Electroosmosis in a Microchannel (COMSOL) The microchannel ABCD in Fig. E12.4.1, with length L = 5 × 10^−4 m and width H = 5× 10^−5 m, contains an electrically conducting liquid whose physical properties are given in Table E12.4.1, in which “S” denotes units of Siemens, also equivalent to ohm^−1. N

Question

Electroosmosis in a Microchannel (COMSOL)

The microchannel ABCD in Fig. E12.4.1, with length [latex]L=5 imes 10^{-4} \mathrm{~m}[/latex] and width [latex]H=5 imes 10^{-5} \mathrm{~m}[/latex], contains an electrically conducting liquid whose physical properties are given in Table E12.4.1, in which "S" denotes units of Siemens, also equivalent to [latex]\mathrm{ohm}^{-1}[/latex]. Note also that a coulomb (C) equals an ampere-second (A s), and that the dielectric constant is also known as the relative permittivity. Electric potentials of zero and [latex]1 \mathrm{~V}[/latex] are applied at the left and right ends [latex]\mathrm{AB}[/latex] and [latex]\mathrm{CD}[/latex], respectively, and we wish to find the resulting liquid velocities, streamlines, and electric potential distribution.

Table E12.4.1 Physical Properties of Liquid
Property	Value	Units	COMSOL Name
Density, ρ	1.000	kg/[latex]m^{3} [/latex]	rho1
Viscosity, η	0.001	kg/m s	eta
Electric conductance, [latex]k_{1}[/latex]	0.11845	S/m	k1
Dielectric constant	80.2	---	epsr
(relative permittivity, [latex]ε_{r} [/latex])		---	---
Permittivity of free
space, [latex]ε_{0}[/latex]	8.85 × [latex]10^{-12} [/latex]	C/V m	eps0
Liquid permittivity, [latex]ε_{w}=ε_{r} ε_{0} [/latex]	7.097 × [latex]10^{-10} [/latex]	C/V m	epsw
Wall zeta potential, [latex]\zeta_{0}[/latex]	−0.0965	V	zet0

Table E12.4.2 Boundary Conditions
Boundary	Navier-Stokes Mode	Electric Curr. Mode
1	Outflow, no backflow	Electric potential
	[latex]p_{0}=0[/latex]	[latex]V_{0}=0[/latex]
2	Inflow/outflow velocity	Electric insulation
	[latex]v_{x}=\left(\varepsilon_{w} \zeta_{0} / \eta ight) \partial V / \partial x[/latex]
	[latex]v_{y}=\left(\varepsilon_{w} \zeta_{0} / \eta ight) \partial V / \partial y[/latex]
3	Inflow/outflow velocity	Electric insulation
	[latex]v_{x}=\left(\varepsilon_{w} \zeta_{0} / \eta ight) \partial V / \partial x[/latex]
	[latex]v_{y}=\left(\varepsilon_{w} \zeta_{0} / \eta ight) \partial V / \partial y[/latex]
4	Neutral (no normal applied stress)	Electric potential
	[latex]\mathbf{n} \cdot\left(-p \mathbf{I}+\eta\left( abla \mathbf{u}+( abla \mathbf{u})^{\mathrm{T}} ight) ight)=0[/latex]	[latex]V_{0}=1.0[/latex]

Accepted Answer

See Chapter 14 for more details about COMSOL Multiphysics. All mouse-clicks are left-clicks (the same as Select) unless specifically denoted as right-clicks (R).

This problem falls into the multiphysics category, in that it involves both simultaneous motion of fluid and electric charge.

Select the Physics

Open COMSOL and Left-click Model Wizard, followed by 2D, Fluid Flow (the little rotating triangle is called a "glyph"), Single Phase Flow, Laminar Flow, Add. While on the Physics Selection panel, L-click AC/DC, Electric Currents (not Electric Currents Shell), Add.
L-click Study, Stationary, Done.

