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18 pages, 425 KiB

Open AccessArticle

Modal Representation of Inertial Effects in Fluid–Particle Interactions and the Regularity of the Memory Kernels

by Giuseppe Procopio and Massimiliano Giona

Fluids 2023, 8(3), 84; https://doi.org/10.3390/fluids8030084 - 28 Feb 2023

Cited by 10 | Viewed by 2171

This article develops a modal expansion (in terms of functions exponentially decaying with time) of the force acting on a micrometric particle and stemming from fluid inertial effects (usually referred to as the Basset force) deriving from the application of the time-dependent Stokes equation to model fluid–particle interactions. One of the main results is that viscoelastic effects induce the regularization of the inertial memory kernels at

t = 0

, eliminating the

1 / \sqrt{t}

-singularity characterizing Newtonian fluids. The physical origin of this regularization stems from the finite propagation velocity of the internal shear stresses characterizing viscoelastic constitutive equations. The analytical expression for the fluid inertial kernel is derived for a Maxwell fluid, and a general method is proposed to obtain accurate approximations of it for generic complex viscoelastic fluids, characterized by a spectrum of relaxation times. Full article

(This article belongs to the Special Issue Recent Advances in Fluid Mechanics: Feature Papers, 2022)

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20 pages, 4197 KiB

Open AccessArticle

A Coupled CFD-DEM Study on the Effect of Basset Force Aimed at the Motion of a Single Bubble

by Huiting Chen, Weitian Ding, Han Wei, Henrik Saxén and Yaowei Yu

Materials 2022, 15(15), 5461; https://doi.org/10.3390/ma15155461 - 8 Aug 2022

Cited by 11 | Viewed by 2501

Abstract

The physical meaning of Basset force is first studied via polynomial approximation and the Fourier series representation method. After compiling the Basset force into the coupling interface with Visual C, a dynamic mathematical model is set up to describe the upward motion behavior of a single bubble by adopting the CFD-DEM method. Afterwards, the coupling interface with Basset force proposed in this study is verified experimentally and shows very good agreements. The initial velocity, releasing depth, bubble size, density ratio and viscosity ratio are studied qualitatively due to their great importance to Basset force. The ratio of Basset force to the sum of Basset force and drag force and to buoyancy,

{\vec{F}}_{B a}

{\vec{F}}_{D}

{\vec{F}}_{B a}

) and |

{\vec{F}}_{B a}

{\vec{F}}_{B}

|, are employed to quantify the contribution of Basset force quantitatively. In addition, some instructive outlooks and recommendations on a further development of appropriate and justifiable use of Basset force are highlighted at last. Full article

(This article belongs to the Special Issue Numerical Simulations in Metal Refining Process)

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12 pages, 609 KiB

Open AccessArticle

Improvement of Mathematical Model for Sedimentation Process

by Ivan Pavlenko, Marek Ochowiak, Praveen Agarwal, Radosław Olszewski, Bernard Michałek and Andżelika Krupińska

Energies 2021, 14(15), 4561; https://doi.org/10.3390/en14154561 - 28 Jul 2021

Cited by 10 | Viewed by 2851

Abstract

In this article, the fractional-order differential equation of particle sedimentation was obtained. It considers the Basset force’s fractional origin and contains the Riemann–Liouville fractional integral rewritten as a Grunwald–Letnikov derivative. As a result, the general solution of the proposed fractional-order differential equation was found analytically. The belonging of this solution to the real range of values was strictly theoretically proven. The obtained solution was validated on a particular analytical case study. In addition, it was proven numerically with the approach based on the S-approximation method using the block-pulse operational matrix. The proposed mathematical model can be applied for modeling the processes of fine particles sedimentation in liquids, aerosol deposition in gas flows, and particle deposition in gas-dispersed systems. Full article

(This article belongs to the Special Issue Computational Heat Transfer and Fluid Mechanics)

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14 pages, 299 KiB

Open AccessArticle

Factorization à la Dirac Applied to Some Equations of Classical Physics

by Zine El Abiddine Fellah, Erick Ogam, Mohamed Fellah and Claude Depollier

Mathematics 2021, 9(8), 899; https://doi.org/10.3390/math9080899 - 18 Apr 2021

Cited by 2 | Viewed by 2285

Abstract

In this paper, we present an application of Dirac’s factorization method to three types of the partial differential equations, i.e., the wave equation, the scattering equation, and the telegrapher’s equation. This method gives results that contribute to a better understanding of physical phenomena by generalizing the Euler and constituent equations. Its application to the wave equation shows that it is indeed a factorization method, since it gives d’Alembert’s solutions in a more general framework. In the case of the diffusion equation, a fractional differential equation has been established that has already been highlighted by other authors in particular cases, but by indirect methods. Dirac’s method brings several new results in the case of the telegraphers’ equation corresponding to the propagation of an acoustic wave in a dissipative fluid. On the one hand, its formalism facilitates the temporal interpretation of phenomena, in particular the density and compressibility of the fluid become temporal operators, which can be “seen” as susceptibilities of the fluid. On the other hand, a consequence of this temporal modeling is the highlighting in Euler’s equation of a term similar to the one that was introduced by Boussinesq and Basset in the equation of the motion of a solid sphere in a unsteady fluid. Full article

(This article belongs to the Special Issue Fractional Calculus in Anomalous Transport Theory)

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