Computational Continuum Mechanics
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description: The journal publishes papers describing recent research and computational results in the field of continuum mechanics. The subject of the journal includes theoretical and numerical methods for solving the problems of mechanics of a deformable solid and fluid mechanics; effective applications of these methods to the study of natural and technogenic phenomena, technological processes, the behavior of machines and structures.
publisher: Institute of Continuous Media Mechanics
language: en
publication name: Computational Continuum Mechanics
issn: 1999-6691
items: The influence of the configuration of travelling magnetic fields on the efficiency of liquid metal stirring in a rectangular cell
items: Hybrid computational scheme for modeling the excitation of elastic waves in an isotropic layer by a piezoceramic actuator of various configurations
items: Modeling of the response of magnetic microgel particles of different structures to an applied magnetic field
items: Methods for identification of material constants associated with the martensite stabilization effect in shape memory alloy models
items: Influence of viscosity on nonlinear resonance of forced oscillations of a bubble in a vibrating liquid
items: Numerical investigation of convective dissolution of fluid CO2 in water during its storage in an underground porous formation
items: Axisymmetric buckling of FGM shells under thermal shock
items: Modeling of hydrogen degradation of a metal pipe under pressure
items: Numerical simulation of the interaction between a supersonic viscous gas flow and a transverse jet injected at an angle from the slot in a flat plate
items: The effect of variable thermophysical properties of the working fluid on the processes of natural convection in a closed partially porous cube with a heater
items: Implementation of С¹-continuous finite element method within strain gradient elasticity for crack problems
items: Natural and forced vibrations of solid bodies with account for damping and prestressed state induced by piezoelectric elements
items: Numerical simulation of two-dimensional flow of a micropolar fluid in an enclosed cavity with the full dissipation function
items: Simulation of high-temperature oxide and sulfide-oxide corrosion of Fe76Cr24 alloy based on a coupled model of chemomechanics
items: Numerical modeling of diurnal oscillation of insolation in the idealized model of general atmospheric circulation
items: Fluid flow in a thick torus driven by distributed forces
items: The influence of biomechanical parameters of an erythrocyte and viscosity of the surrounding fluid on the hydrodynamic interaction with a solid wall
items: Interaction of mechanical, temperature and diffusion fields in a solid cylinder subjected to transient heating
items: Convection in a closed cavity with a large body of neutral buoyancy
items: Modeling the loss of stability and failure of mine shaft tubing supports
items: Numerical simulation of natural vibration of a partially fluid-filled truncated conical shell in a gravity field
items: Numerical study of the motion of a solid particle in the field of ultrasonic standing wave
items: The onset and evolution of nonlinear convective regimes of binary mixtures in multilayer systems simulating synclinal geological folds
items: Some coupled diffusion problems for ternary systems
items: On the development of an approximate theory of direct and inverse problem for an inhomogeneous rectangular region
items: On one solution to the problem of resonant oscillations of a lifting rope taking into account its slippage when winding on the surface of a drum
items: Analysis of the bifurcation structure near the excitation threshold of convection during uniform fluid infiltration through a slightly asymmetric porous chip
items: A magnetic field in the vicinity of a homogeneous magnet filling an axisymmetric, simply connected or multiply connected region
items: Effect of a traveling magnetic field on heat and mass transfer during submerged heater crystal growth
items: Simulation of convective polymerase chain reaction based on multiphase Lagrangian particle tracking model

The influence of the configuration of travelling magnetic fields on the efficiency of liquid metal stirring in a rectangular cell
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description: This paper aims to investigate numerically a vortex flow of liquid metal in a rectangular mould excited by two linear inductors of magnetic fields located at different distances and at different magnetic couplings . The occurrence of magnetic coupling between the inductors is due to their connection via passive ferromagnetic elements, which intensify the electromagnetic interaction between the coils of adjacent inductors leading to significant non-uniformity of electromagnetic field. The stirring efficiency of the flow driven by electromagnetic forces is estimated in terms of the integral parameter of homogeneity of passive impurity, whose concentration directly depends on the evolution of vortex structures of different size. It has been shown that the most effective stirring of liquid metal in the mould is achieved at relatively close distance between the inductors but in the absence of direct magnetic coupling. This ensures consistency between the mould geometry and the topology of electromagnetic field, and, as a result, generates along the mould length a large-scale elongated vortex, facilitating thereby the transport of impurity throughout the bulk of the liquid metal. The vortex itself loses stability and breaks down with a certain periodicity into many small vortex structures. The flow becomes chaotic and stirring is intensified. In the case of strong magnetic coupling between the inductors the resulting flows exhibit pronounced instability and asymmetry responsible for the generation of several vortices along the principal mould direction. One of the vortices can trap the impurity producing a blocking effect, which impedes the transport of impurity across the mould . The study was conducted using numerical modeling. The model was verified using data from the laboratory experiment.
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creator: Ivan Mitropolit
creator: Gennadiy Losev
creator: Vladislav Eltishchev
creator: Ilya Kolesnichenko
rights: Copyright (c) 2026 Computational Continuum Mechanics https://creativecommons.org/licenses/by/4.0
date: 2026-08-05
publication date: 2026-08-05
starting page: 258
ending page: 269
doi: 10.7242/1999-6691/2026.19.2.18

Hybrid computational scheme for modeling the excitation of elastic waves in an isotropic layer by a piezoceramic actuator of various configurations
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description: Modeling of elastic guided wave excitation by surface-mounted or embedded ultrasonic transducers is one of the key problems in the development and verification of nondestructive testing and structural health monitoring methods. To address this challenge, the dynamics of three-dimensional waveguides is analyzed by applying a hybrid numerical-analytical scheme. The problem reduces to studying the dynamic behavior of an isotropic elastic layer subjected to a localized vibration source positioned at an arbitrary distance from the layer boundaries. Within the proposed scheme, the construction of a solution involves two stages. In the first stage, a set of auxiliary numerical solutions is constructed in the vicinity of the transducer. In the second stage, these solutions are coupled with the wave field in the remaining homogeneous part of the waveguide, which is represented as a sum of cylindrical guided waves. Each wave is expressed as the product of the modal waveform along the vertical coordinate by the function, characterizing the intensity of radiation in a given direction. The latter can be represented as a Fourier series expansion in a polar angle, whose coefficients are determined from the condition of continuity of the displacement and stress vectors at the interface between the region containing the vibration source and the remaining part of the waveguide. The proposed scheme is capable of modeling the interaction between the elastic layer and arbitrarily shaped transducer with consideration for physical fields coupling and does not require an independent software implementation of finite element calculations. Moreover, it can be used to evaluate the excitation efficiency of normal modes in a selected frequency range as a function of the propagation direction. Verification of the scheme is performed by comparing its results with the numerical solutions obtained by use of other approaches. The specific features of elastic guided wave excitation by piezoceramic tr…
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creator: Александр Евдокимов
creator: Polina Nets
creator: Artem Eremin
creator: Mikhail Vareldzhan
rights: Copyright (c) 2026 Computational Continuum Mechanics https://creativecommons.org/licenses/by/4.0
date: 2026-08-05
publication date: 2026-08-05
starting page: 240
ending page: 257
doi: 10.7242/1999-6691/2026.19.2.17

