Geodesic
Brownian Motion in the Fluids with Complex Rheology
V. V. Rusakov1, 2 ; Yu. L. Raikher1, 2 ; R. Perzynski3
1 Institute of Continuous Media Mechanics, Russian Academy of Sciences, Ural Branch, 1 Korolyov street, Perm, 614013, Russia
2 Perm National Research Polytechnic University, 29a Komsomol avenue, 614990, Perm, Russia
3 Université Pierre et Marie Curie – PHENIX, UMR 8234, 4 place Jussieu, 75005 Paris, France
Mathematical modelling of natural phenomena, Tome 10 (2015) no. 4, pp. 1-43.

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Theory of Brownian motion of a fine particle in a viscoelastic fluid continuum is developed. The rheology of the embedding medium is described in terms of classical structure (spring-and-damper) schemes. It is shown that a great variety of conceivable ramifications of such structures could be reduced to an effective Jeffreys model: a Maxwell chain shunted by a damper. This scheme comprises just three material parameters: two for viscosities (fast and slow) and one for elasticity. For the thermal motion of a particle in such a fluid, the set of Langevin equations is derived. In the cases of 1D translational and 2D orientational (rotation about fixed axis) motions, the time dependencies of mean-square displacements are found analytically. It is shown that in a Jeffreys fluid, the Brownian motion has three regimes: fast diffusion (short times), slow diffusion (long times) and a crossover of those resulting in virtual localization (dynamic confinement) of the particle. Experimental evidence for that taken from the literature is presented. The developed formalism is applied to the particles with “frozen-in” dipole moments, whose orientational motion is a combination of Brownian diffusion and regular excitation by an AC field. The dynamic susceptibilities are calculated for the ensembles of particles liable for either 2D or 3D rotations. Comparison shows that the out-of-phase part of the susceptibility χ′′(ω) in 3D case differs considerably from that of 2D case. Function χ(ω) for the forced 3D rotary diffusion in a Jeffreys fluid is found for the first time. In our view, it would be useful for realistic theoretical interpretation of microrheology data and the more so for estimating the particle-mediated AC field-induced heating (magnetic hyperthermia) in viscoelastic fluid media.
DOI : 10.1051/mmnp/201510401

V. V. Rusakov 1, 2 ; Yu. L. Raikher 1, 2 ; R. Perzynski 3

1 Institute of Continuous Media Mechanics, Russian Academy of Sciences, Ural Branch, 1 Korolyov street, Perm, 614013, Russia
2 Perm National Research Polytechnic University, 29a Komsomol avenue, 614990, Perm, Russia
3 Université Pierre et Marie Curie – PHENIX, UMR 8234, 4 place Jussieu, 75005 Paris, France 
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V. V. Rusakov; Yu. L. Raikher; R. Perzynski. Brownian Motion in the Fluids with Complex Rheology. Mathematical modelling of natural phenomena, Tome 10 (2015) no. 4, pp. 1-43. doi : 10.1051/mmnp/201510401. http://geodesic.mathdoc.fr/articles/10.1051/mmnp/201510401/

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