1. ME635 Term Project:
The simulation of electro-osmosis flow control system
By Yang He & Jiahua Gu ME635 advisor: Prof. Kishore Pochiraju
ABSTRACT:
The motion of flow in micro-channel can amplify some properties which is tiny on the macro-scale.
electroosmotic velocities can be dramatically apparent when the size of the channel is smaller
than a characteristic length. In this simulation, three different concentrations of KCl solutions were
used as flow, and a packed rectangular microchannel model was established. We studied the
enhancement of electro-osmosis phenomena by applying varying external voltage difference on
the two sides of the packed-channels. We use the nondimensionalized method, flat-plate
approximation and Gauss-Seidel iterative procedure to solve the coupled Poisson equation and
Navier-Stokes equation with the modified micro-flow boundary conditions. By comparing the
micro-channel’s electrical potential distribution and the velocity profile with the macro-channel, we
show the property of electroosmosis phenomena in micro-channel and how to control the velocity
of the flow via exerting voltage.
1. Problem Background
It is well-known that electroosmotic flow in micro-fluidic devices have great applications in medical
research. The better understanding of controlling this flow, the better implement in the drug area. But
the essential part in the field of drug dischargers should meet the requirement of separating fluids in
the atomic level. Unfortunately, mixing fluids in the micron-scale is a bottleneck problem currently. But
it is believed that by controlling the fluids electrically, the fluids can be mixed in micron-scale and can
easily reach to the requirement of separating fluids at atomic level.
When a certain potential applies on the fluid that is running across the porous structure material,
capillary tube, membrane, microchannel, or other fluid conduits, the electroosmotic flow occurs. Based
our knowledge, the velocity of the electroosmotic relies on the porous size of fluid conduits. Also,
compared with the characteristic length scale of the channel, the electroosmotic effect is significantly
obvious in small channel, specifically micron-scale. And the electroosmotic flow plays important role in
chemical separation techniques.
Any solutions with a certain concentration under the applied electric field can produce the mobility of
net electric charge, in this way, the Coulomb force is induced and causes the electroosmotic flow.
Meanwhile, the net fixed electric charge refers to the chemical equilibrium status in the solution
between the electrolyte and a solid surface. The function of the net fixed electric charge is to form a
layer of the mobile ions, we call them electrical double layer or Debye layer. In this sense, the
electroosmotic flow results from the Coulomb force produced by the mobility of net electric charge in
the electric double layer under the applied electric field on two electrodes.

2. 1 packed channel
2 high-voltage power supply with a direct
current micro ammeter
3 positive pole (Pt wire)
4 solvent reservoir
5 two-channel electrode and the negative pole
6 ground electrode
7 flow regulating resistance
7(1) regulate the internal pressure of the capillary
7(2) regulate the flow rate of the output liquid
8 pressure
9 output channel
Figure 1. Schematic diagram of electroosmotic pump setup and components meaning
Figure 2. The SEM image of packed channel.
Figure 1 is the schematic of electroosmotic system based on Chen et. al.’s experiment1
. The meaning of
each component of the system is in the table. Figure 2 shows the SEM image of packed channel
fabricated by the Selvaganapathy et. al2
. In this project, we aims to model the packed channel and
simulate the flow in them.
The configuration of our model is established using SolidWorks (figure3 and figure 4). Note that in order to
show the basic geometry of our model, we deliberately decrease the aspect ratio. But we perform the
calculation in the real ratio, namely 200:1. The whole packed channel composes of 100 sub-channels. Since
the symmetry of individual sub-channel and the same function in electroosmotic process, the constituent
of the packed channel, shown on figure 5, is the target of this simulation. The final result of the simulation
of the packed channel can be obtained by the product of the number of sub-channels with the performance
of single sub-channel.
2. Object and scope
The simulation of electroosmotic system is showed on figure 1 based on Chen et. al.’s experiment3
. The
meaning of each component of the system is: 1 packed channel; 2 high-voltage power supply with a direct
current microammeter; 3 positive pole (Pt wire); 4 solvent reservoir; 5 two-channel electrode and the
negative pole (ground electrode 6); 7 flow regulating resistance [7(1) used to regulate the internal pressure

