Partner: Włodzimierz Bielski

Institute of Geophysics (PL)

Recent publications
1.Bielski W., Wojnar R., Laminar flow past the bottom with obstacles - a porous medium approximation, JOURNAL OF THEORETICAL AND APPLIED MECHANICS, ISSN: 1429-2955, DOI: 10.15632/jtam-pl/151947, Vol.60, No.3, pp.509-520, 2022
Abstract:

We provide a model of a stationary laminar flow in a channel at the bottom of which plants grow in a dense layer. Since this layer of plants is dense, we treat it as a porous medium and we propose to describe the flow in such a medium by Brinkman’s equation. The flow in the fluid layer located above (infiltrated by water) the layers of plants is described by the Stokes equation. We show that such a model gives results consistent with experimental observations. We indicate also that this new model complements the previously given model in which the benthonic plants were considered as a suspension, so the previous model referred to the channel at the bottom of which the plants grew rare. For high permeability of the porous medium, we arrive at the results obtained for the medium filled with a liquid with a suspension.

Keywords:

plants, permeability, gravity driven flow, Darcy-Brinkman’s law, Stokes equation

Affiliations:
Bielski W.-Institute of Geophysics (PL)
Wojnar R.-IPPT PAN
2.Bielski W., Wojnar R., Long gravity waves in a canal with a corrugated bottom in the asymptotic description, JOURNAL OF THEORETICAL AND APPLIED MECHANICS, ISSN: 1429-2955, DOI: 10.15632/jtam-pl/138855, Vol.59, No.3, pp.443-454, 2021
Abstract:

We consider the classic Lagrange long gravitational wave of a homogeneous incompressible fluid in a shallow canal with a corrugated bottom. We use the asymptotic expansion method to find the effective depth of a one-dimensional canal and, hence, the effective wave velocity. A flow in a two-dimensional tank with a corrugated bottom is also studied by this method.

Keywords:

bottom profiles, homogenization approach, effective depth

Affiliations:
Bielski W.-Institute of Geophysics (PL)
Wojnar R.-IPPT PAN
3.Wojnar R., Bielski W., Laminar flow past the bottom with obstacles – a suspension approximation, BULLETIN OF THE POLISH ACADEMY OF SCIENCES: TECHNICAL SCIENCES, ISSN: 0239-7528, DOI: 10.1515/bpasts-2015-0080, Vol.63, No.3, pp.685-695, 2015
Abstract:

From Albert Einstein’s study (1905) it is known that suspension introduced to a fluid modifies its viscosity. We propose to describe the influence of obstacles on the Stokesian flow as a such modification. Hence, we treat the fluid flow through small obstacles as a flow with suspension. The flow is developing past the plane bottom under the gravity force. The spatial distribution of suspension concentration is treated as given, and is regarded as an approximation of different obstacles which modify the fluid flow and change its viscosity. The different densities of suspension are considered, beginning of small suspension concentration until 40%. The influence of suspension concentration on fluid viscosity is analyzed, and Brinkman’s formula as fitting best to experimental data is applied.

Keywords:

Stokes’ flow, Einstein’s suspension, Brinkman’s suspension, non-uniform viscosity, velocity distribution, wastes, plants on the bottom

Affiliations:
Wojnar R.-IPPT PAN
Bielski W.-Institute of Geophysics (PL)
4.Gambin B., Bielski W., Incompressible limit for a magnetostrictive energy functional, BULLETIN OF THE POLISH ACADEMY OF SCIENCES: TECHNICAL SCIENCES, ISSN: 0239-7528, DOI: 10.2478/bpasts-2013-0110, Vol.61, No.4, pp.320-326, 2013
Abstract:

The modern materials undergoing large elastic deformations and exhibiting strong magnetostrictive effect are modelled here by free energy functionals for nonlinear and non-local magnetoelastic behaviour. The aim of this work is to prove a new theorem which claims that a sequence of free energy functionals of slightly compressible magnetostrictive materials with a non-local elastic behaviour, converges to an energy functional of a nearly incompressible magnetostrictive material. This convergence is referred to as a Γ -convergence. The non-locality is limited to non-local elastic behaviour which is modelled by a term containing the second gradient of deformation in the energy functional.

Keywords:

gamma-convergence, incompressibility, magnetostrictive material, second gradient of deformation, existence of minimizers

Affiliations:
Gambin B.-IPPT PAN
Bielski W.-Institute of Geophysics (PL)
5.Bielski W., Gambin B., Relationship between existence of energy minimizers of incompressible and nearly incompressible magnetostrictive materials, REPORTS ON MATHEMATICAL PHYSICS, ISSN: 0034-4877, Vol.66, No.2, pp.147-157, 2010
Abstract:

Models of incompressible and slightly compressible magnetostrictive materials are introduced. They are given by the free energy functionals which depend on magnetization and elastic deformation as well as on their gradients. We demonstrate the existence of minimum of an energy functional for a slightly compressible material. We also prove a theorem on convergence of a sequence of minimizers of less and less compressible material energy functionals to a minimize of energy of incompressible material. Besides the existence of solution of the incompressible magnetostrictive problem is obtained.

