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Fingering instability down the outside Fingering instability down the outside
- f a vertical cylinder
- f a vertical cylinder
Fingering instability down the outside Fingering instability down - - PowerPoint PPT Presentation
Fingering instability down the outside Fingering instability down - - PowerPoint PPT Presentation
Fingering instability down the outside Fingering instability down the outside of a vertical cylinder of a vertical cylinder Linda Smolka Bucknell University joint work with Marc SeGall Workshop on Surfactant Driven Thin Film Flows The
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Background on Gravity-Driven Contact Lines Background on Gravity-Driven Contact Lines
Inclined and Vertical Planes
- Experimental: Huppert, Nature 1982; Silvi & Dussan, Phys. Fluids 1985;
de Bruyn, Phys. Rev A 1992, Jerrett & de Bruyn, Phys. Fluids A 1992, Kondic, SIAM Review 2003.
- Theoretical: Troian, Safran, Herbolzheimer, Joanny, Europhys. Lett. 1989;
Spaid & Homsy, Phys. Fluids 1996; Bertozzi & Brenner, Phys. Fluids 1997.
- Review Article on Thin Films: Craster & Matar, Rev. Mod. Phys. 2009.
Image from Huppert, 1982.
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Background on Gravity-Driven Contact Lines Background on Gravity-Driven Contact Lines
Horizontal Cylinder and Sphere
- Outside Horizontal Cylinder and Sphere:
Takagi and Huppert, JFM 2010. Coating a horizontal cylinder Axisymmetric flow down a sphere
Images from Takagi and Huppert
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Our Study on Gravity-Driven Contact Lines Our Study on Gravity-Driven Contact Lines
H
Vertical Cylinder Consider the dynamics of an axisymmetric contact line. Azimuthal curvature distinguishes a cylindrical substrate from a planar one. Understanding the effects of curvature is useful for coating cylindrical surfaces from wires to conduits. Questions
- Does substrate curvature effect features of the contact line?
- Is there a critical radius above which substrate curvature (= 1/R) is
negligible and the contact line dynamics match those on a vertical plane?
- Is there a critical radius below which fingering is inhibited?
SLIDE 6
Experimental Details Experimental Details
Top View of Experiment Six clear acrylic cylinders (61 cm tall): R = 0.159 - 3.81 cm. Fluids: Glycerin & Silicone oil (1000 cSt ). Flourescent dye and black lights illuminate fluid. Mirrors used to visualize around the cylinder periphery. Fluid delivery: Reservoir cup atop cylinder or fluid pump. Gap height between fluid source and cylinder is 0.105 cm. Framing Rate: 100 frames/s. Image size: 512 x 384 pixels; Pixel resolution: 6 pixels/cm.
Calibration Image Silicone oil - R = 3.81 cm The contact line develops a fingering pattern in all of the experiments.
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Experiments: Fingering Pattern Experiments: Fingering Pattern
- Glycerin partially wets cylinder. Fingers form long straight rivulets with
stationary troughs; same as behavior down an inclined plane*.
- Silicone oil completely wets as tips and troughs travel down the cylinder.
Effect of substrate curvature: Fingers do not form a sawtooth pattern as in inclined plane experiments*. Glycerin Silicone oil
Glycerin
= 58.4 dyn/cm = 8.0 cm2/s
Silicone Oil
= 21.9 dyn/cm = 10.3 cm2/s
* Huppert, Nature 1982; Silvi & Dussan, Phys. Fluids 1985; de Bruyn, Phys. Rev A 1992
cm 3.81 R viscosity kinematic tension surface
= = =
ν
γ
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Experiments: Finger Motion Experiments: Finger Motion
- Glycerin tips travel faster than silicone oil tips.
- Tips and silicone oil troughs travel faster down vertical cylinder than an
inclined plane.*
- Data does not scale with cylinder radius.