3. of the capillary and 7(2) used to regulate the flow rate of the output liquid]; 8 pressure; and 9
output channel. The packed channel is the focus of this simulation. Figure 2 show the SEM image of packed
channel fabricated by the Selvaganapathy et. al4
.
The configuration of our model is established using SolidWorks (figure3 and figure 4). The sub-channel
dimension: width is 2W=100 nm, height is 2H=20m, and length is L =20m (Figure 6).Note that in order
to show the basic geometry of our model, we deliberately decrease the aspect ratio. The whole packed
channel composes of 100 sub-channels. The number of repeated sub-channel also deliberately decreased
in order to show the structure. But we do calculation in the real aspect ratio namely 200:1. The reason why
we choose the high aspect channel as the study object is that the high aspect ratio can provide high level
of electric potential distribution in the fluidic domain. Since the symmetry of individual sub-channel and
the same function in electroosmotic process, the constituent of the packed channel, shown on figure 5, is
the interest of this simulation. We set the Cartesian Coordinates and choose the center of XY plane as the
origin, and that is used in the Model 2’s calculation. In our 2nd
model, the rectangle interest domain has the
mirror symmetry. So the 2nd
model’s calculation region can be simplified to the half of the former area. The
final result of the simulation of the packed channel can be obtain by the product of the number of sub-
channels with the performance of single sub-channel.
In this simulation, we use the Flow Simulation in SolidWorks to give the first model: flow through the macro-
channel driven by steady pressure gradient. The purpose to do that is to set a reference model in contrast
with the micro-channel situation. Then we study the micro-channel use MATLAB’s pde solver, and Simulink
3 ode solver to solve the coupled partial differential equations in the case of micro-channel. Subsequently,
we make a comparison and discuss the difference in two model. Finally, we give some conclusions about
how to control the flow using electroosmotic device based on our modeling.
Figure 3. Schematic of packed channel

4. Figure 4. The section view of the packed channel Figure 5. The study interest region of this simulation
3. Model 1: the original model: flow through the macro-channel driven by steady pressure gradient
First, the original channel is studied using SolidWorks’ Flow simulation. But SolidWorks cannot couple
the electrical field with the motion of the flow. The purpose of this simulation is to demonstrate the
velocity of flow induced by a specific pressure gradient in a magnitude of micrometer. In this model,
there is no voltage added on each side of the channel, the roughness of the internal surface is zero,
and we use water as the flux. Before the simulation, two lids are added at each side of the channel.
The boundary condition is that the pressure add one inner surface of the lid. We define there is
Laminar flow only. The boundary condition is that two inner face of lids are added static pressure.
Figure 6. SolidWorks model of individual channel
Figure 7. lids on the two entrance of the channel and the static pressure added on the calculation
domain

