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SIMPLE JOINT INVERSION LOCALIZED FORMULAE FOR RELAXATION SPECTRUM RECOVERY

Part of: Theoretical approximation of solutions Integral equations, integral transforms Foundations, constitutive equations, rheology Volterra integral equations

Published online by Cambridge University Press:  05 July 2016

R. S. ANDERSSEN ,
A. R. DAVIES ,
F. R. de HOOG  and
R. J. LOY Open 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 equations 65R20: Integral equations 76A05: Non-Newtonian fluids
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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