Modeling of the response of magnetic microgel particles of different structures to an applied magnetic field
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description: A comprehensive model description of the response of magnetopolymer systems to external stimuli that is usually unfeasible in experimental studies is an urgent problem from the viewpoint of potential applications in biomedicine. The response of hydrogels with magnetic inclusions - ferrogels - to an external magnetic field is of a complex magnetic, structural, and mechanical character determined by competing interactions between the components. Therefore it is necessary to provide the desired level of detalization in the selection of a research method . To describe this response by the coarse-grained molecular dynamics method it is suggested to implement the model of a single microgel particle, containing uniformly distributed magnetic nanoparticles, or nanopaerticles organized as core-shell structures. A procedure for particles assembly and bonding is proposed, and the values of the model parameters are determined. They are used to describe the structure, which is typical for magnetic gels - a cross-linked polymer matrix-carrier containing a significant volume of solvent and chemically embedded nanoparticles. It is demonstrated that by varying the number of elastic bonds between model elements and their stiffness one can reproduce the properties of different magnetopolymer systems: from a weakly cross-linked hydrogel precursor to a rigid composite particle. The influence of the magnetic filler parameters and the form of its spatial distribution in the microgel particle on the ground state of the examined object is investigated in the absence of an external magnetic field and under magnetization. It has been shown that non-uniformities of the concentration of magnetic nanoparticles in the microgel in the form of a magnetic core or shell can influence magnetization, cluster formation, and volume changes in the hydrogel matrix.
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creator: Aleksandr Vladimirovich Ryzhkov
rights: Copyright (c) 2026 Computational Continuum Mechanics https://creativecommons.org/licenses/by/4.0
date: 2026-06-01
publication date: 2026-06-01
starting page: 124
ending page: 137
doi: 10.7242/1999-6691/2026.19.1.9

Methods for identification of material constants associated with the martensite stabilization effect in shape memory alloy models
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description: The martensite stabilization effect significantly influences the thermomechanical response of shape memory alloys and must be properly taken into account in numerical modeling. In our previous studies, to describe this effect in TiNi alloys we developed a model using the microstructural approach. The foundation of the model was the hypothesis of damage accumulation at inter-martensitic boundaries. In this work, four approaches to the identification of material constants of this model are considered: manual fitting, the deterministic Nelder--Mead method, a stochastic genetic algorithm, and a neural network approach based on the Deep Sets architecture. The study uses the results of experiments for the Ti50Ni50 and Ti49Ni51, titanium nickelide alloys reported in the literature. These experiments reproduce three characteristic modes of applying preliminary deformation: tensile deformation in the martensitic state, cooling under constant load, and stress-induced martensite transformation. The parameter identification procedure aims to minimize the discrepancy function measuring the difference between the calculated and experimentally observed shifts in the reverse martensitic transformation temperature. A comparison of the accuracy and computational efficiency of the applied methods was carried out for both alloys. The results show that the Nelder--Mead method and the genetic algorithm provide a comparable quality of the experimental data approximation at moderate computational costs and relatively simple parameter setting. The neural network approach demonstrates less accurate agreement between the results of numerical simulation and experiments but provides fast inference after training; however, it requires pre-generation of large training datasets and exhibits increased sensitivity to the initial training conditions. The obtained results confirm the applicability of different methods of automated parameter identification for models of shape memory alloys.
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creator: Timofei Rebrov
creator: Fedor Belyaev
creator: Aleksandr Volkov
creator: Egor Vukolov
creator: Margarita Evard
rights: Copyright (c) 2026 Computational Continuum Mechanics https://creativecommons.org/licenses/by/4.0
date: 2026-08-05
publication date: 2026-08-05
starting page: 172
ending page: 188
doi: 10.7242/1999-6691/2026.19.2.12

Influence of viscosity on nonlinear resonance of forced oscillations of a bubble in a vibrating liquid
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description: The paper presents the results of the study on dynamics of a gas bubble in an oscillating viscous fluid. The focus is on the case when the frequency of external vibrations is close to half the natural frequency of the quadrupole mode of free oscillations, which corresponds to the condition of the nonlinear resonance previously discovered in the inviscid approximation. A viscous potential flow model was used to take into account dissipation. In the framework of this model, the analytical expressions of the main process characteristics were obtained. The solution was expressed as an asymptotic series in a small parameter - the ratio of vibration amplitude to the bubble radius. The first-order theory shows that the bubble oscillates as a whole without deformation. The dependence of the amplitude and the phase shift of these oscillations on viscosity was established. The second-order theory investigates the nonlinear resonance associated with the excitation of the quadrupole deformation mode. It is shown that viscosity does not shift the resonant frequency, which remains half the natural frequency of the quadrupole mode, as in an ideal fluid, but significantly limits the amplitude of resonant oscillations. The results of numerical modeling of 3D axisymmetric bubble oscillations at the resonant frequency show good agreement with the analytical data. These findings are important for understanding and controlling processes in such applications as acoustic flotation, chemical reactors, and medical ultrasound diagnostics.
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creator: Vladimir Vladislavovich Konovalov
creator: Andrey Olegovich Ivantsov
rights: Copyright (c) 2026 Computational Continuum Mechanics http://creativecommons.org/licenses/by-nc/4.0
date: 2026-02-25
publication date: 2026-02-25
starting page: 385
ending page: 399
doi: 10.7242/1999-6691/2025.18.4.28