5. Figure 8. Velocity profile exerted by static pressure Figure 9. Position of electrodes5
The consequence demonstrate that the velocity profile cannot be showed very good. It is because
SolidWorks is not a very useful tool to study micro-size behavior. The common size of Solidworks’
model is about several cm or m, which is too large to show the phenomena in below micro-scale.
4. Model 2: KCl solution flow through the sub-channel driven by electroosmosis
4.1 Mechanism of electroosmisis and the governing equations
The computational domain is depicted in figure 5. For simplicity, the flow is supposed to be hydro-
dynamically fully developed. It is assumed that all physical properties are constant and we assume that
an ideal symmetrical KCl salt which is fully dissolved in water. The mechanism is illustrated on figure 10.
For a steady state, fully developed flow, the velocity u satisfies u(x,y). Also the hydraulic pressure P is a
function of z only and the pressure gradient dP/dz is constant. The pressure gradient in the flow serves
as the initial driving force. If the gravity effect is negligible, the body force is caused by the addition of
the pressure gradient and the electrical field.
The governing equations are:
𝜀 (
𝜕2 𝜓
𝜕𝑥2 +
𝜕2 𝜓
𝜕𝑦2) = 2𝐹𝑧0 𝑐0sinh⁡(
𝑧0 𝐹𝜓
𝑅 𝑢
) (1)
𝜇 (
𝜕2 𝑢
𝜕𝑥2 +
𝜕2 𝑢
𝜕𝑦2) =
𝑑𝑝
𝑑𝑧
− 2𝐹𝑧0 𝑐0sinh⁡(
𝑧0 𝐹𝜓
𝑅 𝑢
)
Δ𝑉
𝐿
(2)
Where Co is the concentration of the KCl solution; u is the velocity; p is the pressure; z0 is the charge
number of the KCl ions, Faraday constant (96500 C mol−1) andV is the voltage difference at two electrode.
The position of electrode is demonstrated on figure 9.
Figure 10. Depiction of
electro-osmotic flow6

6. Equation (1) is the Poisson equation solved for the electric potential field and equation (2) is the
Navier-Stokes equation used to describe the velocity field. The two fields are coupled in a system
shown on Figure 6.
The boundary conditions are:
𝑢 = 0⁡𝑎𝑡⁡𝑦 = 0, 𝑦 = 𝑤𝑐, 𝑥 = 0, 𝑥 = 𝐻 (3)
𝜓 = 𝜁⁡𝑎𝑡⁡𝑦 = 0, 𝑦 = 𝑤𝑐, 𝑥 = 0, 𝑥 = 𝐻⁡ (4)
4.2 Parameters
4.2.1 Zeta potential
In order to set the boundary conditions, we must know the zeta potential of our model. Table 17
gives the
experimentally determined zeta potential values.
Here we choose the 1mM,0.1mM and 0.01mM KCl solution that means the zeta potential is -42e-3 V, -67e-
3 V and -97e-3 V respectively.
4.2.2 Viscosity and density
We refer to two academic papers to determine the viscosity of 1mM, 0.1mM and 0.01mM KCl
solution under the 500 kPa89
. There is no data about our designed concentration.
Figure 11. Relation of Viscosity and concentration

7. Since the viscosities of electrolyte solutions are usually given by the square polynomial equation
𝜇 = 𝐴 + 𝐵𝑐 + 𝐷𝑐2
(5)
We use curving fitting to find out the equation:
𝜇 = 0.0891 − 0.007836𝑐 + 0.004894𝑐2
(6)
Since the concentration is really small in our case. We assume that the viscosities of 1mM, 0.1 mM and
0.01mM KCl solution are the same, namely, 0.0891 mPa/s. Also we plot the density against the
concentration curve and fit it.
Figure 12. Relation between density and concentration
The relation between the concentration and density can be expressed as:
𝜌 = 0.003817𝑐 + 1.001 (7)
Since the KCl solutions we choose have very low concentration, we can assume that the density of all the
solutions are 1.001 g/cm-3.
4.2.3 Permittivity
We plot the concentration related permittivity of KCl electrolyte according to the experiment data in
table2.10 We find the relationship between concetration and permittivity is linear and we give it a linear
regression:
Table 2 Figure 13. Permittivity VS concentration

