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Comparative analysis of two numerical methods for simulation of unsteady natural convection and thermal surface radiation within a differentially heated cubical cavity has been carried out. The considered domain of interest had two isothermal opposite vertical faces, while other walls are adiabatic. The walls surfaces were diffuse and gray, namely, their directional spectral emissivity and absorptance do not depend on direction or wavelength but can depend on surface temperature. For the reflected radiation we had two approaches such as: 1) the reflected radiation is diffuse, namely, an intensity of the reflected radiation in any point of the surface is uniform for all directions; 2) the reflected radiation is uniform for each surface of the considered enclosure. Mathematical models formulated both in primitive variables “velocity–pressure” and in transformed variables “vector potential functions – vorticity vector” have been performed numerically using finite volume method and finite difference methods, respectively. It should be noted that radiative heat transfer has been analyzed using the net-radiation method in Poljak approach.

Using primitive variables and finite volume method for the considered boundary-value problem we applied power-law for an approximation of convective terms and central differences for an approximation of diffusive terms. The difference motion and energy equations have been solved using iterative method of alternating directions. Definition of the pressure field associated with velocity field has been performed using SIMPLE procedure.

Using transformed variables and finite difference method for the considered boundary-value problem we applied monotonic Samarsky scheme for convective terms and central differences for diffusive terms. Parabolic equations have been solved using locally one-dimensional Samarsky scheme. Discretization of elliptic equations for vector potential functions has been conducted using symmetric approximation of the second-order derivatives. Obtained difference equation has been solved by successive over-relaxation method. Optimal value of the relaxation parameter has been found on the basis of computational experiments.

As a result we have found the similar distributions of velocity and temperature in the case of these two approaches for different values of Rayleigh number, that illustrates an operability of the used techniques. The efficiency of transformed variables with finite difference method for unsteady problems has been shown.

Mathematical simulation of unsteady natural convection and thermal surface radiation within a rotating square enclosure was performed. The considered domain of interest had two isothermal opposite walls subjected to constant low and high temperatures, while other walls are adiabatic. The walls were diffuse and gray. The considered cavity rotated with constant angular velocity relative to the axis that was perpendicular to the cavity and crossed the cavity in the center. Mathematical model, formulated in dimensionless transformed variables “stream function – vorticity” using the Boussinesq approximation and diathermic approach for the medium, was performed numerically using the finite difference method. The vorticity dispersion equation and energy equation were solved using locally one-dimensional Samarskii scheme. The diffusive terms were approximated by central differences, while the convective terms were approximated using monotonic Samarskii scheme. The difference equations were solved by the Thomas algorithm. The approximated Poisson equation for the stream function was solved by successive over-relaxation method. Optimal value of the relaxation parameter was found on the basis of computational experiments. Radiative heat transfer was analyzed using the net-radiation method in Poljak approach. The developed computational code was tested using the grid independence analysis and experimental and numerical results for the model problem.

Numerical analysis of unsteady natural convection and thermal surface radiation within the rotating enclosure was performed for the following parameters: Ra = 10³–10⁶, Ta = 0–10⁵, Pr = 0.7, ε = 0–0.9. All distributions were obtained for the twentieth complete revolution when one can find the periodic behavior of flow and heat transfer. As a result we revealed that at low angular velocity the convective flow can intensify but the following growth of angular velocity leads to suppression of the convective flow. The radiative Nusselt number changes weakly with the Taylor number.

The article presents a method for linearization the effective function of material instantaneous deformation in order to generalize the torsional vibration equation to the case of nonlinearly deformable rheologically active shafts. It is considered layered and structurally heterogeneous, on average isotropic shafts made of nonlinearly viscoelastic components. The technique consists in determining the approximate shear modulus by minimizing the root-mean-square deviation in approximation of the effective diagram of instantaneous deformation.

The method allows to estimate analytically values of natural frequencies of layered and structurally heterogeneous nonlinearly viscoelastic shaft. This makes it possible to significantly reduce resources in vibration analysis, as well as to track changes in values of natural frequencies with changing geometric, physico-mechanical and structural parameters of shafts, which is especially important at the initial stages of modeling and design. In addition, the paper shows that only a pronounced nonlinearity of the effective state equation has an effect on the natural frequencies, and in some cases the nonlinearity in determining the natural frequencies can be neglected.

