SQUEEZE FLOW RHEOMETER: THE SOLUTION OF THE INVERSE PROBLEM FOR THE PARAMETERS OF CONSTITUTIVE EQUATION AND WALL SLIP

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SQUEEZE FLOW RHEOMETER THE SOLUTION OF THE
INVERSE PROBLEM FOR THE PARAMETERS OF
CONSTITUTIVE EQUATION AND WALL SLIP

• Hansong Tang Dilhan M. Kalyon
• Stevens Institute of Technology
• 12th JOCG Continuous Mixer and Extruder Users
  Group Meeting
• Indian Head, MD, October 31, 2002

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Compressive Squeeze Flow
F
ßt
SAMPLE
z
h
r
ßb
R
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Outline

• Introduction of the squeeze flow and the squeeze
  flow rheometer
• FEM model of the squeeze flow
• Analytical 1-D lubrication flow based model of
  the squeeze flow
• Comparison of the analytical model with the FEM
  solution for viscoplastic with wall slip
• The definition of the inverse problem for the
  determination of constitutive equation and the
  wall slip parameters.
• Typical solution of the inverse problem.

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COMPRESSIVE SQUEEZE FLOW

• Complex deformation field driven by both shear
  and normal stresses.
• Under small h/R conditions the shear stress
  dominates over the normal stresses and becomes a
  shear flow.
• When the disk surfaces are lubricated with a
  low-viscosity fluid, the normal stresses dominate
  and becomes biaxial extensional flow

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Compressive Squeeze Flow
F
ßt
Uz(z)
h
Ur(r,z)
z
r
ßb
R
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COMPRESSIVE SQUEEZE FLOW

• Radial flow with a r-velocity component, ur(r,
  z).

Shear
Extensional
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COMPRESSIVE SQUEEZE FLOW

• Generate conditions for which either the shear
  stress or the normal stress contribution is
  negligible.
• Characterize the shear viscosity
• Upon lubrication of the surfaces of the disks
  with a lubricant with a relatively low shear
  viscosity, as an biaxial extensional rheometer

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Compressive Squeeze Flow as a Biaxial
Extensional Rheometer
F
ßt
Low viscosity lubricant
SAMPLE
z
h
r
ßb
R
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FEM ANALYSIS
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Compressive Squeeze Flow
ßt
FEM MESH
z
h
r
ßb
R
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Is the elasticity of the melt important?
Predictions vs experimental force measurements
(Zhang et al., 1995) LLDPE with m 5.91x103
Pa.s0.6 and n0.6 without wall slip
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CONCLUSIONS

• Even for high molecular weight polymer melts
  conditions exist for which generalized Newtonian
  fluid is valid.
• A combination of numerical methods and
  experimental results are necessary to determine
  the conditions for which elasticity can be
  neglected

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HERSCHEL BULKLEY 1-D
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Herschel Bulkley Fluid
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1-D or 2-d analysis
n
t
Naviers slip condition
Unit normal vector
Unit tangent vector
Total stress tensor
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Is it possible to simplify the Equations of
Motion to enable analytical methods?

• Allow compressive squeeze flow to be used as a
  rheometer to characterize the shear viscosity
  material function.
• The traditional method is to employ the
  lubrication assumption

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1-D ANALYSIS WITH THE LUBRICATION APPROXIMATION
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Compressive Squeeze Flow
F
ßt
Uz(z)
h
Ur(r,z)
z
r
ßb
R
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1-D SOLUTION WITH DIFFERENT SLIP COEFFICIENTS AT
TWO SURFACES
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DIMENSIONLESS PARAMETERS
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Top disk
?2
Unsheared region
Unsheared region
?1
Bottom disk
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1-D SOLUTION WITH DIFFERENT SLIP COEFFICIENTS AT
TWO SURFACES
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VELOCITY DISTRIBUTION 1-D HERSHEL BULKLEY WITH
SLIP
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TO DETERMINE THE EXTREMUM SOLVE
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TO DETERMINE THE PRESSURE GRADIENT SOLVE
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FORCE DETERMINATION
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For the case where the slip coefficients are
equal, ?b ?t
?1 ?2 ?1 ?2, the velocity profile
becomes symmetrical around z h(t)/2 to give
?2 h(t) -?1 .
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Viscoplastic Fluids

• Analytical solutions of viscoplastic fluids are
  handicapped
• Use of the lubrication assumption and the
  resulting 1-D analysis for viscoplastic fluids
  always predicts an unyielded zone.
• However, by definition the unyielded zone does
  not exist in squeeze flow.

