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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,

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3 d film and droplet flows over topography
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


3d film flow over topography

spin coat

liquid

> 50μm

conformal

liquid

coating

topographic substrate

levelling

period

cure film

solid

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!

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


3d film flow over topography1
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 topography2
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

y

s(x,y)

h(x,y)

L0

x

outflow

a


3d film flow over topography3
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 topography4
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
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:

,

Restrictions: no velocity profiles and internal flow structure

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


Depth-Averaged Formulation for Inertial Film Flows

DAF system of equations:

For Re = 0 DAF ≡ LUB

Boundary conditions:

Inflow b.c.

Outflow (fully developed flow)

Occlusion b.c.


Flow over 3d trench effect of inertia
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


Accuracy of daf approach
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
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
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


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
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 solutions1
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 solutions2

Substrate

Bath

3D FE Navier-Stokes Solutions

Mixing in a Forward Roll Coater Due to Variable Roll Speeds


3d fe navier stokes solutions3
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
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 solutions4
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
Droplet Flows: Bio-pesticides

  • Droplet Flow Modelling and Analysis


Application of bio pesticides
Application of Bio-pesticides

ChangingEU 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

  • 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
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
Droplet spray evaporation time: effect of adjuvant


D roplet size distribution for bio pesticides
Droplet 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
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


Droplet flow over a leaf simple theory
Droplet flow over a leaf: simple theory concentration of adjuvant

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:


Droplet flow over a leaf simple theory vs experiments
Droplet flow over a leaf: simple theory vs. experiments concentration of adjuvant

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:


Droplet flow over a leaf 60 effect of inertia
Droplet flow over a leaf ( concentration of adjuvantθ=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 60 effect of inertia1
Droplet flow over a leaf ( concentration of adjuvantθ=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 60 summary of computations
Droplet flow over a leaf ( concentration of adjuvantθ=60º): summary of computations


Droplet flow over a leaf theory shows small effect of initial velocity
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
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 60 vs under 120 a leaf computation
Droplet flow over ( initial conditionθ=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
Bio-pesticides: initial conclusions initial condition

  • 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!


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