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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.

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3D Film Flow over Topography solid topographic substrate spin coat liquid conformal liquid coating cure film levelling period > 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!

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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.

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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 L0L0 H0H0

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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

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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!

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Depth-Averaged Formulation for Inertial Film Flows 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: 2.Depth-averaging stage to decrease dimensionality of unknown functions by one:, Restrictions: Restrictions: no velocity profiles and internal flow structure

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DAF system of equations: Boundary conditions: 1.Inflow b.c. 2.Outflow (fully developed flow) 3.Occlusion b.c. For Re = 0 DAF ≡ LUB Depth-Averaged Formulation for Inertial Film Flows

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

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

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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

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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.

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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.

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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 topography Glass et al, Pest Management Science, 2010 Need highly resolved grids for 3D flows

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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

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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!

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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

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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.

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

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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

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

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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

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Droplet Flows: Bio-pesticides Droplet Flow Modelling and Analysis

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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

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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

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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?

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Droplet spray e vaporation time: effect of adjuvant Size of droplet s Conce ntratio n (%) Initial mass (mg) Mass fraction left after 10 min (%) Evapor ation time (min) large0130.336.326.3 0.01138.036.624.0 0.1161.048.736.0 small087.313.316.3 0.0192.59.716.0 0.1138.333.325.7

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

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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+adjuvant273.3275.1269.4330.5352.9 water+carrier material 285.9276.1297.3329.2360.8 water+commercial product (biopesticide) 271.0272.8282.6307.5360.6 addition of bio-pesticide does not affect Volume Mean Diameter of the spray

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Droplet flow over a leaf: simple theory 2 nd 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:

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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:

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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

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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

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Droplet flow over a leaf (θ=60º): summary of computations V, mm 3 R, mm Bosinθ Caa, m/sCaa, m/sCaa, m/s Experiment Computation Re=0 Computation Re=10 0.270.40.06000.00030.020.00010.007 101.30.620.0030.130.0050.210.0050.22 201.70.990.0060.240.0100.420.0090.40 301.91.300.0080.330.0120.540.0110.48 402.11.570.0110.480.0140.620.0120.55

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

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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

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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

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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!

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