Reflection and transmission of transient ultrasonic wave in fractal porous material: Application of fractional calculus

Fellah, Zine El Abiddine; Fellah, M.; Ogam, Erick; Berbiche, Amine; Dépollier, Claude

doi:10.1016/j.wavemoti.2021.102804

Cited by 6 publications

(6 citation statements)

References 42 publications

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“…According to Mandelbrot [1,2], these fractal dimensions can give a better understanding of turbulence, star distribution, galaxies, and so on. Fractal methods have become increasingly popular in recent years and have the potential to be useful in a variety of fields and contexts including biology and medicine [3,4], geology and earth sciences [5,6], image analysis [7,8], astronomy [9,10], and acoustics [11][12][13][14][15]. Fluids and rough surfaces which are the scope of our study also exhibit a self-similar "fractal" structure.…”

Section: Introductionmentioning

confidence: 99%

A Generalization of Poiseuille’s Law for the Flow of a Self-Similar (Fractal) Fluid through a Tube Having a Fractal Rough Surface

et al. 2023

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In this paper, a generalization of Poiseuille’s law for a self-similar fluid flow through a tube having a rough surface is proposed. The originality of this work is to consider, simultaneously, the self-similarity structure of the fluid and the roughness of the tube surface. This study can have a wide range of applications, for example, for fractal fluid dynamics in hydrology. The roughness of the tube surface presents a fractal structure that can be described by the surface fractal noninteger dimensions. Complex fluids that are invariant to changes in scale (self-similar) are modeled as a continuous medium in noninteger dimensional spaces. In this work, the analytical solution of the Navier–Stokes equations for the case of a self-similar fluid flow through a rough “fractal” tube is presented. New expressions of the velocity profiles, the fluid discharge, and the friction factor are determined analytically and plotted numerically. These expressions contain fractal dimensions describing the effects of the fractal aspect of the fluid and of that of the tube surface. This approach reveals some very important results. For the velocity profile to represent a physical solution, the fractal dimension of the fluid ranges between 0.5 and 1. This study also qualitatively demonstrates that self-similar fluids have shear thickening-like behavior. The fractal (self-similarity) nature of the fluid and the roughness of the surface both have a huge impact on the dynamics of the flow. The fractal dimension of the fluid affects the amplitude and the shape of the velocity profile, which loses its parabolic shape for some values of the fluid fractal dimension. By contrast, the roughness of the surface affects only the amplitude of the velocity profile. Nevertheless, both the fluid’s fractal dimension and the surface roughness have a major influence on the behavior of the fluid, and should not be neglected.

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Section: Introductionmentioning

confidence: 99%

A Generalization of Poiseuille’s Law for the Flow of a Self-Similar (Fractal) Fluid through a Tube Having a Fractal Rough Surface

et al. 2023

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“…Since fractional calculus is rapidly growing recently with the old models replaced by fractional ones in the light of a diverse choice of fractional derivative definitions, we mainly concentrate on recent studies. For instance, anomalous diffusion processes were investigated by means of fractional models in oil pollution [5], in tumor growth and oncological particularities [6,7], in antioxidant vegetable [8], in the voltage regulator of the power industry [9], in nuclear neutron transport [10], in enhancing low-frequency signal [11], in computer vision [12], in radioactive and transmutation linear chains [13], in optimizing current sequences in lithium-ion batteries [14,15], in structural analysis creep [16], in the transmission dynamics of Nipah virus [17], in chronic hepatitis B-related liver fibrosis [18], in cytokeratin [19], in the link formation of temporal networks [20], in slow decay phenomena of the Tesla Model S battery [21], in the slip flow of nanoparticles [22], and in the ultrasonic propagation of wave in a fractal porous material [23], among many others.…”

Section: Introductionmentioning

confidence: 99%

Liquid Vortex Formation in a Swirling Container Considering Fractional Time Derivative of Caputo

Turkyilmazoglu,

Alofi

2024

Fractal Fract

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This paper applies fractional calculus to a practical example in fluid mechanics, illustrating its impact beyond traditional integer order calculus. We focus on the classic problem of a rigid body rotating within a uniformly rotating container, which generates a liquid vortex from an undisturbed initial state. Our aim is to compare the time evolutions of the physical system in fractional and integer order models by examining the torque transmission from the rotating body to the surrounding liquid. This is achieved through closed-form, time-developing solutions expressed in terms of Mittag–Leffler and Bessel functions. Analysis reveals that the rotational velocity and, consequently, the vortex structure of the liquid are influenced by three distinct time zones that differ between integer and noninteger models. Anomalous diffusion, favoring noninteger fractions, dominates at early times but gradually gives way to the integer derivative model behavior as time progresses through a transitional regime. Our derived vortex formula clearly demonstrates how the liquid vortex is regulated in time for each considered fractional model.

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“…Porous media with self-similar structure, referred to as fractal porous media, are characterized by complex geometries and exhibit unique acoustic properties that are different from those of traditional porous media. Several studies [5,6,8,9] have investigated the acoustic behavior of fractal porous media, including the effects of fractal geometry on acoustic absorption, transmission, and scattering.…”

Section: Introductionmentioning

confidence: 99%

Ultrasonic Propagation in Fractal Porous Material Having Rigid Frame

Fellah,

Bouchendouka,

Martin

et al. 2023

10th ECCOMAS Thematic Conference on Smart Structures and Materials

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This paper discusses the impact of the fractal structure of porous materials on wave behavior. In addition to commonly used parameters such as porosity, tortuosity, viscous and thermal characteristic lengths, a fractal dimension (α) can be introduced to represent the selfsimilarity of the material. The Helmholtz equation for wave propagation in a porous medium can then be modified to depend on non-integer dimensions. The fractal structure affects the wave's speed, attenuation, and phase shifts, with supersonic wave speeds possible for for certain values of the fractal dimension. The equivalent thickness of the material also varies with the fractal dimension becoming larger for low values of the fractal dimension. This provides us with valuable insights into the potential for creating novel metamaterials with exceptional acoustic properties by adjusting the fractal dimension value, enabling the attainment of supersonic velocities and substantial equivalent thicknesses. Understanding the effect of the fractal dimension on wave behavior has important implications for developing effective acoustic materials and can inform research in noise reduction, metamaterials, medicine, seismology, geophysics, and petroleum engineering.

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Reflection and transmission of transient ultrasonic wave in fractal porous material: Application of fractional calculus

Cited by 6 publications

References 42 publications

A Generalization of Poiseuille’s Law for the Flow of a Self-Similar (Fractal) Fluid through a Tube Having a Fractal Rough Surface

A Generalization of Poiseuille’s Law for the Flow of a Self-Similar (Fractal) Fluid through a Tube Having a Fractal Rough Surface

Liquid Vortex Formation in a Swirling Container Considering Fractional Time Derivative of Caputo

Ultrasonic Propagation in Fractal Porous Material Having Rigid Frame

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