respectively.
The value of the equilibrium distribution functions
are calculated through relation (11), in which the
local density ρ and local velocity u have been used.
In this relation, we have
and
.
2
242
122
i
i
isss
eq
iuc
uc uu
ccc
f
w
(11)
The particle transfer velocities and weighting
coefficients for this model are given in table 1. In
this relation, is the sound speed in the fluid and is
equal to
√, where c is the particle speed in
horizontal and vertical directions in the mesh being
∆
∆ 1.
The force term can be written as follows in which
is the force per unit of volume defined
in the macroscopic space.
Table 1 Weighting coefficient and particle
velocities for Model
i wi ix
c iy
c
0 4/9 0 0
1 1/9 +1 0
2 1/9 0 +1
3 1/9 -1 0
4 1/9 0 -1
5 1/36 +1 +1
6 1/36 -1 +1
7 1/36 -1 -1
8 1/36 +1 -1
24
1w
2
ii
i
i
ss
i
F
cc
cF
c
t
u
uc
(12)
Since the LBM is utilized to analyze flow in the
mesoscopic space, the moments of the distribution
functions must be used to calculate macroscopic
quantities such as density and velocity. For instance,
the local density and velocity are obtained through
relations (13) and (14), respectively.
8
0
i
i
f
(13)
8
0
8
0
u2
v2y
iix x
i
iiy
i
c
c
t
f
F
t
f
F
(14)
In the LBM, the pressure is obtained from relation
(15), and the relaxation time is related to the
kinematic viscosity through relation (16).
2
s
P
c
(15)
20.5
s
c
(16)
The BGK collision operator depends only on the
parameter τ, which itself is related only to the
kinematic viscosity υ. If the value of τ is close to 0.5
or it is higher than 1, then using this operator will
cause some problems, for example it can reduce the
simulation accuracy, increase simulation time or
sometimes it can make the simulation unstable. To
solve these problems, the TRT operator has been
used for simulation in this study. This operator is
dependent on two relaxation times and there is no
limitation to determine the relaxation time anymore.
Same as BGK operator, one of these relaxation times
are obtained from the kinematic viscosity, and
another constant is obtained based on the relation
A. Alimoradi and S. Ali Mirbozorgi / JAFM, Vol. 13, No. 6, pp. 1847-1858, 2020.
1851
(17) using a magic number called Λ. In this
simulation, Λ
is assumed.
20.5
0.5 0.5
s
c
(17)
,,
,,
,,
eq
ii
e
ii i
q
i
i
i
f
xc fxt
f
xt f xt
fxtf x
tt t
t
t
t
t
F
(18)
2
2
i
eq eq
eq i
i
i
i
i
ff
f
f
f
f
(19)
The collision function with the presence of this
operator can be observed in relation (18). The
procedure of determination of the new distribution
functions has been indicated in relation (19); in these
relations, ̅ is the distribution function opposite of the
function i.
3.2 Immersed-Boundary Method (IBM)
Since an orthogonal mesh is often used with uniform
distances in rectangular geometries in the LBM, the
geometry analysis of the complex flows, such as the
flow passing over objects, has always been
challenging in this method. For the simulation of the
complex flow geometry using the LBM, secondary
methods can be used. The immersed boundary
method, the ghost fluid method, and the blocking
method are among these methods. In this paper, the
two methods of blocking and immersed boundary
methods have been exploited. It is worth noting that
due to the specific features of LBM, the blocking
technique can be conveniently used.
The immersed-boundary method is based on the
simulation of the flow around the objects (the
immersed boundary) introducing the virtual
volumetric forces F in the equations governing the
fluid flow field. Different models of this method are
based on the procedure of calculation of this force.
