ME410/510CP Spring 2011
Wall-Edge-Vertex Bound Drops Report
Duc Nguyen
INTRODUCTION
The subject of this study is the stability of Wall-Edge-Vertex Bound. The drop, for easy description, is
similar to those drops on the leaves. The drop is also fundamental to many natural and industrial process
mentioned by Chen [1]. In this study, Surface Evolver – Fluid Interface tool (SE-FIT), a program
developed at PSU, is the main tool to study the relationship between the stability the drop and the
variables perturbing the drop such as the drop volume, contact angle, fluid properties and gravity. The
main factors perturbing the drop (making them unstable) are assumed to be combined as a Bond number.
SE-FIT critical value algorithm developed at PSU combine each set the variables as a Bond number. The
algorithm defines the break of the drop to find out the critical bond number above which the drop
becomes unstable. Using the algorithm with fixed corner angle at the tip and, fixed contact angle, the
outcome of interest is to derive the critical Bond number as the function of gravity orientation. The
outcome of this study is also expected as a source to compare with the Bond critical number computed
from the experimental results of drop stability by PSU.
Assumptions and Suggested Model for Bond critical value for Wall-Vertex-Bound Drops
Several assumptions are made for the drop. The contact angle hysteresis plays no role in supporting the
drop as well as in the calculation process. The contact angle is also satisfied uniform conditions. The
gravity orientation is defined by one unit vector and one angle or in other words is placed on a plane. The
thickness of the bound plate is also ignored. The drop could not cover the back side of the bound plate.
Two mathematical assumptions for the drop breakup condition are stated in Chen [1]: Gibb’s inequalities
for pinning edge drops and Finn-Concus capillary surface equilibrium characteristics. Those mathematical
theories are beyond the scope of this report.
Many sub-subjects can be studied for the drop including the correlation between Bond critical number
and critical contact angle, the critical contact angle and gravity orientation, geometry parameters of the
corresponding volume. Many convincing results can be found in Chen’s article. Recall that the solely
subject of this report is only to study to derive the critical Bond number as the function of gravity
orientation. The outcome is suggested and expected as the analogy to Moody chart and Colebrook
equation which means Bond critical number is a function of the other two dimensionless numbers. The
reason of this suggestion is that the variables in this study contain many dimensionless quantity of
geometry such as gravity orientation, corner angle and contact angle.
Surface Energy and Surface Evolver Code development
The main strategy to develop a computation of Bond critical number is to using energy method, i.e. to
setup geometry constraints and surface energy constraints as the adequate inputs for Surface Evolver to
compute Bond critical number. Surface evolver will use those inputs to implement numerical
computations: calculates surface energy, moves the vertices by gradient descent method and finally
perform Hessian to find at least local minimum of the surface. The local minimum state of the drop will
be checked by breakup conditions and critical value searching function in SE-FIT to find out the desired
Bond critical number. In this scope of the study, only the derivation of surface energy and SE-FIT code
will be developed.
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Duc Nguyen
The calculation of surface energy is essentially based on differential analysis. In this study, the abridged
description for surface energy calculation is shown; the complete theory can be found in Chen [2],
technical notes for SE-FIT program.
The coordinate setting for the differential analysis of this problem is described in the figure below. The
simulation is run for the case of half corner angle of 55 degree, contact angle of 0 and 60 degree, 𝜓 = 0
and the gravity orientation 𝜙 varies from 0 to 175 degree with 5 degree increment.
