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

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Model Formulation

#### 2.1. Hydrodynamic Fluctuations

#### 2.2. Power Law Viscoelasticity

#### 2.3. Transverse Velocity Correlation

_{2}) and a solution of 0.02% separan MG500+2% water in glucose (E

_{1}). Chhabra et al. [26] have studied the rheological properties of these fluids and according to their shear stress and normal stress data, S

_{2}would be classified as a weakly elastic fluid with a small λ, whereas E

_{1}is exceedingly elastic and has a large λ. Figure 1 shows the plot of ${\widehat{C}}_{NF}(\overrightarrow{k},t)$ as a function of time t, as given by Equation (20), for the small values λ = 0.03, 0.06, which would correspond to a fluid like S

_{2}.

_{1}, as λ increases from λ = 0.3 to λ = 0.4. However, in this case ${\widehat{C}}_{NF}(\overrightarrow{k},t)$ decays three orders of magnitude faster than for S

_{2}.

## 3. Time Fractional Derivatives

_{2}and for the same parameter values used in Figure 1.

## 4. Dynamic Shear Modulus

_{s}is the shear viscosity, one can see that $\widehat{G}\left(s\right)$ plays the role of a dynamic shear viscosity. It is useful to define the dynamic shear modulus as

_{2}this yields the solid curve for ${G}_{NF}^{\u2033}\left(\omega \right)$ and the dotted curve for ${G}_{F}^{\u2033}\left(\omega \right)$ in Figure 5.

_{1}as shown in Figure 6.

_{2}and E

_{1}, respectively.

_{1}; R is larger than one for the frequency intervals considered. Note, for instance, that R has a maximum value of about R ~ 3.4 at ω = 8.3 × 10

^{7}s

^{−1}and a minimum value of R ~ 1.63 for ω = 1 × 10

^{5}s

^{−1}. This means that fractional fluctuations have a large effect on the transverse velocity fluctuations correlation than on the velocity correlation. These results indicate that the relative changes in R, are not a small effect and might be measurable. The same behavior of R is observed for S

_{2}in Figure 8.

