3-D Film and Droplet Flows over Topography
Several important practical applications: e.g. film flow in the eye, electronics cooling, heat exchangers, combustion chambers, etc...
Focus on: precision coating of micro-scale displays and sensors, Tourovskaia et al,
Nature Protocols, 3, 2006.
Plant disease control
Pesticide flow over leaves, Glass et al,
Pest Management Science, 2010.

3D Film Flow over Topography
For displays and sensors, coat liquid layers over functional topography – light-emitting species on a screen
Key goal: ensure surfaces are as planar as possible – ensures product quality and functionality – BUT free surface disturbances are persistent!
> 50μm spin coat liquid topographic substrate solid cure film levelling period conformal liquid coating
Stillwagon, Larson and
Taylor, J. Electrochem.
Soc.
1987

3D Film Flow over Topography
Key Modelling Challenges :
•3-D surface tension dominated free surface flows are very complex
– Navier-Stokes solvers at early stage of development (see later)
•Surface topography often very small (~100s nm) but influential – need highly resolved grids?
•No universal wetting models exist
•Large computational problems – adaptive multigrid, parallel computing?
•Very little experimental data for realistic 3D flows.

3D Film Flow over Topography
Finite Element methods not as well-established for 3-D free surface flow. Promising alternatives include Level-Set, Volume of
Fluid (VoF ), Lattice Boltzmann etc… but still issues for 3D surface tension dominated flows – grid resolution etc...
Fortunately thin film lubrication low assumptions often valid provided: ε=H0/L0 <<1 and capillary number Ca<<1 y gravity
H
0 x
Enables 3D flow to be modelled by 2D systems of pdes.
a
L
0 s(x,y) h(x,y)

3D Film Flow over Topography
Decre & Baret, JFM, 2003: Flow of Water Film over a Trench Topography
Comparison between experimental free surface profiles and those predicted by solution of the full Navier-Stokes and
Lubrication equations.
Agreement is very good between all data.
Lubrication theory is accurate – for thin film flows with small topography and inertia!

3D Film Flow over Topography
Thin Film Flows with Significant Inertia
Free surfaces can be strongly influenced by inertia: e.g. free surface instability, droplet coalescence,... standard lubrication theory can be extended to account for significant inertia – Depth Averaged
Formulation of Veremieiev et al, Computer & Fluids, 2010.
Film Flows of Arbitrary Thickness over Arbitrary Topography
Need full numerical solutions of 3D Navier-Stokes equations!

Depth-Averaged Formulation for
Inertial Film Flows
1. Reduction of the Navier-Stokes equations by the longwave approximation:
 
H
0
L
0

1 Restrictions:
Ca

1 s
 

1
,
2. Depth-averaging stage to decrease dimensionality of unknown functions by one: u
 x , y , t


1 h f
 s udz v
 x , y , t


1 h f
 s vdz h
 x , y , t
  x , y , t

 s ( x , y )
Restrictions: no velocity profiles and internal flow structure
3. Assumption of Nusselt velocity profile to estimate unknown friction and dispersion terms: u

3 u

 
1 2

2
 v

3 v

 
1 2

2

  z
 h s

Depth-Averaged Formulation for
Inertial Film Flows
DAF system of equations:


 h
 t
Re
Re






 u
 t

 v t


 u
5 h v
5 h
 h
 t
 t

 h
6
5


   


6
5 u



 u
 x
0 u
 x
 y

 v
 x v

 u
 y v





 v

 y






 x





3
Ca
 y


 2


3
Ca h

 2 s


2

 h
 s
 cot

For Re = 0 DAF ≡ LUB

2

 h
 s


 cot


3 u h
2

 h
 s



2

3 v h
2
Boundary conditions:
1. Inflow b.c.
2. Outflow (fully developed flow)
3. Occlusion b.c.
 