Figure 1 Coordinate setting for differential analysis
The volume domain is Ω and the boundary surface is Σ .For an infinitesimal volume dV:
𝑑𝐸𝐺 = 𝜌𝐺𝑥 𝑥𝑑𝑥𝑑𝑦𝑑𝑧 − 𝜌𝐺𝑦 𝑦𝑑𝑥𝑑𝑦𝑑𝑧 − 𝜌𝐺𝑧 𝑧𝑑𝑥𝑑𝑦𝑑𝑧
∴ 𝐸𝐺 = 𝜌 ∰(−𝐺𝑥 𝑥 − 𝐺𝑦 𝑦 − 𝐺𝑍 𝑧) 𝑑𝑥𝑑𝑦𝑑𝑧
Using divergence theorem:
∇𝐹̅ = −𝐺𝑥 𝑥 − 𝐺𝑦 𝑦 − 𝐺𝑧 𝑧
Let 𝐹𝑧 =
𝜕𝐹
𝜕𝑧
1
∴ 𝐹𝑧 = −𝐺𝑥 𝑥𝑧 − 𝐺𝑦 𝑦𝑧 − 2 𝐺𝑧 𝑧 2
1
∴ 𝐸 = 𝐸𝐼 + 𝐸𝐺 = 𝜎𝐴 + 𝜌 ∯ 𝐹̅ 𝑛̅𝑑𝐴 = 𝜎𝐴 + 𝜌 ∯ 𝐹𝑧 𝑑𝑥𝑑𝑦 = 𝜎𝐴 − 𝜌 ∯(𝐺𝑥 𝑥𝑧 + 𝐺𝑦 𝑦𝑧 + 𝐺𝑧 𝑧 2 )𝑑𝑥𝑑𝑦
2
The governing equation for this problem is the energy equation:
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Duc Nguyen
1
𝐸 = 𝜎𝐴 − 𝜌 ∯(𝐺𝑥 𝑥𝑧 + 𝐺𝑦 𝑦𝑧 + 𝐺𝑧 𝑧 2 )𝑑𝑥𝑑𝑦
2
Using the coordinate shown in Figure 1, we have the following system of equation:
𝐺𝑥 = −𝐺𝜙 cosϕ = −Gψ cosψ (1)
𝐺𝑦 = −Gψ sinψ (2)
𝐺𝑧 = −𝐺𝜙 sinϕ (3)
𝐺𝜙 cosϕ = Gψ cosψ (4)
(Gψ sinψ)2 + (𝐺𝜙 sinϕ)2 + (Gψ cosψ)2 = G (5)
𝜋
Let’s change variables: 𝜙 = 𝜙 ′ − 2 ∴ sin 𝜙 = − 𝑐𝑜𝑠𝜙 ′ 𝑎𝑛𝑑 𝑐𝑜𝑠𝜙 = 𝑠𝑖𝑛𝜙 ′ solving the above system of
equation:
𝐺𝑥 = −𝐺
𝐺𝑥 = −𝐺
𝐺𝑥 = 𝐺
𝑐𝑜𝑠𝜓𝑠𝑖𝑛𝜙 ′
= 𝐺𝐹1 (𝜙, 𝜓)
(cos 2 𝜙 + cos 2 𝜓 sin2 𝜙)1/2
𝑠𝑖𝑛𝜓𝑠𝑖𝑛𝜙 ′
1
(cos 2 𝜙 + cos 2 𝜓 sin2 𝜙)2
= 𝐺𝐹2 (𝜙, 𝜓)
𝑐𝑜𝑠𝜓𝑐𝑜𝑠𝜙 ′
= 𝐺𝐹3 (𝜙, 𝜓)
(cos2 𝜙 + cos2 𝜓 sin2 𝜙)1/2
𝜋
2
The phi angle which is the actual tilt angle will be defined by evolver as 𝜙 angle. In figure 1, if 𝜙 = − ,
it will corresponds to the pendent drop case. The corresponding 𝜙 ′ = 0 matches with the settings of
pendent drop, therefore the notation of 𝜙 and 𝜙 ′ can be swapped. The three following parameter will be
used in Surface Evolver code:
𝐺𝑋 = 𝐹1 = −
𝐺𝑌 = 𝐹2 = −
𝐺𝑍 = 𝐹3 =
(cos2 𝜙
𝑐𝑜𝑠𝜓𝑠𝑖𝑛𝜙
+ cos 2 𝜓 sin2 𝜙)1/2
𝑠𝑖𝑛𝜓𝑠𝑖𝑛𝜙
1
(cos 2 𝜙 + cos 2 𝜓 sin2 𝜙)2
𝑐𝑜𝑠𝜓𝑐𝑜𝑠𝜙
(cos 2 𝜙 + cos 2 𝜓 sin2 𝜙)1/2
Substituting into the governing equation
1
1
𝐸 = 𝜎𝐴 − 𝜌 ∯(𝐺𝑥 𝑥𝑧 + 𝐺𝑦 𝑦𝑧 + 𝐺𝑧 𝑧 2 )𝑑𝑥𝑑𝑦 = 𝜎𝐴 − 𝜌𝐺 ∯(𝐹1 𝑥𝑧 + 𝐹2 𝑦𝑧 + 𝐺𝑧 𝑧 2 )𝑑𝑥𝑑𝑦
2
2
Nondimensionalize this equation using length scale 𝐿~𝑉 1/3 suggested by Chen [1]
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Duc Nguyen
𝐸
𝐴 𝜌𝐺𝐿2
1
=
−
∯ (𝐹1 𝑥𝑧 + 𝐹2 𝑦𝑧 + 𝐺𝑧 𝑧 2 ) 𝑑𝑥 ∗ 𝑑𝑦 ∗
2
2
4
𝜎𝐿
𝐿
𝜎𝐿
2
or
1
𝐸 ∗ = 𝐴∗ − 𝐵𝑜 ∗ ∯ (𝐹1 𝑥𝑧 + 𝐹2 𝑦𝑧 + 𝐺𝑧 𝑧 2 ) 𝑑𝑥 ∗ 𝑑𝑦 ∗
2
Dropping the star and neglect term 𝐴∗ as Surface Evolver using part of energy, the following equation is
adequate to be the energy equation for Surface Evolver:
1
𝐸 = −𝐵𝑜 ∯ (𝐹1 𝑥𝑧 + 𝐹2 𝑦𝑧 + 𝐺𝑧 𝑧 2 ) 𝑑𝑥𝑑𝑦
2
The initial geometry is described in the figure 2 below. The code with detail description is also shown.
Figure 2. Initial geometry for the wall-edge-vertex drop
SE-FIT code
//walle.fe
//Author:Duc Nguyen
// Date:6/3/11
//Constraint set up for Wall-Edge-Vertex drop
//parameters
PARAMETER CornerHalfAngle = 55
PARAMETER ContactAngle = 0
PARAMETER GAnglePsi = 0 //Gravity orientation in a plane
PARAMETER GAnglePhi = 90 //Valid for a single run. Re-initialized for the batch run
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//macros
#define WALLT (-cos(ContactAngle*pi/180)) // virtual tension of facet on plane
#define TANA (tan(CornerHalfAngle*pi/180)) // geometry macro
