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Marangoni Effect
Introduction
Marangoni convection occurs when the surface tension of an interface (generally liquid-air) depends on the concentration of a species or on the temperature distribution. In the case of temperature dependence, the Marangoni effect is also called thermo-capillary convection. It is of primary importance in the fields of:
•
Welding
•
Crystal growth
•
Electron beam melting of metals
Direct experimental studies are not easy to do in these systems because the materials are often metals and temperatures are very high. One possibility is to replace the real system with an experimental setup using a transparent liquid at ambient temperatures.
Model Definition
This tutorial describes the 2D stationary behavior of a vessel filled with silicone oil, for which the thermo-physical properties are known. The aim of the study is to compute the temperature field that induces a flow through the Marangoni effect. The model shows this effect using the simple geometry in the figure below.
Governing Equations
A stationary momentum balance equation describes the velocity field and the pressure distribution (Navier–Stokes equations, see the section Incompressible Flow in the COMSOL Multiphysics Reference Manual). To include the heating of the fluid, the fluid flow is coupled to an energy balance.
You can use the Boussinesq approximation to include the effect of temperature on the velocity field. In this approximation, variations in temperature produce a buoyancy force (or Archimedes’ force) that lifts the fluid as described in the sections Gravity and The Boussinesq Approximation in the CFD Module User’s Guide.
The following equation describes the forces that the Marangoni effect induces on the interface (liquid/air):
(1)
Here γ is the temperature derivative of the surface tension (N/(m·K)). Equation 1 states that the shear stress on a surface is proportional to the temperature gradient (Ref. 1).
Notes About the COMSOL Implementation
To solve the momentum and energy balance equations, use the predefined Nonisothermal Flow multiphysics coupling. It automatically couples a Laminar Flow interface for the fluid flow to a Heat Transfer in Fluids interface for the heat transfer by convection and conduction in each direction:
•
The Boussinesq approximation means that an expression including temperature acts as a force in the y direction in the momentum balance.
•
The convective heat transfer depends on the velocities from the momentum balance.
This means that you must solve the coupled system directly using the nonlinear solver.
To impose the condition that the shear stress is proportional to the temperature gradient on the surface, use the Marangoni Effect multiphysics feature in the Multiphysics node.
Results
The Marangoni effect becomes more pronounced as the temperature difference increases:
Figure 1: Marangoni convection with a temperature difference of 0.001 K.
For the very low temperature difference of 0.001 K, the temperature field is almost decoupled from the velocity field. Therefore, the temperature decreases almost linearly from left to right.
Figure 2: Marangoni convection with a temperature difference of 0.05 K.
For the temperature difference of 0.05 K notice how the Marangoni convection influences the flow of fluid and the distribution of temperature. The temperature is no longer decreasing linearly and you can clearly see the advection of the isotherms caused by the flow.
Figure 3: Marangoni convection with a temperature difference of 2 K.
At higher temperature differences (2 K in Figure 3 above), the physical coupling between the temperature and the velocity field is clearly visible. The heat conduction is small compared to the convection, and at the surface the fluid accelerates where the temperature gradient is high.
Reference
1. V.G. Levich, Physicochemical Hydrodynamics, Prentice-Hall, N.J., 1962.
Application Library path: Heat_Transfer_Module/Tutorials,_Forced_and_Natural_Convection/marangoni_effect
Modeling Instructions
From the File menu, choose New.
New
In the New window, click  Model Wizard.
Model Wizard
1
In the Model Wizard window, click  2D.
2
In the Select Physics tree, select Fluid Flow > Nonisothermal Flow > Laminar Flow.
3
Click Add.
4
Click  Study.
5
In the Select Study tree, select General Studies > Stationary.
6
Click  Done.
Geometry 1
Rectangle 1 (r1)
1
In the Geometry toolbar, click  Rectangle.
2
In the Settings window for Rectangle, locate the Size and Shape section.
3
In the Width text field, type 10[mm].
4
In the Height text field, type 5[mm].
5
In the Geometry toolbar, click  Build All.
6
Click the  Zoom Extents button in the Graphics toolbar.
Global Definitions
Parameters 1
1
In the Model Builder window, under Global Definitions click Parameters 1.
2
In the Settings window for Parameters, locate the Parameters section.
3
Click  Load from File.
4
Browse to the model’s Application Libraries folder and double-click the file marangoni_effect_parameters.txt.