* Huppert, Nature 1982; Jerrett & de Bruyn, Phys. Fluids A 1992
Position of tips and troughs follow the power-law scaling
B
) t
- A(t
z
= Glycerin: Red Silicone Oil: Blue
Tips Troughs
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Experiments: Finger Width Experiments: Finger Width
Finger width is invariant to:
- Wetting property of the fluids
- Cylinder radius
Glycerin: Red Silicone Oil: Blue
Number Bond / ) R g (
- B
ˆ
2
= = γ ρ
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Derivation of Lubrication Model Derivation of Lubrication Model
s thicknes film upstream : surface free fluid : ) , , ( radius cylinder : R H t z h θ , 1 /
< < =
R H
ε
H
Consider the motion of an incompressible thin film moving down the outside of a vertical cylinder. In the lubrication approximation, assume which is proportional to the substrate curvature. Nondimensionalize the free boundary problem using the scalings of Evans, Schwartz,
Roy, Phys. Fluids 2004:
z r
w v u e e e u
+ + =
θ
. gH , p p , w U w , v U v , u , t U R t , r R r , z R z , h R h , y R y
2
v U gH U u
= = = = = = = = = = ρ ε ε ε
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Scaled Free Boundary Problem Scaled Free Boundary Problem
( )
1 Re 2 Re 2 Re 1 1
2 2 3 2 2 2 2 2 2 3 2 2
+ ∆ + ∂ ∂ − = ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ ∂ ∂ − − ∆ + ∂ ∂ − = + ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ ∂ ∂ − − ∆ + ∂ ∂ − = − ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ + ∂ ∂ + ∂ ∂
w z p z w w w r v y w u t w UT R u r r v v p r r v u z v w v r v y v u t v UT R v r r u u y p r v z u w u r v y u u t u UT R z w v r u r y r
ε θ ε θ ε ε θ ε ε θ ε θ ε ε ε ε θ ε ε ε ε θ
Continuity and Navier-Stokes equations: Boundary conditions: where:
. , 1
2 2 2 2 2 2 2 2 2 2 2 2
z z r y r y r
∂ ∂ + ∂ ∂ = ∇ ∂ ∂ + ∂ ∂ + ∂ ∂ ∂ ∂ = ∆ θ ε θ ε
( ) ( ) ( ) ( ) ( ) u z h w h r v t h h y O y w h y O v y v h y O h h O y v h y w z h y u p h y w v u y
= ∂ ∂ + ∂ ∂ + ∂ ∂ = = + ∂ ∂ = = + − ∂ ∂ = + ∇ − − − = + ∂ ∂ ∂ ∂ − ∂ ∂ ∂ ∂ − ∂ ∂ + − = = = = = θ ε ε ε ε ε ε ε ε θ ε ε ε
: : : 1
- B
ˆ 1 2 2 2 : :
2 2 2 2 2
. / Bo where Bo /
- B
ˆ , / Re , /
2 2 2
γ ρ ε γ ρ ν ε
gH gR RU R H
= = = = =
−
. 1 Re and 1
2
< < < < ε ε
Lubrication approximation: Assume
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Lubrication Model Lubrication Model
..., ...,
) 1 ( ) ( ) 1 ( ) ( 1
+ + = + + + =
−
u u u
ε ε ε
p p p p ( ) ( ) ( ) ( ) ) 1 (
- B
ˆ 3 1 3 1 1
2 3 3
= ∇ + ∇ ∇ + + ∂ + ∂ +
h h h h h h
z t
ε ε ε
Expand the pressure and velocity fields in powers of epsilon: linearize the free boundary problem and solve for flow variables at Next, substitute flow variables into conservation of mass where to obtain an evolution equation for the film height: Gravity Surface Tension
). ( and ) 1 (
ε
O O
( ) , 1
= ∂ ∂ + ∂ ∂ + ∂ ∂ +
z Q Q t h h
z
θ ε
θ
( )
( )(
) , 1 ,
) 1 ( ) ( ) 1 ( ) (
y d w w y y d w r Q y d v v y d v Q
h
- h
- z
h
- h
- ∫
∫ ∫ ∫
+ + = = + = = ε ε ε
θ
z
∂ ∂ + ∂ ∂ = ∇
z θ
e e
θ
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Experimental Parameters Experimental Parameters
Cylinder Radius Slenderness Parameter Glycerin Glycerin Silicone Oil Silicone Oil R (cm) Re Re 0.159 5.3e-1 1.6e-2 0.536 9.9e-3 1.12 0.318 2.7e-1 3.3e-2 2.14 2.0e-2 4.46 0.635 1.3e-1 7.0e-2 8.55 4.1e-2 17.8 0.953 8.9e-2 8.3e-2 19.3 5.7e-1 40.1 1.27 6.7e-2 1.5e-1 34.2 8.42e-2 71.2 Cylinder Radius Slenderness Parameter Glycerin Glycerin Silicone Oil Silicone Oil . 32 . Bo , 15 . Bo , B / R g
- B
ˆ , / H g Bo , / R Re curvature substrate ~ R / cm) 0.02 0.085 ( thickness film upstream :
- il
silicone glycerin 2 2 2
= = = = = = = ± =
−
- U
H H
ε γ ρ γ ρ ν ε
- Values of and Re satisfy conditions to apply lubrication model.