Keywords:

micromagnetics, magnetostrictive material, incompressible, nearly incompressible materials, existence problem

Affiliations:
Bielski W.-Institute of Geophysics (PL)
Gambin B.-IPPT PAN
6.Bielski W., Kruglenko E., Telega J.J., Young measures and their applications in micromechanics and optimization. I. Mathematical principles., Mathematica Applicanda (Matematyka Stosowana), ISSN: 1730-2668, DOI: 10.14708/ma.v31i45/04.1914, Vol.31, No.45/04, pp.90-138, 2003
Abstract:

The paper is a review of modern mathematical methods for the analysis of continuous problems of nonlinear mechanics and magnetism. The bibliography contains 175 items. The second part of the paper deals with mechanical problems described by nonconvex density energy functions and with numerical methods to solve such problems.

Affiliations:
Bielski W.-Institute of Geophysics (PL)
Kruglenko E.-IPPT PAN
Telega J.J.-IPPT PAN
7.Bielski W., Telega J.J., A non-linear elastic plate model of moderate thickness: Existence, uniquenness and duality, JOURNAL OF ELASTICITY, ISSN: 0374-3535, DOI: 10.1007/BF00041792, Vol.42, No.3, pp.243-273, 1996
Abstract:

Three problems are solved for a nonlinear model of elastic plates with transverse shear deformations. The material of the plate may be anisotropic. An existence theorem is formulated and proved for a class of boundary conditions. A uniqueness theorem is given for small loadings. The dual problem is derived and the minimax or Lagrangian approach is discussed.

Affiliations:
Bielski W.-Institute of Geophysics (PL)
Telega J.J.-IPPT PAN
8.Bielski W., Telega J.J., Tarcie na uskokach geologicznych: modelowanie i wariacyjne sformułowanie zagadnień brzegowo-początkowych, Prace IPPT - IFTR Reports, ISSN: 2299-3657, No.31, pp.1-30, 1994
9.Bielski W., Telega J.J., On Existence of Solutions and Duality for a Model of Non-linear Elastic Plates with Transverse Shear Deformations, Prace IPPT - IFTR Reports, ISSN: 2299-3657, No.35, pp.1-26, 1992

