COMSOL 6.4 - Droplet Breakup in a T-Junction

Emulsions consist of small liquid droplets immersed in another liquid, typically oil in water or water in oil. Emulsions find wide application in the production of food, cosmetics, and pharmaceutical products. The properties and quality of an emulsion typically depend on the size and the distribution off the droplets. This example studies in detail how to create uniform droplets in a microchannel T-junction.

Setting up the model you can make use of the multiphysics coupling feature Laminar Two-Phase Flow, Level Set interface. The model uses the multiphysics coupling wetted wall boundary condition at the solid walls, with a contact angle of 135°. From the results, you can determine the size of the created droplets and the rate with which they are produced.

Model Definition

Figure 1 shows the geometry of the T-shaped microchannel with a rectangular cross section. For the separated fluid elements to correspond to droplets, the geometry is modeled in 3D. Due to symmetry, it is sufficient to model only half of the junction geometry. The modeling domain is shown in Figure 1. The fluid to be dispersed into small droplets, Fluid 2, enters through the vertical channel. The other fluid, Fluid 1, flows from the right to left through the horizontal channel.

Figure 1: The modeling domain of the T-junction.

The problem described is straightforward to set up with the Laminar Two-Phase Flow, Level Set multiphysics coupling feature. The feature couples the Laminar Flow and Level Set physics interfaces.

The Laminar Flow interface sets up a momentum transport equation and a continuity equation. The Level Set interface sets up a level set equation for the level-set variable. The fluid interface is defined by the 0.5 contour of the level-set function.

The Laminar Flow and Level Set interfaces use the following equations:

In the equations above, ρ denotes density (kg/m3), u velocity (m/s), t time (s), μ dynamic viscosity (Pa·s), p pressure (Pa), and Fst the surface tension force (N/m3). Furthermore, ϕ is the level-set function, and γ and ε are numerical stabilization parameters.

The effective density and viscosity are defined as volume averaged,

where ρ1, ρ2, μ1, and μ2 are the densities and viscosities of Fluid 1 and Fluid 2, and the level-set function ϕ acts as the volume fraction of the second phase. Other definitions of the viscosity and pressure are also available. This model uses the Multiphase material to compute the effective material parameters.

Physical Parameters

The two liquids have the following physical properties:


Quantity	Value, Fluid 1	Value, Fluid 2
Density (kg/m3)	1000	1000
Dynamic viscosity (Pa·s)	0.00195	0.00671

The surface tension coefficient is 5 × 10−3 N/m.

Boundary Conditions

At both inlets, Fully developed flow conditions with prescribed volume flow rates are used. At the outflow boundary, the Pressure condition is set. The Wetted wall multiphysics boundary condition applies to all solid boundaries with the contact angle specified as 135° and a slip length equal to 6.5 × 10−5 m. The contact angle is the angle between the fluid interface and the solid wall at points where the fluid interface attaches to the wall. The slip length is the distance to the position outside the wall where the extrapolated tangential velocity component is zero (see Figure 2).

Figure 2: The contact angle, θ, and the slip length, β.

Results and Discussion

Figure 3 shows the fluid interface (the level-set function ϕ = 0.5) and velocity streamlines at various times. The first droplet is formed after approximately 0.03 s.

Figure 3: Velocity streamlines and the phase boundary at t = 0.02 s, 0.04 s, 0.06 s, and 0.08 s.

You can calculate the effective diameter, deff — that is, the diameter of a spherical droplet with the same volume as the formed droplet — using the following expression:

(1)

Here, Ω represents the leftmost part of the horizontal channel, where x < −0.2 mm. In this case, the results show that deff is between 0.11 mm and 0.12 mm. The results are in fair agreement with those presented in Ref. 1.

Reference

1. S. van der Graaf, T. Nisisako, C.G.P.H. Schroën, R.G.M. van der Sman, and R.M. Boom, “Lattice Boltzmann Simulations of Droplet Formation in a T-Shaped Microchannel,” Langmuir, vol. 22, pp. 4144–4152, 2006.

Microfluidics_Module/Two-Phase_Flow/droplet_breakup

Modeling Instructions

From the menu, choose .

New

In the window, click

Model Wizard

In the window, click

In the tree, select > > > .

Click .

Click

In the tree, select > .

Click

Geometry 1

In the window, under click .

In the window for , locate the section.

From the list, choose .

Work Plane 1 (wp1)

In the toolbar, click

In the window for , locate the section.

From the list, choose .

Click

Work Plane 1 (wp1) > Rectangle 1 (r1)

In the toolbar, click

In the window for , locate the section.

In the text field, type 0.1.

In the text field, type 0.4.

Locate the section. In the text field, type 0.1.

Work Plane 1 (wp1) > Rectangle 2 (r2)

In the toolbar, click

In the window for , locate the section.