Definitions
Variables 1
1
In the Definitions toolbar, click  Local Variables.
2
In the Settings window for Variables, locate the Variables section.
3
In the table, enter the following settings:
Name
Expression
Unit
Description
deltaT
T-T_right
K
Excess temperature in model domain
This variable is useful when visualizing the model results.
Materials
Silicone Oil
1
In the Materials toolbar, click  Blank Material.
2
In the Settings window for Material, type Silicone Oil in the Label text field.
3
Locate the Material Contents section. In the table, enter the following settings:
Property
Variable
Value
Unit
Property group
Dynamic viscosity
mu
mu1
Pa·s
Basic
Heat capacity at constant pressure
Cp
Cp1
J/(kg·K)
Basic
Thermal conductivity
k_iso ; kii = k_iso, kij = 0
k1
W/(m·K)
Basic
Laminar Flow (spf)
1
In the Model Builder window, under Component 1 (comp1) click Laminar Flow (spf).
2
In the Settings window for Laminar Flow, locate the Physical Model section.
3
Select the Include gravity checkbox.
4
From the Compressibility list, choose Incompressible flow.
Wall 2
1
In the Physics toolbar, click  Boundaries and choose Wall.
2
Select Boundary 3 only.
3
In the Settings window for Wall, locate the Boundary Condition section.
4
From the Wall condition list, choose Slip.
Pressure Point Constraint 1
1
In the Physics toolbar, click  Points and choose Pressure Point Constraint.
2
Select Point 1 only.
Heat Transfer in Fluids (ht)
1
In the Model Builder window, under Component 1 (comp1) click Heat Transfer in Fluids (ht).
2
In the Settings window for Heat Transfer in Fluids, locate the Physical Model section.
3
In the Tref text field, type T_ref.
Here, T_ref is the reference temperature at which the material properties are evaluated. It is defined in Parameters under Global Definitions.
Initial Values 1
1
In the Model Builder window, under Component 1 (comp1) > Heat Transfer in Fluids (ht) click Initial Values 1.
2
In the Settings window for Initial Values, locate the Initial Values section.
3
In the T text field, type T_right.
Temperature 1
1
In the Physics toolbar, click  Boundaries and choose Temperature.
2
Select Boundary 4 only.
3
In the Settings window for Temperature, locate the Temperature section.
4
In the T0 text field, type T_right.
Temperature 2
1
In the Physics toolbar, click  Boundaries and choose Temperature.
2
Select Boundary 1 only.
3
In the Settings window for Temperature, locate the Temperature section.
4
In the T0 text field, type T_right+DeltaT.
Multiphysics
Nonisothermal Flow 1 (nitf1)
1
In the Model Builder window, under Component 1 (comp1) > Multiphysics click Nonisothermal Flow 1 (nitf1).
2
In the Settings window for Nonisothermal Flow, locate the Material Properties section.
3
Select the Boussinesq approximation checkbox.
4
From the Specify density list, choose Custom, linearized density.
5
In the ρref text field, type rho1.
6
In the αp,0 text field, type alphap1.
Marangoni Effect 1 (mar1)
1
In the Physics toolbar, click  Multiphysics Couplings and choose Boundary > Marangoni Effect.
2
Select Boundary 3 only.
3
In the Settings window for Marangoni Effect, locate the Surface Tension section.
4
In the σ text field, type gamma*T.
Mesh 1
Free Triangular 1
In the Mesh toolbar, click  Free Triangular.
Size 1
1
Right-click Free Triangular 1 and choose Size.
2
In the Settings window for Size, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Boundary.
4
Select Boundary 3 only.
5
Locate the Element Size section. Click the Custom button.
6
Locate the Element Size Parameters section.
7
Select the Maximum element size checkbox. In the associated text field, type 1e-4.
Size 2
1
In the Model Builder window, right-click Free Triangular 1 and choose Size.
2
In the Settings window for Size, locate the Geometric Entity Selection section.
3
From the Geometric entity level list, choose Point.
4
Select Points 2 and 4 only.
5
Locate the Element Size section. Click the Custom button.
6
Locate the Element Size Parameters section.
7
Select the Maximum element size checkbox. In the associated text field, type 2e-5.
Size
1
In the Model Builder window, under Component 1 (comp1) > Mesh 1 click Size.
2
In the Settings window for Size, locate the Element Size section.
3
From the Predefined list, choose Extra fine.
4
Click the Custom button.
