Transient Propagation of Longitudinal and Transverse Waves in Cancellous Bone: Application of Biot Theory and Fractional Calculus

Benmorsli, Djihane; Fellah, Zine El Abiddine; Belgroune, Djema; Ongwen, Nicholas O.; Ogam, Erick; Dépollier, Claude; Fellah, M.

doi:10.3390/sym14101971

Cited by 4 publications

(1 citation statement)

References 60 publications

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“…Asjad et al [48] used the Prabhakar operator to obtain a fractional problem of Jeffrey fluid along a moving vertical plate and the equations for energy, and the Laplace approach solved momentum. Recent definitions of fractional derivatives can be seen in [40,[49][50][51].…”

Section: Introductionmentioning

confidence: 99%

Solution of Water and Sodium Alginate-Based Casson Type Hybrid Nanofluid with Slip and Sinusoidal Heat Conditions: A Prabhakar Fractional Derivative Approach

et al. 2022

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This paper aims to investigate free convection heat transmission in hybrid nanofluids across an inclined pours plate, which characterizes an asymmetrical hybrid nanofluid flow and heat transfer behavior. With an angled magnetic field applied, sliding on the border of walls is also considered with sinusoidal heat transfer boundary conditions. The non-dimensional leading equations are converted into a fractional model using an effective mathematical fractional approach known as the Prabhakar time fractional derivative. Silver (Ag) and titanium dioxide (TiO2) are both considered nanoparticles, with water (H2O) and sodium alginate (C6H9NaO7) serving as the base fluids. The solution of the momentum, concentration, and energy equation is found by utilizing the Laplace scheme, and different numerical algorithms are considered for the inverse of Laplace, i.e., Stehfest and Tzou’s. The graphical analysis investigates the impact and symmetry of significant physical and fractional parameters. Consequently, we surmise that water-based hybrid nanofluid has a somewhat higher velocity than sodium alginate-based hybrid nanofluid. Furthermore, the Casson parameter has a dual effect on the momentum profile. Furthermore, the memory effect reduces as fractional restriction increases for both the velocity and temperature layers. The results demonstrate that increasing the heat transmission in the solid nanoparticle volume fractions enhanced the heat transmission. In addition, the numerical assessment examined the increase in mass and heat transmission, while shear stress was increased with an increase in the Prabhakar fractional parameter α.

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Section: Introductionmentioning

confidence: 99%

Solution of Water and Sodium Alginate-Based Casson Type Hybrid Nanofluid with Slip and Sinusoidal Heat Conditions: A Prabhakar Fractional Derivative Approach

et al. 2022

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A Dynamic Three‐Field Finite Element Model for Wave Propagation in Linear Elastic Porous Media

Campos,

Gracie

2024

Num Anal Meth Geomechanics

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A three‐field finite element (FE) model for dynamic porous media considering the de la Cruz and Spanos (dCS) theory is presented. Due to fluid viscous dissipation terms, wave propagation in the dCS theory yields an additional rotational wave compared to Biot (BT) theory. In addition, introducing porosity as a dynamic variable in the dCS model allows solid‐fluid nonreciprocal interactions. Due to the volume‐averaging technique, the dCS model further accounts for a macroscopic shear modulus and adds a new macroscopic constant. The porous media governing equations are formulated in terms of solid displacement, fluid pressure, and fluid displacement. Space and time convergence rates for the FE dCS model are demonstrated in a one‐dimensional case. A dimensionless analysis performed in the dCS framework led to negligible differences between BT and dCS models except when assuming high fluid viscosity. Domains with small characteristic lengths resulted in BT and dCS damping terms in the same order of magnitude. One‐ and two‐dimensional examples showed that dCS nonreciprocal interactions and the macroscopic shear modulus are responsible for modifying wave patterns. A two‐dimensional injection well simulation with water and slickwater showed higher wave attenuation for the latter. High frequencies in dCS model were noticed to yield more significant changes in wave patterns. The numerical results highlight the contributions of the dCS porous media model and its importance in simulations of laboratory scale experiments, ultrasonic frequencies, and highly viscous fluids.

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Transient high-frequency spherical wave propagation in porous medium using fractional calculus technique

Soltani,

Seyedpour,

Ricken

et al. 2023

Acta Mech

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Transient high-frequency spherical wave propagation in the porous medium is studied using the Biot-JKD theory. The porous media is considered to be a composed of deformable solid skeleton and viscous incompressible fluid inside the pores. In order to treat the fractional proportionality of the dynamic tortuosity to the frequency (or equivalently, to time) due to the viscous interaction between solid and fluid phases, the fractional calculus theory along with the Laplace and Fourier transforms are used to solve the coupled governing partial differential equations of the scaler and vector potential functions obtained from the Helmholtz’s decomposition in the Laplace domain. Both the longitudinal and transverse waves, additionally the reflection and transmission kernels are determined in detail at the Laplace domain. For the Laplace-to-time inversion transform, Durbin’s numerical formula is used and the independence of the results from the involved tuning and accuracy parameters is checked. The effects of the porosity, dynamic tortuosity, characteristics length, etc. on the reflected pressure and stress are investigated. The general pattern of the results is similar to our previous knowledge of wave propagation. Further works and experiments may be conducted in future works for various applications.

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Transient Propagation of Longitudinal and Transverse Waves in Cancellous Bone: Application of Biot Theory and Fractional Calculus

Cited by 4 publications

References 60 publications

Solution of Water and Sodium Alginate-Based Casson Type Hybrid Nanofluid with Slip and Sinusoidal Heat Conditions: A Prabhakar Fractional Derivative Approach

Solution of Water and Sodium Alginate-Based Casson Type Hybrid Nanofluid with Slip and Sinusoidal Heat Conditions: A Prabhakar Fractional Derivative Approach

A Dynamic Three‐Field Finite Element Model for Wave Propagation in Linear Elastic Porous Media

Transient high-frequency spherical wave propagation in porous medium using fractional calculus technique

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