As equations of state of the composite material components, the article considers the equations of nonlinear heredity with instantaneous deformation functions in the form of the Prandtl’s bilinear diagrams. To homogenize the state equations of layered shafts, it is applied the Voigt’s hypothesis on the homogeneity of deformations and the Reuss’ hypothesis on the homogeneity of stresses in the volume of a composite body. Using these assumptions, effective secant and tangential shear moduli, proportionality limits, as well as creep and relaxation kernels of longitudinal, axial and transversely layered shafts are obtained. In addition, it is obtained the indicated effective characteristics of a structurally heterogeneous, on average isotropic shaft using the homogenization method previously proposed by the authors, based on the determination of the material deformation parameters by the rule of a mixture for the Voigt’s and the Reuss’ state equations.

Stability of finite difference schemes of lattice Boltzmann method for modelling of 1D diffusion for cases of D1Q2 and D1Q3 lattices is investigated. Finite difference schemes are constructed for the system of linear Bhatnagar–Gross–Krook (BGK) kinetic equations on single particle distribution functions. Brief review of articles of other authors is realized. With application of multiscale expansion by Chapman–Enskog method it is demonstrated that system of BGK kinetic equations at small Knudsen number is transformated to scalar linear diffusion equation. The solution of linear diffusion equation is obtained as a sum of single particle distribution functions. The method of linear travelling wave propagation is used to show the unconditional asymptotic stability of the solution of Cauchy problem for the system of BGK equations at all values of relaxation time. Stability of the scheme for D1Q2 lattice is demonstrated by the method of differential approximation. Stability condition is written in form of the inequality on values of relaxation time. The possibility of the reduction of stability analysis of the schemes for BGK equations to the analysis of special schemes for diffusion equation for the case of D1Q3 lattice is investigated. Numerical stability investigation is realized by von Neumann method. Absolute values of the eigenvalues of the transition matrix are investigated in parameter space of the schemes. It is demonstrated that in wide range of the parameters changing the values of modulas of eigenvalues are lower than unity, so the scheme is stable with respect to initial conditions.

In this paper, a time-dependent natural convective heat transfer in a closed square cavity filled with non- Newtonian fluid was considered in the presence of an isothermal energy source located on the lower wall of the region under consideration. The vertical boundaries were kept at constant low temperature, while the horizontal walls were completely insulated. The behavior of a non-Newtonian fluid was described by the Ostwald de Ville power law. The process under study was described by transient partial differential equations using dimensionless non-primitive variables “stream function – vorticity – temperature”. This method allows excluding the pressure field from the number of unknown parameters, while the non-dimensionalization allows generalizing the obtained results to a variety of physical formulations. The considered mathematical model with the corresponding boundary conditions was solved on the basis of the finite difference method. The algebraic equation for the stream function was solved by the method of successive lower relaxation. Discrete analogs of the vorticity equation and energy equation were solved by the Thomas algorithm. The developed numerical algorithm was tested in detail on a class of model problems and good agreement with other authors was achieved. Also during the study, the mesh sensitivity analysis was performed that allows choosing the optimal mesh.

As a result of numerical simulation of unsteady natural convection of a non-Newtonian power-law fluid in a closed square cavity with a local isothermal energy source, the influence of governing parameters was analyzed including the impact of the Rayleigh number in the range 104–106, power-law index $n = 0.6–1.4$, and also the position of the heating element on the flow structure and heat transfer performance inside the cavity. The analysis was carried out on the basis of the obtained distributions of streamlines and isotherms in the cavity, as well as on the basis of the dependences of the average Nusselt number. As a result, it was established that pseudoplastic fluids $(n < 1)$ intensify heat removal from the heater surface. The increase in the Rayleigh number and the central location of the heating element also correspond to the effective cooling of the heat source.