Define the Parameters

R-click Global Definitions and select Parameters. In the Settings window to the right of the Model Builder, enter the parameters eta and rho1 in the Name column and enter [latex]0.001[\mathrm{~kg} / \mathrm{m} / \mathrm{s}][/latex] and [latex]1000\left[\mathrm{~kg} / \mathrm{m}^{\wedge} 3 ight][/latex], respectively, in the Expression column. Similarly, add the parameters [latex]k 1=0.11845[\mathrm{~S} / \mathrm{m}][/latex], epsr [latex]=80.2[1][/latex] (note the method of entering a dimensionless quantity), eps [latex]0=8.85 \mathrm{e}-12[\mathrm{C} / \mathrm{V} / \mathrm{m}][/latex] and [latex]z e t 0=-0.0965[\mathrm{~V}] .[/latex] Also, define epsw as the expression epsr*eps0, [latex]L=5.0 \mathrm{e}-4[\mathrm{~m}][/latex] and [latex]H=5.0 \mathrm{e}-05[\mathrm{~m}][/latex].

Establish the Geometry

R-click Geometry 1 and select Rectangle. Enter the values for Width and Height as [latex]L[/latex] and [latex]H[/latex]. Note the default is based on the lower left corner being at the origin, [latex](0,0)[/latex]. L-click Build Selected.

Define the Fluid Properties

R-click the Materials node within Component 1 and select Blank Material. Note that the new material will have the required material parameters based on the physics. Enter the parameters rho1 and eta as the values for Density and Dynamic viscosity. Similarly, enter the parameters [latex]k 1[/latex] and epsr as the Electrical conductivity and Relative permittivity.
Expand Laminar Flow by clicking the little glyph to its left, and select Fluid Properties 1. In the Settings window, ensure that the Fluid Properties are set to From material. Also click on the Equation glyph to see the equations being solved, as in Fig. E12.4.3(a).

Define the Boundary Conditions

Inspect the default boundary Laminar Flow conditions by selecting the Wall 1 node, and note that all are No slip. Add an outlet boundary condition by R-clicking the Laminar Flow node and selecting Outlet. Assign the left boundary segment (#1) by hovering and L-clicking that segment in the Graphics window, noting that Suppress Backflow is checked.
Add two inlet boundary conditions by R-clicking the Laminar Flow node and selecting Inlet. Assign the top and bottom boundary segments (#2 and #3) by hovering and L-clicking those segments in the Graphics window. Ensure the Boundary condition is set to Velocity field, and define the [latex]x[/latex] - and [latex]y[/latex]-components of velocity to be (epsw*zet0/eta)*Vx and (epsw*zet0 /eta) [latex]^{*} V y[/latex], respectively. Note the [latex]V x[/latex] and [latex]V y[/latex] are the two components of the potential gradient from the Electric Currents physics.
Define the neutral boundary condition. R-click the Laminar Flow node and select Open Boundary. Assign the right boundary segment (#4) by hovering and L-clicking that segment in the Graphics window. Note the default case of zero Normal stress.

Define the Electrical Properties

Check the electrical properties. Click the Electric Currents glyph and then select Current Conservation. In the Settings window, ensure that the Electrical conductivity and Relative permittivity are set to From material. Check the Equations glyph to see the equations are being solved, as in Fig. E12.4.3(b).
Inspect the default boundary condition for Electric Currents by selecting the Electric Insulation 1 node and noting that all four boundaries are insulated. Add the left-hand potential condition by R-clicking the Electric Currents node and selecting Electric Potential. Then L-click Electric Potential 1, and in the Settings window, assign the left boundary segment (#1) by hovering and L-clicking that segment in the Graphics window. Ensure that the Electric potential is set to zero. Similarly, by selecting Electric Potential and then clicking Electric Potential 2, define the right-hand boundary (#4) as having an Electric potential 1.0 [V].

