Rosen Tenchev

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Finite element analysis of non-Newtonian polymer
flows

• Rosen Tenchev
• School of Computing, University of Leeds

Research is part of MUPP2 Microscale Polymer
Processing - 2nd collaborative project
coordinated at the IRC in Polymer Science and
technology at the University of Leeds)
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Contents
1. Non-Newtonean fluids - introduction
2. Governing equation
3. Finite element discretization
4. Validation and examples
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Newtonian fluid
y
? - viscosity
(gases, liquids and solutions of low molecular
weight)
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• Non-Newtonian fluids
• - Time-independent fluids
• - Time-dependent fluids
• - Viscoelastic fluids

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Time-independent fluids
Bingham plastic
Dilatant (Shear-thickening) ngt1
Newtonian n1
Shear stress
Pseudoplastic (Shear-thinning) nlt1
Bingham fluids paints, ketchup, toothpaste
(yield stress must be exceeded to start the
flow) Shear thinning macromolecular fluids
(molecules are progressively aligned) Shear
thickenning suspension of solids at high solids
content (liquid lubricates motion)
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Time-dependent fluids
High molecular weight polymers
Thixotropic
Rheopectric
Shear stress
Shear stress
Shear rate
Shear rate
Hysteresis loops
(time history dependant)
Viscoelastic fluids
Deformation is gradually recovered when stress is
removed
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Viscoelastic fluids
Shear stress
Shear strain
Solid or fluid ? Depends on the observation
time scale!
Shear stress
Shear strain
0
0
time
time
Rheology Study of flow And deformation.
Elastic solid
Pure viscous fluid
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Governing equations
Equation of motion - Navier-Stokes equations,
stress divergence form
Stress decomposition deviatoric stress s
hydrostatic pressure P
Deviatoric stress decomposition
T - the extra stress due to polymer
deformation
Flow of high molecular weight polymers (melts or
solutions). Simplifying assumptions
No body force (gravity)
i.e. Stoke flow.
Slow flow and very high viscosity
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Mass conservation (continuity equation)
Incompressible fluid
Energy conservation (heat transfer equation)
Not used. Flow is at constant temperature.
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Polymer equations
Oldroyd B model
In 2D a system of 3 equations In 3D a
system of 6 equations
Rolie-Poly multi-mode model
n number of modes
Required material properties
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Relaxing the incompressibility equation
Mean strain (volumetric strain)
Mean stress
Constitutive equation
k bulk modulus
Relaxed incompressibility (penalty method)
k ? ? for incompressibility ? penalty number,
?gtgt1
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/ - of the Penalty Method
- or not exactly fulfilling the
incompressibility condition
(there are no ideally incompressible fluids)
General fluid flow
- not suitable for iterative solvers (high
condition number due to ?).
Polymer flow
best computational speed via the use of
direct solvers (LU decomposition).
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System of equations
In 3D system of 3 equations
Four mode Rolie-Poly 4 systems of 6 equations
in long hand (2D, one mode)
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Galerkins weighted residual method
Stoke equation
etc.
Similary for the y eq.
reduce integration
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Polymer equations (Petrov-Galerkin )
etc
Backwards Euler
where
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The finite element program
1. Mesh generation - built-in, file
driven, 2D and 3D - input from an
external mesh generator
2. Mesh and boundary condition visualisation
(model check)
3. FE solution - choice of
formulation Penalty method
(memory restrictions to keep K-1)
Velocity - pressure formulation (iterative
solution, no K-1) - choice of elements
linear or quadratic elements
triangles or quadrilaterals in
2D, tetr., hex., prisms in 3D.
4. Run time visualization of principal results
5. Post processing of all results
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Geometry input data 1. Nodes coordinates
2. SE definition
Mesh generation Mesh density and grading
defined for each SE.
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Checking input data - prescribed
boundary inflow
Zero x and y velocity (no slip boundary)
¼ model
Plane of symmetry
Zero y velocity (line of symmetry)
Plane of symmetry
Parabolic x velocity
Bi-parabolic x velocity
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Study of element performance
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Validation Comparison with other programs
Our FEM program (Eulerean formulation)
A finite volume program (based on Lagrangian
formulation)
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A Test Problem
Comparison with experimental results
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41 contraction flow (a quarter model)
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Polymer Axx
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Velocity profiles experiment v FEA
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Velocity profiles experiment v FEA
u mm/s
Black line - experiment Blue dots 3D
FEM Magenta dash 2D FEM.
x mm
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Birefringence ?1 -?2
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Birefringence plot 2D v 3D
Course mesh. Disturbances spreading downstream
from the corner
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Finer mesh. Disturbances are decreased
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Experiment
FE results
Birefringence ?1 -?2
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Future work
1. Studying and resolving some stability issues
at higher velocities
2. Extending the element library
3. Memory optimization for large 3D problems
(bandwith optimized symm. storage of K in Stoke
eq. has only 5 non-zeroes, factorized K-1
about 50. Changing to u-P formulation and
iterative solver for Stoke eq.)
4. Other material models suitable for molecularly
different polymer
5. Modelling flows with free surfaces
(moving mesh formulation)