Instytut Podstawowych Problemów Techniki

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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.

Słowa kluczowe:

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

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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.

Słowa kluczowe:

bottom profiles, homogenization approach, effective depth

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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.

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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 zero-th approximation.

Słowa kluczowe:

porosity, Darcy's law, Stokes' equation, successive iterations, Born's approximation

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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.

Słowa kluczowe:

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

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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.

For (i) we give a method and find the effective thermal conductivity for the model of the liver tissue, and for the point (ii) we present numerical and analytical treatment of the problem, and indicate the principal difference of heat propagation in linear and nonlinear media. In linear media, as it is well known, the range of the heat field is infinite for all times t > 0, and in nonlinear media it is finite.

Pennes’ equation, which should characterize the heat propagation in the living tissue, is in general a quasi-nonlinear partial differential equation, and consists of three terms, one of which describes Fourier’s heat diffusion with conductivity being a function of temperature T. This term is just a point of our analysis.

We show that a nonlinear character of the medium (heat conductivity dependent on the temperature) changes in qualitative manner the nature of heat transfer. It is proved that for the heat source concentrated initially (t = 0) at the space point, the range of heated region (for t > 0) is finite. The proof is analytical, and illustrated by a numerical experiment.

Słowa kluczowe:

heat transport, asymptotic homogenisation, effective heat conductivity

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Strain-induced 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.

Słowa kluczowe:

Nanowires, II-VI semiconductors, Magnesium, Band gap, Quantum effects

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The thermal effects of a stationary Stokesian flow through an elastic micro-porous medium are compared with the entropy produced by Darcy’s flow. A micro-cellular elastic medium is considered as an approximation of the elastic porous medium. It is shown that after asymptotic two-scale 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 micro-porous systems the seepage of both types of fluids is described by the same equations.

Słowa kluczowe:

porous media, Onsager’s principle, entropy, heat production

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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.

An analysis of the state of strain and stress in the core-shell nanowire within linear elasticity, using an analogy with thermal stresses is presented, in the similar way as it is applied, e.g. in hygro-mechanics. The suitable system of the differential Lame-Navier’s type equations is derived, and its solution for the axially symmetric problem is given. The jump of stress at the core-shell boundary is determined.

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One-dimensional 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.

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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.

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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.

Słowa kluczowe:

difference-differential limit, diffusion coefficient, Smoluchowski’s equation, Einstein’s formula, limit transitions, infinite velocity paradox

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The aim of this paper is to analyze different contributions and differents points of view concerning the meaning of the piezoelectric effect in the bone. It is now obvious that this effect is overhelming in dry bone. In wet bone more important are streaming potentials.

Słowa kluczowe:

bone, piezoelectricity

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The problem of homogenization of a piezoelectric periodic composite in which thermal effects are taken into account is treated by a method of 2-scale asymptotic expansions. Albeit a nonlinear entropy-temperature relation is used, the local problems are formulated in a way similar to that of a linear theory. Also, effective material coefficients are in majority the same as those obtained in a linearized theory. The difference appears in the homogenized thermoelastic, thermoelectric and specific heat coefficients only. A Francfort type result on shift of the initial conditions for a homogenized problem is also obtained.

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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.

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On the basis of the paper [1] two topics are discussed. Firstly, exact formulae for the homogenized coeffi-cients of a layered thermopiezoelectric composite are derived. Secondly, by applying the Ritz method, the local problems are solved approximately. Specific cases are also examined and illustrated.

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In [9] the formulae were derived for the effective moduli of a piezoelectric composite exhibiting fine periodic structure. The static case considered was studied by the method of F-convergence, cf.[l]. Next, in [10] these results were extended by investigating the dynamic behaviour. The same formulae for the effective moduli were obtained by using the method of Bloch expansions. The aim of this contribution is to present our preliminary results concerning homogenization of the equations of the linear thermopiezoelectricity. To such a problem the method of F-convergence is not applicable. Therefore the relevant analysis was performed by using the method of two-scale asymptotic developments. Our analysis leads to interesting conclusions related to the effective moduli and to the change of the initial condition for the temperature of the homogenized body.

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Surface waves in an isotropic homogeneous semi-infinite space are studied within the linear thermoelasticity with two relaxation times. The boundary of the semi-space is assumed to be free of stresses and through it there is a heat exchange with environment. Associated dispersion relation is derived and its numerical and asymptotic analysis are presented. The analogy of our results with those of a theory with one relaxation time is discussed. Also, Rayleigh's classical result for isothermal elastodynamics, and Lockett's thermoelastic solution corresponding to low frequencies and small wave vectors are recovered.

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Two uniqueness theorems of linear thermoelasticity with one relaxation time corresponding to natural initial boundary value problems described in terms of two different pairs of thermoelastic variables: displacement-heat flux and stress-temperature are proved.

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The effect of the internal reflection at Brewster's angle is applied to obtain a photoelastic pattern for stresses in a two-dimensional layer. The reflection itself acts as an analyzer, therefore only one Polaroid — the polarizer is needed in an experiment. The explanation of the pattern formation is presented, using the generalised Jones matrices.
The possibility of the application of the method to stress separation, on basis of the oblique incidence method is pointed out.

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The kinetic equation for the dilute Lorentz gas of particles with repulsive-attractive potential, is derived. For that purpose the distribution function is decomposed into two parts: the one corresponding to the bounded motion of the marked particle and the second corresponding to its unbounded motion; only the second part, as representing the diffusion, is considered.

After the appropriate modification of the projection operator, a general kinetic equation for the diffusion part of the distribution function is obtained. The low density limit of the scattering operator of this equation is found. The regions of bounded motion are excluded in the integration over position space.

For the square-well potential, the Laplace-transformed kinetic equation is also given in full detail.

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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.

Słowa kluczowe:

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

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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 micro-periodic cellular medium, and in circumstances justified by biological reasons, the mathematical methods of homogenisation developed for micro-periodic 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, [1, 2, 3]. 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 oxygen-rich 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.
We have evaluated the dependence of effective conductivity λeff for the composite consisting of the basic cells arranged in a two-dimensional periodic system and built of the collagen capillaries filled with the water. Analytical and numerical results are going to be verified by measurement of temperature using magnetic resonance imaging (MRI) and through measurement of backscattered ultrasound waves.

Słowa kluczowe:

liver tissue, Pennes’ equation, heat transport, asymptotic homogenization, effective coefficients

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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 micro-periodic medium, and the mathematical methods of homogenisation developed for micro-periodic media can be applied to determine some overall properties of the tissue. Pennes’ equation of heat propagation in biological tissue is a quasi-nonlinear 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.

Słowa kluczowe:

Pennes’ equation, micro-periodic structure, effective conductivity

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