Create the Mesh and Solve the Problem

L-Click the Mesh node and L-click Build All, using the coarser element size---this is not a problem requiring high definition, and we can thereby more easily display the mesh, as in Fig. E12.4.4 (Mesh Statistics gives 1,548 elements).
Expose the solver setting by R-clicking the Study node, and select Show Default Solver. Within the Solver Configurations node, expand the Solution Configurations by L-clicking the glyph to the left of Solution 1. Set the scaling option by L-clicking the Dependent Variables 1 glyph, selecting Electric potential, and locating the Scaling options in the Settings window. Change the method from Automatic to None by L-clicking the drop-down dialog---an essential step for the following none-too-obvious reason. Namely, for accuracy in solving simultaneous equations, COMSOL tries to scale the dependent variables so that they are all of order 1. The scaling is accomplished by dividing each variable by a reference quantity. But in our present problem, because there is no applied potential difference in the [latex]y[/latex] direction, there is no flow in that direction, so that [latex]v_{y}=0[/latex] and the scaling is impossible. If this precaution were not taken, the attempted solution would not converge. (Example 12.5, which has a flow in the [latex]y[/latex]-direction, is not so restricted.)
Save the model file as E12.5.mph or similar, because it will be needed at the beginning of Example 12.5, to avoid repeating all the steps up to here.
R-click the Study 1 node and select Compute. You can see the development of the solution under the Progress tab at the bottom of the Graphics window, but it is very fast --- so "re-Compute" if you missed it. A uniform light green (not reproduced here) indicates a constant velocity magnitude throughout the region (as expected). By clicking anywhere on the plot and looking at the indicated value below the Graphics window, or observing the color bar at the right, we discover that the magnitude of the velocity is [latex]1.37 imes 10^{-4} \mathrm{~m} / \mathrm{s}[/latex], in complete agreement with the results obtained from Eqn. 12.22 (away from the walls).

[latex]u=-\frac{\varepsilon E_{y}}{\mu}(\zeta-\phi)=-\frac{\varepsilon \zeta E_{y}}{\mu}\left(1-e^{-x / \lambda_{D}} ight),[/latex] (12.22)

Display the Results

R-click the Results node and select 2D Plot Group. R-click the newly created 2D Plot Group 4 and select Arrow Surface. Note the default values and L-click Plot at the top of the Settings window. For better reproduction, we change the Color from Red to Black, the Arrow base to Center, use the slider to increase the Scale factor to [latex]0.08[/latex], and reduce the Number of [latex]y[/latex] grid points from 15 to 10 , giving Fig. E12.4.5.
R-click the Results node and select 2D Plot Group. R-click the newly created Plot Group 5, L-click Rename, and enter the name Streamline Group. R-click the Streamline Group and select Streamline. In the Streamline Settings panel, select the Positions options, select Start point controlled, and change the Entry method to Coordinates. To define start points on the left edge, set the [latex]x[/latex]-coordinate to the single value [latex]L[/latex]. For clarity, we wish to decrease the number of [latex]y[/latex] grid points from 15 to 10 . For the [latex]y[/latex]-values, use the range build icon to the right of the field by L-clicking. In the panel that opens, change the entry method to Number of Values. Enter [latex]0.0[/latex] and [latex]H[/latex] as the Start and Stop values, with 10 as the Number of values. L-click Replace. Note that this process simply builds entries for the range function, and users could directly enter the range function as an option. Change Color to Black. L-click Plot, giving Fig. E12.4.6.
To obtain a plot of the equipotentials, first R-click the Results node and select 2D Plot Group. R-click the new 2d Plot Group 6 and select Contour. In the Expression field, enter the variable representing the potential, [latex]V[/latex]. Note the Setting defaults and L-click Plot, resulting in Fig. E12.4.7. The full-color color bar (only a small portion of which is shown here, in black) extends from [latex]0.03[/latex] to [latex]0.97 \mathrm{~V}[/latex].

Results and Discussion

Fig. E12.4.5 shows the velocity vectors, represented as arrows. In agreement with electroosmotic theory, the velocity is constant everywhere. Also, the liquid, which has a negative zeta potential [latex]\zeta_{0}[/latex] (and hence an excess of positive ions next to the walls), flows uniformly in the direction of decreasing electric potential---from right to left.

Fig. E12.4.6 shows the streamlines as parallel straight lines pointing in the negative [latex]x[/latex]-direction, in agreement with the uniform velocity vectors.

Fig. E12.4.7 shows a series of equally spaced equipotentials, varying from [latex]1 \mathrm{~V}[/latex] at the right-hand boundary [latex]\mathrm{CD}[/latex] to zero [latex]\mathrm{V}[/latex] at the left-hand boundary [latex]\mathrm{AB}[/latex].

This example has intentionally been a simple one physically, so we could concentrate on the COMSOL steps needed in a problem involving two sets of physics---fluid flow and electric fields. Step 13, on scaling, is particularly important. The next example, which follows immediately, shows how these COMSOL principles can be extended to somewhat more involved geometry and boundary conditions.

Question 12.4: Electroosmosis in a Microchannel (COMSOL) The microchannel A...

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