3-D Film and Droplet Flows over Topography

Plant disease control

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.

3D Film Flow over Topography

solid

topographic substrate

spin coatliquid

conformalliquid

coating

cure film

levellingperiod

> 50μm

Stillwagon, Larson and Taylor, J. Electrochem. Soc. 1987

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!

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

Enables 3D flow to be modelled

by 2D systems of pdes.

x

y

h(x,y)

s(x,y)

gravity

inflow

outflow

L0

H0

3D Film Flow over 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!

Decre & Baret, JFM, 2003: Flow of Water Film over a Trench Topography

3D Film Flow over TopographyThin 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

100 LH 1Ca

1. Reduction of the Navier-Stokes equations by the long-wave approximation:

3. Assumption of Nusselt velocity profile to estimate unknown friction and dispersion terms:

2213 uu 2213 vvh

sz

2. Depth-averaging stage to decrease dimensionality of unknown functions by one:

f

s

udzh

tyxu1

,, f

s

vdzh

tyxv1

,,

,

),(,,,, yxstyxftyxh

Restrictions: 1, yxs

Restrictions: no velocity profiles and internal flow structure

23

cot25

6

5Re

22

3

h

ushsh

Caxy

uv

x

uu

t

h

h

u

t

u

2

23 3

cot25

6

5Re

h

vshsh

Cayy

vv

x

vu

t

h

h

v

t

v

0

y

vh

x

uh

t

h

1,0,

3

2,, 0xhvu

0,,,, ,0

pp wylx hvuy

hvux

DAF system of equations:

Boundary conditions:

1.Inflow b.c.

2.Outflow (fully developed flow)

3.Occlusion b.c.

For Re = 0 DAF ≡ LUB

2

tan,

Syx nh 0, , yxvu

Depth-Averaged Formulation for Inertial Film Flows

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

bow wavesurge

comet tail

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

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, comet-tail 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 topographyGlass et al, Pest Management Science, 2010

Need highly resolved grids for 3D flows

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

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.

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 Solutions

Gravity-driven flow of a water film over a trench topography: comparison between free surface predictions

3D FE Navier-Stokes vs DAF 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

3D FE Navier-Stokes Solutions

Droplet Flows: Bio-pesticides

Droplet Flow Modelling and Analysis

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 evaporation time: effect of adjuvant

Size of droplets

Concentration (%)

Initial mass (mg)

Mass fraction left after 10 min (%)

Evaporation time (min)

large 0 130.3

36.3 26.3

0.01 138.0

36.6 24.0

0.1 161.0

48.7 36.0

small 0 87.3 13.3 16.30.01 92.5 9.7 16.00.1 138.

333.3 25.7

Droplet size distribution for bio-pesticides

Teejet XR110 05 nozzle with 0.8bar

Matabi 12Ltr Elegance18+ knapsack sprayer

VMD of the bio-pesticide spray depending on the concentration of adjuvant

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

addition of bio-pesticide does not affect Volume Mean Diameter of the spray

Droplet flow over a leaf: simple theory

2nd Newton’s law in x direction:

Stokes drag:

Terminal velocity:

Velocity:

Contact angle hysteresis:

Relaxation time:

theoretical expressions from Dussan (1985):

Volume of smallest droplet that can move:

Droplet flow over a leaf: simple theory vs. experiments

Podgorski, Flesselles, Limat (2001) experiments:

Dussan (1985) theory:

Le Grand, Daerr & Limat (2005), experiments:

47V10 silicon oil drops flowing over a fluoro-polymer FC725 surface:

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, mm3

R, mm

BosinθCa a, m/s Ca a, m/s Ca a, m/s

ExperimentComputation

Re=0Computation

Re=10

0.27 0.4 0.06 0 00.000

30.02

0.0001

0.007

10 1.3 0.62 0.003 0.13 0.005 0.21 0.005 0.2220 1.7 0.99 0.006 0.24 0.010 0.42 0.009 0.4030 1.9 1.30 0.008 0.33 0.012 0.54 0.011 0.4840 2.1 1.57 0.011 0.48 0.014 0.62 0.012 0.55

Droplet flow over a leaf: theory shows small effect of initial velocity

Relaxation time:

Initial velocity:

Velocity:

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

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

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!