Behavior of a Droplet on a Thin Liquid FilmBehavior of a Droplet on a Thin Liquid Film Ellen R. Peterson Carnegie Mellon University Department of Mathematical Sciences Center for Nonlinear

Behavior of a Droplet on a Thin Liquid Film
Ellen R. Peterson
Carnegie Mellon UniversityDepartment of Mathematical Sciences
Center for Nonlinear Analysis
February 23, 2012
Funded by National Science Foundation
Ellen R. Peterson (Carnegie Mellon) Spreading Droplets February 23, 2012 1 / 35

Introduction
Project Motivation
Goal: Improve treatment for Cystic Fibrosis- use surfactant to spread medicine over mucus in lungs
Research Group:
CMU Physics: Steve Garoff, Fan Gao
CMU Chemical Engineering:Todd Przybycien, Robert Tilton, Roomi Kalita, Ramankur Sharma, AmsulKhanal
U. Pitt Medical Center: Tim CorcoranQuestions:
How does the fluid spread? Where is the surfactant? How do multiple droplets interact?

Introduction
Background on Spreading
substrate
ruid
Dz
H
HD

Surfactant
Background on Spreading Surfactant
Surfactant spreading on liquid subphase:
Model: Gaver and Grotberg (1990)
ht +1r
(
12 rh
2()r)
r= 1
r
(
13 rh
3hr
)
r 1
r
(
13 rh
3(
1r(rhr )r
)
r
)
r
t +1r(rh()r )r =
1r
(
12 rh
2hr)
r 1
r
(
12 rh
2(
1r(rhr )r
)
r
)
r+ 1
r(rr )r
Asymptotic spreading behavior/similarity solutions:Jensen, Grotberg (1992), Jensen (1994)
ht 1
r
(
12 rh
2r)
r= 0 t
1
r(rhr )r = 0
Similarity scaling:
h(r , t) = H(), (r , t) =1
t12
G (), =r
t14
Similarity solutions:
H() = 22, G () = 18log , =
r
t14

Surfactant
Numerical similarity solutions: ERP, Shearer (2012)Equations:
ht r0(t)
r0(t)h =
1
r0(t)2
(
12h2
)
t r0(t)
r0(t) =
1
r0(t)2(h)
r0 = 1
r0h(1, t)(1, t)
=r
r0(t)
Solutions:
h(r , t) = H(), (r , t) =1
t12
G(), =r
r0(t)
H() = 22, G() = 1
8log , =
r
t14
0 0.2 0.4 0.6 0.8 1 1.20
0.5
1
1.5
2
r/r0(t)
h
0 0.2 0.4 0.6 0.8 1 1.20
1
2
3
4
r/r0(t)
H G
1000

Surfactant
Inner Solution: ERP, Shearer (2011)
h(r , t) = t29
(
H0 +1
18G0
(
r
t16
)2
+ A
(
r
t19
)2)
+ O(r4)
(r , t) = t49
(
G0 +
H0
(
r
t16
)2)
+ O(r4)
0 0.1 0.2 0.3 0.4
0.02
0.04
0.06
0.08
0.1
0.12
h(
,t)
0 0.2 0.4 0.6
0.55
0.6
0.65
t4/9
(,t)

Surfactant
Experimental GoalsExperimental spreading:Bull et al (1999), Fallest et al (2010)
Visualize the height of the film (using laser line)
Visualize the location of the surfactant (fluorescence)
Match the experimental data to the PDE model
0 20 400
1
2
H [m
m]
0 20 400
2
4
[
g/c
m2]
r [mm]
(a) t = 1 sec
0 20 400
1
2
H [m
m]
0 20 400
2
4
[
g/c
m2]
r [mm]
(b) t = 2 sec
0 20 400
1
2
H [m
m]
0 20 400
2
4
[
g/c
m2]
r [mm]
(c) t = 3 sec
0 20 400
1
2
H [m
m]
0 20 400
2
4
[
g/c
m2]
r [mm]
(d) t = 6 sec
ring
Karen Danielss Lab NC State University
expmovie.mp4Media File (video/mp4)

Surfactant
Spreading Behavior
100
101
101
102
r 0 [m
m]
=0.25
0 20 400
1
2
H [m
m]
0 20 400
2
4
[
g/c
m2]
r [mm]
(b) t = 2 sec
tracking
101
100
101
101
102
t [sec]
r M [m
m]
=0.295
0 20 400
1
2
H [m
m]
0 20 400
2
4
[
g/c
m2](b) t = 2 sec
tracking
r [mm]
t [sec]