Numerical investigation of convective dissolution of fluid CO2 in water during its storage in an underground porous formation
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description: The study addresses an important environmental problem related to harmful effects of greenhouse gases on Earth's temperature and climate and the need to reduce anthropogenic carbon dioxide emissions into the atmosphere. One-sided density-driven convection developed inside a porous formation during dissolution of supercritical CO2 fluid in water is investigated in reference to the technologies of carbon dioxide storage in underground water-saturated formations. The hydrodynamic model includes equations of continuity, motion (based on the Darcy's law), convection-diffusion and a non-linear equation of state that relates the density of fluid medium with the concentration of CO2. A fluid is described as a single-phase two-component medium with variable viscosity. Based on the data available in the literature, the physical properties of the CO2 + H2O system were determined in the ranges of temperature T=307.7 - 342.8 К and pressure P=8.07 - 31.42 MPa. Numerical simulation of hydrodynamic processes in this system was carried with the aid of the developed finite-difference numerical code. In particular, the code was employed to study how the characteristics of convective dissolution depend on the temperature and the pressure. It has been found that with increase of these parameters the convective dissolution of CO2 fluid in water is more intensive. For example, the time during which half of CO2 dissolves in water and is transported down by a convective flow is reduced almost twice. Thus, an increase in temperature and pressure ensures more reliable storage of carbon dioxide in underground water-saturated formations.
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creator: Elena Soboleva
rights: Copyright (c) 2026 Computational Continuum Mechanics https://creativecommons.org/licenses/by/4.0
date: 2026-08-05
publication date: 2026-08-05
starting page: 201
ending page: 213
doi: 10.7242/1999-6691/2026.19.2.14

Axisymmetric buckling of FGM shells under thermal shock
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description: The paper deals with the problem of vibrations and dynamic buckling of FGM shells of revolution subjected to thermal shock. A finite-element algorithm is developed for studying geometrically nonlinear vibration of shells under unsteady heat conduction through the shell wall. A two-node shell finite element is formulated taking into account transverse shear deformation based on the first-order shear deformation theory by S. Timoshenko. To avoid the shear locking effect, the strains of the element are approximated using an exact analytical solution of the bending problem obtained in the framework of the Timoshenko beam theory. The reference surface used is the surface, passing through the centers of mass of the transverse fibers, which makes it possible to separate the inertial terms corresponding to translational motion and rotation. Equations of motion of the finite-element model are solved numerically using a step-by-step method based on the implicit Newmark integration scheme. At each time step of integration, temperature loads and stiffness characteristics of the shell are determined by numerically solving the unsteady heat conduction problem formulated with account of temperature-dependent properties of the material. Deformed configuration of the shell at each time step is determined using the Newton-Raphson iterative procedure. The proposed numerical algorithm is verified by comparing the numerical solution of test problems with the data available in the literature. Dynamic snap-through buckling of shallow spherical and conical shells, experiencing a sudden rise of temperature at the inner surface, is studied. The effect of material properties, geometrical parameters, and the rate of heating of the inner surface on the dynamic loss of stability is discussed.
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creator: Stanislav Levyakov
rights: Copyright (c) 2026 Computational Continuum Mechanics https://creativecommons.org/licenses/by/4.0
date: 2026-08-05
publication date: 2026-08-05
starting page: 143
ending page: 160
doi: 10.7242/1999-6691/2026.19.2.10

Modeling of hydrogen degradation of a metal pipe under pressure
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description: Modeling of hydrogen effect on the strength of a metal structure is an urgent problem. The hazard of detrimental action of hydrogen lies in the fact that it often has no visible external signs and reveals itself only at the instant of structure failure. The effect of hydrogen on the metal changes its mechanical properties and, as a result, leads to a redistribution of stresses inside the structure. This, in turn, affects the diffusion of hydrogen, creating a complex interrelation that must be taken into account when calculating the strength of real structures. The paper presents a mathematical model for computing the process of degradation of pipe material under pressure of hydrogen-containing medium. Two continuums are considered: a deformable body and freely diffusing hydrogen, which, under certain conditions, attaches to metal atoms and becomes bound hydrogen. In this case, hydrogen absorbs energy from the crystal lattice, which causes deterioration of mechanical properties of the metal. Since the process of interaction between hydrogen and metal proceeds very slowly, it is considered as quasi-stationary. The mutual influence of hydrogen diffusion and stress-strain state of the pipe on the penetration of hydrogen into the metal and degradation of its mechanical properties is modeled numerically. An iterative algorithm has been developed to calculate the change in the stress-strain state of the pipe weakened by hydrogen. Numerical calculations have been carried out to evaluate mechanical stresses, concentrations of free and bound hydrogen in the pipe, as well as changes in the modulus of elasticity of the metal as a result of damage accumulation.. The calculations show that some combinations of hydrogen concentration and pressure on the pipe wall can lead to the formation of pores and circumferential cracks in the pipe.
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creator: Tatiana Zinovieva
creator: Alexander Belyaev
rights: Copyright (c) 2026 Computational Continuum Mechanics https://creativecommons.org/licenses/by/4.0
date: 2026-08-05
publication date: 2026-08-05
starting page: 189
ending page: 200
doi: 10.7242/1999-6691/2026.19.2.13

Numerical simulation of the interaction between a supersonic viscous gas flow and a transverse jet injected at an angle from the slot in a flat plate
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description: This paper presents the results of a comprehensive numerical study of the interaction between a supersonic air flow with a Mach number of 3.5 and a stationary jet injected through a slot in a streamlined plate. The mathematical model is based on the Reynolds-averaged Navier–Stokes equations, which are solved by the finite volume method. In the first stage, methodological calculations were performed to substantiate the choice of optimal parameters for the computational grid, the convective flow approximation scheme, and the turbulence model. It was found that the best fit to the experimental data is achieved by combining the Spalart–Allmaras turbulence model allowing for compressibility and the second-order AUSM+up scheme. After validating the computational methodology, a systematic parametric study was conducted to examine the influence of two factors: the relative pressure in the jet (in the range from 8.74 to 63.5) and the injection angle (from –30 to 30◦). It is shown that increasing the jet intensity leads to a nearly linear increase in the length of the separation zone in front of the slot, the depth of jet penetration in the cross flow, and the resulting force acting on the plate. The highest injection efficiency, characterized by the gain factor, is achieved at a relative pressure of 17.12; its further increase is accompanied by an enhancement of the shock-wave interaction and a decrease in efficiency. The study of the influence of injection angle revealed a pronounced asymmetry between the concurrent and countercurrent injected jets. The countercurrent injection leads to the formation of extensive separation zones, intense shock waves, and significant total pressure losses. By contrast, the concurrent injection forces the jet against the surface, weakens the shock-wave structure, and reduces losses.
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creator: Elena Viktorovna Babich
creator: Elizaveta Vladimirovna Kolesnik
rights: Copyright (c) 2026 Computational Continuum Mechanics https://creativecommons.org/licenses/by/4.0
date: 2026-06-01
publication date: 2026-06-01
starting page: 95
ending page: 107
doi: 10.7242/1999-6691/2026.19.1.7

