Micromixer

This example studies a laminar static mixer with two parallel sets of split-reshape- recombine mixing elements. The following image shows the geometry of a single mixing element.

Figure 1: The micromixer splits the incoming fluid in the direction perpendicular to the interface separating the two fluid layers. After recombining them, the mixer stacks the two flows on top of each other, resulting in four fluid layers.

Each mixing element doubles the number of fluid layers, resulting in a fast mixing process. This technique is suited for laminar flow mixing and has small pressure losses. In this example, the mixing structure consists of two parallel sets of mixing elements, where each set is two elements long. You measure the mixing quality with the relative variance of the concentration profile, S, calculated as

where Kx is the yz-plane intersecting the mixing structure at coordinate x, and

is the mean concentration.

Model Definition

Domain Equations

The fluid flow is described by the Navier–Stokes equations

where ρ denotes the density (kg/m3), u the velocity (m/s), μ denotes the viscosity (N·s/m2), and p represents the pressure (Pa). The modeled fluid is water with a viscosity of 1·10−3 N·s/m2 and a density of 1000 kg/m3.

The mass flux is given by diffusion and convection, and the resulting mass balance is

where D denotes the diffusion coefficient (m2/s) and c gives the concentration (mol/m3). The modeled species has a diffusion coefficient of 5 · 10−9 m2/s.

Boundary Conditions

At the inlet the model assumes fully developed laminar flow. It sets the velocity to a parabolic profile with a mean velocity of 0.01 m/s. At the outlet, the model sets the pressure to zero. All other boundaries have the no slip condition

The inlet concentration has a discontinuous profile where the upper half has a concentration of 27 mol/m3 and the lower half is pure water. The boundary condition is defined such that

Results

The figure below shows the final concentration profile. The concentration values in some points of the solution are slightly negative; this is due to the numerical method.

Figure 2: Following three split-reshape-recombine cycles, the outflow has eight fluid layers.

Calculating the relative variance of the concentration at the inlet and outlet, the model set up according to the instructions below reaches a mixing quality of 0.20. For perfect mixing this value would be 0, and a value of 1 means no mixing at all.

Notes About the COMSOL Implementation

The default discretization for the flow equations in the Laminar Flow interface is based on P1+P1 elements. This means that linear elements are used for both velocity and pressure. In this model, the maximum cell Reynolds number for the flow is of the order 10−2, which means that it is beneficial to use a P2+P1 discretization (second order for velocity and first order for pressure).

The default discretization for the concentration in the Transport of Diluted Species interface is linear. This is more robust than a higher-order discretization but requires a refinement of the mesh in order to capture the concentration gradients. This model includes mixing layers with high concentration gradients. In this case it is more computationally efficient to use quadratic elements and a coarser mesh.

COMSOL_Multiphysics/Fluid_Dynamics/micromixer

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

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Click .

In the tree, select .

Click .

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Click

Geometry 1

The model geometry is available as a parameterized geometry sequence in a separate MPH-file. If you want to create it from scratch yourself, you can follow the instructions in the Appendix — Geometry Modeling Instructions section. Otherwise, insert the geometry sequence as follows:

In the window, expand the node.

Right-click and choose .

Browse to the model’s Application Libraries folder and double-click the file micromixer_geom_sequence.mph.

The application’s Application Library folder is shown in the section immediately before the current section. Note that the path given there is relative to the COMSOL Application Library root, which for a standard installation on Windows is C:\\Program Files\COMSOL\COMSOL56\Multiphysics\applications.

In the toolbar, click

Click the

button in the toolbar.

Global Definitions

Parameters 1

In the window, under click .

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Value	Description
c0	27[mol/m^3]	27 mol/m³	Inlet concentration
D	4.5e-9[m^2/s]	4.5E-9 m²/s	Diffusion coefficient
h_max	0.1[mm]	1E-4 m	Mesh element size parameter
U_mean	10[mm/s]	0.01 m/s	Mean velocity
a	1.4[mm]	0.0014 m	Inlet width
alpha	36*U_mean/a^4	9.3711E10 1/(m³·s)	Laminar velocity profile normalization constant

The normalization constant alpha applies to a quadratic inlet of the specified side length.

Definitions

Now define a smoothed step function that you will later use to impose a concentration gradient.

Step 1 (step1)

In the toolbar, click

and choose .

In the window for , click to expand the section.

In the text field, type 1e-4.

Proceed to define global variables for the concentration variances at the outlet and inlet as well as their ratio, S_outlet, which gives a measure of the mixing quality. For this purpose, you need two nonlocal average couplings.

Average 1 (aveop1)

In the toolbar, click

and choose .

In the window for , type aveop_inlet in the text field.

Locate the section. From the list, choose .

From the list, choose .

Average 2 (aveop2)

In the toolbar, click

and choose .

In the window for , type aveop_outlet in the text field.

Locate the section. From the list, choose .

From the list, choose .

Variables 1

In the toolbar, click

In the window for , locate the section.

In the table, enter the following settings:


Name	Expression	Unit	Description
Varc_inlet	aveop_inlet((c-c0/2)^2)	mol²/m^6	Concentration variance, inlet
Varc_outlet	aveop_outlet((c-c0/2)^2)	mol²/m^6	Concentration variance, outlet
S_outlet	Varc_outlet/Varc_inlet		Relative concentration variance, outlet

Add Material

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to open the window.

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to close the window.

Materials

Water, liquid (mat1)

By default, the first material you add applies to all domains, so you do not need to make any further material settings.

For microfluidic applications, using higher order elements is an efficient way to increase the spatial resolution.

Laminar Flow (spf)

In the window, under click .

In the window for , click to expand the section.

From the list, choose .

Click to collapse the section.

Transport of Diluted Species (tds)

In the window, under click .

In the window for , click to expand the section.

From the list, choose .

Click to collapse the section.

Laminar Flow (spf)

Initial Values 1

In the window, under click .

In the window for , locate the section.

Specify the u vector as


U_mean	x
0	y
0	z

Inlet 1

In the toolbar, click

and choose .

In the window for , locate the section.

From the list, choose .

Locate the section. Click the button.

Specify the u0 vector as


alpha(a/2+y)(a/2-y)(a/2+z)(a/2-z)	x
0	y
0	z

The inlet, of side a, is centered around the origin in the yz-plane; thus, the expression for ux is zero at the channel walls, as required for a fully developed laminar flow profile.

Outlet 1

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