5
Locate the Element Size Parameters section. In the Maximum element growth rate text field, type 1.1.
6
Click  Build All.
Study 1
Step 1: Stationary
1
In the Model Builder window, under Study 1 click Step 1: Stationary.
2
In the Settings window for Stationary, click to expand the Study Extensions section.
3
Select the Auxiliary sweep checkbox.
4
Click  Add.
5
In the table, enter the following settings:
Parameter name
Parameter value list
Parameter unit
DeltaT (Excess temperature on the left boundary)
1e-3 5e-2 2
K
6
In the Study toolbar, click  Compute.
Results
To show the temperature field as a surface plot along with overlaid temperature contours and the velocity field using arrows, follow the steps given below.
Result Templates
1
In the Results toolbar, click  Result Templates to open the Result Templates window.
2
Go to the Result Templates window.
3
In the tree, select Study 1/Solution 1 (sol1) > Heat Transfer in Fluids > Isothermal Contours (ht).
4
Click the Add Result Template button in the window toolbar.
5
In the Results toolbar, click  Result Templates to close the Result Templates window.
Results
Isothermal Contours (ht)
1
In the Settings window for 2D Plot Group, locate the Data section.
2
From the Parameter value (DeltaT (K)) list, choose 0.001.
Contour 1
1
In the Model Builder window, expand the Isothermal Contours (ht) node, then click Contour 1.
2
In the Settings window for Contour, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 (comp1) > Definitions > Variables > deltaT - Excess temperature in model domain - K.
3
Locate the Coloring and Style section. From the Coloring list, choose Uniform.
4
From the Color list, choose Black.
5
Clear the Color legend checkbox.
Isothermal Contours (ht)
In the Model Builder window, click Isothermal Contours (ht).
Arrow Surface 1
1
In the Isothermal Contours (ht) toolbar, click  Arrow Surface.
2
In the Settings window for Arrow Surface, locate the Coloring and Style section.
3
From the Color list, choose Black.
Isothermal Contours (ht)
In the Model Builder window, click Isothermal Contours (ht).
Surface 1
1
In the Isothermal Contours (ht) toolbar, click  Surface.
2
In the Settings window for Surface, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Component 1 (comp1) > Definitions > Variables > deltaT - Excess temperature in model domain - K.
3
Locate the Coloring and Style section. From the Color table list, choose HeatCameraLight.
4
In the Isothermal Contours (ht) toolbar, click  Plot.
5
Click the  Zoom Extents button in the Graphics toolbar.
Isothermal Contours (ht)
The Marangoni effect becomes more pronounced as the temperature difference increases. Visualize this by changing the Parameter value selection.
1
In the Model Builder window, click Isothermal Contours (ht).
2
In the Settings window for 2D Plot Group, locate the Data section.
3
From the Parameter value (DeltaT (K)) list, choose 0.05.
4
In the Isothermal Contours (ht) toolbar, click  Plot.
5
Click the  Zoom Extents button in the Graphics toolbar.
6
From the Parameter value (DeltaT (K)) list, choose 2.
7
In the Isothermal Contours (ht) toolbar, click  Plot.
Follow these steps to visualize the importance of the Marangoni effect on the convection cell.
Convection Cell
1
In the Results toolbar, click  2D Plot Group.
2
In the Settings window for 2D Plot Group, type Convection Cell in the Label text field.
3
Click to collapse the Data section.
Surface 1
1
In the Convection Cell toolbar, click  Surface.
2
In the Settings window for Surface, locate the Expression section.
3
In the Expression text field, type T.
4
Locate the Coloring and Style section. From the Color table list, choose HeatCameraLight.
Convection Cell
In the Convection Cell toolbar, click  Streamline.
Streamline 1
1
In the Settings window for Streamline, locate the Expression section.
2
Select the Description checkbox. In the associated text field, type Velocity field (m/s).
3
Locate the Streamline Positioning section. From the Positioning list, choose Magnitude controlled.
4
Locate the Coloring and Style section. Find the Point style subsection. From the Type list, choose Arrow.
Color Expression 1
Right-click Streamline 1 and choose Color Expression.
Convection Cell
1
In the Settings window for 2D Plot Group, click to expand the Data section.
2
From the Parameter value (DeltaT (K)) list, choose 0.001.
3
In the Convection Cell toolbar, click  Plot.
4
Click  Plot Next.
5
Click  Plot Next.