The effect of variable thermophysical properties of the working fluid on the processes of natural convection in a closed partially porous cube with a heater
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description: The present paper deals with a numerical investigation of natural convection and heat transfer in a closed cubical partially porous cavity with a local heat-generating cubical element at the bottom wall. The working fluid is an incompressible fluid with temperature-dependent viscosity and thermal conductivity. The mathematical model is based on the Boussinesq approximation and a system of equations written in terms of «vector potential - vorticity vector - temperature» variables. The governing equations are solved numerically using the second-order finite difference method and locally one-dimensional Samarskii scheme for time integration. The developed numerical approach is verified by comparing the obtained results with the known solutions of the classical problem of natural convection in a differentially heated cubic cavity. A parametric analysis is carried out to examine the effects of temperature dependence of viscosity and thermal conductivity of the fluid, as well as the height of the porous layer on the flow structure and heat transfer. The results demonstrate that temperature-dependent viscosity causes the intensification of convective motion near the heat source and enhances the general fluid circulation in the cavity. An increase in the parameter of thermal conductivity variation leads to a more uniform temperature distribution and a reduction of local thermal gradients. It has been shown that the combined effect of variable viscosity and thermal conductivity changes the average Nusselt number and the thermal regime of the heat-generating element. The hydrodynamics of the process can be significantly improved by introducing a porous layer, which allows decreasing of overheating of the energy source. These findings serve as a basis for selecting coolants with a specified temperature dependence of rheological properties that provide an intensive heat removal near the fuel elements. In particular, the use of fluids with a pronounced decrease of viscosity at h…
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creator: Marina Astanina
creator: Mikhail Sheremet
rights: Copyright (c) 2026 Computational Continuum Mechanics https://creativecommons.org/licenses/by/4.0
date: 2026-08-05
publication date: 2026-08-05
starting page: 228
ending page: 239
doi: 10.7242/1999-6691/2026.19.2.16

Implementation of С¹-continuous finite element method within strain gradient elasticity for crack problems
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description: Strain gradient elasticity (SGE) offers the possibility of regularizing a wide class of problems in classical solid mechanics, including those involving cracks, sharp corners, concentrated loads, dislocations, etc. Within the framework of SGE, it is assumed that the strain energy density depends not only on the strains themselves but also on their spatial derivatives, which imposes additional smoothness requirements on the strain and stress fields realized in SGE solutions. This work presents an implementation of a С¹-continuous finite element method, based on a simplified strain gradient elasticity theory, in the Abaqus/CAE software suite. Standard and enriched three-node elements are presented; 5th-order polynomials are used as shape functions. The implementation was carried out using a user subroutine UEL and a custom user interface. Based on calculations performed for mode I, mode II, and mixed-mode crack opening problems, the mesh convergence of the obtained solutions is investigated, and estimates of the stress distribution near the crack tip are provided. The computational efficiency of the enriched elements compared to the standard ones is also demonstrated, due to their faster convergence rate. The use of enriched elements allows for obtaining the amplitude coefficients of the asymptotic SGE solutions, which, in turn, enable the calculation of the J-integral and the application of energy criteria from linear fracture mechanics while accounting for the size effect.
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creator: Kirill Shelkov
creator: Yury Solyaev
rights: Copyright (c) 2026 Computational Continuum Mechanics https://creativecommons.org/licenses/by/4.0
date: 2026-06-01
publication date: 2026-06-01
starting page: 59
ending page: 75
doi: 10.7242/1999-6691/2026.19.1.5

Natural and forced vibrations of solid bodies with account for damping and prestressed state induced by piezoelectric elements
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description: The widespread use of thin-walled structures subjected to undesirable dynamic phenomena necessitates the development of effective methods for controlling their vibrational characteristics to prevent resonance. A promising approach to solving this problem is the application of smart materials, particularly, piezoelectric elements, enabling the implementation of passive, active, and hybrid damping systems. This work investigates the natural and forced vibrations of piecewise-homogeneous electro-elastic and electro-viscoelastic bodies using the finite element method. A distinctive feature of the study is the consideration of the prestressed and predeformed state caused by an embedded piezoelectric element as a result of the reverse piezoelectric effect due to DC voltage supply. A mathematical formulation of the problem, interpreting the prestress state in terms of geometric stiffness, and accounting for viscous and Rayleigh damping mechanisms is developed based on the principle of virtual displacements. Using a plate with an embedded piezoelectric element as an example, the influence of the control voltage and the type of damping on the complex natural frequencies and amplitude-frequency responses is studied. It has been established that both viscous and Rayleigh damping have slight effect on the shift of resonant frequencies caused by the prestressed state. The results of solution of the forced vibration problem allow us to quantify a decrease in the vibration amplitude of the plate in the prestressed state caused by the piezoelectric element in the case of external excitation with frequency corresponding to the resonance frequency of the unloaded plate. The results presented demonstrate the influence of the prestressed state on the amplitude-frequency response of a plate with closely spaced natural frequencies at different values of the electric potential on the piezoelectric element.
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creator: Aleksandr Olegovich Kamenskikh
rights: Copyright (c) 2026 Computational Continuum Mechanics http://creativecommons.org/licenses/by-nc/4.0
date: 2026-03-11
publication date: 2026-03-11
starting page: 400
ending page: 410
doi: 10.7242/1999-6691/2025.18.4.29

Numerical simulation of two-dimensional flow of a micropolar fluid in an enclosed cavity with the full dissipation function
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description: The paper studies numerically a steady two-dimensional flow of incompressible micropolar fluid in a square cavity with a moving top wall. The steady solution is obtained by the time-marching method - time integration of the full system of unsteady Eringen equations. A distinctive feature of this work is a detailed calculation and analysis of the total dissipation function, which includes three mechanisms: classical viscous dissipation, dissipation due to microrotation gradients, and dissipation resulting from the interaction between macroscopic vorticity and the microrotation field. A numerical algorithm is developed based on the Chorin–Temam projection method; its correctness is verified by comparison with the classical Newtonian limit. A parametric study is carried out over wide ranges of the governing dimensionless parameters - the Reynolds number, the micropolarity parameter, and the microscale parameter. It is shown that the integral dissipation increases with the Reynolds number and the micropolarity parameter, although there are local non-monotonicities due to a nonlinear interaction of dissipative mechanisms. The influence of the microscale parameter is found to be considerably weaker than that of the other two parameters. It is shown that as the Reynolds number increases, the thickness of the boundary layer at the moving wall decreases, and the microrotation velocity maximum diminishes and is localized in the near-wall layers. Analysis of the velocity and microrotation distributions confirms the formation of thin boundary layers near solid walls, where dissipation is maximal. Micropolar effects lead to a marked increase in dissipation compared with a classical Newtonian fluid, which demonstrates the importance of considering the internal rotational degrees of freedom in the energy balance. The results may find application in microfluidics and modeling of complex fluid flows.
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creator: Kristina Gubareva
creator: Evgenii Prosviryakov
rights: Copyright (c) 2026 Computational Continuum Mechanics https://creativecommons.org/licenses/by/4.0
date: 2026-08-05
publication date: 2026-08-05
starting page: 161
ending page: 171
doi: 10.7242/1999-6691/2026.19.2.11

