Ryszard Wojnar, Ph.D. 

Doctoral thesis
1973  Liniowa mechanika statystyczna uogólnionej dyfuzji
 1315 
Recent publications
1.  Wojnar R., Kinetic equation for the dilute Boltzmann gas in an external field, ACTA PHYSICA POLONICA B, ISSN: 05874254, DOI: 10.5506/APhysPolB.49.905, Vol.49, No.5, pp.905920, 2018 Abstract: We report a kinetic equation for an auxiliary distribution function f(k,v1,t) which yields the intermediate scattering function Is(k,t). To this end, the projection operator proposed by Stecki was applied. The scattering operator was given in explicit form in the limit of low density gas. The general kinetic equation was next specialized for the case of Lorentz gas. Affiliations:
 
2.  Wojnar R., Heuristic derivation of Brinkman's seepage equation, Technical Sciences, ISSN: 15054675, Vol.20, No.4, pp.359374, 2017 Abstract: Brinkman’s law is describing the seepage of viscous fluid through a porous medium and is more acuratethan the classicalDarcy’s law.Namely, Brinkman’s law permitsto conform the flow through a porous medium to the free Stokes’ flow. However, Brinkman’s law, similarly as Schro¨dinger’s equation was only devined. Fluid in its motion through a porous solid is interacting at every point with the walls of pores, but the interactions of the fluid particles inside pores are different than the interactions at the walls, and are described by Stokes’ equation. Here, we arrive at Brinkman’s law from Stokes’ flow equation making use of successive iterations, in type of Born’s approximation method, and using Darcy’s law as a zeroth approximation. Keywords:porosity, Darcy’s law, Stokes’ equation, successive iterations, Born’s approximation Affiliations:
 
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: 02397528, DOI: 10.1515/bpasts20150080, Vol.63, No.3, pp.685695, 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, nonuniform viscosity, velocity distribution, wastes, plants on the bottom Affiliations:
 
4.  Gambin B., Kruglenko E., Gałka A.A.^{♦}, Wojnar R., Macroscopic thermal properties of quasilinear cellular medium on example of the liver tissue, COMPUTER ASSISTED METHODS IN ENGINEERING AND SCIENCE, ISSN: 22993649, Vol.22, No.4, pp.329346, 2015 Abstract: There are two main topics of this research: (i) one topic considers overall properties of a nonlinear cellular composite, treated as a model of the liver tissue, and (ii) the other topic concerns the propagation of heat in the nonlinear medium described by the homogenised coefficient of thermal conductivity. heat transport, asymptotic homogenisation, effective heat conductivity Affiliations:
 
5.  Wojnar P.^{♦}, Zieliński M.^{♦}, Janik E.^{♦}, Zaleszczyk W.^{♦}, Wojciechowski T.^{♦}, Wojnar R., Szymura M.^{♦}, Kłopotowski Ł.^{♦}, Baczewski L.T.^{♦}, Pietruchik A.^{♦}, Wiater M.^{♦}, Kret S.^{♦}, Karczewski G.^{♦}, Wojtowicz T.^{♦}, Kossut J.^{♦}, Straininduced energy gap variation in ZnTe/ZnMgTe core/shell nanowires, APPLIED PHYSICS LETTERS, ISSN: 00036951, DOI: 10.1063/1.4873355, Vol.104, pp.16311115, 2014 Abstract: Straininduced changes of ZnTe energy gap in ZnTe/ZnMgTe core/shell nanowires arising from lattice mismatch between the core and the shell semiconductor are studied by means of optical methods. It is shown that the increase of the Mg content in the shell, as well as the increase of the shell thickness result in an effective redshift of the near band edge photoluminescence from ZnTe nanowire cores, which reflects directly the decrease of energy gap under tensile strain conditions. The conclusions are supported by theoretical calculations in terms of the valence force field model. The observed change of ZnTe energy gap can be as large as 120 meV with respect to the unstrained conditions and can be tuned in a continuous manner by adjusting shell parameters, which open a path towards an effective band gap engineering in these structures. Keywords:Nanowires, IIVI semiconductors, Magnesium, Band gap, Quantum effects Affiliations:
 