In the text field, type 0.1.

Locate the section. In the text field, type -0.7.

Work Plane 1 (wp1) > Plane Geometry

In the toolbar, click

Work Plane 1 (wp1) > Polygon 1 (pol1)

In the toolbar, click

In the window for , locate the section.

From the list, choose .

Locate the section. In the table, enter the following settings:


xw (mm)	yw (mm)
0	0.2
0.1	0.2

Click

Work Plane 1 (wp1) > Polygon 2 (pol2)

In the toolbar, click

In the window for , locate the section.

In the table, enter the following settings:


xw (mm)	yw (mm)
0.1	0
0.1	0.1

Click

Extrude 1 (ext1)

In the window, under > right-click and choose .

In the window for , locate the section.

In the table, enter the following settings:


Distances (mm)
0.05

Click

Form Union (fin)

In the window, click .

In the window for , click

The geometry should look like in Figure 1.

Multiphysics

Two-Phase Flow, Level Set 1 (tpf1)

In the window, under > click .

In the window for , locate the section.

Click

Materials

Fluid 1

In the window, under > > click .

In the window for , type Fluid 1 in the text field.

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Density	rho	1e3[kg/m^3]	kg/m³	Basic
Dynamic viscosity	mu	1.95e-3[Pa*s]	Pa·s	Basic

Fluid 2

In the window, under > > click .

In the window for , type Fluid 2 in the text field.

Locate the section. In the table, enter the following settings:


Property	Variable	Value	Unit	Property group
Density	rho	1e3[kg/m^3]	kg/m³	Basic
Dynamic viscosity	mu	6.71e-3[Pa*s]	Pa·s	Basic

Definitions

Step 1 (step1)

In the toolbar, click

and choose .

In the window for , locate the section.

In the text field, type 1e-3.

Click to expand the section. In the text field, type 2e-3.

Integration 1 (intop1)

Add a nonlocal integration coupling that you will use to calculate the effective droplet diameter according to Equation 1 in the section.

In the toolbar, click

and choose .

In the window for , locate the section.

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
V1	0.4e-6/3600*step1(t[1/s])[m^3/s]	m³/s	Volume flow, inlet 1
V2	0.2e-6/3600*step1(t[1/s])[m^3/s]	m³/s	Volume flow, inlet 2
d_eff	2(intop1((phils>0.5)(x<-0.2[mm]))3/(4pi))^(1/3)	m	Effective droplet diameter

Multiphysics

Two-Phase Flow, Level Set 1 (tpf1)

In the window, under > click .

In the window for , locate the section.

Select the checkbox.

From the list, choose . In the σ text field, type 5e-3[N/m].

Level Set (ls)

Level Set Model 1

In the window, under > click .

In the window for , locate the section.

In the γ text field, type 0.1[m/s].

In the εls text field, type 5e-6.

Multiphysics

Wetted Wall 1 (ww1)

In the window, under > click .

Select Boundaries 2–4, 6, 7, 9, 10, and 16–20 only.

In the window for , locate the section.

From the list, choose . In the β text field, type 5e-6[m].

Locate the section. In the θw text field, type 3*pi/4[rad].

Level Set (ls)

Initial Values, Fluid 2

In the window, under > click .

Select Domain 3 only.

For Domains 1, 2 and 4, the default initial value settings apply.

Laminar Flow (spf)

In the window, under click .

Inlet 1

In the toolbar, click

and choose .

Select Boundary 22 only.

In the window for , locate the section.

From the list, choose .

Locate the section. Click the button.

In the V0 text field, type V1.

Inlet 2

In the toolbar, click

and choose .

Select Boundary 12 only.

In the window for , locate the section.

From the list, choose .

Locate the section. Click the button.

In the V0 text field, type V2.

Outlet 1

In the toolbar, click

and choose .

Select Boundary 1 only.

Symmetry 1

In the toolbar, click

and choose .

Select Boundaries 5, 13, 14, and 21 only.

Level Set (ls)

In the window, under click .

Inlet 1

In the toolbar, click

and choose .

Select Boundary 12 only.

In the window for , locate the section.

From the list, choose φ.

Inlet 2

In the toolbar, click

and choose .

Select Boundary 22 only.

Outlet 1

In the toolbar, click

and choose .

Select Boundary 1 only.

Symmetry 1

In the toolbar, click

and choose .

Select Boundaries 5, 13, 14, and 21 only.

Mesh 1

Mapped 1

In the toolbar, click

and choose .

Select Boundaries 2, 7, 10, and 16 only.

Distribution 1

Right-click and choose .

Select Edge 3 only.

In the window for , locate the section.

In the text field, type 160.

Distribution 2

In the window, right-click and choose .

Select Edges 1 and 9 only.