This method was first proposed by (Peskin 1972) to
simulate the flow around the heart valve. The
integration of the two methods of LBM and the
immersed-boundary method (LB-IBM) was first
suggested by (Feng and Michaelides 2004). From
that year on, investigations have been conducted to
solve this method problems and to improve it. For
instance, (Feng and Michaelides 2004) proposed a
penalty method to calculate the force, and (Wu and
Shu 2009) introduced the implicit velocity correction
method. This method can simulate an object
precisely without the fluid penetrating it, however,
due to the implicit nature of this method and the need
for solving the equations at all points simultaneously,
the simulation of this method is costly and very time-
consuming. In 2011, (Kang and Hassan 2011)
removed the problems associated with the velocity
correction method and introduced the Multi-Direct-
Forcing method. In this method, construction and
inversion of the matrix of coefficients are no longer
necessary and the process of solving the equations is
performed iteratively. While requiring less time to
simulate, this method can produce the results of the
implicit velocity correction method accurately.
In the IBM, two series of points are used: Eulerian
points and Lagrangian points. In the Eulerian points,
in which the location, velocity, and the force is
defined respectively as ,
,
,, ,
,
, , and ,
,
, , the main
flow equations are solved. It worth mentioning that,
, and , replace
and
in relation (7) and the
LBM, respectively.
In the Lagrangian points specifying the boundary of
the object and their location, velocity, and force
with
,
, and
, respectively, the specific equations of
this method, such as the equation for determining the
virtual force, are solved. In the case of the rigidity of
the object, the distances of these points remain the
same and fixed during the solution process. Since
Eulerian and Lagrangian points do not necessarily
have a direct relationship with each other, a mutual
relation must be introduced between these two series
of points so that the fluid and boundary of the virtual
body feel each other presence. Peskin introduced this
relationship using the Dirac delta function so that this
function has already become a cornerstone for all
IBMs. In this method, first, the velocity of the
Lagrangian points is calculated using the velocity
interpolation from Eulerian points by the Dirac delta
function. Having the velocity at the Lagrangian
points, and after calculating the partial forces at these
points, these forces are distributed at the Eulerian
points to create the required force field to simulate
the boundary of the object in the fluid flow. The
relation between velocity interpolation and force
distribution is written respectively as relations (20)
and (21).
,
,
,,
,,
v
uu ,
v,
kijijk
k
i
ii
j
ij jjk
X
X
X
X
(20)
,,
,
,
,
,
Fx Fx
Fy Fy
,
,
i
ij k ij k
i
j
ij
j
jkik
XX
XX
(21)
In these relations, Δ is indicative of the Dirac delta
function, written as relations (22) and (23) for the
two-dimensional problems.
A. Alimoradi and S. Ali Mirbozorgi / JAFM, Vol. 13, No. 6, pp. 1847-1858, 2020.
1852
,
,,
,
ij k
ij k ij k
XX
x
yxy
(22)
11cos 0 2
42
02
(23)
In the IBM method, the Lagrangian force is
calculated in different ways, each being the origin of
a particular method. In the current study, the Multi-
Direct-Forcing method proposed by Kang and
Hassan has been used to calculate this force, in which
the force is calculated according to the following
algorithm.
1. Set iteration counter 0
2. Finding the uncorrected velocity from relation
(24) and interpolating the Lagrangian
velocities from Eulerian points
,
,
u
v
m
ij x
m
ij y
c
fc
f
(24)
3. 1
4. Calculating the Lagrangian forces from
relation (25)
1
1
v
2
2
uu
v
m
k
m
dm
kk
dm
k
kk
F
t
x
Fy
t
(25)
5. Spreading the Lagrangian forces to obtain the
Eulerian ones.
6. Correct Eulerian velocity with relation (26)
,
1
,,
,
1
,,
Fx
uu 2
Fy
vv 2
m
ij
mm
ij ij
m
ij
mm
ij ij
t
t
(26)
7. Repeat steps 3-7 until m reaches a pre-defined
value
8. Calculate the total correction force with relation
(27)
1
,
1
,,
,
Max
Max
mm
ij
m
mm
ij
m
ij
ij
F
xFx
F
yFy
(27)
As previously stated, the blocking technique is
another technique helping the LBM to analyze flow;
this technique is often used to model fixed rigid
bodies in the flow path. In this method, the body is
modeled as an object with boundaries coinciding
with the mesh points and introduced with the Zou-
He boundary condition model, besides, the velocity
values of the points inside this block -in the section
regarding the computation of macroscopic
quantities- are set to be zero. In this paper, as shown
in Fig. 6, the step added in the channel of the
micropumps have been modeled using this method
and the rotor modeled using the IBM.
