The efficient and accurate solution of continuous thin film flow over localised surface patterning and past occlusions.

1
The efficient and accurate solution of continuous
thin film flow over localised surface patterning
and past occlusions.

• YC Lee, HM Thompson and PH Gaskell
• School of Mechanical Engineering, University of
  Leeds, UK

2
Contents

• Motivation
• The Lubrication Approach
• Numerical Method
• Adaptive Multigrid
• Results
• Conclusions

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Motivation

• Continuous thin film flows over surface patterns
  and topography have many practical applications
• Manufacture of photographic film and ink-jet
  media
• Deposition of coatings and printings
• Photolithographic production of printed circuit
  and displays
• Micro-fluidic devices
• Redistribution of liquid films on biological
  surfaces
• Heat exchangers
• Important to understand and predict associated
  flow phenomena.

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Motivation

• Problem considered here are gravity-driven flow
• over simple and complex surface patterning made
  up from combination of simple primitives
• past solid occlusions

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The Lubrication Approach
Lubrication equations - asymptotic expansion of
the Navier-Stokes equations in terms of ?
H0/L0. Valid for small free surface gradients

• Scaling (Aksel (2000))
• H0 (3?Q/?g sin?) 1/3 film thickness for the
  fully developed film flow down an inclined plane,
  U0surface velocity
• L0 (?H0/3 ?g sin?) 1/3 Capillary length
• Dimensionless groups
• ? H0/L0 ltlt 1, Ca ?U0/?.

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The Lubrication Approach

• Lubrication equations

Note Equation (1) ensures mass conservation and
equation (2) gives pressure due to Capillary and
hydrostatic terms. Possible to account for
variable viscosity (due to temperature,
concentration etc) and surface tension. Here
focus on flow of water films with
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Numerical Method

• Discretisation

Finite Difference discretisation Control
Volumes centred at grid vertices. Time
integration using Crank-Nicolson.
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Numerical Method
Method of Solution Non-linear lubrication
equations are challenging to solve numerically.
Usual approach is to use semi-implicit
Alternating Direction Implicit methods -
Time-Splitting. Our approach use Full Multigrid
(FMG) and Full Approximation Storage (FAS)
remove longer wavelength errors by relaxation on
coarser grid levels.
G0 9x9 G1 17x17 G2 33x33 G3 65x65 etc
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Numerical Method
Advantage Efficiency of the non-adaptive
Multigrid approach (1) CPU time for a given
number of unknowns, N, is O(N). (2)
Implicit good stability (3) Lends itself to
adaptive mesh refinement
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Numerical Method
2-D 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 can be very accurate.
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Numerical Method
Error is classified by the maximum difference
between the Navier-Stokes and lubrication profiles
Shows how error increases with Reynolds number
(Re) and topography height. Contours show
recirculation in the Navier-Stokes solutions for
Re15 (top) and Re0.15 (bottom) where errors are
respectively 13 and 8.5.
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Adaptive Multigrid

• Purpose To capture the effects of distribution
  of small, isolated features efficiently. Use of
  coarse grids in region of smooth flow.
• Areas requiring local grid refinement are
  identified using the local truncation error
  analysis.

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Adaptive Multigrid
Numerical Analysis Re-write discretised
lubrication equations on the kth grid in the
form where are the
unknown h and p variables on grid k and the
superscripts (n1), n indicate values of these
variables at end of (n1)st and nth time steps
respectively. Residual
measures the error in satisfying the
discretised equations on grid level k.
Relative Truncation Error where
is a Restriction Operator from Grid k to Grid
k-1. Large values of indicate local
refinement needed.
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Adaptive Multigrid
Interface boundaries Adaptive refinement
proceeds by conserving numerical flux per Control
Volume (CV) area at coarse and locally-refined
regions. O grid vertex on fine grid k ? - grid
vertex on next coarsest grid k-1 ? - ghost node
at adaptive boundary
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Adaptive Multigrid
Preserving flux Adaptive refinement proceeds
by conserving numerical flux per Control Volume
(CV) area at coarse and locally-refined
regions. For lubrication equations flux across
CV defined by (Lee et al (2006)) where

or At boundary of local refinement interface
conserving flux per area Enables values of h
and p at ghost nodes to be determined as
Dirichlet conditions for the adaptive solution.
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Results
Flow of thin water films of asymptotic thickness
100 µm, at constant flow rate 1.64 x 10-6 m2/s.
Capillary length LC0.78mm, N0.122 (gravity
little influence on free surface) Results
obtained using a FMG V(2,2) cycle with a 9x9
coarse grid, and finest grid level, k5, of mesh
size 1/256. Flow domain extends over 40
Capillary lengths in each direction 39mm x 39mm
square domain.
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Results

• Simple circuit case

Circuit overall length 10.9mm, width 3.6mm, 30µm
height. Performed on a 17x17 global coarse grid
refinement performed over next four finer levels
if Resultant adaptive mesh and free-surface
profile at steady-state
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Results

• Effects of circuit orientation on amplitude of
  free surface disturbance.
• Varying orientation of the simple circuit
  pattern to the direction of flow
• 0 to 90 degrees
• 0o 45o 90o

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Results

• Maximum free-surface disturbance occurs at 20
  degrees from both ends (0o and 90o).
• Maximum amplitude at 70o (8.8)
• Minimum amplitude at 40o (6.5)

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Results

• Complex circuit case
• 78mm x 78mm square domain.
• Retaining similar circuit dimensions.
• Shows flexibility of adaptive scheme
• CPU time less than 10 than those performed
  non-adaptively.

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Results

• Skew elliptical occlusion case
• Skew elliptical occlusions of semi-major axis
  145mm and semi-minor axis 72mm
• Rectangular domain 14.4mm x 7.2mm
• Capillary length LC0.36mm, N0.026
• Asymptotic film thickness 10mm
• Global coarse grid 65x33
• Up to 5 refinement levels, mesh size 1/1024

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Results

• Large and wide disturbance upstream and
  downstream.
• Requires large domain to negate boundary
  influences.
• Requires fine resolution to model problem
  accurately due to small occlusion.
• Free surface Iso-contour

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Results

• To achieve grid independence solution, need very
  fine grid levels.
• Adaptive approach require only 14440 nodes
  compared to over 2 million nodes if performed
  non-adaptively.

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Results

• Inlet wave flow past circular occlusion
• Transient problem shows evolution of adaptive
  grids.
• 78mm x 78mm square domain.
• Circular occlusion radius 1.95mm

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Results
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Conclusions

• Continuous thin film flow over surface patterning
  and past occlusions arise in many applications.
• Lubrication approach can yield valuable insight.
• Adaptive multigrid is very efficient and provides
  the ability to effectively solve flow past
  topography and occlusion distributions.
• Little experimental data is currently available.