Simulation of high-temperature oxide and sulfide-oxide corrosion of Fe76Cr24 alloy based on a coupled model of chemomechanics
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description: The operation of gas turbine engine parts at elevated temperatures in aggressive environments containing oxygen and sulfur compounds leads to the formation of a corrosive film that changes the mechanical characteristics and stress state in the surface layer. This, in turn, affects the rate of propagation of the corrosion product. High-temperature salt corrosion of metal alloys is accompanied by inter-diffusion and chemical reactions between metal components and aggressive substances, swelling of the material and the occurrence of rheological strains. For this reason the description of this process must be carried out within the framework of a coupled formulation. For the theoretical description of the process, a model based on the classical thermodynamics of irreversible processes is used. It includes the equations of mass balance of the components and the equilibrium equation for the material as a whole, and takes into account the elastic-viscous-plastic behavior of corrosion products at elevated temperatures. The COMSOL Multiphysics finite element package was used to calculate the processes of high-temperature oxidation and sulfide-oxide corrosion of a two-component iron and chromium alloy. Analysis of the results obtained showed the formation of a layered structure of a corrosion film, the advance of the front of compressive residual stresses and corrosion deep into the material, and stress relaxation near the surface. A comparison of geometrically linear and nonlinear formulations of the model, considering or ignoring the influence of stresses on diffusion flows and chemical reaction rates, as well as plastic and viscous strains, revealed a significant effect of each of these factors on the propagation velocity of the corrosion front and on the stress-strain state of the material.
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creator: Ilya Ernstovich Keller
creator: Dmitrii Sergeevich Dudin
rights: Copyright (c) 2026 Computational Continuum Mechanics https://creativecommons.org/licenses/by/4.0
date: 2026-06-01
publication date: 2026-06-01
starting page: 76
ending page: 94
doi: 10.7242/1999-6691/2026.19.1.6

Numerical modeling of diurnal oscillation of insolation in the idealized model of general atmospheric circulation
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description: The Earth's rotation around its axis causes the sunlit region to constantly shift in the opposite direction. The resulting diurnal oscillations of the surface temperature can influence the structure and dynamics of the general circulation and midlatitude baroclinic waves. This paper presents a numerical study of the influence of diurnal insolation oscillations on the formation and characteristics of large-scale flows using a numerical analogue of the laboratory model of atmospheric general circulation. Two main regimes were considered: with regular and irregular baroclinic waves. Diurnal insolation oscillations were modeled using a localized, moving heat source. It is shown that for both regimes of baroclinic waves, the transition from stationary, zonally uniform heating to non-stationary, zonally inhomogeneous heating does not lead to qualitative changes of the mean flow structure or the distribution of pulsations. This only leads to quantitative changes, the magnitude of which does not exceed 15%. The hypothesis that a moving heat source would lead to an increase in pulsation energy was not confirmed. Moreover, the energy of meridional velocity pulsations, which characterizes the energy of baroclinic waves, decreases in the case of a moving heat source. A relatively weak effect of the transition from a stationary, zonally uniform heating regime to a non-stationary, localized one is explained by the fact that disturbances caused by the movement of the heat source are both spatially and frequency-localized.
subject: Articles
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creator: Andrei Anatol′yevich Gavrilov
creator: Andrei Nikolayevich Sukhanovskii
rights: Copyright (c) 2026 Computational Continuum Mechanics http://creativecommons.org/licenses/by-nc/4.0
date: 2026-03-13
publication date: 2026-03-13
starting page: 5
ending page: 15
doi: 10.7242/1999-6691/2026.19.1.1

Fluid flow in a thick torus driven by distributed forces
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description: In this article, a laminar, steady-state flow of incompressible fluid in a curvilinear channel with a circular cross-section is studied numerically. The influence of the bulk force on the stationary flow is analyzed through the consideration of its three main natural causes: a pressure gradient in the channel, a uniform action of external forces, and an analog of the force generated by the uniform deceleration of a rotating channel (inertia force). The key parameter used to describe flows in a toroidal channel is the Dean number, which takes into account the influence of the channel curvature and flow intensity. Direct numerical simulations at all possible channel curvatures and Reynolds numbers up to 300 showed that to describe all features of flow in thick toroidal channels, it is insufficient to use the Dean number alone. It has been found that the position of the longitudinal velocity maximum depends on the Reynolds number and curvature, and the type of force. At low values of the Reynolds number, that is, less than 40, the maximum may shift towards the outer axis of the torus, whereas at high values, it shifts only away from it. The simulation revealed the splitting of the velocity maximum , as well as the occurrence of a third local maximum ; these effects arise only under a gradient force and at relatively high curvatures. It has been shown that the Dean number is insufficient to fully describe the flow at high curvatures. For the inertial force, the Dean number unambiguously determines the position of the velocity maximum but fails to describe the intensity of the secondary flow at the Reynolds numbers of 50 and higher. At a constant force, the Dean number is applicable only up to certain values of the Reynolds number, while for a pressure gradient, the use of the Dean number is absolutely meaningless. It has been established that the intensity of the secondary flow reaches a maximum at the Reynolds numbers from 50 to 100 (with nondimensionalization performe…
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creator: Anton Viktorovich Chupin
rights: Copyright (c) 2026 Computational Continuum Mechanics http://creativecommons.org/licenses/by-nc/4.0
date: 2026-03-13
publication date: 2026-03-13
starting page: 411
ending page: 423
doi: 10.7242/1999-6691/2025.18.4.30

The influence of biomechanical parameters of an erythrocyte and viscosity of the surrounding fluid on the hydrodynamic interaction with a solid wall
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description: The nature of blood flow in capillaries, arterioles, and venules largely depends on the mechanical interactions between red blood cells and vessel walls . A theoretical understanding of the principles of hydrodynamic interaction between blood cells and vessel walls is essential for a comprehensive biophysical description of blood flow in the microcirculatory system, as well as for the design of medical microfluidic devices for processing blood samples. In this study, numerical methods are used to investigate the behavior of a single red blood cell (erythocyte) near a solid flat surface under conditions of shear flow of a viscous Newtonian fluid. The aim of the study is to identify the pattern of erythrocyte motion depending on its biomechanical parameters and hydrodynamic conditions. To address these issues, we use a numerical model that combines particle dynamics and the Lattice Boltzmann method with the Bhatnagar-Gross-Krook collision integral. The biomechanical model of the red blood cell takes into account tensile and bending elasticity, as well as the forces that maintain the membrane area and the cell volume. Computer simulation of red blood cell trajectories under various hydrodynamic conditions shows that the velocity of cell repulsion from a solid wall is linearly dependent on the shear rate. It varies non-monotonically as the red blood cell moves away from the wall, and decreases at greater distances. The lift effect in the model is most pronounced when the red blood cell moves in a “tank-treading” mode at high shear stresses. The dependence of the lift velocity on the elasticity and volume of the cell is quantified by varying the biomechanical parameters of the erythrocyte.
subject: Articles
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creator: Aleksey Vyacheslavovich Belyaev
rights: Copyright (c) 2026 Computational Continuum Mechanics http://creativecommons.org/licenses/by-nc/4.0
date: 2026-03-13
publication date: 2026-03-13
starting page: 485
ending page: 499
doi: 10.7242/1999-6691/2025.18.4.35