#define SINA (sin(CornerHalfAngle*pi/180)) // geometry macro
#define COSA (cos(CornerHalfAngle*pi/180)) // geometry macro
#define GSTUFF (sqrt(cos(GAnglePsi*pi/180)^2+sin(GAnglePsi*pi/180)^2*sin(GAnglePhi*pi/180)^2))
#define GX (-cos(GAnglePsi*pi/180)*sin(GAnglePhi*pi/180)/GSTUFF) //Energy equation’s terms
#define GY (-sin(GAnglePsi*pi/180)*sin(GAnglePhi*pi/180)/GSTUFF) //Energy equation’s terms
#define GZ (cos(GAnglePsi*pi/180)*cos(GAnglePhi*pi/180)/GSTUFF) //Energy equation’s terms
PARAMETER BOND = 1.5 //Valid for a single run. Re-initialized for the batch run
//constraints
constraint wall // wall constraint
formula: z=0
energy
e1: 0
e2: WALLT*x
e3: 0
constraint pinning_edge_1 nonnegative // pinning edge constraint
formula: y + x*TANA = 0
constraint pinning_edge_2 nonpositive // pinning edge constraint
formula: y - x*TANA = 0
//quantities
//integrand:
quantity grav energy method facet_vector_integral //compute surface integral in energy equation
vector_integrand:
q1: 0
q2: 0
q3: -BOND*(GX*x*z + GY*y*z + 0.5*GZ*z^2)
quantity xmc info_only method facet_vector_integral //compute geometry information (optional)
vector_integrand:
q1: 0
q2: 0
q3: z*x
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quantity ymc info_only method facet_vector_integral ////compute geometry information (optional)
vector_integrand:
q1: 0
q2: 0
q3: z*y
quantity zmc info_only method facet_vector_integral //compute geometry information (optional)
vector_integrand:
q1: 0
q2: 0
q3: 0.5*z^2
//defining vertices
vertices
1 0.0 0.0 0.0 constraint wall pinning_edge_1 pinning_edge_2 // 4 vertices on plane
2 2*COSA -2*SINA 0.0 constraint wall pinning_edge_1
3 2*COSA 2*SINA 0.0 constraint wall pinning_edge_2
4 2*COSA 0.0 1.0
5 0.0 0.0 0.0
6 3*COSA -3*SINA 0.0 fixed
7 3*COSA 3*SINA 0.0 fixed
//defining edges
edges
1 1 2 constraint wall pinning_edge_1
2 2 3 constraint wall pinning_edge_1 pinning_edge_2
3 3 1 constraint wall pinning_edge_2
4 14
5 24
6 34
//Wall edge
7 5 6 no_refine fixed color green
8 5 7 no_refine fixed color green
//defining facet comply with right hand rule
faces
1 2 6 -5 tension 1 grav xmc ymc zmc frontcolor lightblue backcolor red
2 1 5 -4 tension 1 grav xmc ymc zmc frontcolor lightblue backcolor red
3 4 -6 3 tension 1 grav xmc ymc zmc frontcolor lightblue backcolor red
//4 7 8 //
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//body
bodies
1 1 2 3 volume 1 density 0
//read
read
r //automated refined
r
r
//show
show
R //reset view angle
15u25r2d // comfortable viewangle
Results
Before analyzing the data, the computation time of SE-FIT program should also be reported. The average
time for calculate a critical Bond number is about 30 minutes. Sessile drops especially take huge amount
of time, can be up to one hour. Some optimizing method is found during calculation such as initializing
reasonable tolerance and initial guess of Bond critical number.
