32 research outputs found

### Revisiting the Low-Frequency Dipolar Perturbation by an Impenetrable Ellipsoid in a Conductive Surrounding

This contribution deals with the scattering by a metallic ellipsoidal target, embedded in a homogeneous conductive medium, which is stimulated when a 3D time-harmonic magnetic dipole is operating at the low-frequency realm. The incident, the scattered, and the total three-dimensional electromagnetic fields, which satisfy Maxwellâ€™s equations, yield low-frequency expansions in terms of positive integral powers of the complex-valued wave number of the exterior medium. We preserve the static Rayleigh approximation and the first three dynamic terms, while the additional terms of minor contribution are neglected. The Maxwell-type problem is transformed into intertwined potential-type boundary value problems with impenetrable boundary conditions, whereas the environment of a genuine ellipsoidal coordinate system provides the necessary setting for tackling such problems in anisotropic space. The fields are represented via nonaxisymmetric infinite series expansions in terms of harmonic eigenfunctions, affiliated with the ellipsoidal system, obtaining analytical closed-form solutions in a compact fashion. Until nowadays, such problems were attacked by using the very few ellipsoidal harmonics exhibiting an analytical form. In the present article, we address this issue by incorporating the full series expansion of the potentials and utilizing the entire subspace of ellipsoidal harmonic eigenfunctions

### Low-frequency dipolar excitation of a perfect ellipsoidal conductor

International audienceThis paper deals with the scattering by a perfectly conductive ellipsoid under magnetic dipolar excitation at low frequency. The source and the ellipsoid are embedded in an infinite homogeneous conducting ground. The main idea is to obtain an analytical solution of this scattering problem in order to have a fast numerical estimation of the scattered field that can be useful for real data inversion. Maxwell equations and boundary conditions, describing the problem, are firstly expanded using low-frequency expansion of the fields up to order three. It will be shown that fields have to be found incrementally. The static one (term of order zero) satisfies the Laplace equation. The next non-zero term (term of order two) is more complicated and satisfies the Poisson equation. The order-three term is independent of the previous ones and is described by the Laplace equation. They constitute three different scattering problems that are solved using the separated variables method in the ellipsoidal coordinate system. Solutions are written as expansions on the few analytically known scalar ellipsoidal harmonics. Details are given to explain how those solutions are achieved with an example of numerical results

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### On the Spheroidal Semiseparation for Stokes Flow

Many heat and mass transport problems involve particle-fluid systems, where the assumption of Stokes flow provides a very good approximation for representing small particles embedded within a viscous, incompressible fluid characterizing the steady, creeping flow. The present work is concerned with some interesting practical aspects of the theoretical analysis of Stokes flow in spheroidal domains. The stream function Ïˆ, for axisymmetric Stokes flow, satisfies the well-known equation E4Ïˆ=0. Despite the fact that in spherical coordinates this equation admits separable solutions, this property is not preserved when one seeks solutions in the spheroidal geometry. Nevertheless, defining some kind of semiseparability, the complete solution for Ïˆ in spheroidal coordinates has been obtained in the form of products combining Gegenbauer functions of different degrees. Thus, the general solution is represented in a full-series expansion in terms of eigenfunctions, which are elements of the space kerE2 (separable solutions), and in terms of generalized eigenfunctions, which are elements of the space kerE4 (semiseparable solutions). In this work we revisit this aspect by introducing a different and simpler way of representing the aforementioned generalized eigenfunctions. Consequently, additional semiseparable solutions are provided in terms of the Gegenbauer functions, whereas the completeness is preserved and the full-series expansion is rewritten in terms of these functions.Peer Reviewe

### Connection Formulae between Ellipsoidal and Spherical Harmonics with Applications to Fluid Dynamics and Electromagnetic Scattering

The environment of the ellipsoidal system, significantly more complex than the spherical one, provides the necessary settings for tackling boundary value problems in anisotropic space. However, the theory of LamÃ© functions and ellipsoidal harmonics affiliated with the ellipsoidal system is rather complicated. A turning point would reside in the existence of expressions interlacing these two different systems. Still, there is no simple way, if at all, to bridge the gap. The present paper addresses this issue. We provide explicit formulas of specific ellipsoidal harmonics expressed in terms of their counterparts in the classical spherical system. These expressions are then put into practice in the framework of physical applications

### On the Perturbation of the Three-Dimensional Stokes Flow of Micropolar Fluids by a Constant Uniform Magnetic Field in a Circular Cylinder

