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
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 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. gravity H0 inflow 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!
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 • 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 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.
Substrate Bath 3D FE Navier-Stokes Solutions Mixing in a Forward Roll Coater Due to Variable Roll Speeds
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 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 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 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
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: theory shows small effect of initial velocity Velocity: 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!