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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

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

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

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

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

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.

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.

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