Download General Exam Preparation

John A. Smythe
University of Washington
Concept is to model the collapse and spread of a water droplet on a surface. This general case
will then be modified to apply to liquid characteristics that are consistent with the materials of
interest in this work. It follows then that mass tracking and solvent evaporation must be
included. The final step is to apply the model to a non-planar surface. The first surface will be
the case of a small depression just beyond the initial edge of the droplet.
Assumption is that the beginning radius is larger than the balance between the hydrostatic force
and the Laplace pressure.

  g 1
1

Consolidation gives:

g
 1 
The diameter (or capillary length) is approximately 2.7 mm for water in air. This just provides
a lower bound for the droplet size.
The second concept is the surface energy for boundary 4. This is assumed to be sufficiently
high that the droplet (water for the first case) will wet the surface at least to some degree. The
initial intent is to examine conditions where the droplet relaxes but does not grow in r such that
it will need to interact with boundary 6. Mass is also conserved and will need to be tracked as
the interface moves.
The contact edge is expected to “freeze” in a state known as an
advancing front.
5
3
Subdomain 2 (air)
6
Droplet
subdomain
3
z
4
r
1
Subdomain 1 (substrate)
7
2
Initial condition
Note: boundary 4 can also be the edge of the domain space. Substrate is only needed if a
heating or cooling effect is considered. The initial cases will just have a boundary at the line
labeled 4.
Gen Exam Preparation
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John A. Smythe
University of Washington
5
3
Subdomain 2 (air)
z
Collapsed droplet
6
4
r
Subdomain 1 (substrate)
1
7
2
After small amount of time (< 1 Sec)
The collapse event (dz and dr) is expected to be driven by body force of gravity and the
advancing contact front. Velocity in r (ux) is non zero for cases of r not equal zero. Velocity
in z (uz) is non zero for all cases of r until equilibrium conditions are met.
The interfacial free energies for the three phases (Young circa 1830) as associated with the
equilibrium. It is important to note that this relation does not take in to account volume. This
may not be a significant limitation because the edge of the droplet in contact with solid surface
should dominate. The capillary length is only a specific case of a spherical droplet in free
space.
 lv cos    sv   sl
cos  
γlv
 sv   sl
 lv
θ
γsl
γsv
Where γlv, γsv and γsl refer to the interfacial energies of the liquid-vapor, solid-vapor and solidliquid interfaces respectively.
Gen Exam Preparation
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John A. Smythe
University of Washington
The fluid flow for water and air are both to be governed by incompressible Navier-Stokes.
Though air is compressible, the pressure in this case is constant.


 u

  u  u      u  u T  p  Fst  g
 t

u  0
Level-set is intended to be used to define the liquid-gas interface as it moves towards
equilibrium. The governing equations (vector short form) for two phase flow subdomain
settings:




u
 u  u     I   u  u T  F  g  n
t
u  0
The surface tension force acting on the interface between liquid and gas is σκδn.
Curvature κ depends on second derivatives of Φ , n is the unit normal to the interface and δ is a
Dirac delta function concentrated at the interface..


1   
 u       

t




The variables available for Sources/Sinks are the surface tension coefficient σ (N/m), gravity gz
(m/s2), volume forces Fr (N/m3) and Fz (N/m3). The level set has variables γ (gamma) that is
used as the reinitialization parameter (m/s) and ε (epsilon) that is the parameter that controls
the interface thickness (m).
Simplified test for model structure development is using the following geometry showing the
droplet after initial wetting of the surface.
Gen Exam Preparation
3 of 13
John A. Smythe
University of Washington
4
3
Subdomain 2 (air)
6
7
1
z
Droplet
2
r
5
Subdomain 1 (droplet)
Geom and sub domains with Boundary numbers
Boundary settings:
Governing equation for type Interior boundary and condition Initial fluid interface (boundary
7):
 pI  u  u n  0
T
Governing equation at top (4) and right (6) boundary (no slip) is u=0.
Boundary 2 and 5 governing equation (Wall, Wetted wall)