Fig. 3. A sample of generated mesh and
Blocking and IB techniques
4. VALIDATION RESULTS
To validate the results, first, the problem of the flow
around a cylinder and then the simulation results of
an I-shaped micropump are compared with the
results of the studies carried out by Abdelgawad et
al. 2004 and Da Silva et al. 2007 The solution
domain for the problem of the flow around a cylinder
can be observed in Fig. 4. The streamlines and
vortices behind the cylinder have been demonstrated
for a flow with Re = 40 in Fig. 5.
Due to the use of the Multi-Direct-Forcing method,
penetration towards inside the fixed cylinder can be
witnessed, however, as illustrated in table 2, the
vortex length obtained is sufficiently consistent with
the results reported by other researchers. Taking into
account these results, and as mentioned in reference
(Kang and Hassan 2011), it can be claimed that the
Multi-Direct-Forcing method (MDF) can produce
results similar to those of the Implicit-Direct-Forcing
method (IDF), but by spending less time. This time
difference in simulation is noticeable, especially for
problems with moving or rotating objects. For
example, for the current simulation with stationary
boundary (for the IDF the process of building and
reversing the coefficient matrix was done only once
out of the main loop), the simulation for the MDF
method required about 1152 seconds while for the
IDF, simulation time was about 1030 seconds. In this
situation, where the matrix is built only once, the
required computation time for the IDF method is
about 100 seconds slower than the MDF method.
However, if there is a moving or a flexible boundary
where you need to rebuild and reverse the matrix for
each step of the time, the IDF method will require
A. Alimoradi and S. Ali Mirbozorgi / JAFM, Vol. 13, No. 6, pp. 1847-1858, 2020.
1853
much more time and resources to produce the similar
results that the MDF method can produce in a much
shorter time.
Once the problem of the flow around a fixed cylinder
was investigated, the numerical analysis of the flow
in the micropump shown in Fig. 1 was performed and
the results were compared with the findings of other
studies. Figure 6 illustrates the streamlines around
the rotor of this micropump. Given the rotor direction
and its proximity to the bottom wall, finally a pure
flow has been created to the right with the rotation of
the rotor. This is while two relatively large vortex
flows have been formed above the rotor.
Fig. 4. Geometry of flow past a circular cylinder
problem.
Fig. 5. Geometry of flow past a circular
cylinder problem.
Table 2 Comparison between vortex length in
present work and other studies
Wu & Shu (Wu and
Shu 2009) 1.86 4.62
Shu et al. (Shu, Liu et
al. 2007) 1.8 4.4
Dennis & Chang
(Dennis and Chang
1970)
1.88 4.69
Fornberg (Fornberg
1980) 1.82 4.48
He & Doolen (He and
Doolen 1997) 1.84 4.49
Present Work 1.84 4.57
Fig. 6. Streamlines in a simple I-Shaped viscous
micropump.
Regarding the interpretation of the cause of the
vortices, it should be said that if the length of the
channel of this micropump is divided into three zones
of the inlet boundary before the vortex flow, the
vortex flow zone, and the zone after the vortex flow to
the outlet boundary, the negative pressure gradient is
the agent for the fluid motion in the first and third
zones, and rotation of the moving wall (rotor) in the
second zone is the fluid motion factor, As shown in
Fig. 7. This is while the presence of a negative
pressure gradient in the first and second zones causes
a positive pressure gradient in the vortex region.
When the fluid moves to the right from the top of the
rotor due to the rotor rotation, it cannot continue its
way to the right easily in the presence of a positive
pressure gradient, and thus diverges to the upper wall.
