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 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

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.

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

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.

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.

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:

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.

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 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 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 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 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 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

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 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

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:

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:

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 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

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

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!