- Wide range of in experiments.
- B
ˆ
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Steady-State Traveling Wave Solutions of Lubrication Model Steady-State Traveling Wave Solutions of Lubrication Model
Predictions from model (right graph):
- Influence of substrate curvature negligible for small as capillary ridge height
and shape converge to that for a traveling wave down a vertical plane.
- When substrate curvature and capillary effects influence behavior.
Details:
- Traveling waves modeled with a precursor film ahead of the contact line, b=0.07.
- Left graph: Parameters taken from experiments.
- Left graph: Characteristic coordinate
z - Ut)is scaled by R.
Silicone Oil Glycerin: Red Silicone Oil: Blue
) / 1 ~ ( R
ε
max
h
ε
), 10 (
1
−
≥ O ε
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Linear Stability Analysis Linear Stability Analysis
Experimental parameters used to compute dispersion curves. Influence of Capillary Effects The range of unstable modes, most unstable mode and maximum growth rate are larger for silicone oil. The contact line for silicone oil is more unstable to fingering than glycerin. Perturbed solution:
) iq t exp( ) (
θ β ξ ϕ + + = h
h
Red: R=0.16 cm Black: R=0.32 cm Blue: R=0.64 cm Red: R=0.95 cm Black: R=1.27 cm Blue: R=3.81 cm
Glycerin: Dashed Silicone Oil: Solid
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Prediction from Lubrication Model Prediction from Lubrication Model
Maximum Growth Rate
Collapse of data: Maximum growth rate scales with the Bond number for
42 .
- B
ˆ ~ *
β
Glycerin: Red Silicone Oil: Blue
*
β
1.3.
- B
ˆ
>
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Stability: Theory and Experiments Stability: Theory and Experiments
Periodicity of Cylinder Periphery: # of fingers must be integer-valued. Linear Approximation in : The # of fingers in a pattern and wave number, q, are equivalent. For fingers to develop, Interpretation: The contact line is stable when unstable modes are < 1. Otherwise, contact line is unstable. Results: Excellent agreement between linear stability theory and experiments.
Scaling of q with Bond #: The # of fingers increases nearly linearly with cylinder circumference.
1.
≥
q
45 .
- B
ˆ ~ q
Glycerin - ; Silicone oil – Red: Most unstable mode (LSA) Blue: Cutoff mode (LSA) Solid: Experimental data Shaded: Unstable modes
0.6
ρ γ /
025 . R
- r
0.6
- B
ˆ
> >
Prediction from Stability Theory: The contact line down a vertical cylinder is unstable for Can we predict when no fingering occurs?
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Finger Wavelength: Theory and Experiments Finger Wavelength: Theory and Experiments
Symbols: Experimental data; Solid/Dashed colored lines: Cut-off/Most unstable wavelength from LSA (contact line is stable below solid curve); Black dashed line: Most unstable wavelength for a vertical plane, Spaid and Homsy, Phys. Fluids 1996.