6.  Wojnar R., Flow of Stokesian fluid through a cellular medium and thermal effects, BULLETIN OF THE POLISH ACADEMY OF SCIENCES: TECHNICAL SCIENCES, ISSN: 02397528, DOI: 10.2478/bpasts20140031, Vol.62, No.2, pp.321329, 2014 Abstract: The thermal effects of a stationary Stokesian flow through an elastic microporous medium are compared with the entropy produced by Darcy’s flow. A microcellular elastic medium is considered as an approximation of the elastic porous medium. It is shown that after asymptotic twoscale analysis these two approaches, one analytical, starting from Stoke’s equation and the second phenomenological, starting from Darcy’s law give the same result. The incompressible and linearly compressible fluids are considered, and it is shown that in microporous systems the seepage of both types of fluids is described by the same equations. Keywords:porous media, Onsager’s principle, entropy, heat production Affiliations:
 
7.  Wojnar R., Wojnar P.^{♦}, Kret S.^{♦}, Elastic State Induced Energy Gap Variation in ZnTe/ZnMgTe Core/Shell Nanowires, TECHNISCHE MECHANIK, ISSN: 02323869, Vol.34, pp.233245, 2014 Abstract: The zinc telluride (ZnTe) nanowires grown recently are covered with the ZnMgTe shell. As a result of addition of magnesium the ZnMgTe lattice is expanded with respect to pure ZnTe lattice. From the lattice mismatch between the ZnMgTe shell and ZnTe nanowire core the internal strain and stress are created. Depending on the shell thickness and the Mg content in the shell the optical emission exhibits a considerable energy shift. To estimate this effect, at least qualitatively, the elastic state of the nanowire is calculated.
 
8.  Wojnar R., Random Walk, Diffusion and Wave Equation, ACTA PHYSICA POLONICA B, ISSN: 05874254, DOI: 10.5506/APhysPolB.44.1067, Vol.44, No.5, pp.10671084, 2013 Abstract: Onedimensional random walk is analyzed. First, it is shown that the classical interpretation of random walk reaching Lord Rayleigh’s analysis should be completed. Further, an attention is called to the fact that the parabolic diffusion is not an unique interpretation, but also the wave (or hyperbolic) equation can be deduced. It depends on the accepted scale of the length of step h and duration of the step τ in the walk, whether Fick–Smoluchowski’s diffusion or a wave process is obtained. Only additional arguments, such as positivity of distribution function or positivity of the entropy growth, can help to choose the proper physical model. Also, the infinite diffusion velocity paradox in connection with Einstein’s formula is explained. Affiliations:
 
9.  Wojnar R., Rayleigh's Distribution, Wigner's Surmise and Equation of the Diffusion, ACTA PHYSICA POLONICA A, ISSN: 05874246, DOI: 10.12693/APhysPolA.123.624, Vol.123, No.3, pp.624628, 2013 Abstract: After summaries on Rayleigh's distribution and Wigner's surmise, the time evolution of Rayleigh Wigner's statistics is studied and a suitable diffusion type equation is proposed. Also the variance and kurtosis of time evolution of Rayleigh's distribution are calculated. Obtained results maybe useful in description of physical, social and biological processes. Affiliations:
 
10.  Wojnar R., Random walk, RayleighKac’ scheme and diffusion equation, REPORTS ON MATHEMATICAL PHYSICS, ISSN: 00344877, Vol.72, No.3, pp.321332, 2013 Abstract: Nonsymmetric random walk is studied and amelioration of existing schemes of its interpretation is proposed. It is shown that for a nonsymmetric random walk not only the analog of the drift force appears, but in comparison with the symmetric random walk the diffusion coefficient is modified. The infinite velocity paradox is also discussed. Keywords:differencedifferential limit, diffusion coefficient, Smoluchowski’s equation, Einstein’s formula, limit transitions, infinite velocity paradox Affiliations:
 
