# Vol 23, No 3 (2019)

**Year:**2019**Articles:**10**URL:**https://journals.eco-vector.com/1991-8615/issue/view/1233

### Closed vortex lines in fluid and gas

#### Abstract

Continuous fluid and gas flows with closed vortex tubes are investigated. The circulation along the vortex line of the ratio of the density of the resultant of all forces (applied to the fluid or gas) to the density of the fluid or gas is considered. It coincides with the circulation (along the same vortex line) of the partial derivative of the velocity vector with respect to time and, therefore, for stationary flows, it is equal to zero on any closed vortex line. For non-stationary flows, vortex tubes are considered, which remain closed for at least a certain time interval. A previously unknown regularity has been discovered, consisting in the fact that at, each fixed moment of time, such circulation is the same for all closed vortex lines that make up the vortex tube. This regularity is true for compressible and incompressible, viscous (various rheologies) and non-viscous fluids in a field of potential and non-potential external mass forces. Since this regularity is not embedded in modern numerical algorithms, it can be used to verify the numerical calculations of unsteady flows with closed vortex tubes by checking the equality of circulations on different closed vortex lines (in a tube). The expression for the distribution density of the resultant of all forces applied to fluid or gas may contain higher-order derivatives. At the same time, the expression for the partial derivative of the velocity vector with respect to time and the expression for the vector of vorticity (which is necessary for constructing the vortex line) contain only the first derivatives; which makes it possible to use new regularity for verifying the calculations made by methods of high and low orders simaltaniously.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2019;23(3):407-416

407-416

### On a mathematical model of non-isothermal creeping flows of a fluid through a given domain

#### Abstract

We study a mathematical model describing steady creeping flows of a non-uniformly heated incompressible fluid through a bounded 3D domain with locally Lipschitz boundary. The model under consideration is a system of second-order nonlinear partial differential equations with mixed boundary conditions. On in-flow and out-flow parts of the boundary the pressure, the temperature and the tangential component of the velocity field are prescribed, while on impermeable solid walls the no-slip condition and a Robin-type condition for the temperature are used. For this boundary-value problem, we introduce the concept of a weak solution (a pair “velocity-temperature”), which is defined as a solution to some system of integral equations. The main result of the work is a theorem on the existence of weak solutions in a subspace of the Cartesian product of two Sobolev's spaces. To prove this theorem, we give an operator interpretation of the boundary value problem, derive a priori estimates of solutions, and apply the Leray-Schauder fixed point theorem. Moreover, energy equalities are established for weak solutions.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2019;23(3):417-429

417-429

### On the correctness of boundary value problems for the mixed type equation of the second kind

#### Abstract

In this paper, the intervals of change in the exponent of the degree of degeneration of a mixed-type equation with characteristic degeneration are established. The first boundary problem and the modified boundary problem (analogue of the Keldysh problem) with the conditions of periodicity are correctly set. In the case of the first problem, a criterion for the uniqueness of its solution is manifested. It is shown that the solution of the analogue of the Keldysh problem is unique up to a term of a linear function. Solutions are constructed as the sum of series of eigenfunctions of the corresponding one-dimensional spectral problem. In justifying the convergence of a series representing the solution of the first boundary-value problem, the problem of small denominators of a more complex structure arises in the class of regular solutions of this equation than in previously known works. The estimate on separation from zero is established with the corresponding asymptotic. Based on this estimate, sufficient conditions are found for the boundary functions to substantiate the uniform convergence of the series and their derivatives up to the second order inclusive.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2019;23(3):430-451

430-451

### Continuum approach to high-cycle fatigue. The finite life-time case with stochastic stress history

#### Abstract

In this paper, we consider continuum approach for high-cycle fatigue in the case where life-time is finite. The method is based on differential equations and all basic concepts are explained. A stress history is assumed to be a stochastic process and this leads us to the theory of stochastic differential equations. The life-time is a quantity, which tells us when the breakdown of the material happens. In this method, it is naturally a random variable. The basic assumption is, that the distribution of the life-time is log-normal or Weibull. We give a numerical basic example to demonstrate the method.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2019;23(3):452-463