, ) / ( , 6 . 12
3 / 1
g H l l
ρ γ λ = =
Results: Excellent agreement between theory and experiments. Predictions from Model:
- Substrate curvature is negligible when and influence behavior when
- Larger surface tension increases finger wavelength (glycerin
silicone oil
( ), 10
2
−
≤ O ε
R / 1 ~
ε
Glycerin Silicone oil
( ). 10
1
−
= O ε
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Conclusions Conclusions
Model Predictions:
- Substrate curvature is a negligible effect when (shape and stability of
the contact line down a vertical cylinder matches that down a vertical plane) and substrate curvature is an important effect when
- Model identifies a critical Bond number for fingering.
We’ve presented an experimental and analytical study on the dynamics of an axisymmetric contact line moving down a vertical circular cylinder. Model accurately predicts the wavelength and number of fingers that form in experiments for a variety of cylinder sizes and for two fluids.
( )
2
10
−
≤ O ε
( ). 10
1
−
= O ε
Future Directions:
- Analyze fingering for thicker films when H ~ R (flow down a wire). Non-trivial!
- Study effect of inertia on fingering dynamics.
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Is there a model that allows H~R? Is there a model that allows H~R?
)) ( /( H R g L
+ = ρ γ
, 1 / ) (
< < + =
L H R
ε
H
Craster & Matar, On viscous beads flowing down a vertical fibre, JFM 2006. Derive a 1D model in the long-wavelength limit with L is the capillary length
- r equivalently the Bond number << 1.
. H) g(R , p p , w w ?, v , u , t t , z z , r H) (R
2
v U gL U U u L U L r
+ = = = = = = = + = ρ ε
z r
w v u e e e u
+ + =
θ
The Continuity Equation sets the scaling on the azimuthal velocity.
1
= ∂ ∂ + ∂ ∂ + + ∂ ∂
z w v r r u r u
θ
1 ?
= ∂ ∂ + ∂ ∂ + + + ∂ ∂ +
z w L U v r H R r u r u H R U
θ ε
! v U v
ε =
For all the terms to be O(1) requires that promising Naïve Idea: Develop a 2D model using C&M’s scalings. Free surface at r = S= R+h
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Leading Order Free Boundary Problem Leading Order Free Boundary Problem
Continuity and Navier-Stokes equations (dropping hat notation): with boundary conditions at the free surface r = S:
( ) ( ) ( )
u S S v S S S S v r v u S w r w S S S u S v S r u S S w S S r w S S S S S S S p
t z
= + + = − − ∂ ∂ + ∂ ∂ + ∂ ∂ + ∂ ∂ − + ∂ ∂ − ∂ ∂ = ∂ ∂ + − ∂ ∂ + − + + =
θ θ θ θ θ θ θ θ θ θ
θ θ θ θ ε
z 2 2 zz 2 3 2 2 2 2 2
wS , 1 1 1 2 2 , , S
- 1
. 1 1 1 , 1 , , 1
2 2 2 2 2
+ ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ = ∂ ∂ = ∂ ∂ = ∂ ∂ + ∂ ∂ + + ∂ ∂ θ θ θ
w r r w r r w z p p r r p z w v r r u r u
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Leading Order Free Boundary Problem Leading Order Free Boundary Problem
Continuity and Navier-Stokes equations: Normal BC at r = S:
. 1 1 1 , 1 , , 1
2 2 2 2 2
+ ∂ ∂ + ∂ ∂ + ∂ ∂ = ∂ ∂ = ∂ ∂ = ∂ ∂ = ∂ ∂ + ∂ ∂ + + ∂ ∂ θ θ θ
w r r w r r w z p p r r p z w v r r u r u ) , ( t z p p =
( ) ( ) ( )
, S
- 1
zz 2 3 2 2 2 2 2
ε
θ θ θ θ θ
+ − + + =
S S S S S S S p Depends only
- n z, t.
For fingering, need azimuthal dependence in free surface, S. Mathematically inconsistent! Naïve!
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