Numerical study of unsteady mixed convection in an open partially porous horizontal channel with a heatgenerating source was performed. The outer surfaces of horizontal walls of finite thickness were adiabatic. In the channel there was a Newtonian heat-conducting fluid with a temperature-dependent viscosity. The discrete heatconducting and heat-generating source is located inside the bottom wall. The temperature of the fluid phase was equal to the temperature of the porous medium, and calculations were performed using the local thermal equilibrium model. The porous insertion is isotropic, homogeneous and permeable to fluid. The Darcy–Brinkman model was used to simulate the transport process within the porous medium. Governing equations formulated in dimensionless variables “stream function – vorticity – temperature” using the Boussinesq approximation were solved numerically by the finite difference method. The vorticity dispersion equation and energy equation were solved using locally one-dimensional Samarskii scheme. The diffusive terms were approximated by central differences, while the convective terms were approximated using monotonic Samarskii scheme. The difference equations were solved by the Thomas algorithm. The approximated Poisson equation for the stream function was solved separately by successive over-relaxation method. Optimal value of the relaxation parameter was found on the basis of computational experiments. The developed computational code was tested using a set of uniform grids and verified by comparing the results obtained of other authors.

Numerical analysis of unsteady mixed convection of variable viscosity fluid in the horizontal channel with a heat-generating source was performed for the following parameters: $\mathrm{Pr} = 7.0$, $\varepsilon = 0.8$, $\mathrm{Gr} = 10^5$, $C = 0-1$, $10^{-5} < \mathrm{Da} < 10^{-1}$, $50 < \mathrm{Re} < 500$, $\delta = l/H = 0.6-3$. Distributions of the isolines of the stream function, temperature and the dependences of the average Nusselt number and the average temperature inside the heater were obtained in a steady-state regime, when the stationary picture of the flow and heat transfer is observed. As a result we showed that an addition of a porous insertion leads to an intensification of heat removal from the surface of the energy source. The increase in the porous insertion sizes and the use of working fluid with different thermal characteristics, lead to a decrease in temperature inside the source.

We comsider a model of the dynamics a political system of several parties, accompanied and controlled by the growth of social tension. A system of nonlinear ordinary differential equations is proposed with respect to fractions and an additional scalar variable characterizing the magnitude of tension in society the change of each party is proportional to the current value multiplied by a coefficient that consists of an influx of novice, a flow from competing parties, and a loss due to the growth of social tension. The change in tension is made up of party contributions and own relaxation. The number of parties is fixed, there are no mechanisms in the model for combining existing or the birth of new parties.

To study of possible scenarios of the dynamic processes of the model we derive an approach based on the selection of conditions under which this problem belongs to the class of cosymmetric systems. For the case of two parties, it is shown that in the system under consideration may have two families of equilibria, as well as a family of limit cycles. The existence of cosymmetry for a system of differential equations is ensured by the presence of additional constraints on the parameters, and in this case, the emergence of continuous families of stationary and nonstationary solutions is possible. To analyze the scenarios of cosymmetry breaking, an approach based on the selective function is applied. In the case of one political party, there is no multistability, one stable solution corresponds to each set of parameters. For the case of two parties, it is shown that in the system under consideration may have two families of equilibria, as well as a family of limit cycles. The results of numerical experiments demonstrating the destruction of the families and the implementation of various scenarios leading to the stabilization of the political system with the coexistence of both parties or to the disappearance of one of the parties, when part of the population ceases to support one of the parties and becomes indifferent are presented.

This model can be used to predict the inter-party struggle during the election campaign. In this case necessary to take into account the dependence of the coefficients of the system on time.

Non-smooth optimization often arises in many applied problems. The issues of developing efficient computational procedures for such problems in high-dimensional spaces are very topical. First-order methods (subgradient methods) are well applicable here, but in fairly general situations they lead to low speed guarantees for large-scale problems. One of the approaches to this type of problem can be to identify a subclass of non-smooth problems that allow relatively optimistic results on the rate of convergence. For example, one of the options for additional assumptions can be the condition of a sharp minimum, proposed in the late 1960s by B. T. Polyak. In the case of the availability of information about the minimal value of the function for Lipschitz-continuous problems with a sharp minimum, it turned out to be possible to propose a subgradient method with a Polyak step-size, which guarantees a linear rate of convergence in the argument. This approach made it possible to cover a number of important applied problems (for example, the problem of projecting onto a convex compact set). However, both the condition of the availability of the minimal value of the function and the condition of a sharp minimum itself look rather restrictive. In this regard, in this paper, we propose a generalized condition for a sharp minimum, somewhat similar to the inexact oracle proposed recently by Devolder – Glineur – Nesterov. The proposed approach makes it possible to extend the class of applicability of subgradient methods with the Polyak step-size, to the situation of inexact information about the value of the minimum, as well as the unknown Lipschitz constant of the objective function. Moreover, the use of local analogs of the global characteristics of the objective function makes it possible to apply the results of this type to wider classes of problems. We show the possibility of applying the proposed approach to strongly convex nonsmooth problems, also, we make an experimental comparison with the known optimal subgradient method for such a class of problems. Moreover, there were obtained some results connected to the applicability of the proposed technique to some types of problems with convexity relaxations: the recently proposed notion of weak $\beta$-quasi-convexity and ordinary quasiconvexity. Also in the paper, we study a generalization of the described technique to the situation with the assumption that the $\delta$-subgradient of the objective function is available instead of the usual subgradient. For one of the considered methods, conditions are found under which, in practice, it is possible to escape the projection of the considered iterative sequence onto the feasible set of the problem.