After the collision with the upper wall, part of it
deviates to the right and the other part to the left, so
that a stagnation point is formed on the upper wall at
the right side of the rotor. The backflow near the upper
wall cannot continue to the left, as it collides with the
main flow and deviates towards the center of the
channel to satisfy the principle of mass conservation
so that another stagnation point is formed beside the
upper wall and before the rotor. The backflow beside
the two stagnation points eventually leads to the
formation of a vortex region as shown in the figure.
However, the flow diverged from the vicinity of the
stagnation point before the rotor cannot completely
pass over the rotor, and therefore part of it creates a
vortex center before the rotor. Accordingly, the
diverging flow near the stagnation point after the rotor
cannot completely pass near the upper wall, hence
creating another vortex center at the right side of the
rotor. If the distance between the rotor and the upper
wall increase, the flow through this distance is
facilitated and the two vortex centers unify and only
one vortex is formed above the rotor.
Fig. 7. Pressure change in centerline of an I-
shaped viscous micropump.
Re 20Re 40
A. Alimoradi and S. Ali Mirbozorgi / JAFM, Vol. 13, No. 6, pp. 1847-1858, 2020.
1854
Figure 8 depicts the variations in the outlet
dimensionless average velocity
∗
relative to the
channel width
∗
, and the average velocity can be
obtained through relation (28).
0
*
1
H
uuydy
H
u
uU
(28)
These results have been obtained with values
of 0.5, ∆
∗
0.5,0.9. is a number
between 0 and 1 and a controlling parameter for ,
which are linked by the relation
1 . With a constant and, decreases with
an increase in . Following these results, the
maximum average velocity occurs in a width of
∗
1.53. It should be noted that in this case,
∗
also increases with increasing
∗
.
Fig. 8. Comparison of average fluid velocity
based on channel height in this work and
(Abdelgawad et al. 2004).
Figure 9 illustrates the comparison of variations of
the maximum flow rate
∗
in terms of distance
from the wall
∗
with the corresponding results in
the study by Da Silva et al., in which the flow rate is
calculated using relation (29).
** *
QuH
(29)
Fig. 9. Comparison of maximum average fluid
velocity and maximum flow rate based on rotor
distance in this work and (Da Silva et al. 2007).
Fig. 10. Comparison of optimum channel height
based on rotor distance from bottom wall in this
work and (Da Silva et al. 2007).
Based on these results, the flow rate is maximized
when
∗
0.025 . It should be noted that in
calculating the maximum flow rate, the optimum
channel width associated with each
∗
is always
used. It can be seen in Fig. 10 that the optimum
channel width increases with increasing
∗
. The
optimum channel width is a width in which the outlet
flow rate is maximum. In this simulation, the values
of the Reynolds number and pressure are assumed to
be
1 and ∆
∗
1, respectively.
As shown in Figs. 8 to 10, the results of the numerical
analysis of the present study are highly consistent
with the results of other studies.
5. R
ESULTS
The vortex flow above the rotor is one of the main
barriers against passing the flow from a viscous
micropump, as the width of the main flow decreases
with increasing the size of this vortex, ultimately
leading to the reduced amount of fluid transferred
and the decreased outlet flow rate.
The primary idea in this research was to reduce the
negative effect of the vortex flows by adding a wall
in the channel. For example, two vortices formed on
the rotor of an I-shaped micropump can be observed
in Fig. 6, in which 1, ∆ 1, and
∗
0.025.
In the I-Shaped viscous micropump with a step
above the rotor, which is shown in Fig. 2, the step
is located above the rotor and its height is measured
relative to the upper wall. As demonstrated in Fig.
10, the added wall can affect the flow behavior at
the site of the two stagnation points by controlling
the flow and reduce the size of the vortices above
the rotor and eventually increase the flow passage
width. It is worth mentioning that the height of the
added wall is only a small part of the channel
width, however, it affects the flow width in such a
way that it easily compensates for the decrease in
channel width at the rotor site. These changes can
be observed in streamlines by comparing Figs. 6
and 11.