Figure 3 and 4 below describe the numerical results of critical Bond number for different 𝜙 angle of
gravity orientation. The data is separately plot for the case of the pendent drop to 𝜙 = 90 (Figure 3) and
the case from 𝜙 = 90 to 𝜙 = 175 of almost sessile drop (Figure 4). The interesting outcome of separate
plot is that the data results in accurate fitted curves.
7
0 deg Contact Angle
6
60 deg Contact Angle
Bocr
5
4
3
2
1
0
0.00
0.20
0.40
0.60
0.80
1.00
Sinφ
Figure 3 Critical Bond Number of Wall-edge-Vertex-Drop, 0 < 𝜙 < 90, contact angle of 0 and 60 degree
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90
60 deg Contact Angle
80
0 deg Contact Angle
70
Bocr
60
50
40
30
20
10
0
0.00
0.20
0.40
0.60
0.80
1.00
Sinφ
Figure 4 Critical Bond Number of Wall-edge-Vertex-Drop, 90 < 𝜙 < 175, contact angle of 0 and 60
degree
The analyses of data are based on figure 3 and figure 4. The data and fit curves form a chart analogous to
the kind of chart like Moody chart. On the left vertical axis is a dimensionless number of critical Bond
number; On the right vertical axis is a dimensionless quantity of contact angle ; On the horizontal axis is
a dimensionless quantity of sin 𝜙.
For 0 < 𝜙 < 90, the function of critical Bond number to 𝑠𝑖𝑛 𝜙 is fit to a second order polynomial
function of sin 𝜙.
For 0 degree contact angle: 𝐵𝑜𝑐𝑟 = 2.57 sin2 𝜙 − 5.74 sin 𝜙 + 6.39
For 60 degree contact angle: 𝐵𝑜𝑐𝑟 = 1.49 sin2 𝜙 − 3.51 sin 𝜙 + 4.07
For 90 < 𝜙 < 175, the function of critical Bond number to 𝑠𝑖𝑛 𝜙 is fit to a power law function of sin 𝜙.
For 0 degree contact angle: 𝐵𝑜𝑐𝑟 = 3.47 sin1.28 𝜙
For 60 degree contact angle: 𝐵𝑜𝑐𝑟 = 2.19 sin1.28 𝜙
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Let’s call the critical Bond number for pendent drop is 𝐵𝑜𝑐𝑟0 , contact angle is 𝜃, half corner angle is 𝛼,
gravity orientation is 𝜙. Interestingly, each fit reveals an independent value. For 0 < 𝜙 < 90, the critical
Bond number is the vertical intercept. For 90 < 𝜙 < 175, the critical Bond number is the order of power.
The most important conclusion is stated follow:
For contact angle less than 60 degree, it is convinced to suggest the two equations for different case of
gravity orientation:
For 0 < 𝜙 < 90: 𝐵𝑜𝑐𝑟 = sin2 𝜙 − 𝐹(𝜃, 𝛼, 𝜙) sin 𝜙 + 𝐵𝑜𝑐𝑟0 .
where 𝐹(𝜃, 𝛼, 𝜙) is a dimensionless number.