8. It is obvious that the permittivity has the linear relation with the concentration as
𝜖 = 78.3 − 8.851𝑐 (8)
So the permittivity is 78.3.
4.3 dimensionless parameters
The scale of the domain is about hundred nanometer, therefore the direct use of the parameter in
calculation will be complicated and not efficiency. 11
Here we make all the parameter and governing
equations nondimensionalized as follow:
 Debye-Huckel parameter:
𝑘2
= 2𝑧0
2
𝑒2
𝑐0 𝑁𝐴/𝜖𝑘 𝑏 𝑇 (9)
k2=1.083e13; 1.083e12; 1.083e11 for 1mM, 0.1mM and 0.01mM KCl solution respectively.
Note that 1/k is the Debye length d: 3.039e-7; 9.609e-7; 3.039e-6 for 1mM, 0.1mM and 0.01mM
KCl solution respectively. Debye length means the thickness of EDL (electrical double layer), that will
be discussed later when we use the micro-fluidics phenomena to modify the boundary conditions.
 Hydraulic diameter of the rectangular domain:
𝐷 = 4𝐻𝑊(𝐻 + 𝑊) (10)
D=2e-7.
 Dimensionless zeta potential:
𝜁̅ = 𝑧𝑒𝜁/𝑘 𝑏 𝑇 (11)
Zeta1=-1.6338; -2.6063; -3.7733 for 1mM,0.1mM and 0.01mM KCl solution respectively.
Now we can give the dimensionless Poisson equation:
𝜕2Ψ
𝜕𝑋2 +
𝜕2Ψ
𝜕𝑌2 = 𝐾2
sinh⁡(Ψ) (12)
Where X=x/D, Y=y/D, K=kD andΨ = 𝑧𝑒𝜓/𝑘 𝑏 𝑇.
K2=4.332e-1; 4.332e-2; 4.332e-3 for 1mM,0.1mM and 0.01mM KCl solution respectively.
The boundary condition is:
𝑌 = 0⁡⁡⁡⁡⁡⁡⁡⁡
𝜕Ψ
𝜕𝑌
= 0; ⁡⁡⁡𝑌 =
𝐻
𝐷
⁡⁡⁡⁡⁡⁡⁡⁡Ψ = 𝜁̅ (13)
𝑋 = 0⁡⁡⁡⁡⁡⁡⁡⁡
𝜕Ψ
𝜕𝑋
= 0; ⁡⁡⁡𝑋 =
𝑊
𝐷
⁡⁡⁡⁡⁡⁡⁡⁡Ψ = 𝜁̅⁡⁡ (14)
 Define the reference velocity is U= 1e-4 m/s;
 Reference Reynolds number:
𝑅 𝑒 = 𝜌𝐷𝑈/𝜇 (15)
The coupled dimensionless Navier-Stokes equation is more complicated because the pressure gradient
and the electrical potential field impact on the velocity field simultaneously. The reference velocity, the
initial pressure gradient and the external electric field need to be adjusted to make the parameters of
Navier-Stokes equation have a same order of magnitude to obtain the prime control.
𝜕2 𝑢̅
𝜕𝑋2 +
𝜕2 𝑢̅
𝜕𝑌2 =
𝐷2
𝜇𝑈
𝑑𝑝
𝑑𝑍
+ ΔV
𝐷2
𝜇𝑈
(2𝑧𝑒𝐶0 𝑁𝐴)sinh⁡(Ψ) (15)
In our calculation, we choose 10 V as the external electric potential and 1e6 Pa/m as the pressure
gradient. Therefore the parameter value is:
𝐷2
𝜇𝑈
𝑑𝑝
𝑑𝑍
=4.489;
ΔV
𝐷2
𝜇𝑈
(2𝑧𝑒𝐶0 𝑁𝐴)=8.659; 8.659e-1; 8.659e-2 for 1mM,0.1mM and 0.01mM KCl solution respectively.
The boundary conditions is:

