Włodzimierz Bielski

Flows in random porous media: effective models

Abstract The aim of this paper is to provide a synthesis of results related to modeling transport in random media. Both single and two-phase flows have been considered. Special emphasis has been put on methods of stochastic homogenization.

Exact controllability of anisotropic elastic bodies

The aim of this contribution is to study the exact controllability of linear, anisotropic elastic bodies by applying Lions’ Hilbert Uniqueness Method.

Controllability and Stabilization in Elasticity, Heat Conduction and Thermoelasticity: Review of Recent Developments

The aim of this paper is to review developments in exact and approximate controllability as well as stabilization of elastic, thermoelastic, and thermo-viscoelastic bodies. Heat equations are also discussed.

Numerical Modelling of Electric Current Flow in Nanocrystalline 2D Carbon-Palladium Structures via Homogenization Method

We present a model of electric current flow through a two-dimensional palladium-carbon nanocomposite material and study the electrical conductivity of such material. The asymptotic homogenization theory and the finite element method are applied to analyse and solve the problem. The results of numerical computations are compared with the experimental data.

Modelling of a Time Dependent Electron Diffusion Problem for Nanocrystalline One‐dimensional Carbon‐Palladium Structures via Homogenization

Analytical and numerical phenomena of the electron resistance in the time dependent electron flow in Pd nanocrys‐tals embedded in a carbonaceous matrix are studied. The asymptotic homogenization theory and Finite Element Method are applied to analyse and solve the problem.

Stokes Flow Through a Tube with Wavy Wall

We propose a study of the flow in a tube with wavy wall adopting Malevich - Mityushev - Adler’s method, and find a correction to Hagen-Poiseuille’s solution. The problem is to be solved by expanding the velocity and pressure fields in Fourier series involving an infinite set of unknown coefficients. The boundary surface is expanded in Taylor’s series. A perturbation expansion in terms of the powers of the small parameter \(\varepsilon \) of the full set of Stokes’ equations yields a cascade of boundary value problems which are solved at each step in closed form. Even in the first order approximation \(O(\varepsilon )\), new results are obtained.

Effective conductivity in two-dimensional two-component structures: macroscopic isotropy

In this paper we use the mathematical methods of the homogenization theory to model the electrical conductivity of a two component nanostructure. We consider here a nanocomposite material in the form of a thin plate of negligible thickness compared to the diameter. Hence we assume that our problem is two-dimensional. As an example of such material we choose a carbon-palladium nanocomposite. We use the homogenization theory to study our problem, because of a complex microgeometry of the nanostructures. We show that the effective coefficient, under some assumptions, may be equal to a geometric mean of the coefficients of both components.

Selected Theoretical Methods in Solid Earth Physics: Contribution from the Institute of Geophysics PAS

Solid Earth Physics, including Seismology, Physics of the Earth, Earth Magnetism to name a few more topical disciplines, strongly relies on mathematical and numerical possibilities of modeling very complex physical processes ongoing in the Earth interior. Tremendous progress in geophysical instrumentation and still increasing quality and quantity of observational data also prompts for advanced processing methods in order to get more reliable interpretations. The goal of this chapter is to review some contributions from the Institute of Geophysics, Polish Academy of Sciences (IGF PAS) to physical and mathematical concepts used in Solid Earth Physics. We have selected some topics which are general enough to be interesting for a wide range of readers, leaving many topical issues uncovered in this review.

Numerical Modelling of Mechanical Properties of C‐Pd Film by Homogenization Technique and Finite Element Method

The nanomechanical properties of nanostructural carbonaceous‐palladium films are studied. The nanoindentation experiments are numerically using the Finite Element Method. The homogenization theory is applied to compute the properties of the composite material used as the input data for nanoindentation calculations.

Variational formulation of the transient motion of a rolling cylinder undergoing plane finite deformations

Abstract This paper is concerned with a variational formulation of the problem of the transient motion of a rolling cylinder undergoing plane finite deformations. Frictionless motion is described by one quasi-variational inequality. If friction is taken into account then the formulation is available in the form of an evolution implicit variational inequality coupled with a quasi-variational inequality.