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Relative difference between the force values for
the 1-D and 2-D for a power law fluid with
n0.45.
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Radial velocity profiles predicted by the 1-D and
2-D analyses with wall slip n0.45, ß 1.0,
h/R0.01
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Radial velocity predicted by the 1-D and 2-D
analyses of a power law fluid with wall slip
n0.45, ß1.0, h/R0.25
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Radial velocity predicted by the 1-D and 2-D
analyses of a power law fluid with wall slip
n0.45, ß5.0, h/R0.01.
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Radial velocity predicted by the 1-D and 2-D
analyses of a power law fluid with wall slip
n0.45, ß5.0, h/R0.25.
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HDPE DuPont 2005, Zhang et al. 1995 data,
m44,900 Pa-s 0.25 , n0.25, ß5.13E-15m/(Pa
2.35 s)
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Force values for the 1-D and 2-D analyses vs.
yield stress for Herschel-Bulkley fluids with
n0.45 without wall slip.
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Radial velocity profiles predicted by the 1-D and
2-D analyses for Herschel-Bulkley fluid without
wall slip n0.45, ?y25 gap-to-radius ratio
h/R0.01.
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Radial velocity profiles predicted by the 1-D and
2-D analyses for Herschel-Bulkley fluid without
wall slip n0.25, ?y25 gap-to-radius ratio
h/R0.25.
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Radial velocity profiles predicted by the 1-D and
2-D analyses for Herschel-Bulkley fluid without
wall slip n0.45, ?y75 gap-to-radius ratio
h/R0.01.
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Radial velocity profiles predicted by the 1-D and
2-D analyses for Herschel-Bulkley fluid without
wall slip n0.45, ?y75 gap-to-radius ratio
h/R0.25.
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CONCLUSIONS

• Under some conditions lubrication assumption
  holds for viscoplastic fluids and/or wall slip
  h/R lt 0.05.
• Analytical models of the compressive squeeze flow
  of viscoplastic materials with Naviers wall
  slip can be used in the relatively low h/R range.

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CONCLUSIONS

• Accuracy of 1-D analysis decreases
• -with increasing yield stress
• -increasing Naviers wall slip coefficient
• Validity of the 1-D analysis should be
  determined with numerical analysis.

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CONCLUSIONS

• Numerical analysis is necessary to determine the
  conditions under which
• -the elasticity of the melt can be
  neglected
• -range in which the lubrication assumption
  is valid.

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CONCLUSIONS

• At proper squeeze flow conditions compressive
  squeeze flow can be analyzed with analytical
  models.
• Under such conditions use as a shear rheometer
  and for Naviers wall slip condition

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Need to develop further

• However, in all these analyses one starts with
  known parameters and solves for the pressure,
  force etc.
• The parameters can only be determined from
  analytical solution for Power Law fluid without
  slip (two parameters).
• Can one use the squeeze flow as a rheometer
  without trial and error for viscoplastic with
  wall slip?

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Inverse problem

• Inverse problem
• Objective function
• Error control

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Inverse problem

• Deepest descent method
• where

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Inverse problem

• Conjugate method
• Suppose
• Then
• Orthogonality and conjugacy conditions

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Inverse problem

• Sensitivity

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Conclusion of the probing of the inverse solution

• It is not possible to determine all five
  parameters using the inverse approach
• However, if three parameters are known using
  other methods then the remaining two parameters
  can be determined in a unique fashion from the
  analysis of a single squeeze from squeeze flow.

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Numerical example

• Squeeze flow

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Numerical example

• Capillary flow

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Conclusion

• The same inverse analysis can be done on flow
  through a capillary die
• The conclusion is the same as that of the squeeze
  flow, only two parameters at a time can be
  determined in a unique fashion using solely
  capillary flow
• Thus, need to combine multiple rheometers for
  adequate solution of the inverse problem.

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Numerical example

• Global minimum solution

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Application

• Pure PDMS material

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Application

• Identified parameters

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Application

• PDMS with 20 and 40 fillers

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Application

• Identified parameters
• 20 PDMS
• 40PDMS

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Conclusion

• Inverse problem has advantages
• Squeeze and capillary flows together will
  likely provide the best methodology of
  determining the parameters of wall slip and
  viscoplasticity.

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Acknowledgement

• We acknowledge with gratitude the support of
  TACOM/ARDEC, NSWC/IH, BMDO/IST (ONR), DARPA, PBMA
  and various companies including Unilever,
  Duracell, Henkel-Loctite, GPU, and MPR which made
  this investigation possible.