9. 𝑌 = 0⁡⁡⁡⁡⁡⁡⁡⁡
𝜕𝑢̅
𝜕𝑌
= 0; ⁡⁡⁡𝑌 =
𝐻
𝐷
⁡⁡⁡⁡⁡⁡⁡⁡𝑢̅ = 0 (16)
𝑋 = 0⁡⁡⁡⁡⁡⁡⁡⁡
𝜕𝑢̅
𝜕𝑋
= 0; ⁡⁡⁡𝑋 =
𝑊
𝐷
⁡⁡⁡⁡⁡⁡⁡⁡𝑢̅ = 0⁡⁡ (17)
4.4 Gauss-Seidel iterative procedure
As for equation (1) there is a nonlinear term in the partial differential equation, namely, sinh⁡(
𝑧0 𝐹𝜓
𝑅 𝑢
).
This part needs to be convert into a linear one to get an efficiency calculation. First, we study the
mathematic property of the y=sinh(c*x) equation. The Gauss-Seidel iterative procedure can convert
the nonlinear y=sinh(c*x) term into the linear one.
sinh(Ψ) = cosh(Ψ∗) (Ψ − Ψ∗) + sinh⁡(Ψ∗
) (18)
The procedure is that, first we can guess a potential * and put it into equation which is linear one now
as a term of the partial differential equation. After solve the partial differential equation we can get the
solution  and we treat it as * to put into the equation again. Repeat the loop until we have a
converged solution. Therefore we can see from the whole process the linearization improves the
convergence of the iterative procedure.
4.5 Electric field by Simulink
Since the aspect ratio of the channel is much higher than 1. The effect of the top and the bottom
surface on the velocity and electric potential fields are negligible. In this case, the governing equations
can be further simplified:
𝜕2Ψ
𝜕𝑋2 = 𝐾2
sinh⁡(Ψ) (19)
Here we use a very interesting method to solve equ(19). We treat the simulation time as width of our
model. And we set the fixed time step 0.005 and use 3ODE solver. Note that we cannot control
boundary conditions 𝑋 = 𝑤⁡
𝜕Ψ
𝜕𝑋
= 0 and 𝑋 = 𝑤⁡⁡
𝜕𝑢̅
𝜕𝑋
= 0 . Hopefully, the velocity profile and the
electric distribution tends to have an rapidly increased slope when near the X=w boundary. Moreover,
the Simulink model is a first step calculation, the result of the Simulink model can guide us to set our
pde model.
Figure 13. diagram of the Simulink model 1

10. We set the Simulink model to solve this problem, the MATLAB code to control this is:
% distribution_v_1
% parameteres of three KCl solution with different concentration
K0=[0.4332 4.332e-2 4.332e-3];
W=0.5;
zeta0=[-1.6338 -2.6063 -3.7733];
figure (1);
hold on;
for jj=1:3
K2=K0(jj);
zeta=zeta0(jj);
% Set the guessed Psi
Psi0.signals.values=-1.3.*jj.*ones(1,1001)';
Psi0.time=[0:0.0005:0.5]';
subplot(1,3,jj);
for ii=1:30 % control the number of iteration
sim('Psi1');
Psi0=Psi;
plot(Psi.time,Psi.signals.values);
end
end
hold off
Figure 14. Plot of dimensionless length VS dimensionless electric potential
We also study the influence of the guessed electrical potential value. We choose the 0.001mM KCl group
to do the calculation with different guessed initial values. All groups have a constant iteration number so
that the convergence can be easily recognized by comparison. The result are shown on figure 15.
We find if we choose a guessed value just as much as the boundary, the solution can get converged very
quickly. That is very useful when we need to do approximate calculation and can solve considerable time.
And this is used in our last PDE model.

