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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:
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Welding
•
Crystal growth
•
Electron beam melting of metals
Direct experimental studies are not easy to carry out 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 Incompressible Flow). 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 Gravity and The Boussinesq Approximation sections 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 Preset Studies for Selected Physics Interfaces>Stationary.
6
Click Done.
Geometry 1
Rectangle 1 (r1)
1
On the Geometry toolbar, click Primitives and choose 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
Click Build Selected.
Global Definitions
Parameters
1
On the Home toolbar, click Parameters.
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
On the Home toolbar, click Variables and choose 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
Material 1 (mat1)
1
On 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
Name
Value
Unit
Property group
Dynamic viscosity
mu
mu1
Pa·s
Basic
Thermal conductivity
k
k1
W/(m·K)
Basic
Heat capacity at constant pressure
Cp
Cp1
J/(kg·K)
Basic
Ratio of specific heats
gamma
1
1
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 check box.
4
From the Compressibility list, choose Incompressible flow.
5
Find the Reference values subsection. 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.
Wall 2
1
On 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
On the Physics toolbar, click Points and choose Pressure Point Constraint.
2
Select Point 1 only.
Heat Transfer in Fluids (ht)
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, type T_right in the T text field.
3
In the Model Builder window, click Heat Transfer in Fluids (ht).
Temperature 1
1
On 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
On 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
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
From the Specify density list, choose Custom, linearized density.
4
In the ρref text field, type rho1.
5
In the αp text field, type alphap1.
Marangoni Effect 1 (me1)
1
On the Physics toolbar, click Multiphysics 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
On the Mesh toolbar, click Free Triangular.
Size 1
1
In the Model Builder window, under Component 1 (comp1)>Mesh 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. Select the Maximum element size check box.
7
In the associated text field, type 1e-4.
Size 2
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 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. Select the Maximum element size check box.
7
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 Settings window for Stationary, click to expand the Study extensions section.
2
Locate the Study Extensions section. Select the Auxiliary sweep check box.
3
Click Add.
4
In the table, enter the following settings:
Parameter name
Parameter value list
Parameter unit
DeltaT
1e-3 5e-2 2
K
5
On the Home toolbar, click Compute.
Results
Velocity (spf)
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.
Isothermal Contours (ht)
1
In the Model Builder window, under Results 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 0.001.
Contour
1
In the Model Builder window, expand the Isothermal Contours (ht) node, then click Contour.
2
In the Settings window for Contour, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Model>Component 1>Definitions>Variables>deltaT - Excess temperature in model domain.
3
Locate the Coloring and Style section. From the Coloring list, choose Uniform.
4
On the Isothermal Contours (ht) toolbar, click Arrow Surface.
Isothermal Contours (ht)
1
In the Model Builder window, under Results click Isothermal Contours (ht).
2
Click Surface.
Surface 1
1
In the Model Builder window, under Results>Isothermal Contours (ht) click Surface 1.
2
In the Settings window for Surface, click Replace Expression in the upper-right corner of the Expression section. From the menu, choose Model>Component 1>Definitions>Variables>deltaT - Excess temperature in model domain.
3
On the Isothermal Contours (ht) toolbar, click Plot.
4
Click the Zoom Extents button on 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, under Results 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
On the Isothermal Contours (ht) toolbar, click Plot.
5
Click the Zoom Extents button on the Graphics toolbar.
6
From the Parameter value (DeltaT (K)) list, choose 2.
7
On the Isothermal Contours (ht) toolbar, click Plot.