Modern engineering technology involves the micropolar magnetohydrodynamic flow of magnetic fluids. Here, we consider a colloidal suspension of non-conductive ferromagnetic material, which consists of small spherical particles that behave as rigid magnetic dipoles, in a carrier liquid of approximately zero conductivity and low-Reynolds number properties. The interaction of a 3D constant uniform magnetic field with the three-dimensional steady creeping motion (Stokes flow) of a viscous incompressible micropolar fluid in a circular cylinder is investigated, where the magnetization of the ferrofluid has been taken into account and the magnetic Stokes partial differential equations have been presented. Our goal is to apply the proper boundary conditions, so as to obtain the flow fields in a closed analytical form via the potential representation theory, and to study several characteristics of the flow. In view of this aim, we make use of an improved new complete and unique differential representation of magnetic Stokes flow, valid for non-axisymmetric geometries, which provides the velocity and total pressure fields in terms of easy-to-find potentials. We use these results to simulate the creeping flow of a magnetic fluid inside a circular duct and to obtain the flow fields associated with this kind of flow

### Dipolar Excitation of a Perfectly Electrically Conducting Spheroid in a Lossless Medium at the Low-Frequency Regime

The electromagnetic vector fields, which are scattered off a highly conductive spheroid that is embedded within an otherwise lossless medium, are investigated in this contribution. A time-harmonic magnetic dipolar source, located nearby and operating at low frequencies, serves as the excitation primary field, being arbitrarily orientated in the three-dimensional space. The main idea is to obtain an analytical solution of this scattering problem, using the appropriate system of spheroidal coordinates, such that a possibly fast numerical estimation of the scattered fields could be useful for real data inversion. To this end, incident and scattered as well as total fields are written in a rigorous low-frequency manner in terms of positive integral powers of the real-valued wave number of the exterior environment. Then, the Maxwell-type problem is converted to interconnected Laplaceâ€™s or Poissonâ€™s equations, complemented by the perfectly conducting boundary conditions on the spheroidal object and the necessary radiation behavior at infinity. The static approximation and the three first dynamic contributors are sufficient for the present study, while terms of higher orders are neglected at the low-frequency regime. Henceforth, the 3D scattering boundary value problems are solved incrementally, whereas the determination of the unknown constant coefficients leads either to concrete expressions or to infinite linear algebraic systems, which can be readily solved by implementing standard cut-off techniques. The nonaxisymmetric scattered magnetic and electric fields follow and they are obtained in an analytical compact fashion via infinite series expansions in spheroidal eigenfunctions. In order to demonstrate the efficiency of our analytical approach, the results are degenerated so as to recover the spherical case, which validates this approach

### Comparison of differential representations for radially symmetric Stokes flow

Papkovich and Neuber (PN), and Palaniappan, Nigam, Amaranath, and
Usha (PNAU) proposed two different representations of the
velocity and the pressure fields in Stokes flow, in terms of
harmonic and biharmonic functions, which form a practical tool
for many important physical applications. One is the
particle-in-cell model for Stokes flow through a swarm of
particles. Most of the analytical models in this realm consider
spherical particles since for many interior and exterior flow
problems involving small particles, spherical geometry provides a
very good approximation. In the interest of producing
ready-to-use basic functions for Stokes flow, we calculate the
PNAU and the PN eigensolutions generated by the appropriate
eigenfunctions, and the full series expansion is provided. We
obtain connection formulae by which we can transform any solution
of the Stokes system from the PN to the PNAU eigenform. This
procedure shows that any PNAU eigenform corresponds to a
combination of PN eigenfunctions, a fact that reflects the
flexibility of the second representation. Hence, the advantage of
the PN representation as it compares to the PNAU solution is
obvious. An application is included, which solves the problem of
the flow in a fluid cell filling the space between two concentric
spherical surfaces with Kuwabara-type boundary conditions

### The 3D Happel model for complete isotropic Stokes flow

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

### The Avascular Tumour Growth in the Presence of Inhomogeneous Physical Parameters Imposed from a Finite Spherical Nutritive Environment

A well-known mathematical model of radially symmetric tumour growth is revisited in the present work. Under this aim, a cancerous spherical mass lying in a finite concentric nutritive surrounding is considered. The host spherical shell provides the tumor with vital nutrients, receives the debris of the necrotic cancer cells, and also transmits to the tumour the pressure imposed on its exterior boundary. We focus on studying the type of inhomogeneity that the nutrient supply and the pressure field imposed on the host exterior boundary, can exhibit in order for the spherical structure to be supported. It turns out that, if the imposed fields depart from being homogeneous, only a special type of interrelated inhomogeneity between nutrient and pressure can secure the spherical growth. The work includes an analytic derivation of the related boundary value problems based on physical conservation laws and their analytical treatment. Implementations in cases of special physical interest are examined, and also existing homogeneous results from the literature are fully recovered