11. Figure 15 influence of different guessed initial value
4.6 Velocity profile
The Navier-Stokes equation can also be simplified as:
𝜕2 𝑢̅
𝜕𝑋2 =
𝐷2
𝜇𝑈
𝑑𝑝
𝑑𝑍
+ ΔV
𝐷2
𝜇𝑈
(2𝑧𝑒𝐶0 𝑁𝐴)sinh⁡(Ψ) (20)
We designate
𝐷2
𝜇𝑈
𝑑𝑝
𝑑𝑍
as A and ΔV
𝐷2
𝜇𝑈
(2𝑧𝑒𝐶0 𝑁𝐴) as B.
Figure 16. Diagram of the Simulink model 2

12. Figure 17. Velocity profile of flow for 1mM, 0.1mM and 0.01mM KCl solution
The MATLAB code control the Simulink is:
% velocity1
% parameteres of three KCl solution with different concentration
K0=[0.4332 4.332e-2 4.332e-3];
W=0.5;
zeta0=[-1.6338 -2.6063 -3.7733];
A=4.489;
B0=[8.659 8.659e-1 8.659e-2];
v0=0;
figure (1);
hold on;
for jj=1:3
K2=K0(jj);
B=B0(jj);
zeta=zeta0(jj);
% Set the guessed Psi
Psi0.signals.values=-1.3.*jj.*ones(1,1001)';
Psi0.time=[0:0.0005:0.5]';
subplot(1,3,jj);
for ii=1:30 % control the number of iteration
sim('v1');
Psi0=Psi;
plot(v.time,v.signals.values);
end
end
hold off
4.7 Modification of nano-scale flow boundary conditions and results
The most significant difference between the macro-channel and nano-channel is the surface velocity. In
macro-channel, that velocity always be zero. However surface velocity of a nano-channel has a relatively
large value. Thus the boundary conditions of our model need to be modified.

13. Figure 18. Slip length Figure 19. Relation between slip length and Debye length
The so called Navier boundary condition is12
:
𝑣𝑠 = 𝑏𝜕 𝑥 𝑣𝑧 (21)
Where vs is the slip velocity, b is the slip length:
𝑏 = 𝜇/⁡𝜅 (22)
The friction coefficient of a flow at laminar condition can be expressed as
𝜅 = 64/𝑅 𝑒 (23)
Where Reference Reynolds number: 𝑅 𝑒 = 𝜌𝐷𝑈/𝜇.
Using equation (15), (21) and (22), we can calculate the slip length b.
Then the slip velocity can be calculated since:
𝑣𝑠 =
𝜁
1+
𝑏
𝜆 𝐷
(24)
The efficiency of the slip length is the ratio b/W. In our case the ratio can be relatively tremendous
since the channel is at nano-scale.
Another element that influence the surface velocity is led by the external electric field.
Conventionally we ignore this part of velocity at that position. This part of velocity can be expressed
as:
𝑣 𝐸 = −
𝜖𝜁
𝜇
𝐸 (25)
Figure 20. Velocity profile with modified boundary conditions

14. 4.8 PDE model
In this model we choose the KCl solution with 1e-3 Molarity as the flow in the channel.
First, we define the calculation domain as R1. The domain is a quarter of the dimensionless cross-section
of the channel due to its symmetry.
Figure 20. Calculation domain of PDE model
Then, we set the boundary condition. Here we treat the right edge and the bottom edge are the surface.
So the coupled (equ12 and equ15) boundary conditions at this place is in Dirichlet (red) form. Another
two edges have the Neumann (blue) boundary condition.
Figure 21. Schematic of boundary conditions of PDE model
The parameters of two coupled partial differential equations have been discussed before. But we also
need to deal with the nonlinear sinh() part. Previously we use Gauss-Seidel iterative procedure to
convert it to a linear part, and we have a finding about how to give the initial guessed value. That is used
in our PDE parameter setting to simplify the question.
The mesh we set is:
Figure 22. Schematic of the mesh of PDE model