In the window for , locate the section.

In the text field, type 20.

Distribution 3

Right-click and choose .

Select Edges 12 and 28 only.

In the window for , locate the section.

From the list, choose .

In the text field, type 25.

In the text field, type 4.

Distribution 4

Right-click and choose .

Select Edges 24 and 27 only.

In the window for , locate the section.

From the list, choose .

In the text field, type 20.

In the text field, type 3.

Select the checkbox.

Mapped 1

Right-click and choose .

Swept 1

In the toolbar, click

In the window for , click to expand the section.

Select Boundaries 2, 7, and 10 only.

Distribution 1

Right-click and choose .

In the window for , locate the section.

In the text field, type 10.

Click

Study 1

Step 2: Time Dependent

In the window, under click .

In the window for , locate the section.

In the text field, type range(0,2.5e-3,0.08).

Click to expand the section. Select the checkbox.

This choice means that the window will show a surface plot of the volume fraction of while solving, and this plot will be updated at each output time step.

Solution 1 (sol1)

Manually tune the solver sequence for optimal performance and accuracy.

In the toolbar, click

In the window, expand the node.

In the window, expand the > > > node, then click .

In the window for , locate the section.

From the list, choose .

In the text field, type 0.1.

In the window, under > > click .

In the window for , click to expand the section.

Find the subsection. In the text field, type 1.

In the toolbar, click

Results

Isosurface 1

Right-click and choose .

In the window for , locate the section.

In the text field, type ls.Vf1.

Locate the section. From the list, choose .

In the text field, type 0.5.

Locate the section. From the list, choose .

From the list, choose .

In the toolbar, click

Volume Fraction

In the window, under click .

In the window for , locate the section.

From the list, choose .

In the text field, type Volume Fraction.

Slice 1

Right-click and choose .

In the window for , locate the section.

In the text field, type ls.Vf1.

Surface

In the window, right-click and choose .

Volume Fraction of Fluid 1 (ls)

Now, the first default plot group shows a slice plot of the velocity combined with a contour plot of the volume fraction of fluid 1, and the second plot group shows the volume fraction of fluid 1 as a slice plot. Follow these steps to reproduce the series of velocity field plots shown in Figure 3.

Slice 1

In the window, expand the node.

Right-click and choose .

Isosurface 1

In the window, click .

In the window for , locate the section.

From the list, choose .

Isosurface 2

Right-click > > and choose .

Deformation 1

In the window, right-click and choose .

In the window for , locate the section.

In the text field, type -2*y.

Locate the section.

Select the checkbox. In the associated text field, type 1.

Streamline 1

In the window, right-click and choose .

In the window for , locate the section.

In the text field, type 8.

Select Boundaries 12 and 22 only.

Locate the section. Find the subsection. From the list, choose .

Select the checkbox. In the associated text field, type 2e-3.

Color Expression 1

Right-click and choose .

In the window for , locate the section.

From the list, choose .

Clear the checkbox.

Streamline 2

In the window, under > right-click and choose .

Deformation 1

In the window, right-click and choose .

In the window for , locate the section.

In the text field, type -2*y.

Locate the section.

Select the checkbox. In the associated text field, type 1.

Surface 1

In the window, right-click and choose .

In the window for , locate the section.

From the list, choose .

Selection 1

Right-click and choose .

Select Boundaries 3 and 17–20 only.

Surface 2

In the window, under > right-click and choose .

Deformation 1

In the window, expand the node.

Right-click and choose .

In the window for , locate the section.

In the text field, type 0.05.

Locate the section.

Select the checkbox. In the associated text field, type 1.

Selection 1

In the window, click .

In the window for , locate the section.

Click to select the

toggle button.

Select Boundaries 3–6, 9, 13, 14, and 17–21 only.

Volume Fraction of Fluid 1 (ls)

In the window, under click .

In the window for , locate the section.

From the list, choose .

Locate the section. Clear the checkbox.

In the toolbar, click

Compare the resulting plot with the upper-left plot in Figure 3.To reproduce the remaining three plots, plot the solution for the time values 0.04, 0.06, and 0.08 s.

Global Evaluation 1

Next, evaluate the effective droplet diameter computed according to Equation 1.

In the toolbar, click

In the window for , click in the upper-right corner of the section. From the menu, choose > > > .

Click

Table 1

Go to the window.

The result, roughly 0.12 mm, is displayed in the table in the window.

Results

Finally, generate a movie of the moving fluid interface and the velocity streamlines.

Volume Fraction of Fluid 1 (ls)

In the window, under click .

Animation 1

In the toolbar, click

and choose .

COMSOL Multiphysics generates the movie. To play the movie, click the button in the toolbar. If you want to export a movie in GIF, Flash, or AVI format, select from the list.