Interaction of mechanical, temperature and diffusion fields in a solid cylinder subjected to transient heating
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description: The article examines the phenomena caused by the mutual influence of mechanical, temperature and diffusion fields in a continuous orthotropic multicomponent cylinder under the action of unsteady external perturbations set on the cylinder surface. The cylinder does not experience volumetric perturbations. It is assumed that the physico-mechanical processes occurring in the cylinder depend only on the radial coordinate. The mathematical formulation involves a system of coupled nonstationary thermodiffusion equations in a cylindrical coordinate system. The employed heat and mass transfer model accounts for the finite velocity of thermal and diffusion wave propagation. The kinematic relations for thermal and diffusion flows are constructed based on the generalized Lord–Shulman theory. The proposed algorithm for solving a polar-symmetric problem is based on the integral representation of the desired fields in the form of time-dependent convolutions of Green's functions and the functions, describing surface thermomechanodiffusion perturbations. The Green's functions are found by applying the time-domain Laplace transform and the Fourier–Bessel series expansion, which makes it possible to reduce the initial boundary value problem to a system of linear algebraic equations with respect to the coefficients of the Fourier series. The solutions of this system are rational functions, and therefore the inverse Laplace transformation is carried out analytically using the theory of residues and tables of operational calculus. Using a three-component cylinder as an example, a numerical analysis of the interaction between mechanical, temperature and diffusion fields under external unsteady thermal action is performed and the obtained results are compared with the solutions to known problems of thermoelasticity and mechanodiffusion. The effect of the finite velocity of propagation of thermal and diffusion fluxes on the kinetics of heat and mass transfer in continuous media under diffe…
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creator: Andrei Vladimirovich Zemskov
creator: Danila Olegovich Kazimirov
creator: Anatoliy Vladimirovich Malashkin
rights: Copyright (c) 2026 Computational Continuum Mechanics http://creativecommons.org/licenses/by-nc/4.0
date: 2026-04-04
publication date: 2026-04-04
starting page: 16
ending page: 30
doi: 10.7242/1999-6691/2026.19.1.2

Convection in a closed cavity with a large body of neutral buoyancy
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description: Rayleigh-Benard convection in a square cell with a sufficiently large, heat-insulating body of neutral buoyancy is studied numerically for two moderate values of the Rayleigh number Ra=105 and Ra=106. The emphasis is placed on changes in the dynamics of bodies and convective flows observed at a significant increase in body size. In the absence of a body, stationary two-roll convection is formed in the cell at Ra=105, and at Ra=106 a periodic oscillatory regime occurs. The appearance of a freely floating circular body in a cell changes the flow behavior, which is qualitatively independent of the body size, but fundamentally different for two considered values of the Rayleigh number. Thus, at Ra=105, in the presence of a body, a stable large-scale circulation is formed, occupying the entire cell. The movement of the body is irregular, stopping at two diagonally located corners of the cell. At Ra=106, the body movement scenario changes: a free-floating body performs quasi-stable oscillatory movements, moving back and forth along a trajectory representing the letter Pi rotated by 90 degrees. The introduction of a floating body leads to a decrease in the intensity of convective flow and convective heat transfer. It is shown that the values of the Reynolds and Nusselt numbers decrease significantly with increasing body size. The interaction between body and temperature boundary layers causes noticeable temporary fluctuations in heat fluxes, resulting in quasi-periodic fluctuations in the average temperature of the fluid. The most pronounced correlation between the temperature and the vertical coordinate of body position is observed in the case of large bodies and a higher value of the Rayleigh number.
subject: Articles
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creator: Sergey Anatol′yevich Filimonov
creator: Andrey Anatol′yevich Gavrilov
creator: Andrey Nikolayevich Sukhanovskii
creator: Andrey Yur′yevich Vasiliev
creator: Petr Gotlobovich Frick
rights: Copyright (c) 2026 Computational Continuum Mechanics http://creativecommons.org/licenses/by-nc/4.0
date: 2026-03-11
publication date: 2026-03-11
starting page: 373
ending page: 384
doi: 10.7242/1999-6691/2025.18.4.27

Modeling the loss of stability and failure of mine shaft tubing supports
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description: The structural integrity and deformation reliability of mine shaft units play a crucial role in mining operations. A structural failure can lead to water ingress into the shaft and subsequent flooding. The deformation strength and reliability of a mine shaft depends on the efficiency and accuracy of the tubing support manufacturing process, which consists of several stages. This paper presents the results of numerical modeling of the deformation of a tubing ring under the effect of uneven pressure, which can occur during plugging. The following factors are considered in the modeling: contact interaction between tubing sections, contact interaction between the tubing and concrete, and the influence of the initial stress-strain state of the tubing formed during tightening of bolted connections. The problem is formulated within the framework of elastic-plastic deformation with account of finite strains. Based on the numerical solution, the conditions under which the loss of stability and failure of the tubing ring are possible were determined. A significant dependence was found between the pressure causing tubing failure and the area over which this pressure is distributed. This dependence is not monotonic and reaches a minimum when the applied pressure is distributed over the area of two tubing segments. The pressure in this case is approximately ≈1.4 MPa, i.e., it is in the range of realistic plugging pressures, which can reach 3 MPa. The obtained dependences allow assessing the feasibility of critical deformation states in tubing support, and they can be used to select optimal plugging modes when constructing mine shafts.
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creator: Aleksey Petrovich Shestakov
creator: Igor′ Nikolayevich Shardakov
creator: Irina Olegovna Glot
rights: Copyright (c) 2026 Computational Continuum Mechanics http://creativecommons.org/licenses/by-nc/4.0
date: 2026-03-13
publication date: 2026-03-13
starting page: 424
ending page: 432
doi: 10.7242/1999-6691/2025.18.4.31

