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Entropy 2018, 20(1), 28; https://doi.org/10.3390/e20010028

Fractional Time Fluctuations in Viscoelasticity: A Comparative Study of Correlations and Elastic Moduli

1
Instituto de Física, Universidad Nacional Autónoma de México, Apdo. Postal 20-364, Cd. de México 01000, Mexico
2
Sistema Nacional de Investigadores, Mexico City 03940, Mexico
3
Proyecto Universitario de Fenómenos Lineales y Mecánica (FENOMEC), Universidad Nacional Autónoma de México, Apdo. Postal 20-364, Cd. de México 01000, Mexico
4
Departamento I. P. H., Universidad Autónoma Metropolitana, Iztapalapa, Apdo. Postal 55-534, Cd. de México 09340, Mexico
*
Author to whom correspondence should be addressed.
Received: 27 November 2017 / Revised: 25 December 2017 / Accepted: 3 January 2018 / Published: 11 January 2018
(This article belongs to the Special Issue Mesoscopic Thermodynamics and Dynamics)
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Abstract

We calculate the transverse velocity fluctuations correlation function of a linear and homogeneous viscoelastic liquid by using a generalized Langevin equation (GLE) approach. We consider a long-ranged (power-law) viscoelastic memory and a noise with a long-range (power-law) auto-correlation. We first evaluate the transverse velocity fluctuations correlation function for conventional time derivatives C ^ N F ( k , t ) and then introduce time fractional derivatives in their equations of motion and calculate the corresponding fractional correlation function. We find that the magnitude of the fractional correlation C ^ F ( k , t ) is always lower than the non-fractional one and decays more rapidly. The relationship between the fractional loss modulus G F ( ω ) and C ^ F ( k , t ) is also calculated analytically. The difference between the values of G ( ω ) for two specific viscoelastic fluids is quantified. Our model calculation shows that the fractional effects on this measurable quantity may be three times as large as compared with its non-fractional value. The fact that the dynamic shear modulus is related to the light scattering spectrum suggests that the measurement of this property might be used as a suitable test to assess the effects of temporal fractional derivatives on a measurable property. Finally, we summarize the main results of our approach and emphasize that the eventual validity of our model calculations can only come from experimentation. View Full-Text
Keywords: fluctuations; elastic moduli; correlation functions fluctuations; elastic moduli; correlation functions
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This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. (CC BY 4.0).
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Rodríguez, R.F.; Salinas-Rodríguez, E.; Fujioka, J. Fractional Time Fluctuations in Viscoelasticity: A Comparative Study of Correlations and Elastic Moduli. Entropy 2018, 20, 28.

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