On solvability of initial boundary-value problems of micropolar elastic shells with rigid inclusions

The problem of dynamics of a linear micropolar shell with a finite set of rigid inclusions is considered. The equations of motion consist of the system of partial differential equations (PDEs) describing small deformations of an elastic shell and ordinary differential equations (ODEs) describing the motions of inclusions. Few types of the contact of the shell with inclusions are considered. The weak setup of the problem is formulated and studied. It is proved a theorem of existence and uniqueness of a weak solution for the problem under consideration.

Models of activated recrystallization on isolated grain boundaries in polycrystals

New phenomenological models of recrystallization of a polycrystalline material in two regimes are proposed taking into account the finite width of grain boundaries. The solutions are obtained in an analytical form for the initial-boundary value problems formulated. They describe the distributions of the concentration of impurities diffusing from the surface coating, both in the grain boundary and in the grain itself in the recrystallized region. The speed of the recrystallization front movement is indicated, which agrees with the types of the corresponding kinetic dependences observed in experiments.

On the Unique Solvability of Incomplete Cauchy Type Problems for a Class of Multi-Term Equations with the Riemann–Liouville Derivatives

Incomplete Cauchy-type problems are considered for linear multi-term equations solved with respect to the highest derivative in Banach spaces with fractional Riemann–Liouville derivatives and with linear closed operators at them. Some new existence and uniqueness theorems for solutions are presented explicitly and the analyticity of the solutions of the homogeneous equations are also shown. The asymmetry of the Cauchy-type problem under study is expressed in the presence of a so-called defect, which shows the number of lower-order initial conditions that should not be set when setting the problem. As applications, our abstract results are used in the study of a class of initial-boundary value problems for multi-term equations with Riemann–Liouville derivatives in time and with polynomials of a self-adjoint elliptic differential operator with respect to spatial variables.

Comment on “Tensional homeostasis at different length scales” by D. Stamenović and M. L. Smith, Soft Matter, 2021, 17, 10274–10285, DOI: 10.1039/D0SM01911A

Assessing potential mechanical homeostasis requires appropriate solutions to the initial-boundary value problems that define the biophysical situation of interest and appropriate definitions of what is meant by homeostasis, including its range.

Investigation of dynamic processes in pressure measurement systems for gas-liquid media

Initial-boundary value problems for systems of differential equations are considered, which are mathematical models of the mechanical system "pipeline - pressure sensor". In such a system, to mitigate the effects of vibration accelerations and high temperatures, the sensor is located at a certain distance from the engine and is connected to it via a pipeline. The "pipeline - pressure sensor" system is designed to measure pressure in gas-liquid media, for example, to control the pressure of the working medium in the combustion chambers of engines. On the basis of the proposed models, the joint dynamics of the sensitive element of the pressure sensor and the working medium in the pipeline is studied. To describe the motion of the working medium, linear models of fluid and gas mechanics are used, to describe the dynamics of a sensitive element, linear models of the mechanics of a deformable solid are applied. Analytical and numerical methods for solving initial-boundary value problems under study are presented. The numerical study of the initial-boundary value problem was carried out on the basis of the Galerkin method. In analytical study using the introduction of averaged characteristics, the solution of the original two-dimensional problem is reduced to the study of a one-dimensional model, whose further study made it possible to reduce the solution of the problem to the study of a differential equation with a deviating argument. Also, a numerical experiment is carried out and an example of calculating the deflection of the sensor’s moving element is presented.

Modeling of filtration processes in periodic porous media

Modeling of dynamic processes of diffusion and filtration of liquids in porous media are discussed. The media are formed by a large number of blocks with low permeability, and separated by a connected system of faults with high permeability. The modeling is based on solving initial boundary value problems for parabolic equations of diffusion and filtration in porous media. The structure of the media leads to the dependence of the equations on a small parameter. Assertions on the solvability and regularity of such problems and the corresponding homogenized convolution problems are considered. The statements are actual for the numerical solution of this problem with guaranteed accuracy that is necessary to model the considered processes.

Mixed initial-boundary value problems describing motions of Maxwell fluids with linear dependence of viscosity on the pressure

Some mixed initial-boundary value problems are analytically studied. They correspond to unsteady motions of the incompressible upper-convected Maxwell (IUCM) fluids with linear dependence of viscosity on the pressure between infinite horizontal parallel plates. The fluid motion is generated by the upper plate that applies time-dependent shear stresses to the fluid. Exact solutions are established for the dimensionless velocity and nontrivial shear stress fields using a suitable change of the spatial variable and the Laplace transform technique. They are presented as sum of the steady-state and transient components and are used to determine the required time to reach the permanent state. Comparisons between exact and numerical solutions indicate an excellent agreement. Analytical solutions for the unsteady motion of the same fluids induced by an exponential shear stress on the boundary are obtained as limiting cases of the general solutions. Moreover, the steady-state solutions corresponding to the ordinary IUCM fluids performing the initial motions are provided by means of asymptotic approximations of standard Bessel functions. Finally, spatial variation of starting solutions and the influence of physical parameters on the fluid motion are graphically underlined and discussed.