Numerical simulation of natural vibration of a partially fluid-filled truncated conical shell in a gravity field
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description: In this article, the behavior of circular truncated conical shells partially filled with an ideal fluid is studied taking into account the gravitational effects on its free surface. The mathematical formulation of the problem of the elastic body dynamics is developed using the variational principle of virtual displacements and the relations of the classical theory of shells, which are based on the Kirchhoff-Love hypotheses. The behavior of the compressible fluid is described by the equations of a potential theory, which together with the boundary conditions are converted to a weak form using the Bubnov-Galerkin method. The hydrodynamic pressure exerted by the fluid on the inner surface of the shell is calculated using the linearized Bernoulli equation. The problem is solved in the framework of axisymmetric formulation based on a semi-analytical version of the finite element method. The natural frequencies are calculated using the QR transform. The identification of coupled vibration modes in the full frequency spectrum is accomplished through an iterative dynamic condensation procedure based on the Muller method. The reliability of the results obtained within the framework of the developed numerical model is confirmed by comparison with known data using examples of elastic and rigid shells. The influence of the cone angle on the lower frequencies of sloshing and coupled vibration modes is analyzed for shells with different boundary conditions (rigidly clamped, simply supported and cantilevered shells), different linear dimensions and levels of filling of the shell with fluid. It has been demonstrated that the character of the change in lower frequencies is determined by the magnitude of the angle at the apex and is determined by its geometric parameters for coupled modes and is independent of them in the case of sloshing modes.
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creator: Sergey Arkad′yevich Bochkarev
creator: Aleksandr Nikolayevich Senin
rights: Copyright (c) 2025 Computational Continuum Mechanics http://creativecommons.org/licenses/by-nc/4.0
date: 2025-12-14
publication date: 2025-12-14
starting page: 354
ending page: 367
doi: 10.7242/1999-6691/2025.18.3.26

Numerical study of the motion of a solid particle in the field of ultrasonic standing wave
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description: This numerical study examines the dynamics of a solid particle in a viscous fluid within a closed cavity under the influence of an ultrasonic standing wave. The study comprises two stages. First, the distribution of acoustic pressure in the quiescent fluid is calculated, ignoring disturbances induced by the particle or the flow. Two ultrasonic source configurations are considered: a planar ultrasonic vibrator extending throughout the entire lower boundary and a finite-size ultrasonic source. For a planar vibrator, the distribution of acoustic pressure of a harmonic plane wave is derived analytically from the one-dimensional Helmholtz equation. For finite-size sources, the same equation is numerically solved using the finite element method to account for the complex structure of the acoustic field and edge effects near the ultrasonic source. At the second stage, the computed pressure distribution is used to determine the spatial distribution of the Gor'kov potentials and the resulting acoustic radiation force, acting on the spherical particle of a specified size. The particle trajectory in the field of the ultrasonic standing wave is determined by solving numerically the equation of particle motion, taking into account the contributions from the added mass, acoustic radiation force, and drag force. The drag coefficient for the Reynolds numbers ranging from 0.1 to 0.8×10³ is calculated using the Schiller-Naumann correlation. Numerical simulations are conducted for different initial positions of the particle. The calculated data are consistent with the known concept of the solid particle motion in the field of ultrasonic standing wave, according to which a solid particle under the influence of the acoustic radiation forces tends to get into the region of the Gor'kov potential minimum, involving the standing wave nodes. Numerical simulation of the solid particle motion in the field of ultrasonic standing wave induced by the finite - size ultrasonic radiator allowed us t…
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creator: Ivan Olegovich Sboev
creator: Tat′yana Petrovna Lyubimova
creator: Konstantin Anatol′yevich Rybkin
rights: Copyright (c) 2026 Computational Continuum Mechanics http://creativecommons.org/licenses/by-nc/4.0
date: 2026-03-13
publication date: 2026-03-13
starting page: 433
ending page: 450
doi: 10.7242/1999-6691/2025.18.4.32

The onset and evolution of nonlinear convective regimes of binary mixtures in multilayer systems simulating synclinal geological folds
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description: This paper studies three-dimensional convective regimes of a binary mixture in a system of three porous bent layers, simulating a synclinal geological fold under the influence of a geothermal temperature gradient. The layers are assumed to have identical porosities and different permeabilities. A mixture of tetralin and dodecane taken in equal proportions is considered as the fluid saturating the porous medium. The components of the mixture represent the groups of hydrocarbons found in oil fields. The numerical study is performed to evaluate the linear stability of the mechanical equilibrium of a binary mixture in the inclined porous layer saturated with liquid in a gravity field under the influence of a strictly vertical temperature gradient. The convection threshold found is compared with the threshold obtained for the system with non-linear stability. In three-dimensional nonlinear calculations for a three-layer mixture, the permeabilities of all three layers are varied. Here it is assumed that the permeabilities of the outer layers are identical and always lower than the permeability of the inner layer. It was found that when the permeability of the inner layer is much higher than the permeabilities of the outer layers, the flow is localized in the inner layer. When the permeabilities of the outer and inner layers are sufficiently close, the flow is localized near the outer boundaries of the fold, even if the inner layer is more permeable than the outer layers. As the supercriticality increases, the formation of longitudinal rolls is observed in the initially plane-parallel flow (within each fold limb), and the flow becomes spiral. A further increase in supercriticality leads to a predominance of the longitudinal roll component in the flow and an increase in the number of longitudinal rolls along the fold limb (increase in the wave number of longitudinal rolls).
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creator: Ivan Shubenkov
creator: Tatyana Lyubimova
rights: Copyright (c) 2026 Computational Continuum Mechanics http://creativecommons.org/licenses/by-nc/4.0
date: 2026-04-04
publication date: 2026-04-04
starting page: 31
ending page: 42
doi: 10.7242/1999-6691/2026.19.1.3

Some coupled diffusion problems for ternary systems
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description: Diffusion plays an important role in creating of new composite materials because it and accompanying phenomena participate in the formation of so-called transition layers. This paper presents possible boundary conditions for diffusion problems involving non--ideal contact between materials. A discontinuity in concentrations is also possible in the presence of ideal contact, due to differences in the mobilities of elements in different materials. Non-ideal contact leads to singularities in the boundary conditions, which may be due to various physical causes and leads to the concept of diffusion resistance at the boundary. It is shown that, within the framework of thermodynamics, the condition with an exponential dependence of the discontinuity in concentrations on temperature is substantiated. An algorithm for numerically solving the conjugate diffusive problem was developed based on component-wise separation of equations and an implicit second--order difference scheme, which is analogous to separation by physical processes. The boundary conditions were also approximated to second order using a Taylor series representation of the unknown quantities over small spatial steps (different for different regions) in the vicinity of the boundary. Using the problem of concentration redistribution between two materials as an example, the convergence of the algorithm and the consistency of the resulting concentration distributions at different points in time are illustrated. The parameters chosen for illustrations comply with thermodynamic constraints. Diffusion resistance at the interface between materials affects the concentration distribution and the rate at which equilibrium is established. Cross-diffusion coefficients do not alter the qualitative influence of imperfect contact on concentration redistribution. Such problems and the proposed algorithm can be useful in modeling the synthesis of composite materials, as well as in problems involving welding, soldering, and coat…
subject: Articles
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creator: Sergey Yevgen′yevich Povernov
creator: Anna Georgiyevna Knyazeva
rights: Copyright (c) 2026 Computational Continuum Mechanics http://creativecommons.org/licenses/by-nc/4.0
date: 2026-03-13
publication date: 2026-03-13
starting page: 451
ending page: 467
doi: 10.7242/1999-6691/2025.18.4.33