452-463

### On plane thermoelastic waves in hemitropic micropolar continua

#### Abstract

The paper deals with the coupled heat transport and dynamic equations of the hemitropic thermoelastic micropolar continuum formulated in terms of displacements, microrotations and temperature increment which are to be determined in applied problems. The mechanism of thermal conductivity is considered as simple thermodiffusion. Hemitropic constitutive constants are reduced to a minimum set nevertheless retaining hemitropic constitutive behaviour and thermoelastic semi-isotropy. Solutions of thermoelastic coupled equations in the form of propagating plane waves are studied. Their spatial polarizations are determined. An algebraic bicubic equation for the determination of wavenumbers is obtained. It is found that for a coupled thermoelastic wave actually there are exactly three normal complex wavenumbers. Athermal wave is also investigated. Spatial polarizations in this case form (together with the wave vector) a spatial trihedron of mutually orthogonal directions. For an athermal wave there are (depending on the case) either two real normal wavenumbers or single wavenumber.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2019;23(3):464-474

464-474

### Compliance functions of electromagnetoelastic piezoelectric and piezomagnetic half-plane and half-space with functionally graded or layered coatings

#### Abstract

The paper addresses to the construction of the compliance functions of axissymmetric and plane contact problems of electromagnetoelasticity for semi-infinite piezoelectric piezomagnetic solids with functionally graded or piece-wise homogeneous coatings. Materials of coating and substrate are assumed to be transversely isotropic. Computation of the compliance functions are reduced to the solution of two-point boundary value problems for a system of ordinary differential equations with variable coefficients are obtained using integral transformation technique. Boundary conditions of these systems describe distributed tangential or normal mechanical loading or action of an electrical of magnetic fields. Dual integral equations and their systems are obtained for contact problems on indentation by an insulating and conductive punches with kernel transforms equal to compliance functions. Asymptotic behavior of the compliance functions is analyzed. Specially designed approximations for the kernel transforms are constructed based on the analysis of their properties. These approximations make it possible to construct the solutions of the approximated systems of dual integral equations in a closed analytical form. Numerical results illustrating all 10 independent compliance functions are provided for different materials of coating and substrate and different types of variation of properties in depth of the coating. It is shown that in the case of absence of the tangential mechanical loading all the compliance functions are positive. Conditions of existing sign alternating compliance functions corresponding tangential mechanical loading are analyzed. The differences between the properties of the compliance functions, corresponding to homogeneous and functionally graded coatings are illustrated.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2019;23(3):475-496

475-496

### The effect of elevated temperature and tensile force loading on the relaxation of residual stresses in surface-hardened elements of the rod structure under creep conditions

#### Abstract

A mathematical model for the relaxation of residual stresses in surface-hardened cylindrical elements of statically indefinable rod systems under creep conditions with elevated temperature and tensile force loading was developed. The following problems were solved during the modeling: reconstruction of the stress-strain state in a cylindrical rod after the surface treatment by microspheres; consideration of the influence of temperature loading on the magnitude and the fields of residual stresses due to the temperature dependence of Young's modulus; calculation of relaxation of residual stresses in hardened elements of the system under the influence of elevated temperature and tensile force loading under creep conditions; analysis of the final residual stresses after creep and reduced temperature and tensile force unloading. @@The problems were solved within the first two stages of creep of the system of material elements. For a detailed analysis a three-element statically indefinable system with hardened elements at the temperature of 20°C and an operating temperature of 675°C made of ZhS6U alloy was used. @@To implement the solutions of the problems mentioned, numerical algorithms were developed using discretization by spatial and temporal coordinates and using the method of time steps. For a posteriori estimation of the convergence and stability of the numerical method the numerical results were compared for large values of the calculation time with the asymptotic values of the stress-strain state characteristics corresponding to the steady-state creep stage obtained by the analytical method. The results obtained by both approaches are consistent. @@The results of calculations were illustrated the kinetics of residual stresses in all three rods of the system during creep under the influence of elevated temperature and tensile force loading, starting from the moment of their formation after hardening. It was shown that a stepwise change in the magnitude and the distribution of residual stresses occurs only due to the “instantaneous” temperature heating of the elements of the rod structure due to the temperature dependence of the Young's modulus. It was also established by calculations that the relaxation of residual stresses in the most loaded rods system is much slower than in less loaded ones. To illustrate the main results obtained in this paper, we plotted the distribution of residual stresses along the depth of the hardened layer.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2019;23(3):497-524