A. Alimoradi and S. Ali Mirbozorgi / JAFM, Vol. 13, No. 6, pp. 1847-1858, 2020.
1855
Fig. 11. Streamlines in a viscous micropump of
Fig. 2.
If the graph of the effect of the step height on the flow
rate is plotted for this micropump, Fig. 12 will be
obtained; this figure shows the positive effect of
adding the step in the channel on the outlet flow rate,
and as shown in this figure, in case of having a
certain height, it can increase the outlet flow rate.
Fig. 12. Effect of height of step in flowrate of
viscous micropump of Fig 2.
If the outlet flow rate of this micropump in the step-
free mode and in the optimal mode, in which the flow
rate is maximum, is plotted in different channel
widths, Fig. 13 will be obtained. As indicated in this
figure, in the step-free mode (simple I-shaped
micropump), the outlet flow will be maximum, and
then will start to decrease due to the reduction of the
flow passage width by the vortex flow, however, by
adjusting the step in the second micropump, an
optimum size can be found for which the formed
vortices will have the minimum size and as a result,
the flow passage width will increase and this will
increase the outlet flow rate. The plot of optimal step
height to obtain maximum flow rate based on
channel width can be observed in Fig. 14.
According to this graph, it can be seen that up to the
channel width of ∗1.6, the optimal step height is
zero, making the output flow rate be the same in two
cases of with and without a step. However, with
increasing the channel width, an increase in the step
height can be useful to reduce the vortex flow size in
addition to increasing the outlet flow rate. This
increase in the outlet flow rate is such that in the
channel width of ∗3.7, the outlet flow rate of the
micropump with an optimum height, has an increase
of 150% relative to the outlet flow rate in the step-
free state.
Figure 15 has been obtained by plotting the power
consumption diagram of the simple I-shaped
micropump and its peer in the presence of an
optimum step.
Fig. 13. Comparison of maximum and primary
flow rate in a viscous micropump of Fig. 2.
Fig. 14. Optimal step height for maximum flow
rate based on channel height in a viscous
micropump of Fig. 2.
As can be seen in this figure, surprisingly, the power
consumption for both micro-pumps is almost the
same, so that they vary by a maximum of 0.15% in
high values of the channel width. Thus, it can be
concluded that the step addition does not affect the
fluid velocity right around the rotor, and only by
controlling the volume of fluid in the vortices (by
reducing it), it can increase the flow passage width
between the rotor wall and the step wall, so that a
higher flow rate is achieved at the same power
consumption rate.
If the pressure difference of the two ends of the
micropump is equal to zero, i.e. ∆∗0, the
micropump will transfer the maximum flow rate that
it can produce. If the graph of the maximum flow rate
of the micropumps of Figs. 1 and 2 is plotted, Fig. 16
will be obtained. As can be observed, the maximum
flow rate can be increased by adjusting the step
height.
A comparison can be observed in Fig. 17 between the
maximum pressure differences produced by the
micropumps of Figs. 1 and 2. As it is clear, in the
simple I-shaped micropump, the maximum pressure
difference is maximized with a channel width
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1856
increase, and then the maximum pressure difference
has a downward trend. However, in the micropump
of Fig. 2, by adjusting the step height, the maximum
Fig. 15. Comparison between the dimensionless
rotor power that viscous micropump Fig. 1 and
Fig. 2 need to generate flow.
Fig. 16. Comparison of the maximum flow rate
that viscous micropumps of Fig 1 and Fig. 2 can
produce in the absence of counter pressure.
pressure difference can be produced in all widths of
the channel. In Fig. 18, the height of the optimal step
height to produce the maximum pressure difference
can be observed.
If the free width
∗
∗
∗
at the rotor section
(the width between the lower and upper walls at the
rotor section in this micropump) is calculated, it
can be observed that the free width will be equal to
∗
1.2
. This means that the free width should be
1.2 to produce the maximum pressure difference.
Regarding this result, Fig. 19 has been obtained
plotting the maximum pressure difference
produced taking into account several channel
widths based on the free width; the values of this
graph are the same as Fig. 17 for the simple I-
shaped micro-pump.
