Vladimir Mityushev

Generalized Clausius–Mossotti Formula for Random Composite with Circular Fibers

An important area of materials science is the study of effective dielectric, thermal and electrical properties of two phase composite materials with very different properties of the constituents. The case of small concentration is well studied and analytical formulas such as Clausius–Mossotti (Maxwell–Garnett) are successfully used by physicists and engineers. We investigate analytically the case of an arbitrary number of unidirectional circular fibers in the periodicity cell when the concentration of the fibers is not small, i.e., we account for interactions of all orders (pair, triplet, etc.). We next consider transversely-random unidirectional composite of the parallel fibers and obtain a closed form representation for the effective conductivity (as a power series in the concentration v). We express the coefficients in this expansion in terms of integrals of the elliptic Eisenstein functions. These integrals are evaluated and the explicit dependence of the parameter d, which characterizes random position of the fibers centers, is obtained. Thus we have extended the Clausius–Mossotti formula for the non dilute mixtures by adding the higher order terms in concentration and qualitatively evaluated the effect of randomness in the fibers locations. In particular, we have proven that the periodic array provides extremum for the effective conductivity in our class of random arrays (“shaking” geometries). Our approach is based on complex analysis techniques and functional equations, which are solved by the successive approximations method.

Longitudinal Permeability of Spatially Periodic Rectangular Arrays of Circular Cylinders I. A Single Cylinder in the Unit Cell

We study the longitudinal permeability of a spatially periodic rectangular array of circular cylinders, when a Newtonian fluid is flowing at low Reynolds number along the cylinders. The longitudinal component of the velocity obeys a Poisson equation which is transformed into a functional equation. This equation can be solved by the method of successive approximations. The major advantage of this technique is that the permeability of the array can be expressed analytically in terms of the radius of the cylinders and of the aspect ratio of the unit cell.

Riemann-Hilbert Problems for Multiply Connected Domains and Circular Slit Maps

A conformal mapping of a multiply connected circular domain onto a complex plane with circular slits is obtained. No restriction on the location of the boundary circles is assumed. The mapping is derived in terms of the uniformly convergent Poincaré series by solution to a Riemann-Hilbert boundary value problem.

Conductivity of a two-dimensional composite containing elliptical inclusions

We develop a method of functional equations to derive analytical approximate formulae for the effective conductivity tensor of the two-dimensional composites with elliptical inclusions. The sizes, the locations and the orientations of the ellipses can be arbitrary. The analytical formulae contain all the above geometrical parameters in symbolic form.

Transport properties of two–dimensional composite materials with circular inclusions

We consider the transport properties of a two–dimensional, two–component, composite medium made from a collection of non–overlapping, identical, circular discs, imbedded in an otherwise uniform host. Both components are isotropic conductors, but the position of the inclusions is arbitrary. The study is based on the analytic properties of such composite materials described by Bergman, Bergman and Dunn, and Milton, and the homogenization theory of random media advanced by Golden and Papanicolaou and Jikovet al. The crucial point of our study is application of the method of functional equations.

Optimal Distribution of the Nonoverlapping Conducting Disks

Conducting nonoverlapping identical disks are embedded in a two-dimensional background. The set of disks is infinite. The disks are distributed in such a way that the obtained composite is macroscopically isotropic. Let the conductivity of inclusions be higher than the conductivity of the matrix. It is proved that the hexagonal (triangular) lattice of disks possess the minimal effective conductivity when the concentration is not high.

A fast algorithm for computing the flux around non-overlapping disks on the plane

Conductivity of n non-overlapping disks embedded in a two-dimensional background can be investigated by the method of images which is based on the successive application of the inversions with respect to circles. For closely placed disks, the classical method of images yields slowly convergent series. The method of images can be treated as an application of successive approximations to a system of functional equations. In this paper, modified functional equations are deduced to essentially accelerate the convergence. Highlights? The method of images is treated as iterations applied to functional equations. ? Modified functional equations are deduced to essentially accelerate the convergence. ? A fast method of images is constructed.

Scalar Riemann–Hilbert Problem for Multiply Connected Domains

We solve the scalar Riemann–Hilbert problem for circular multiply connected domains. The method is based on the reduction of the boundary value problem to a system of functional equations (without integral terms). In the previous works, the Riemann–Hilbert problem and its partial cases such as the Dirichlet problem were solved under geometrical restrictions to the domains. In the present work, the solution is constructed for any circular multiply connected domain in the form of modified Poincare series.

ℝ-linear and Riemann–Hilbert Problems for Multiply Connected Domains

The ℝ-linear problem with constant coefficients for arbitrary multiply connected domains has been solved. The method is based on reduction of the problem to a system of functional equations for a circular domain and to integral equations for a general domain. In previous works, theℝ-linear problem and its partial cases such as the Riemann –Hilbert problem and the Dirichlet problem were solved under geometrical restrictions to the domains. In the present work, the solution is constructed for any circular multiply connected domain in the form of modified Poincar ´e series. Moreover, the modified alternating Schwarz method has been justified for an arbitrary multiply connected domain. This extends application of the alternating Schwarz method, since in the previous works geometrical restrictions were imposed on locations of the inclusions. The same concerns Grave’s method which was worked out before only for simple closed algebraic boundaries or for a collection of confocal boundaries.

Closed-form evaluation of two-dimensional static lattice sums

Closed-form formulae for the conditionally convergent two-dimensional (2D) static lattice sums S2 (for conductivity) and T2 (for elasticity) are deduced in terms of the complete elliptic integrals of the first and second kind. The obtained formulae yield asymptotic analytical formulae for the effective tensors of 2D composites with circular inclusions up to the third order in concentration. Exact relations between S2 and T2 for different lattices are established. In particular, the value S2=π for the square and hexagonal arrays is discussed and T2=π/2 for the hexagonal is deduced.