0
 u , v , h
 x

0


 x
 u , v , h
 x
 l p
  h
 
 

 n

2
3

, 0 ,

1

 y
 u , v , h

 tan

S


2 y

0 , w p

0

Flow over 3D trench: Effect of
Inertia
Gravitydriven flow of thin water film: 130µm ≤ H
0 trench topography: sides 1.2mm, depth 25µm
≤ 275µm over surge bow wave comet tail

Accuracy of DAF approach
Gravitydriven flow of thin water film: 130µm ≤ H
0 stepdown topography: sides 1.2mm, depth 25µm
≤ 275µm over 2D
Max % Error vs Navier-Stokes (FE)
Error ~1-2% for
Re=50 and s
0
≤0.2
s
0
=step size/H
0

Free Surface Planarisation
Noted above: many manufactured products require free surface disturbances to be minimised – planarisation
Very difficult since comet-tail disturbances persist over length scales much larger than the source of disturbances
Possible methods for achieving planarisation include:
• thermal heating of the substrate, Gramlich et al (2002)
• use of electric fields

Electrified Film Flow
Gravity-driven, 3D Electrified film flow over a trench topography
Assumptions:
• Liquid is a perfect conductor
• Air above liquid is a perfect dielectric
Film flow modelled by Depth Averaged Form
Fourier series separable solution of Laplace’s equation for electric potential coupled to film flow by Maxwell free surface stresses.

Electrified Film Flow
Effect of Electric Field Strength on Film Free Surface
No Electric Field With Electric Field
Note: Maxwell stresses can planarise the persistent, comettail disturbances.

Computational Issues
Real and functional surfaces are often extremely complex.
Multiply-connected circuit topography:
Lee, Thompson and Gaskell,
International Journal for
Numerical Methods in Fluids ,
2008

Flow over a maple leaf
Need highly resolved grids for
3D flows topography
Glass et al, Pest
Management Science , 2010

Adaptive Multigrid Methods
• Full Approximation Storage (FAS) Multigrid methods very efficient.
• Spatial and temporal adaptivity enables fine grids to be used only where they are needed.
E.g. Film flow over a substrate with isolated square, circular and diamondshaped topographies
Free Surface Plan View of Adaptive Grid

Parallel Multigrid Methods
Parallel Implementation of Temporally Adaptive Algorithm using:
• Message Passing Interface (MPI)
• Geometric Grid Partitioning
Combination of Multigrid O(N) efficiency and parallel speed up very powerful!

3D FE Navier-Stokes Solutions
Lubrication and Depth Averaged Formulations invalid for flow over arbitrary topography and unable to predict recirculating flow regions
As seen earlier important to predict eddies in many applications:
E.g. In industrial coating

3D FE Navier-Stokes Solutions
Mixing phenomena
E.g. Heat transfer enhancement due to thermal mixing, Scholle et al,
Int. J. Heat Fluid Flow, 2009.

3D FE Navier-Stokes Solutions
Mixing in a Forward Roll Coater Due to Variable Roll Speeds
Substrate
Bath

3D FE Navier-Stokes Solutions
•Commercial CFD codes still rather limited for these type of problems
•Finite Element methods are still the most accurate for surface tension dominated free surface flows – grids based on Arbitrary Lagrangian
Eulerian ‘Spine’ methods
Spine Method for 2D Flow Generalisation to 3D flow

3D FE Navier-Stokes vs DAF
Solutions
Gravity-driven flow of a water film over a trench topography: comparison between free surface predictions

3D FE Navier-Stokes Solutions
Gravity-driven flow of a water film over a trench topography: particle trajectories in the trench
3D FE solutions can predict how fluid residence times and volumes of fluid trapped in the trench depend on trench dimensions

Droplet Flows: Bio-pesticides

Application of Bio-pesticides
Changing EU legislation is limiting use of chemically active pesticides for pest control in crops.
Bio-pesticides using living organisms (nematodes, bacteria etc...) to kill pests are increasing in popularity but little is know about flow deposition onto leaves
Working with Food & Environment Research Agency in York and
Becker Underwood Ltd to understand the dominant flow mechanisms

Nematodes
Nematodes are a popular bio-pesticide control method - natural organisms present in soil typically up to 500 microns in length.
• Aggressive organisms that attack the pest by entering body openings
• Release bacteria that stops pest feeding – kills the pest quickly
• Mixed with water and adjuvants and sprayed onto leaves

What do we want to understand?
• Why do adjuvants improve effectiveness – reduced evaporation rate?
•
How do nematodes affect droplet size distribution?
• How can we model flow over leaves?
• How does impact speed, droplet size and orientation affect droplet motion?