According to this graph, it can be claimed that the
maximum pressure difference produced in this
micro-pump depends only on the free width value in
case of a fixed Reynolds number and the distance
from the wall, and this graph can be used to easily
specify the maximum pressure difference for all the
different channel widths.
Fig. 17. Comparison of the maximum outlet
pressure that viscous micropump of Fig. 1 and
Fig. 2 can produce.
Fig. 18. Optimal step height for maximizing the
outlet pressure that a viscous micropump of Fig.
2 can produce based on channel height.
As mentioned before, this can be categorized as a
new type of I-Shape viscous micropumps. Therefore,
as same as other new pumps, it is necessary to
include 2 sets of graphs which are the P-Q graph or
“The Pump Performance Curve” and W-Q graph or
“The Power Curve”. According to the Fig. 20 which
is the P-Q curve, this new micropump can withstand
more outlet pressure while producing the same flow
rate as the normal I-Shape micropump. One can see
that at the end of the graph, where there is no outlet
pressure, both types of micropump can produce
nearly the same flow rate. This can be explained by
the absence of the counter-pressure which will
reduce the maximum pressure at the right side of the
rotor, that micropump will produce to overcome the
outlet pressure, and by doing that more fluid can flow
at this area which ultimately will reduce the size of
the vortex that will develop over the rotor. So, when
there is no outlet pressure, the vortex is smaller than
there is some outlet pressure, but it still can be
A. Alimoradi and S. Ali Mirbozorgi / JAFM, Vol. 13, No. 6, pp. 1847-1858, 2020.
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smaller with adjusting the step height. On the other
hand, because it’s a small vortex, it has minimum
influence over the flow geometry, so the difference
between the normal I-Shape and the I-Shape with
step micropump flow production is low.
Fig. 19. Maximum Pressure that a viscous
micropump of Fig. 2 can produce based on the
gap between its bottom and top channel walls at
the location of the rotor.
Fig. 20. Comparison between performance
curves of Normal I-Shape and I-Shape with step
viscous micropump.
Fig. 21. Comparison between Power
requirement curves of Normal I-Shape (N) and
I-Shape with step (WS) viscous micropump.
Figure 21 shows the W-Q curve. As it can be
observed in this graph, the normal I-Shape
micropump requires more power than the I-Shape
with step micropump while producing the same flow
rate. This can be explained with the help of Fig. 13
and 15. According to these graphs by increasing the
channel height, power consumption by rotor will
decrease, and the flow rate in the new viscous
micropump increases while in the normal one, this
increase will decrease the outlet flow rate. Therefore,
in the new micropump, by increasing channel height,
one can adjust the flow rate to be the same as the
normal one, while the rotor requires less power.
6. C
ONCLUSION
In this paper, by exploiting the blocking technic and
immersed boundary method, two I-Shape viscous
micropump –a normal I-Shape micropump and a new
I-Shape micropump with an additional step in the
micropump channel- were simulated by using the
Lattice Boltzmann Method.
In this novel model, by adjusting the step height, the
micropump can manipulate the flow in such a way
that it can reduce the size of vortices, which are
located at the top of the rotor, and as a result, it can
increase the main flow pass width and this will
increase the outlet flow rate. Besides, by adjusting
the step height, one can manipulate the maximum
pressure which the micropump can produce. The
results show that the extra step doesn’t change the
fluid velocity which is located in the vicinity of the
rotor. Therefore the extra step doesn’t change the
rotor energy consumption, but by increasing the
main flow pass width it can transfer more fluids than
before. For example, by setting the channel height
∗
3.7 and embedding a step with a height of
∗
1.5, this novel micropump can produce a 150%
more flow rate than the normal I-Shape viscous
micropump.
The results show that by knowing the free width, one
can find the maximum pressure which the
micropump can produce, or one can adjust this
maximum pressure by adjusting the free pass width.
Besides, by using this new structure for micropump,
the designers can also use bigger channel heights
which were not efficient in the original design.
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