15. The whole code to build and solve the PDE model is listed as follow:
function pdemodel
[pde_fig,ax]=pdeinit;
pdetool('appl_cb',2);
set(ax,'DataAspectRatio',[37.5 1 2.5]);
set(ax,'PlotBoxAspectRatio',[1 1 1]);
set(ax,'XLim',[-15 15]);
set(ax,'YLim',[-0.5 0.30000000000000004]);
set(ax,'XTickMode','auto');
set(ax,'YTickMode','auto');
pdetool('gridon','on');
% Geometry description:
pderect([-10.000000000000002 9.9999999999999982 0.25 -0.25],'R1');
set(findobj(get(pde_fig,'Children'),'Tag','PDEEval'),'String','R1')
% Boundary conditions:
pdetool('changemode',0)
pdesetbd(4,'neu',2,...
char('0','0','0','0'),...
char('0','0'))
pdesetbd(3,'dir',2,...
char('1','0','0','1'),...
char('-1.6338','2.234'))
pdesetbd(2,'dir',2,...
char('1','0','0','1'),...
char('-1.6338','2.234'))
pdesetbd(1,'neu',2,...
char('0','0','0','0'),...
char('0','0'))
% Mesh generation:
setappdata(pde_fig,'Hgrad',1.3);
setappdata(pde_fig,'refinemethod','regular');
setappdata(pde_fig,'jiggle',char('on','mean',''));
setappdata(pde_fig,'MesherVersion','preR2013a');
pdetool('initmesh')
pdetool('refine')
pdetool('refine')
pdetool('refine')
pdetool('jiggle')
pdetool('jiggle')
% PDE coefficients:
pdeseteq(1,'1.0',...
char('-1.1163','0.0','0.0','0'),...
char('0.7572','-22.3.*u(1)-10.9696'),...
char('1.0','0.0','0.0','1.0'),'0:10','0.0','0.0','[0 100]')
setappdata(pde_fig,'currparam',...
['1.0 ';'0.0 ';'-1.1163 ';'0.0 ';'0.7572';'1.0 ';'0.0';'0.0';'1.0';...
'0.0 ';'0';'-22.3.*u(1)-10.9696';'0.0';'1.0'])
% Solve parameters:
setappdata(pde_fig,'solveparam',...
char('0','3264','10','pdeadworst',...
'0.5','longest','1','1e-4','','fixed','inf'))
% Plotflags and user data strings:
setappdata(pde_fig,'plotflags',[2 1 2 1 1 1 8 1 0 0 0 1 1 0 1 0 0 1]);
setappdata(pde_fig,'colstring','');
setappdata(pde_fig,'arrowstring','');
setappdata(pde_fig,'deformstring','');

16. setappdata(pde_fig,'heightstring','');
% Solve PDE:
pdetool('solve')
The result is shown on from Figure 22 to figure 25.
Figure 22. Electrical potential contour
Figure 23. 3D electrical potential distribution Figure 24. 3D velocity profile

17. Figure25. Velocity distribution contour
5. Conclusion
We tried to use SolidWorks to establish the first model to see the fluidic phenomena, but the result is not
good. Then we tried the numerical method to build our model. The interest domain is a quarter of the
cross-section area of the channel. All the parameter used in our model are from academic references. In
fact, there is no direct data can be used as the parameter. So we made the curve fitting according to the
existed data and the equations developed by other’s theoretical research. All the parameters, governing
equations and boundary equations are converted into the non-dimensional form. We use a simplification
to make our governing equations can be solved as coupled ordinary partial equations due to the high
aspect ratio of our calculation domain. A guessed electrical potential was put into our equations to have
the Gauss-Seidel iterative procedure which can transform the non-linear part into a linear one.
Meanwhile we discussed the influence of the initial guessed electrical potential value and found a not
only convenient but also relatively precise method which is applied in our PDE model. Then all boundary
conditions were modified according the slip length and Debye length in our case. The solutions describing
the nano-scale electrical potential and velocity profile can be compared with the solutions without
modification of boundary condition. Finally, we gave the PDE model based on the knowledge of our
former works. Overall, the basic idea in our modeling is that we use simplification to show as much
information as we can, then those information can be a guide to help us to build complicated model. This
model is fundamental and original, further study can be developed as more detailed situation adding in
our model.

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