List of chapters in recent monographs
1.
622
Bielski W., Wojnar R., Applicable Solutions in Nonlinear Dynamical Systems, rozdział: Gravity waves in channels with corrugated bottom: asymptotic approaches, Wydawnictwo Politechniki Łódzkiej, Łódź, pp.75-86, 2019
2.
594
Wojnar R., Bielski W., Modern Problems in Applied Analysis. Trends in Mathematics, rozdział: Gravity Driven Flow Past the Bottom with Small Waviness, Birkhauser – Springer International Publishing AG, part of Springer Nature 2018, Piotr Drygaś and Sergei Rogosin (Eds.), pp.181-202, 2018
3.
595
Bielski W., Wojnar R., Dynamical Systems in Theoretical Perspective. Springer Proceedings in Mathematics & Statistics, rozdział: Stokes Flow Through a Tube with Wavy Wall, Springer International Publishing AG, part of Springer Nature 2018, Jan Awrejcewicz (Ed.), 248, pp.379-390, 2018
4.
566
Bielski W., Wojnar R., Vibration, Control and Stability of Dynamical Systems - DSTA 2017, rozdział: Stokes flow through a tube with wavy wall, Jan Awrejcewicz, Marek Kaźmierczak, Jerzy Mrozowski, Paweł Olejnik (Eds.), Department of Automation, Biomechanics and Mechatronics, ARSA Druk i Reklama, Łódź, pp.83-94, 2017
5.
426
Bielski W., Kowalczyk P., Wojnar R., The Book of Abstracts of the Numerical Heat Transfer 2015 - Eurotherm Seminar No. 109, rozdział: Two-temperature heat transfer in metal films, Institute of Thermal Technology, Silesian University of Technology, Gliwice, Poland and Institute of Heat Engineering, Warsaw University of Technology, Warsaw, Poland, Editors: Andrzej J. Nowak, Jerzy Banaszek, and Bozidar Sarler, pp.29-30, 2015
6.
427
Bielski W., Kowalczyk P., Wojnar R., The Conference Proceedings of the Numerical Heat Transfer 2015 - Eurotherm Seminar No. 109, rozdział: Two-temperature heat transfer in metal films, Institute of Thermal Technology, Silesian University of Technology, Gliwice, Poland and Institute of Heat Engineering, Warsaw University of Technology, Warsaw, Poland, Editors: Andrzej J. Nowak, Jerzy Banaszek, and Bozidar Sarler, pp.57-67, 2015
7.
428
Bielski W., Wojnar R., PCM-CMM-2015 - 3rd Polish Congress of Mechanics & 21st Computer Methods in Mechanics, rozdział: Laminar flow past the bottom with obstacles - from suspension to porous medium, Polish Society of Theoretical and Applied Mechanics, Drukarnia WIB Piotr Winczewski, Gdańsk, Poland, Editors: Michał Kleiber, Tadeusz Burczyński, Krzysztof Wilde, Jarosław Górski, Karol Winkelmann, and Łukasz Smakosz, Vol. 1, 1, pp.207-208, 2015
8.
359
Wojnar R., Bielski W., Complex Analysis and Potential Theory with Applications, rozdział: Flow in the canal with plants on the bottom, Cambridge Scientific Publishers, T. Aliev Azerogly, A. Golberg, S.V. Rogosin (Eds.), pp.167-183, 2014
9.
155
Bielski W., Wojnar R., Analytic Methods of Analysis and Differential Equations, rozdział: Homogenisation of flow through double scale porous medium, Cambridge Scientific Publishers, Cambridge UK, Kilbas A.A., Rogosin S.V. (Eds.), pp.27-43, 2008
10.
219
Kruglenko E., Gambin B., Bielski W., Elektromagnetyczne techniki w ochronie zdrowia, rozdział: Materiały nieściśliwe magnetosprężyste jako model tkanek z mikromagnetycznymi cząstkami, CIOP PIB (Warszawa), Krawczyk A., Duraj A. (Eds.), pp.96-109, 2007
11.
179
Bielski W., Gambin B., Kruglenko E., Nonsmooth/Nonconvex Mechanics with applications in Engineering II. NNMAE206, A volume dedicated to the memory of Professor P.D.Panagiotopoulos, rozdział: Selected problems of rigid and deformable micromagnetics, ZITI, Baniotopoulos C.C. (Ed.), pp.69-76, 2006

Conference abstracts
1.Bielski W., Kowalczyk P., Wojnar R., Two-temperature heat transfer in a metal and a longitudinal elastic wave generation, DSTA 2021, 16th INTERNATIONAL CONFERENCE Dynamical Systems Theory and Applications, 2021-12-06/12-09, Łódź (PL), pp.382-383, 2021
Abstract:

A new version of the classic Danilowskaya's problem is presented. To describe the formation of a longitudinal elastic wave in the metal by a laser beam, we use a two-temperature theory of heating such a metal. In this theory of thermoelasticity, there are two temperatures simultaneously, the temperature of the electron gas Te and the temperature of the ionic lattice Ti, and both are functions of position and time. The energy transferred by electrons to the lattice per unit volume of the crystal per unit time is proportional to the difference Te - Ti. We give the equations of energy conservation and entropy production, whose thermodynamic consistency is verified. The equations of the problem are quasi-linear, since the specific heat of the electron gas in the considered temperature range depends on the temperature (degenerate electron gas). The one-dimensional process of transmitting thermal energy to the crystal lattice and the formation of mechanical wave is analyzed by the numerical method and illustrated in the pictures for various exemplary cases.

Keywords:

electron gas, photon-electron interaction, energy balance, entropy growth

Affiliations:
Bielski W.-Institute of Geophysics (PL)
Kowalczyk P.-University of Warsaw (PL)
Wojnar R.-IPPT PAN
2.Bielski W., Wojnar R., Brinkman's regulatization of Darcian seepage, PCM-CMM, 4th Polish Congress of Mechanics, 23rd International Conference on Computer Methods in Mechanics, 2019-09-08/09-12, Kraków (PL), pp.396-MS12-CR, 2019
3.Bielski W., Wojnar R., Plane flow through the porous medium with chessboard-like distribution of permeability, 2nd Workshop on Porous Media, 2018-06-28/06-30, Olsztyn (PL), pp.7, 2018
4.Bielski W., Kowalczyk P., Wojnar R., Thermal stresses and two temperature heat transfer, SolMech 2018, 41st SOLID MECHANICS CONFERENCE, 2018-08-27/08-31, Warszawa (PL), pp.234-235, 2018
5.Bielski W., Gambin B., An estimation of wave velocities in geomaterials, SolMech 2010, 37th Solid Mechanics Conference, 2010-09-06/09-10, Warszawa (PL), pp.240-241, 2010