## 5. Discussion

## Author Contributions

## Conflicts of Interest

## References

- Zwanzig, R. Time-correlation functions and transport coefficients in statistical mechanics. Annu. Rev. Phys. Chem.
**1964**, 16, 67–102. [Google Scholar] [CrossRef] - Onsager, L.; Machlup, S. Fluctuations and Irreversible Processes. Phys. Rev.
**1953**, 91, 1505. [Google Scholar] [CrossRef] - Green, M.S. Markoff Random Processes and the Statistical Mechanics of Time-Dependent Phenomena. II. Irreversible Processes in Fluids. J. Chem. Phys.
**1954**, 22, 398–413. [Google Scholar] [CrossRef] - Fox, R.F. The generalized Langevin equation with Gaussian fluctuations. J. Math. Phys.
**1977**, 18, 2331–2336. [Google Scholar] [CrossRef] - Fox, R.F. Gaussian stochastic processes in physics. Phys. Rep.
**1978**, 48, 179–283. [Google Scholar] [CrossRef] - Grigolini, P.; Rocco, A.; West, B.J. Fractional calculus as a macroscopic manifestation of randomness. Phys. Rev. E
**1999**, 59, 2603–2613. [Google Scholar] [CrossRef] - West, B.J. Fractional calculus view of complexity: A tutorial. Rev. Mod. Phys.
**2014**, 86, 1169–1184. [Google Scholar] [CrossRef] - West, B.J. Physiology, Promiscuity and Prophecy at the Millennium: A Tale of Tails; Studies of Nonlinear Phenomena in the Life Sciences; World Scientific: Singapore, 1999; Volume 7. [Google Scholar]
- Rodríguez, R.F.; Fujioka, J.; Salinas-Rodríguez, E. Fractional fluctuations effects on the light scattered by a viscoelastic suspension. Phys. Rev. E
**2013**, 88, 022154. [Google Scholar] [CrossRef] [PubMed] - Rodríguez, R.F.; Fujioka, J.; Salinas-Rodríguez, E. Fractional correlation functions in simple viscoelastic liquids. Physica A
**2015**, 427, 326–340. [Google Scholar] [CrossRef] - Rodríguez, R.F.; Fujioka, J. Generalized hydrodynamic correlations and fractional memory functions. J. Non-Equilib. Thermodyn.
**2015**, 40, 295–305. [Google Scholar] [CrossRef] - Glöckle, W.G.; Nonnenmacher, T.F. Fox function representation of non-Debye relaxation processes. J. Stat. Phys.
**1993**, 71, 741–757. [Google Scholar] [CrossRef] - Rocco, A.; West, B.J. Fractional calculus and the evolution of fractal phenomena. Physica A
**1999**, 265, 535–546. [Google Scholar] [CrossRef] - Uchaikin, V.V. Fractional Derivatives for Physicists and Engineers; Springer: Berlin, Germany, 2013; Volume I. [Google Scholar]
- Santamaría-Holek, I.; Rubí, J.M.; Gadomski, A. Thermokinetic approach of single particles and clusters involving anomalous diffusion under viscoelastic response. J. Phys. Chem. B
**2007**, 111, 2293–2298. [Google Scholar] [CrossRef] [PubMed] - Mainardi, F. Fractional Calculus and Waves in Linear Viscoelasticity; Imperial College Press: London, UK, 2010. [Google Scholar]
- Ferry, J.C. Viscoelastic Properties of Polymers, Chapter 1, 3rd ed.; Wiley: New York, NY, USA, 1980. [Google Scholar]
- Wang, C.H.; Fischer, E.W. Density fluctuations, dynamic light scattering, longitudinal compliance, and stress modulus in a viscoelastic medium. J. Chem. Phys.
**1985**, 82, 632–639. [Google Scholar] [CrossRef] - Wang, C.H. Depolarized Raleigh-Brillouin scattering of shear waves and molecular reorientation in a viscoelastic liquid. Mol. Phys.
**1980**, 41, 541–565. [Google Scholar] [CrossRef] - Kubo, R.; Toda, M.; Hashitsume, N. Statistical Physics II, Nonequilibrium Statistical Mechanics; Springer: Berlin, Germany, 1985. [Google Scholar]
- Leptos, K.C.; Guasto, J.S.; Gollub, J.P.; Pesei, A.I.; Goldstein, R.E. Dynamics of enhanced tracer diffusion in suspension of swimming eukaryotic microorganisms. Phys. Rev. Lett.
**2009**, 103, 198103. [Google Scholar] [CrossRef] [PubMed] - Eckardt, B.; Zammert, S. Non-normal tracer diffusion from stirring by swimming microorganisms. Eur. Phys. J. E
**2012**, 35, 96–97. [Google Scholar] [CrossRef] [PubMed] - Zaid, I.M.; Dunkel, J.; Yeomans, J.M. Lévy fluctuations and mixing in dilute suspensions NASA/TP-1999-209424/REVI; Tof algae and bacteria. J. R. Soc. Interface
**2011**, 8, 1314–1331. [Google Scholar] [CrossRef] [PubMed] - Jaishankar, A.; McKinley, G.H. Power-law rheology in the bulk and at the interface: Quasi-properties and fractional constitutive equations. Proc. R. Soc. A
**2013**, 469. [Google Scholar] [CrossRef] - Lorenzo, C.F.; Hartley, T.T. Generalized Functions for the Fractional Calculus; NASA/TP-1999-209424/REVI; The National Aeronautics and Space Administration: Washington, DC, USA, 1999; pp. 1–17.
- Chhabra, R.P.; Uhlherr, P.H.T.; Boger, D.V. The influence of fluid elasticity on the drag coefficient for creeping flow around a sphere. J. Non-Newton. Fluid Mech.
**1980**, 6, 187–199. [Google Scholar] [CrossRef] - Greaves, G.N.; Greer, A.L.; Lakes, R.S.; Rouxel, T. Poisson’s ratio and modern materials. Nat. Mater.
**2011**, 10, 823–837. [Google Scholar] [CrossRef] [PubMed] - Caputo, M.; Cametti, C. Diffusion with memory in two cases of biological interest. J. Theor. Biol.
**2008**, 254, 697–703. [Google Scholar] [CrossRef] [PubMed] - Podlubny, L. Fractional Differential Equations; Academic Press: San Diego, CA, USA, 1999. [Google Scholar]
- Kilbas, A.; Srivastava, H.M.; Trujillo, J.J. Theory and Applications of Fractional Differential Equations; Elsevier: Amsterdam, The Netherlands, 2006. [Google Scholar]
- Samko, S.G.; Kilbas, A.A.; Marichev, O.I. Fractional Integrals and Derivatives; Gordon and Beach: Amsterdam, The Netherlands, 1993; pp. 483–498. [Google Scholar]
- Berne, B.J.; Pecora, R. Dynamic Light Scattering; Wiley: New York, NY, USA, 1976; Chapter 10. [Google Scholar]
- Rodríguez, R.F.; Fujioka, J.; Salinas-Rodríguez, E. Fractional effects on the light scattering properties of a simple binary mixture. J. Non-Equilib. Thermodyn.
**2017**. [Google Scholar] [CrossRef] - Gadomski, A.; Kruszewska, N. On clean grain-boundaries involving growth of nonequilibrium crystalline-amorphous superconducting materials addressed by a phenomenological viewpoint. Eur. Phys. J. B
**2012**, 85, 416–428. [Google Scholar] [CrossRef] - Gadomski, A. Nucleation-and-growth problem in model lipid membranes undergoing subgel phase transitions is a problem of time scale. Eur. Phys. J. B
**1999**, 9, 569–571. [Google Scholar] [CrossRef]

