1. 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.
Pesticide flow over leaves, Glass et al,
Pest Management Science, 2010.
Plant disease control

2. 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!
solid
topographic substrate
spin coat
liquid
conformal
coating
cure film
levelling
period
> 50μm
One of the major industrial applications is the deposition and flow of continuous thin liquid films over man-made or naturally occurring functional substrates containing regions of micro-scale topography. The main problem of this application occurs during the levelling or planarization stage, when the free surface of the coated film is desired to be kept as planar as possible, then it hardens and creates an artificial layer over given substrate. For example, you can see the capillary waves for thin film flow over thermoelectric micro-device that have to be levelled during the planarisation stage. In this work we would like to investigate how the free surface of the thin film is influenced by inertia of the fluid and compare capabilities of different mathematical formulations to capture the inertia effects.
Stillwagon, Larson and Taylor, J. Electrochem. Soc. 1987

3. 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.
One of the major industrial applications is the deposition and flow of continuous thin liquid films over man-made or naturally occurring functional substrates containing regions of micro-scale topography. The main problem of this application occurs during the levelling or planarization stage, when the free surface of the coated film is desired to be kept as planar as possible, then it hardens and creates an artificial layer over given substrate. For example, you can see the capillary waves for thin film flow over thermoelectric micro-device that have to be levelled during the planarisation stage. In this work we would like to investigate how the free surface of the thin film is influenced by inertia of the fluid and compare capabilities of different mathematical formulations to capture the inertia effects.

4. 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
Enables 3D flow to be modelled
by 2D systems of pdes.
gravity H0
inflow
One of the major industrial applications is the deposition and flow of continuous thin liquid films over man-made or naturally occurring functional substrates containing regions of micro-scale topography. The main problem of this application occurs during the levelling or planarization stage, when the free surface of the coated film is desired to be kept as planar as possible, then it hardens and creates an artificial layer over given substrate. For example, you can see the capillary waves for thin film flow over thermoelectric micro-device that have to be levelled during the planarisation stage. In this work we would like to investigate how the free surface of the thin film is influenced by inertia of the fluid and compare capabilities of different mathematical formulations to capture the inertia effects. y
s(x,y)
h(x,y) L0 x
outflow a

5. 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!
One of the major industrial applications is the deposition and flow of continuous thin liquid films over man-made or naturally occurring functional substrates containing regions of micro-scale topography. The main problem of this application occurs during the levelling or planarization stage, when the free surface of the coated film is desired to be kept as planar as possible, then it hardens and creates an artificial layer over given substrate. For example, you can see the capillary waves for thin film flow over thermoelectric micro-device that have to be levelled during the planarisation stage. In this work we would like to investigate how the free surface of the thin film is influenced by inertia of the fluid and compare capabilities of different mathematical formulations to capture the inertia effects.

6. 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!
One of the major industrial applications is the deposition and flow of continuous thin liquid films over man-made or naturally occurring functional substrates containing regions of micro-scale topography. The main problem of this application occurs during the levelling or planarization stage, when the free surface of the coated film is desired to be kept as planar as possible, then it hardens and creates an artificial layer over given substrate. For example, you can see the capillary waves for thin film flow over thermoelectric micro-device that have to be levelled during the planarisation stage. In this work we would like to investigate how the free surface of the thin film is influenced by inertia of the fluid and compare capabilities of different mathematical formulations to capture the inertia effects.

7. Depth-Averaged Formulation for Inertial Film Flows
Reduction of the Navier-Stokes equations by the long-wave approximation:
Restrictions:
2. Depth-averaging stage to decrease dimensionality of unknown functions by one: ,
The process of derivation of the Depth-Averaged Form includes reduction of the Navier-Stokes equations by the long-wave approximation, then averaging them along the depth of the film and finally assumption of self-similar velocity profile to estimate unknown terms.
Restrictions: no velocity profiles and internal flow structure
3. Assumption of Nusselt velocity profile to estimate unknown friction and dispersion terms:

8. Depth-Averaged Formulation for Inertial Film Flows
DAF system of equations:
For Re = 0 DAF ≡ LUB
Obtained system of equations for 3 unknown functions – averaged streamwise and spanwise velocities and film thickness is closed by appropriate boundary conditions. As you can see there are convective terms in the momentum equations, that represent inertial forces and will cause a difficulty during the numerical resolution. For zero Reynolds number or without this convective terms the DAF system and the lubrication approximation are equivalent. In this case an explicit forms of the averaged velocities are obtained and are substituted into the equation for the conservation of mass that results in one forth order partial differential equation.
Boundary conditions:
Inflow b.c.
Outflow (fully developed flow)
Occlusion b.c.

9. Flow over 3D trench: Effect of Inertia
Gravity-driven flow of thin water film: 130µm ≤ H0 ≤ 275µm over trench topography: sides 1.2mm, depth 25µm
surge
bow wave
comet tail

10. Accuracy of DAF approach
Gravity-driven flow of thin water film: 130µm ≤ H0 ≤ 275µm over 2D step-down topography: sides 1.2mm, depth 25µm
Max % Error vs Navier-Stokes (FE)
Error ~1-2% for Re=50 and s0 ≤0.2
s0=step size/H0

11. 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

12. 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.

13. 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, comet-tail disturbances.

14. 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
Need highly resolved grids for 3D flows
Flow over a maple leaf topography
Glass et al, Pest Management Science, 2010

15. 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 diamond-shaped topographies
Free Surface Plan View of Adaptive Grid

16. 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!

17. 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

18. 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.

19. 3D FE Navier-Stokes Solutions
Mixing in a Forward Roll Coater Due to Variable Roll Speeds
Substrate
Bath

20. 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

21. 3D FE Navier-Stokes vs DAF Solutions
Gravity-driven flow of a water film over a trench topography: comparison between free surface predictions

22. 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

23. Droplet Flows: Bio-pesticides
Droplet Flow Modelling and Analysis

24. 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

25. 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

26. 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?

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

28. Droplet size distribution for bio-pesticides
Matabi 12Ltr Elegance18+ knapsack sprayer
Teejet XR nozzle with 0.8bar

29. 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
Substance
Dv50 (μm)
c = 0%
c= 0.01%
c = 0.03%
c = 0.1%
c = 0.3%
water+adjuvant
273.3
275.1
269.4
330.5
352.9
water+carrier material
285.9
276.1
297.3
329.2
360.8
water+commercial product (biopesticide)
271.0
272.8
282.6
307.5
360.6

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

31. 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:
droplet flow is governed by this law:
Le Grand, Daerr & Limat (2005), experiments:

32. 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

33. 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

34. Droplet flow over a leaf (θ=60º): summary of computations
V, mm3
R, mm
Bosinθ Ca
a, m/s
Experiment
Computation Re=0
Computation Re=10
0.27
0.4
0.06
0.0003
0.02
0.0001
0.007 10
1.3
0.62
0.003
0.13
0.005
0.21
0.22 20
1.7
0.99
0.006
0.24
0.010
0.42
0.009
0.40 30
1.9
1.30
0.008
0.33
0.012
0.54
0.011
0.48 40
2.1
1.57
0.014
0.55

35. Droplet flow over a leaf: theory shows small effect of initial velocity
Relaxation time:

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

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

38. 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!