Flow of fractal fluid in pipes: Non-integer dimensional space approach

Tarasov, Vasily E.

doi:10.1016/j.chaos.2014.06.008

Cited by 42 publications

(22 citation statements)

References 73 publications

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“…An interesting alternative to the statistical approach of the discrete system is to start from the dynamics defined on a space with fractional dimension [21][22][23][24][25]. The classical Euler fluid would be the 2D limit of the dynamics in such spaces.…”

Section: General Commentsmentioning

confidence: 99%

Self-organization of the vorticity field in two-dimensional quasi-ideal fluids: The statistical and field-theoretical formulations

Spineanu

Vlad

2015

Chaos, Solitons & Fractals

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Section: General Commentsmentioning

confidence: 99%

Self-organization of the vorticity field in two-dimensional quasi-ideal fluids: The statistical and field-theoretical formulations

Spineanu

Vlad

2015

Chaos, Solitons & Fractals

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“…Differentiation in non-integer dimensional space is considered in [31]- [32], but in these papers only a scalar Laplacian for NIDS was suggested. Recently a generalization of differential operators of first order (gradient, divergence, curl operators) and the vector Laplacian has been proposed in [22,33]. The suggested vector calculus for NIDS allows us to expand the range of applications of continuum models of isotropic fractal materials.…”

Section: Introductionmentioning

confidence: 99%

On fractional and fractal formulations of gradient linear and nonlinear elasticity

Aifantis

2019

Acta Mech

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In this paper we consider extensions of the gradient elasticity models proposed earlier by the second author to describe materials with fractional non-locality and fractality using the techniques developed recently by the first author. We derive a generalization of threedimensional continuum gradient elasticity theory, starting from integral relations and assuming a weak non-locality of power-law type that gives constitutive relations with fractional Laplacian terms, by utilizing the fractional Taylor series in wave-vector space. In the sequel we consider non-linear field equations with fractional derivatives of non-integer order to describe nonlinear elastic effects for gradient materials with power-law long-range interactions in the framework of weak non-locality approximation. The special constitutive relationship that we elaborate on, can form the basis for developing a fractional extension of deformation theory of gradient plasticity. Using the perturbation method, we obtain corrections to the constitutive relations of linear fractional gradient elasticity, when the perturbations are caused by weak deviations from linear elasticity or by fractional gradient non-locality. Finally we discuss fractal materials described by continuum models with non-integer dimensional spaces. Using the recently suggested vector calculus for non-integer dimensional spaces, we consider problems of fractal gradient elasticity.

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“…However the physical and geometrical aspects beyond this replacement remain hidden and only in very recent contributions the authors had provided some physical insights beyond fractional operators in mechanics, material sciences and heat transport [6][7][8][9][10][11][12][13][14]. The recent achievements in the physics of fractional-order operators had also used by the authors to introduce some connections among the exponent of power-law decay of time dependent phenomena as stress relaxation in biological and polymeric materials [15] and the scaling exponent of non-euclidean geometry.…”

Section: Introductionmentioning

confidence: 99%

Laminar flow through fractal porous materials: the fractional-order transport equation

Alaimo¹,

Zingales²

2015

Communications in Nonlinear Science and Numerical Simulation

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a b s t r a c tThe anomalous transport of a viscous fluid across a porous media with power-law scaling of the geometrical features of the pores is dealt with in the paper. It has been shown that, assuming a linear force-flux relation for the motion in a porous solid, then a generalized version of the Hagen-Poiseuille equation has been obtained with the aid of RiemannLiouville fractional derivative. The order of the derivative is related to the scaling property of the considered media yielding an appropriate mechanical picture for the use of generalized fractional-order relations, as recently used in scientific literature.

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Flow of fractal fluid in pipes: Non-integer dimensional space approach

Cited by 42 publications

References 73 publications

Self-organization of the vorticity field in two-dimensional quasi-ideal fluids: The statistical and field-theoretical formulations

Self-organization of the vorticity field in two-dimensional quasi-ideal fluids: The statistical and field-theoretical formulations

On fractional and fractal formulations of gradient linear and nonlinear elasticity

Laminar flow through fractal porous materials: the fractional-order transport equation

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