**Figure 1.**Plot of ${\widehat{C}}_{NF}(\overrightarrow{k},t)$ as given by Equation (20), versus time (in seconds) t, for S

_{2}with λ = 0.03 and 0.06. We choose the following material properties values: T = 295 K, G

_{0}= 1.154 kg/ms, ρ

_{0}= 971 kg/m

^{3}[27].

**Figure 2.**The non-fractional correlation ${\widehat{C}}_{NF}(\overrightarrow{k},t)$ given by Equation (20), as a function of time t for E

_{1}with λ = 0.3 and 0.4. Here ρ

_{0}= 1414 kg/m

^{3}, G

_{0}= 17.3 kg/ms, T = 292 K.

**Figure 3.**The transverse velocity correlation function ${\widehat{C}}_{F}(\overrightarrow{k},t)$, as given by Equation (27), for λ = 0.03 and μ = 0.9 and μ = 0.95. The solid curve is ${\widehat{C}}_{NF}(\overrightarrow{k},t)$ and is included for a reference comparison.

**Figure 4.**This figure shows the behavior of ${\widehat{C}}_{F}(\overrightarrow{k},t)$, Equation (27), for E

_{1}with λ = 0.3 and μ = 0.9, 0.95. Same material values as in Figure 2.

**Figure 5.**The non-fractional ${G}_{NF}^{\u2033}\left(\omega \right)$ (---) and fractional ${G}_{F}^{\u2033}\left(\omega \right)$ (...) loss moduli for S

_{2}with λ = 0.05. Same material parameters as in Figure 1.

**Figure 6.**Behavior of ${G}_{NF}^{\u2033}\left(\omega \right)$ (---) and ${G}_{F}^{\u2033}\left(\omega \right)$ (...) for E

_{1}with λ = 0.3. Same material parameters as in Figure 2.

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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.
https://doi.org/10.3390/e20010028

**AMA Style**

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.
https://doi.org/10.3390/e20010028

**Chicago/Turabian Style**

Rodríguez, Rosalío F., Elizabeth Salinas-Rodríguez, and Jorge Fujioka.
2018. "Fractional Time Fluctuations in Viscoelasticity: A Comparative Study of Correlations and Elastic Moduli" *Entropy* 20, no. 1: 28.
https://doi.org/10.3390/e20010028