Mechanical properties of eukaryotic cells play an important role in life cycle conditions and in the development of pathological processes. In this paper we discuss the problem of parameters identification and verification of viscoelastic constitutive models based on force spectroscopy data of living cells. It is proposed to use one-dimensional continuous wavelet transform to calculate the relaxation function. Analytical calculations and the results of numerical simulation are given, which allow to obtain relaxation functions similar to each other on the basis of experimentally determined force curves and theoretical stress-strain relationships using wavelet differentiation algorithms. Test examples demonstrating correctness of software implementation of the proposed algorithms are analyzed. The cell models are considered, on the example of which the application of the proposed procedure of identification and verification of their parameters is demonstrated. Among them are a structural-mechanical model with parallel connected fractional elements, which is currently the most adequate in terms of compliance with atomic force microscopy data of a wide class of cells, and a new statistical-thermodynamic model, which is not inferior in descriptive capabilities to models with fractional derivatives, but has a clearer physical meaning. For the statistical-thermodynamic model, the procedure of its construction is described in detail, which includes the following. Introduction of a structural variable, the order parameter, to describe the orientation properties of the cell cytoskeleton. Setting and solving the statistical problem for the ensemble of actin filaments of a representative cell volume with respect to this variable. Establishment of the type of free energy depending on the order parameter, temperature and external load. It is also proposed to use an oriented-viscous-elastic body as a model of a representative element of the cell. Following the theory of linear thermodynamics, evolutionary equations describing the mechanical behavior of the representative volume of the cell are obtained, which satisfy the basic thermodynamic laws. The problem of optimizing the parameters of the statisticalthermodynamic model of the cell, which can be compared both with experimental data and with the results of simulations based on other mathematical models, is also posed and solved. The viscoelastic characteristics of cells are determined on the basis of comparison with literature data.

Muscle contraction is controlled by Ca2+ ions via regulatory proteins, troponin and tropomyosin, associated with thin actin filaments in sarcomeres. Depending on the Ca2+ concentration, the thin filament rearranges so that tropomyosin moves along its surface, opening or closing access to actin for the motor domains of myosin molecules, and causing contraction or relaxation, respectively. Numerous point amino acid substitutions in tropomyosin are known, leading to genetic pathologies — myo- and cardiomyopathies caused by changes in the structural and functional properties of the thin filament. The results of molecular dynamics modeling of a fragment of a thin filament of cardiac muscle sarcomeres formed by fibrillar actin and wildtype tropomyosin or with amino acid substitutions: the double stabilizing substitution D137L/G126R and the cardiomyopathic substitution S215L are presented. For numerical calculations, we used a new model of a thin filament fragment containing 26 actin monomers and 4 tropomyosin dimers, with a refined structure of the region of overlap of neighboring tropomyosin molecules in each of the two tropomyosin strands. The simulation results showed that tropomyosin significantly increases the bending stiffness of the thin filament, as previously found experimentally. The double stabilizing replacement D137L/G126R leads to a further increase in this rigidity, and the replacement S215L, on the contrary, leads to its decrease, which also corresponds to experimental data. At the same time, these substitutions have different effects on the angular mobility of the actin helix and only slightly modulate the angular mobility of tropomyosin cables relative to the actin helix and the population of hydrogen bonds between negatively charged tropomyosin residues and positively charged actin residues. The results of the verification of the new model demonstrate that its quality is sufficient for the numerical study of the effect of single amino acid substitutions on the structure and dynamics of thin filaments and study the effects leading to dysregulation of muscle contraction. This model can be used as a useful tool for elucidating the molecular mechanisms of some genetic diseases and assessing the pathogenicity of newly discovered genetic variants.