Comparison of differential representations for radially symmetricStokes flow

Dassios, George; Vafeas, Panayiotis

doi:10.1155/s1085337504306044

Cited by 3 publications

(4 citation statements)

References 9 publications

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“…The above results (9) to (13) with (14) to (19) correspond to the leading term of a stream function ψ (LT) , which admits…”

Section: Analytical Approachmentioning

confidence: 92%

“…The volume of the fluid cell is chosen so that the solid volume fraction in the cell coincides with the volume fraction of the swarm. The appropriate boundary conditions, resulting from these assumptions, can be used to determine the flow fields as full series expansion via the Papkovich‐Neuber representation, which represents the velocity and the total pressure fields in terms of harmonic functions and is proved to be widely applicable to spherical and spheroidal geometry. The same problem is solved numerically for the corresponding 3D case, where the numerical implementation has been incorporated via the finite volumes method, whereas the obtained linear systems were approximated by applying the well‐known successful over‐relaxation concept.…”

Section: Introductionmentioning

confidence: 99%

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Validation method for the systematization of results based on a similarity concept

Gavriil

Vafeas

Kanavouras

et al. 2018

Math Methods in App Sciences

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The development of a functional methodological approach is presented here, to clarify a globally valid way of evaluating the precision of mathematical modeling of physical and/or chemical processes. Starting from the description of the system, a phenomenon accompanied by a disclaiming hypothesis is investigated, against which the knowledge is accumulated with time. Moreover, the possibility of the evolution of any phenomenon being interrupted when a parameter overpasses a critical threshold, after which the hypothesis is not any more valid, is introduced. This possibility should be obtained through the dependence of a selected macroscopic quantity (marker) on a specific parameter. To apply this methodology, the problem of Stokes flow through a granular medium of spheroidal grains has been selected as an indicative case study. The prolate spheroidal configuration is considered, since the results for the oblate spheroid can be recovered via a simple transformation. Therein, the three‐dimensional flow fields are initially constructed analytically via the Papkovich‐Neuber differential representation, which provides the velocity and pressure fields in terms of harmonic spheroidal eigenfunctions. Next, under the Kuwabara‐type spheroidal 2D unit cell concept, the above expressions degenerate to the axisymmetric case, and the full solution is then obtained, keeping the leading terms of the series, which are adequate for most engineering applications for specific aspect ratio of the spheroids. In the sequel, the aforementioned problem is solved numerically for a 3D extension of the same model, where this numerical solution has been achieved by using the finite volumes method, while the resulting linear systems were approximated by applying the well‐known successful over‐relaxation concept. Finally, outcomes by both models have been compared via the above methodology, resulting to objective and reliable accuracy criteria.

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“…The above results (9) to (13) with (14) to (19) correspond to the leading term of a stream function ψ (LT) , which admits…”

Section: Analytical Approachmentioning

confidence: 92%

Section: Introductionmentioning

confidence: 99%

Validation method for the systematization of results based on a similarity concept

Gavriil

Vafeas

Kanavouras

et al. 2018

Math Methods in App Sciences

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“…Most of the material here is borrowed from [17], and it stays in close accord with contents of well-known reference books [15] and [9]. Let us consider the spherical coordinate system…”

Section: Appendix a The Vector Spherical Harmonics And Associated Mamentioning

confidence: 99%

Low-Frequency Solution for a Perfectly Conducting Sphere in a Conductive Medium With Dipolar Excitation

Vafeas¹,

Perrusson²,

Lesselier³

2004

PIER

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Abstract-This contribution concerns the interaction of an arbitrarily orientated, time-harmonic, magnetic dipole with a perfectly conducting sphere embedded in a homogeneous conductive medium. A rigorous low-frequency expansion of the electromagnetic field in positive integral powers (jk) n , k complex wavenumber of the exterior medium, is constructed. The first n = 0 vector coefficient (static or Rayleigh) of the magnetic field is already available, so emphasis is on the calculation of the next two nontrivial vector coefficients (at n = 2 and at n = 3) of the magnetic field. Those are found in closed form from exact solutions of coupled (at n = 2, to the one at n = 0) or uncoupled (at n = 3) vector Laplace equations. They are given in compact fashion, as infinite series expansions of vector spherical harmonics with scalar coefficients (for n = 2). The good accuracy of both in-phase (the real part) and quadrature (the imaginary part) vector components of the diffusive magnetic field are illustrated by numerical computations in a realistic case of mineral exploration of the Earth by inductive means. This canonical representation, not available yet in the literature to this time (beyond the static term), may apply to other practical cases than this one in geoelectromagnetics, whilst it adds useful reference results to the already ample library of scattering by simple shapes using analytical methods.

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“…This way, the mathematical formulation of any physical problem is significantly simplified. Many efficient methods have been developed in order to solve this kind of problems in spherical and spheroidal coordinates, considering axial symmetry inherited by the geometry, such as numerical computation [1,6,10] and stream-function techniques [2,3,11,13] or other analyticfunction methods [4,5,7,15]. Nevertheless, 3D flows have not been extensively faced.…”

Section: Introduction Stokes Flowmentioning

confidence: 99%

The 3D Happel model for complete isotropic Stokes flow

Dassios

Vafeas

2004

International Journal of Mathematics and Mathematical Sciences

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The creeping flow through a swarm of spherical particles that move with constant velocity in an arbitrary direction and rotate with an arbitrary constant angular velocity in a quiescent Newtonian fluid is analyzed with a 3D sphere-in-cell model. The mathematical treatment is based on the two-concentric-spheres model. The inner sphere comprises one of the particles in the swarm and the outer sphere consists of a fluid envelope. The appropriate boundary conditions of this non-axisymmetric formulation are similar to those of the 2D sphere-in-cell Happel model, namely, nonslip flow condition on the surface of the solid sphere and nil normal velocity component and shear stress on the external spherical surface. The boundary value problem is solved with the aim of the complete Papkovich-Neuber differential representation of the solutions for Stokes flow, which is valid in non-axisymmetric geometries and provides us with the velocity and total pressure fields in terms of harmonic spherical eigenfunctions. The solution of this 3D model, which is self-sufficient in mechanical energy, is obtained in closed form and analytical expressions for the velocity, the total pressure, the angular velocity, and the stress tensor fields are provided

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Comparison of differential representations for radially symmetricStokes flow

Cited by 3 publications

References 9 publications

Validation method for the systematization of results based on a similarity concept

Validation method for the systematization of results based on a similarity concept

Low-Frequency Solution for a Perfectly Conducting Sphere in a Conductive Medium With Dipolar Excitation

The 3D Happel model for complete isotropic Stokes flow

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