11.  Wojnar R., Student’s tdistribution versus Zeldovich–Kompaneets solution of diffusion problem, ACTA PHYSICA POLONICA A, ISSN: 05874246, Vol.121, No.2B, pp.133136, 2012  
12.  Wojnar R., Twolevel diffusion, random walk and uniqueness, REPORTS ON MATHEMATICAL PHYSICS, ISSN: 00344877, Vol.68, No.1, pp.8596, 2011  
13.  Wojnar R., Boussinesq equation for flow in an aquifer with time dependent porosity, BULLETIN OF THE POLISH ACADEMY OF SCIENCES: TECHNICAL SCIENCES, ISSN: 02397528, Vol.58, No.1, pp.165170, 2010  
14.  Wojnar R., Kinetic equation for a gas with attractive forces as a functional equation, Annales Universitatis Paedagogicae Cracoviensis Studia Mathematica, ISSN: 2081545X, Vol.8, pp.91116, 2009  
15.  Wojnar R., Structural control in tissue development, JOURNAL OF THEORETICAL AND APPLIED MECHANICS, ISSN: 14292955, Vol.43, No.4, pp.805812, 2005  
16.  Telega J.J., Wojnar R., Main Polish Historical and Modern Sources on Applied Mechanics, Applied Mechanics Reviews, ISSN: 00036900, DOI: 10.1115/1.3101933, Vol.49, No.8, pp.401432, 1996 Abstract: The aim of thikocvzeń article is to present the list of books by Polish scientists on applied mechanics in a rather broad sense. An effort was made to cover the last century. Previous Polish sources, until 1874, were collected by Kucharzewski (1894). A list of the main journals in applied mechanics is also given, essentially covering the period starting after the Second World War. Affiliations:
 
17.  Gałka A.^{♦}, Telega J.J., Wojnar R., Thermodiffusion in Heterogeneous Elastic Solids and Homogenization, Prace IPPT  IFTR Reports, ISSN: 22993657, No.14, pp.176, 1993  
18.  Gałka A.^{♦}, Wojnar R., Dynamiczne naprężenia cieplne w półprzestrzeni sprężystej wywołane przez impuls laserowy, Prace IPPT  IFTR Reports, ISSN: 22993657, No.25, pp.153, 1993  
19.  Wojnar R., O jednoznaczności rozwiązań naprężeniowych równań ruchu typu BeltramiegoMichella, Prace IPPT  IFTR Reports, ISSN: 22993657, No.63, pp.111, 1972  
20.  Wojnar R., Twierdzenie o jednoznaczności dla pewnego układu naprężeniowych równań ruchu liniowej teorii sprężystości, Prace IPPT  IFTR Reports, ISSN: 22993657, No.70, pp.110, 1972  
21.  Wojnar R., O wyznaczeniu naprężeń zmiennych w czasie z obrazów elastooptycznych, Prace IPPT  IFTR Reports, ISSN: 22993657, No.33, pp.112, 1970 
List of chapters in recent monographs
1. 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.181202, 2018  
2. 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.379390, 2018  
3. 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.8394, 2017  
4. 426  Bielski W.^{♦}, Kowalczyk P.^{♦}, Wojnar R., The Book of Abstracts of the Numerical Heat Transfer 2015  Eurotherm Seminar No. 109, rozdział: Twotemperature 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.2930, 2015  
5. 393  Rylko N.^{♦}, Wojnar R., Geometry, Integrability, Mechanics and Quantization, rozdział: Resurgence edge effects in composites: fortuity and geometry, Ivailo M. Mladenov, Mariana Hadzhilazova and Vasyl Kovalchuk (Editors), Avangard Prima, Sofia, pp.342349, 2015  
6. 395  Wojnar R., Geometry, Integrability, Mechanics and Quantization, rozdział: Bohmian picture of the wave function and the gauge invariance, Ivailo M. Mladenov, Mariana Hadzhilazova and Vasyl Kovalchuk (Editors), Avangard Prima, Sofia, pp.411424, 2015  
7. 427  Bielski W.^{♦}, Kowalczyk P.^{♦}, Wojnar R., The Conference Proceedings of the Numerical Heat Transfer 2015  Eurotherm Seminar No. 109, rozdział: Twotemperature 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.5767, 2015  
8. 428  Bielski W.^{♦}, Wojnar R., PCMCMM2015  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.207208, 2015  
9. 429  Wojnar R., PCMCMM2015  3rd Polish Congress of Mechanics & 21st Computer Methods in Mechanics, rozdział: Modelling polycrystalline structure of collagen fibrils dense packing by the most uniform concentric pattern, 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.269270, 2015  
10. 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.167183, 2014  
11. 61  Wojnar R., Piezoelectric nanomaterials for biomedical applications. Nanomedicine and Nanotoxicology, rozdział: Piezoelectric phenomena in biological tissues, Springer, Gianni Ciofani and Arianna Menciassi (Eds.), pp.173185, 2012  
12. 24  Wojnar R., Biomechanics of hard tissues: Modeling, testing and materials, rozdział: Bone and cartilage  its structure and physical properties, WileyVCH Verlag GmbH&Co. KGaA, Weinheim, pp.176, 2010  
13. 175  Wojnar R., More Progresses In Analysis: Proceedings of the 5th International Isaac Congres, rozdział: Strains in tissue development: a vortex description, World Scientific Publishing Co., Begehr H.G.W., Nicolosi F. (Eds.), pp.12711281, 2009  
14. 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.2743, 2008 
Conference papers
1.  Wojnar R., Complex dielectric coefficients and maxwell’s averaging, 8th International ISAAC Congress, 20110822/0827, Moscow (RU), pp.17, 2011 
Conference abstracts
1.  Bielski W.^{♦}, Wojnar R., Plane flow through the porous medium with chessboardlike distribution of permeability, 2nd Workshop on Porous Media, 20180628/0630, Olsztyn (PL), pp.7, 2018  
2.  Wojnar R., Movement of coincidence grain boundaries with sigma = 7,13,19,...,49,...,91,...: from isotropy to anisotropy, SolMech 2018, 41st SOLID MECHANICS CONFERENCE, 20180827/0831, Warszawa (PL), pp.250251, 2018  
3.  Bielski W.^{♦}, Kowalczyk P.^{♦}, Wojnar R., Thermal stresses and two temperature heat transfer, SolMech 2018, 41st SOLID MECHANICS CONFERENCE, 20180827/0831, Warszawa (PL), pp.234235, 2018  
4.  Gambin B., Kruglenko E., Wojnar R., Macroscopic thermal properties of quasilinear cellular medium on example of the liver tissue, NHT 2015, Numerical Heat Transfer 2015 – Eurotherm Seminar No. 109, 20150927/0930, Warszawa (PL), pp.177178, 2015 Abstract: After discovery of strong sonar systems, it was realized that the high intensity ultrasound waves can be dangerous for biological organisms. This observation led to research in tissue heating effects. The liver tissue from mathematical point of view can be considered as a microperiodic cellular medium, and in circumstances justified by biological reasons, the mathematical methods of homogenisation developed for microperiodic media can be applied to determine some overall properties of the tissue. Fourier’s heat diffusion term in Pennes’ equation is the point of departure in our analysis, . The liver, the largest internal organ in the human body, is connected to two large blood vessels, the hepatic artery and the portal vein. The hepatic artery carries oxygenrich blood from the aorta, whereas the portal vein carries blood rich in digested nutrients from the entire gastrointestinal tract and also from the spleen and pancreas. These blood vessels subdivide into small capillaries known as liver sinusoids, which then lead to a lobule. A hepatic lobule is a small division of the liver defined at the histological scale. The lobules are arranged into an hexagonal lattice. liver tissue, Pennes’ equation, heat transport, asymptotic homogenization, effective coefficients Affiliations:
 
