On modelling of thermodynamic interaction of particles suspended in two-dimensional medium

Analytical description of temperature distribution in a medium with foreign inclusions is difficult due to the complicated geometry of the problem, so asymptotic and numerical methods are usually used to model thermodynamic processes in heterogeneous media. To be convinced in convergence of these methods the authors consider model problem about two identical round particles in infinite planar medium with temperature gradient which is constant at infinity. Authors refine multipole expansion of the solution obtained earlier by continuing it up to higher powers of small parameter, that is nondimensional radius of thermodynamically interacting particles. Numerical approach to the problem using ANSYS software is described; in particular, appropriate choice of approximate boundary conditions is discussed. Authors ascertain that replacement of infinite medium by finite-sized domain is important source of error in FEM. To find domain boundaries in multiple inclusions’ problem the authors develop “fictituous particle” method; according to it the cloud of particles far from the center of the cloud acts approximately as a single equivalent particle of greater size and so may be replaced by it. Basing on particular quantitative data the dependence of domain size that provides acceptable accuracy on thermal conductivities of medium and of particles is explored. Authors establish series of numerical experiments confirming convergence of multipole expansions method and FEM as well; proximity of their results is illustrated, too.

On the link between mean square-radii and high-order toroidal moments

Multipole expansions of the source play an important role in a broad range of disciplines in modern physics, ranging from the description of exotic states of matter to the design of nanoantennas in photonics. Within the context of the latter, toroidal multipoles, a third group of multipoles complementing the well-known electric and magnetic ones, have been widely investigated since they lead to the formation of non-radiating sources. In the last years, however, the photonics community has brought to light the existence of a fourth type of multipoles that is commonly overlooked. Currently, different groups have provided different mathematical expressions to describe such sources, and they have been coined with different names; on the one hand mean-square radii, and on the other hand, as high order toroidal moments. Despite their clear physical similarity, a formal relation between the two has not yet been established. While explicit formulas for both types have been derived, they are not expressed in the same basis, and therefore it is not possible to draw a clear physical connection between them. In this contribution, we will bridge this gap and rigorously derive the connection between the two representations, taking as an example the cases of the nth order mean square radius of the electric dipole and the nth order electric toroidal dipole. Our results conclusively show that both types of representations are exactly equivalent up to a prefactor.

Multipole expansion of integral powers of cosine theta

Legendre polynomials form the basis for multipole expansion of spatially varying functions. The technique allows for decomposition of the function into two separate parts with one depending on the radial coordinates only and the other depending on the angular variables. In this work, the angular function $$\cos ^k \theta$$ cos k θ is expanded in the Legendre polynomial basis and the algorithm for determining the corresponding coefficients of the Legendre polynomials is generated. This expansion together with the algorithm can be generalized to any case in which a dot product of any two vectors appears. Two alternative multipole expansions for the electron–electron Coulomb repulsion term are obtained. It is shown that the conventional multipole expansion of the Coulomb repulsion term is a special case for one of the expansions generated in this work.

Squirmer rods as elongated microswimmers: flow fields and confinement

We characterize simulated flow fields of active squirmer rods in bulk and Hele-Shaw geometry using hydrodynamic multipole expansions.

Multipole expansions for time-dependent charge and current distributions in quasistatic approximation

We propose a consistent approach to the definition of electric, magnetic and toroidal multipole moments. Electric and magnetic fields are split into potential, vortex and radiative terms, with the latter ones dropped off in the quasistatic approximation. The potential part of the electric field, the vortex parts of the magnetic field and vector potential contain gradients of scalar functions. Formally introducing magnetic and toroidal analogs of the electric charge, we apply multipole expansions for those scalars. Closed-form expressions are derived in an arbitrary order for electric, magnetic and toroidal multipoles, which constitute a full system for expansions of the electromagnetic field.