Fractional Time Fluctuations in Viscoelasticity: A Comparative Study of Correlations and Elastic Moduli
Instituto de Física, Universidad Nacional Autónoma de México, Apdo. Postal 20-364, Cd. de México 01000, Mexico
Sistema Nacional de Investigadores, Mexico City 03940, Mexico
Proyecto Universitario de Fenómenos Lineales y Mecánica (FENOMEC), Universidad Nacional Autónoma de México, Apdo. Postal 20-364, Cd. de México 01000, Mexico
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
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
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
is always lower than the non-fractional one and decays more rapidly. The relationship between the fractional loss modulus
is also calculated analytically. The difference between the values of
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.
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MDPI and ACS Style
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.
Rodríguez RF, Salinas-Rodríguez E, Fujioka J. Fractional Time Fluctuations in Viscoelasticity: A Comparative Study of Correlations and Elastic Moduli. Entropy. 2018; 20(1):28.
Rodríguez, Rosalío F.; Salinas-Rodríguez, Elizabeth; Fujioka, Jorge. 2018. "Fractional Time Fluctuations in Viscoelasticity: A Comparative Study of Correlations and Elastic Moduli." Entropy 20, no. 1: 28.
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