5.  Wojnar R., Gambin B., Thermal properties of biomaterials on the example of the liver, PCMCMM 2015, 3rd Polish Congress of Mechanics and 21st Computer Methods in Mechanics, 20150908/0911, Gdańsk (PL), pp.267268, 2015 Abstract: Lionel Smith Beale, FRS, (1828–1906), a physician and microscopist in an evocative comparison wrote that the liver resembles a magnificent tree with its trunk and branches, with myriad of leaves, synthesizing and detoxifying. The liver in a human is about the size of football, equipped in a circulatory system and is made of about one million primary lobules which are almost identical, like the leaves of the tree. Therefore, the liver from mathematical point of view can be considered as a microperiodic medium, and the mathematical methods of homogenisation developed for microperiodic media can be applied to determine some overall properties of the tissue. Pennes’ equation of heat propagation in biological tissue is a quasinonlinear partial differential equation with coefficients depending on temperature T. It consists of three terms, one of which describes Fourier’s heat diffusion, with the diffusion coefficient depending on T. This term is a subject of this contribution. Keywords:Pennes’ equation, microperiodic structure, effective conductivity Affiliations:
 
6.  Wojnar R., Budowa kolagenu: geometria i fizyka, III National Conference of Nano and Micromechanics, 20120704/0706, Warszawa (PL), pp.169170, 2012 