Droplet spray e vaporation time: effect of adjuvant
Size of droplet s
Conce ntratio n (%) large 0
Initial mass
(mg)
Mass fraction left after
10 min
(%)
130.3 36.3
0.01
138.0 36.6
0.1
161.0 48.7
small 0 87.3
13.3
0.01
92.5
9.7
0.1
138.3 33.3
Evapor ation time
(min)
26.3
24.0
36.0
16.3
16.0
25.7

D roplet size distribution for bio-pesticides
Matabi 12Ltr
Elegance18+ knapsack sprayer
Teejet XR110 05 nozzle with 0.8bar

VMD of the bio-pesticide spray depending on the concentration of adjuvant addition of bio-pesticide does not affect Volume Mean
Diameter of the spray
Dv50 ( μm)
Substance c = 0% c= 0.01% c = 0.03% c = 0.1% c = 0.3%
269.4
330.5
352.9
water+adjuvant 273.3
275.1
water+carrier material water+commercial product
(biopesticide)
285.9
271.0
276.1
272.8
297.3
282.6
329.2
307.5
360.8
360.6

Droplet flow over a leaf: simple theory
2 nd Newton’s law in x direction: theoretical expressions from Dussan (1985):
Stokes drag:
Contact angle hysteresis:
Velocity:
Terminal velocity:
Volume of smallest droplet that can move:
Relaxation time:

Droplet flow over a leaf: simple theory vs. experiments
47V10 silicon oil drops flowing over a fluoro-polymer FC725 surface:
Dussan (1985) theory:
Podgorski, Flesselles,
Limat (2001) experiments:
Le Grand, Daerr & Limat
(2005), experiments: droplet flow is governed by this law:

Droplet flow over a leaf ( θ=60º): effect of inertia
For: V=10mm3, R=1.3mm, terminal velocity=0.22m/s
Lubrication theory Depth averaged formulation

Droplet flow over a leaf ( θ=60º): effect of inertia
For: V=20mm3 R=1.7mm terminal velocity=0.45m/s
Lubrication theory Depth averaged formulation

Droplet flow over a leaf ( θ=60º): summary of computations
V, mm 3
R, mm
Bosinθ
Ca a, m/s
Experiment
Ca a, m/s
Computation
Re=0
Ca a, m/s
Computation
Re=10
0.27
0.4
0.06
0 0 0.0003
0.02
0.0001
0.007
10 1.3
0.62
0.003
0.13
0.005
0.21
0.005
0.22
20 1.7
0.99
0.006
0.24
0.010
0.42
30 1.9
1.30
0.008
0.33
0.012
0.54
40 2.1
1.57
0.011
0.48
0.014
0.62
0.009
0.011
0.012
0.40
0.48
0.55

Droplet flow over a leaf: theory shows small effect of initial velocity
Velocity:
Initial velocity:
Relaxation time:

Droplet flow over a leaf: computation of influence of initial condition
V=10mm3 R=1.3mm
a=0.22m/s
Bosin θ=0.61
v0=0.69m/s
Bosin θ init =1.57
V=10mm3 R=1.3mm a=0.22m/s
Bosin θ=0.61
v0=1.04m/s
Bosin θ init =2.49
this is due to the relaxation of the droplet’s shape

Droplet flow over ( θ=60º) vs. under
( θ=120º) a leaf: computation
V=20mm3
R=1.7mm a=0.45m/s
Bosin θ=0.99
θ=60º
V=20mm3
R=1.7mm a=0.45m/s
Bosin θ=0.99
θ=120º

Bio-pesticides: initial conclusions
 Addition of carrier material or commercial product (bio-pesticide) does not affect the Volume Mean Diameter of the spray.
 Dynamics of the droplet over a leaf are governed by gravity,
Stokes drag and contact angle hysteresis; these are verified by experiments.
 Droplet’s shape can be adequately predicted by lubrication theory, while inertia and initial condition have minor effect.
 Simulating realistically small bio-pesticide droplets is extremely computationally intensive: efficient parallelisation is needed ( see e.g. Lee et al (2011), Advances in Engineering Software )
BUT probably does not add much extra physical understanding!