497-524

### Stochastic models of just-in-time systems and windows of vulnerability in terms of the processes of birth and death

#### Abstract

The paper proposes a method for constructing models based on the analysis of birth and death processes with linear growth in semimartingale terms. Based on this method, stochastic models of simple just-in-time systems (analyzed in the theory of productive systems) and windows of vulnerability (widely discussed in risk theory) are considered. The main results obtained in the work are presented in terms of the average values of the time during which the processes reach zero values. At the same time, they are considered and used in the study of assessment models for local times of the processes. Here, simple Markov processes with a linear growth of intensities (perhaps, depending on time) are analyzed. At the same time, the obtained and used estimates are of theoretical interest. Thus, for example, the average value of the stopping time, at which the process reaches zero, depends on functions such as the harmonic number and the remainder term for the logarithmic function in the Taylor theorem. As the main result, the method of mathematical modeling of just-in-time systems and windows of vulnerability is proposed. The semimartingale description method used here should be considered as the first step of such a modeling, since, being a trajectory method, it allows diffusion (including non-Markov processes) generalizations when constructing stochastic models of windows of vulnerability and just-in-time. In the theoretical part of the article, we formulate statements for the average values of the local time and the stopping times when the birth and death processes reach a given value. This allows us to uniformly present estimates for the models of the just-in-time system and for windows of vulnerability, the result for which is given in the form of a limit theorem. The main results are formulated as theorems and lemmas. The proofs use semimartingale methods.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2019;23(3):525-540

525-540

### Mathematical modeling of coalescence and breakage of droplets and bubbles in an isotropic turbulent flow: A review

#### Abstract

This review devoted to the theoretical analysis, calculation, and modeling of the processes of merging and breakage of droplets and bubbles in an isotropic turbulent flow. We have analyzed a number of studies on these issues. The problems of determining the minimum and maximum sizes of droplets and bubbles, as well as breakage and merging frequencies, which are associated with the solution of the diffusion equation of mass transfer, are considered. The merging of droplets is considered as a result of the thinning of the interfacial film formed by two drops as a result of their collision. A mathematical description of the refinement of the interfacial film, taking into account the Marangoni effect, is proposed. Analysis of many studies, including our own, showed that, depending on the scale of turbulent pulsations, the extreme size, as well as the frequencies of coalescence and breakage of droplets and bubbles, depend on the specific dissipation energy in the turbulent flow, on their sizes and on the physical properties of the particles and the medium. Important parameters that provide aggregative stability of a liquid-liquid or liquid-gas type dispersion medium to breakage, deformation and fusion are the surface tension coefficient and energy dissipation, the physical properties of the medium and particles, and in an isotropic turbulent flow the ratio of the surface coefficient tension to specific energy dissipation. Problems related to the evolution of the particle distribution function in time and size under isotropic turbulence using solutions of the Fokker-Planck stochastic equation for continuous variation of the sizes of droplets and bubbles and the integro-differential kinetic equation of coalescence and fragmentation for jump-like changes in particle sizes are also considered. A set of analytical solutions of these equations for particular cases is proposed. A more in-depth analysis based on the mathematical laws of the transport phenomena makes it possible in the standard way to calculate such systems in an approximation, such as continuous, with an infinitely small jump. It is shown that the deterministic description of these phenomena without taking into account their stochastic nature is incomplete and can lead to significant deviations from the true nature of the above processes. The results obtained are compared with the existing experimental data on coalescence and breakage of droplets and bubbles, which showed satisfactory agreement with the calculated values.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2019;23(3):541-581

541-581

### Stability and convergence of difference schemes for the multi-term time-fractional diffusion equation with generalized memory kernels

#### Abstract

In this paper, a priori estimate for the corresponding differential problem is obtained by using the method of the energy inequalities. We construct a difference analog of the multi-term Caputo fractional derivative with generalized memory kernels (analog of L1 formula). The basic properties of this difference operator are investigated and on its basis some difference schemes generating approximations of the second and fourth order in space and the $ (2{-}\\alpha_0) $-th order in time for the generalized multi-term time-fractional diffusion equation with variable coefficients are considered. Stability of the suggested schemes and also their convergence in the grid $ L_2 $-norm with the rate equal to the order of the approximation error are proved. The obtained results are supported by numerical calculations carried out for some test problems.

**Journal of Samara State Technical University, Ser. Physical and Mathematical Sciences**. 2019;23(3):582-597

582-597