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SIMPLE JOINT INVERSION LOCALIZED FORMULAE FOR RELAXATION SPECTRUM RECOVERY

Part of:Integral equations, integral transformsVolterra integral equationsTheoretical approximation of solutionsFoundations, constitutive equations, rheology

Published online by Cambridge University Press:  05 July 2016

R. S. ANDERSSEN,
A. R. DAVIES,
F. R. de HOOG and
R. J. LOYOpen the ORCID record for R. J. LOY[Opens in a new window]
R. S. ANDERSSEN*
Affiliation:
CSIRO Data61, GPO Box 664, Canberra, ACT 2601, Australia email Bob.Anderssen@csiro.au, Frank.deHoog@csiro.au
A. R. DAVIES
Affiliation:
Mathematics, University of Cardiff, Cardiff, UK email DaviesR@cardiff.ac.uk
F. R. de HOOG
Affiliation:
CSIRO Data61, GPO Box 664, Canberra, ACT 2601, Australia email Bob.Anderssen@csiro.au, Frank.deHoog@csiro.au
R. J. LOY
Affiliation:
Mathematical Sciences Institute, Australian National University, Canberra, ACT 2601, Australia email Rick.Loy@anu.edu.au
*
✉ Bob.Anderssen@csiro.au
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Abstract

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In oscillatory shear experiments, the values of the storage and loss moduli, $G^{\prime }(\unicode[STIX]{x1D714})$ and $G^{\prime \prime }(\unicode[STIX]{x1D714})$ , respectively, are only measured and recorded for a number of values of the frequency $\unicode[STIX]{x1D714}$ in some well-defined finite range $[\unicode[STIX]{x1D714}_{\text{min}},\unicode[STIX]{x1D714}_{\text{max}}]$ . In many practical situations, when the range $[\unicode[STIX]{x1D714}_{\text{min}},\unicode[STIX]{x1D714}_{\text{max}}]$ is sufficiently large, information about the associated polymer dynamics can be assessed by simply comparing the interrelationship between the frequency dependence of $G^{\prime }(\unicode[STIX]{x1D714})$ and $G^{\prime \prime }(\unicode[STIX]{x1D714})$ . For other situations, the required rheological insight can only be obtained once explicit knowledge about the structure of the relaxation time spectrum $H(\unicode[STIX]{x1D70F})$ has been determined through the inversion of the measured storage and loss moduli $G^{\prime }(\unicode[STIX]{x1D714})$ and $G^{\prime \prime }(\unicode[STIX]{x1D714})$ . For the recovery of an approximation to $H(\unicode[STIX]{x1D70F})$ , in order to cope with the limited range $[\unicode[STIX]{x1D714}_{\text{min}},\unicode[STIX]{x1D714}_{\text{max}}]$ of the measurements, some form of localization algorithm is required. A popular strategy for achieving this is to assume that $H(\unicode[STIX]{x1D70F})$ has a separated discrete point mass (Dirac delta function) structure. However, this expedient overlooks the potential information contained in the structure of a possibly continuous $H(\unicode[STIX]{x1D70F})$ . In this paper, simple localization algorithms and, in particular, a joint inversion least squares procedure, are proposed for the rapid recovery of accurate approximations to continuous $H(\unicode[STIX]{x1D70F})$ from limited measurements of $G^{\prime }(\unicode[STIX]{x1D714})$ and $G^{\prime \prime }(\unicode[STIX]{x1D714})$ .

Keywords

relaxation spectrum approximationoscillatory shear datajoint inversionnumerical differentiationrheology

MSC classification

Primary:45L05: Theoretical approximation of solutions
Secondary:45D05: Volterra integral equations65R20: Integral equations76A05: Non-Newtonian fluids

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 58 , Issue 1 , July 2016, pp. 1 - 9
Copyright
© 2016 Australian Mathematical Society 

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