Surfactant
Surfactant Initial Condition
Using time scale t = t :
0 2 4 6
0.5
1
1.5
r
h
0 2 4 60
0.5
1
1.5
2
r
0 2 4 60
0.5
1
1.5
2
r
0 2 4 6
0
0.5
1
1.5
2
r
0 2 4 6
0.5
1
1.5
r
h
0 2 4 60
0.5
1
r
0 2 4 6
0.5
1
1.5
h
r
0 2 4 60
0.2
0.4
0.6
0 2 4 60
0.2
0.4
0.6
0.8
1
r
0 2 4 6
0.5
1
1.5
r
h
r
Surfactant at
t=9
Height at
t=9
Initial Conditions

Spreading Droplet
How do we predict the spreading behavior?
Spreading Parameter: (Harkins 1952)
S = F DF D
Complete spreading (S > 0)- Spreading on a Deep Layer:
Dipietro, Huh, & Cox, 1978; DiPietro& Cox, 1980; Foda & Cox, 1980
- Spreading Power Law:Fraaije & Cazabat, 1989
Lens-shape (S < 0)- Numerical axisymmetric solution,
static lens :Pujado & Scriven, 1971
D
DF
F
S>0
S

Spreading Droplet
Assumptions
Newtonian drop on Newtonian fluid Fluids immiscible Lubrication Approximation H
D

Spreading Droplet
Deriving Droplet Spreading Equations
pDr =
DuDzz , p
DFr =
DFuDFzz , pFr =
FuFzz
pDz =
Dg , pDFz =
DFg , pFz =
Fg
pD(a, t) = patm
Darr , p
DF (b, t) = pD(b, t) DFbrr , pF (h, t) = patm
Fhrr
u =0Fu =0DF
u =0Fz
DFu =uD
u =0Dz
u = uD DDF DFz z
h(r,t)
a(r,t)
b(r,t)

Spreading Droplet
Droplet Spreading Equations - axisymmetric
ht =1
r
[
r
(
Bohr (
1
r(rhr )r
)
r
)
h3
]
r
bt =1
r
[
r
(
b3
(
Bo(1 )br DF(
1
r(rbr )r
)
r
)
+
(
Boar D(
1
r(rar )r
)
r
)
b2
2(3a b)
)]
r
at =1
r
{
r
[(
Bo(1 )br DF(
1
r(rbr )r
)
r
)
b2
2(3a b)
+
(
1
(a b)3 + a3)(
Boar D(
1
r(rar )r
)
r
)]}
r
Bo: Bond number: density ratio: viscosity ratioD : surface tension of drop and airDF : surface tension of drop and fluidrelated work: Kriegsmann & Miksis 2003Ellen R. Peterson (Carnegie Mellon) Spreading Droplets February 23, 2012 13 / 35

Equilibrium Solutions
Bohr (
1
r(rhr )r
)
r
= 0
Boar D(
1
r(rar )r
)
r
= 0
Bo(1 )br DF(
1
r(rbr )r
)
r
= 0.
If 6= 1 these equations are Bessel equations. Well assume < 1
h(r) = Ch + ChhI0
(Bor
)
+ ChhhK0
(Bor
)
a(r) = Ca + CaI0
(
Bo
Dr
)
+ CaaaK0
(
Bo
Dr
)
b(r) = Cb + CbbI0
(
Bo(1 )DF
r
)
+ CbbbK0
(
Bo(1 )DF
r
)
Related studies: Kriegsmann & Miksis (2003), Pujado & Scriven (1971)Ellen R. Peterson (Carnegie Mellon) Spreading Droplets February 23, 2012 14 / 35

Boundary ConditionsEdge Conditions:
ar (0, t) = 0, br (0, t) = 0, hr (L, t) = 0
Continuity of Interface:
a(r0, t) = h(r0, t), b(r0, t) = h(r0, t)
Continuity of Pressure:(
pDF = pF |r=r0)
h Da DFb = 0
r=r0
Height of Film:
2
r0
0
b(r , t)rdr + 2
L
r0
h(r , t)rdr = MassF
Location of r0(t):
MassD = 2
r0
0
(a(r) b(r))r dr
h(r,t)
a(r,t)
b(r,t)
r (t)0

Boundary Conditions
Balance of Surface Tension Forces:
F cos F +D cos D +DF cos DF = 0
F sin F +D sin D +DF sin DF = 0
Apply approximations:
F hr , D ar , DF br
F(
1 +2h2r2
+ . . .
)
D(
1 +2a2r2
+ . . .
)
DF(
1 +2b2r2
+ . . .
)
= 0
F(
hr +3h3r3
+ . . .
)
D(
ar +3a3r3
+ . . .
)
DF(
br +3b3r3
+ . . .
)
= 0
O(1) : 1 D DF = 0= S O(2)
O() : hr Dar DFbr = 0O(2) : h2r Da2r DFb2r = 2
F
D
2-DF
D
F
DF
Ellen R. Peterson (Carnegie Mellon) Spreading

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Behavior of a Droplet on a Thin Liquid FilmBehavior of a Droplet on a Thin Liquid Film Ellen R. Peterson Carnegie Mellon University Department of Mathematical Sciences Center for Nonlinear