For 90 < 𝜙 < 175: 𝐵𝑜𝑐𝑟 = 𝐹(𝜃, 𝛼, 𝜙) sin𝐹(𝛼,𝜙) 𝜙
where 𝐹(𝜃, 𝛼, 𝜙) is a dimensionless number, 𝐹(𝛼, 𝜙) is another dimensionless number independent of
contact angle.
Concluding Remarks
1. For pendent drops, pinning-edge contact conditions, the drop profile does not affect the critical
Bond contact number. This result is confirmed by comparing the result of this report with Chen
[1] and the result of other friends from class ME510CP.
2. For same contact line profile (geometry), the less contact angle requires more volume to make the
drop unstable.
3. Regretfully, the simulation contact angle is equal to 175 does not perform well to produce
convincing result to show here. The Bond critical fluctuates for different contact angles. Since
this case does not produce any consistent result, the critical Bond number as the function of
contact angle and gravity orientation is still not fully examined. None of the transition is
investigated. In the worst case scenario, SE-FIT code in this report could also be wrong end
destroy the whole report.
4. Fixing contact angle from 0 to 60 degree, half corner angle is 55 degree: For those drop close to
the pendent drop, the critical Bond number increase with a second order polynomial function. For
those drop close to the pendent drop, the critical Bond number increase with a power function.
5. It can be predicted that the magnitude of Bond number depends on contact angle, half corner
angle and gravity orientation while the rate of change of Bond number depends on only half
corner angle and gravity orientation (independent with contact angle).
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Reference
Y.Chen et al: “The Shape and Stability of Wall-Bound and Wall-Edge-Bound Drops and
Bubbles” Microgracity sci. technol ,XVII-4, 2004, pp 14-23.
2. Y.Chen et al: SE-FIT1064 Manual, 2011, pp 57-62.
1.
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Appendix
Appendix A Raw Data
HalfCornerAngle=55
ContactAngle=0
ContactAngle=60
Phi
Bo(cr) SinPhi Phi
Bo(cr) SinPhi
0 6.32
0.00
0
4.07
0.00
5 5.92
0.09
5
3.79
0.09
10 5.49
0.17
10
3.5
0.17
15 5.18
0.26
15
3.27
0.26
20
4.7
0.34
20
3.02
0.34
25 4.41
0.42
25
2.87
0.42
30 4.19
0.50
30
2.7
0.50
35 3.93
0.57
35
2.55
0.57
40 3.69
0.64
40
2.46
0.64
45 3.66
0.71
45
2.33
0.71
50
3.5
0.77
50
2.26
0.77
55 3.35
0.82
55
2.19
0.82
60 3.33
0.87
60
2.13
0.87
65 3.26
0.91
65
2.09
0.91
70 3.24
0.94
70
2.08
0.94
75 3.24
0.97
75
2.05
0.97
80 3.21
0.98
80
2.06
0.98
85 3.26
1.00
85
2.06
1.00
90 3.29
1.00
90
2.1
1.00
95 3.38
1.00
95
2.16
1.00
100 3.41
0.98
100
2.21
0.98
105
3.6
0.97
105
2.29
0.97
110 3.78
0.94
110
2.4
0.94
115
4
0.91
115
2.53
0.91
120 4.23
0.87
120
2.69
0.87
125 4.56
0.82
125
2.92
0.82
130 4.99
0.77
130
3.14
0.77
135 5.54
0.71
135
3.48
0.71
140 6.27
0.64
140
3.92
0.64
145 7.21
0.57
145
4.47
0.57
150 8.57
0.50
150
5.33
0.50
155 10.67
0.42
155
6.5
0.42
160 13.83
0.34
160
8.54
0.34
165 19.81
0.26
165 12.13
0.26
170 32.93
0.17
170 20.31
0.17
175 77.19
0.09
175 50.68
0.09
Table A1. Critical Bond number for half corner angle equal to 55 deg, contact angle equal to 0 an 60,
gravity orientation from 0 to 175
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Appendix B Example screenshot for SE-FIT
Figure B1. Initial geometry and volume
Figure B2. The shape of the pendent drop before break up (contact angle is 60 deg,𝐵𝑜𝑐𝑟 = 4.07)
Figure B3. The shape of the pendent drop after break up (contact angle is 60 deg, 𝐵𝑜𝑐𝑟 = 4.42)
12