n  u  0, t   pI   u  u 
T
n  0
Ffr   /  u, n  n interface  cos()
In this case, β is the slip length (m) and θ is the contact angle (rad). Noting that 360 degrees (a
circle) is 2π radians. The terms are defined in the coefficients tab in the boundary settings
dialog when type is Wall and condition is Wetted wall.
Gen Exam Preparation
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John A. Smythe
University of Washington
Wall
θ
θ = contact angle
Radians
Degrees
π/6
30
π/3
60
2π/3
120
5π/6
150
π
180
Boundary 1 and 3 are type Symmetry boundary and condition Axial symmetry.
Governing equation is:
r=0
Boundary 4 and 6 are type Wall and condition No slip.
Governing equation is:
u=0
Boundary 7 is the initial fluid interface. Governing equation (vector short form) is:
 pI  u  u n  0
T
Gen Exam Preparation
5 of 13
John A. Smythe
University of Washington
The geometry for mod2_2-16 has an initial elliptical shape with intercepts at z = 2.5e-3 m and r
= 4e-3 m. Reference points for tracking variables are also shown.
6
3
2
1
5
4
Geom 1 with sub domains and reference points.
mod2_2-16
Non-conservative level set, tolerance 0.01, epsilon ~7e-4.
Gen Exam Preparation
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John A. Smythe
University of Washington
mod2_2-16
ST 0.02
Non-conservative level set, tolerance 0.01, epsilon 7.295e-4. Vf of fluid 2 is phi.
ST 0.02
mod2_2-16
Non-conservative level set, tolerance 0.01, epsilon 7.295e-4.
Gen Exam Preparation
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John A. Smythe
University of Washington
2
4
mod2_2-16
The geometry for sh2-17 has an initial spherical shape with intercepts at z = 3e-3 m and r = 3e3 m. Reference points for tracking variables are also shown.
3
6
2
1
Gen Exam Preparation
5
4
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John A. Smythe
University of Washington
sh2-17
Non--conservative level set, tolerance 0.01, epsilon 7.295e-4.
sh2-17
Non--conservative level set, tolerance 0.01, epsilon 7.295e-4.
Gen Exam Preparation
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John A. Smythe
University of Washington
sh2-17
Density tracking shows accumulation of mass at point 5 which is the bottom right corner. The
wetted wall conditions are permitting accumulation as the droplet is oscillating.
sh2-17
Non--conservative level set, tolerance 0.01, epsilon 7.295e-4.
Gen Exam Preparation
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John A. Smythe
University of Washington
sh2-17
Non--conservative level set, tolerance 0.01, epsilon 7.295e-4.
The next objective was to understand the mass loss to point 5 and to reduce the change in phi.
The tolerance was reduced from 0.01 to 0.005, epsilon was reduced to 3e-4 and a conservative
level set was used. It shows an initial shift in z and then remains somewhat stable.
Gen Exam Preparation
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John A. Smythe
University of Washington
The results still show the apparent pinning at the initial r intercept of the initial fluid interface.
Gen Exam Preparation
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John A. Smythe
University of Washington
Appendix A
Initial setup in COMSOL
Chemical Engineering Module
Momentum Transport
Multiphase Flow
Level Set Two-Phase Flow, Laminar
Transient analysis (or Transient initialization)
Dependent variables: u, v, p, phi
Use of Get Initial Value
Solve > Get Initial Value
Then use Plot Parameters, select Min/Max tab and then enter the Expression to use. In this
case, it was 5*epsilon/gamma.
After initial solution is obtained, the analysis mode needs to be set back to transient
initialization and time field needs to be updated in Solver Parameters>Time stepping area and
Times edit field.
Initialization is done using just a few time steps such as 0:1e-4:3e-3. This is sufficient to
define the initial level set. The solution is then stored and used to run a time of interest. A
relatively large mesh size is run initially to confirm expected behavior in a short time.
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