On the development of an approximate theory of direct and inverse problem for an inhomogeneous rectangular region
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description: The paper is devoted to the development of an approximate theory for solving direct and inverse problems as applied to an inhomogeneous planar domain. The domain is statically deformed under the action of a certain mechanical load. In the direct problem, given the known law of inhomogeneity of material parameters, boundary conditions, and the geometry of the domain, it is required to determine the displacement field. In the inverse problem, it is necessary to reconstruct the law of inhomogeneity of the elastic characteristic using additional data on the measured displacements of a certain unloaded part of the domain’s boundary. Most often, such problems for inhomogeneous bodies are solved numerically – for example, using the finite element method (FEM). However, from the perspective of solution analysis and their application to inverse problems, greater interest lies in: 1) developing simplified models and 2) finding approximate analytical solutions. An approach is proposed for constructing analytical expressions for the components of the two‑dimensional displacement field in the direct problem by introducing a stress function. A comparative analysis is carried out of the solutions obtained using the proposed approximate theory, the Bernoulli–Euler and Timoshenko 1D beam models, and the FEM in a 2D implementation. The comparison demonstrated good accuracy of the obtained analytical representations for various aspect ratios of rectangular domains and inhomogeneity laws. Based on the developed approximate theory and the Timoshenko beam model, an approach is proposed for solving the inverse problem of reconstructing the 1D law of inhomogeneity for the elastic characteristic. Explicit formulas are derived for determining the sought characteristics through the displacement functions specified on one face of a rectangular domain. The results of computational experiments are presented, demonstrating the effectiveness of the proposed approaches for solving direct and invers…
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creator: Rostislav Nedin
creator: Victor Yurov
rights: Copyright (c) 2026 Computational Continuum Mechanics https://creativecommons.org/licenses/by/4.0
date: 2026-06-01
publication date: 2026-06-01
starting page: 108
ending page: 123
doi: 10.7242/1999-6691/2026.19.1.8

On one solution to the problem of resonant oscillations of a lifting rope taking into account its slippage when winding on the surface of a drum
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description: In this paper, we develop a mathematical model and methods for analyzing a non-classical problem of longitudinal oscillations of a rope, the upper end of which is wound on a drum, and a load is fixed to its second end. Taking into account the slippage of rope elements along the drum rim allows one to adequately describe the real dynamic processes in the system. Representing the problem of rope oscillations with moving boundaries as a differential equation with non-integrable boundary conditions provides a non-classical generalization of the hyperbolic problem. The linearization of boundary conditions by the method of averaged estimates simplifies the initial statement of the problem. The construction of equivalent integro-differential equations with symmetric and time-dependent kernels, as well as with time-varying integration limits, creates the basis for applying the apparatus of integral equations. Consideration of cases both with and without slippage provides a comprehensive analysis of the dynamic characteristics of the system. Reducing the integro-differential equation to a dimensionless form by introducing new variables unifies the mathematical model. The solution of the obtained integro-differential equation without taking into account slippage using a combination of the approximate method for constructing solutions of integro-differential equations and the Kantorovich-Galerkin method demonstrates the acceptable level of accuracy for calculations. The expression for the amplitude of oscillations corresponding to the n-th dynamic mode, obtained by the asymptotic method, is an analytical basis for analyzing the resonance properties. The study of the phenomenon of steady-state resonance and passage through resonance of the system by numerical methods and the developed software package has practical value for the dynamic calculation of lifting devices.
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creator: Vladislav L′vovich Litvinov
creator: Maksim Vladimirovich Shamolin
creator: Kristina Vladislavovna Litvinova
rights: Copyright (c) 2025 Computational Continuum Mechanics http://creativecommons.org/licenses/by-nc/4.0
date: 2025-12-14
publication date: 2025-12-14
starting page: 332
ending page: 345
doi: 10.7242/1999-6691/2025.18.3.24

Analysis of the bifurcation structure near the excitation threshold of convection during uniform fluid infiltration through a slightly asymmetric porous chip
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description: Fluid-saturated porous scaffolds are crucial components of flow perfusion bioreactors, which act as microfluidic chips. These devices are used to grow cell cultures in a constant, steady-state environment through maintaining a continuous flow of nutrient-rich solution. Cell proliferation often occurs in non-isothermal conditions, which under gravity can lead to the onset of thermal convection. In this paper, we consider the problem of convection excitation in a porous chip heated from below, across which uniform pumping of an incompressible fluid is carried out. It has been known that incorporating a fluid-saturated porous structure into highly conductive impermeable medium leads to the fact that the boundary value problem takes on the cosymmetry property for any simply connected 2-D shape of the domain. This causes branching of one-parameter families of heterogeneous equilibrium states. Note however that even weak fluid pumping across the porous medium leads to cosymmetry breaking and the type of dynamic behavior of the system in this case depends on the domain symmetry. If under reflection the pumping domain is symmetric about the vertical axis, the resulting convection will be stationary. In the asymmetric domain, periodic oscillations are excited. In this paper, we investigate the bifurcation structure near the convection excitation threshold occurring in a weakly asymmetric domain. The main focus is on the transition from stationary to oscillatory convection. The study of the system includes the following issues: finding the basic state of the system analytically; linear analysis of its stability using the Galerkin method; analysis of weakly nonlinear solutions near the first bifurcation point using the method of multiple time scales. It has been shown that branching of solutions near the excitation threshold of convection in a weakly non-cosymmetric system of equations defined in a weakly asymmetric domain is of surprisingly complex nature. The transition proc…
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creator: Dmitriy Anatol’yevich Bratsun
rights: Copyright (c) 2026 Computational Continuum Mechanics https://creativecommons.org/licenses/by/4.0
date: 2026-06-01
publication date: 2026-06-01
starting page: 43
ending page: 58
doi: 10.7242/1999-6691/2026.19.1.4

A magnetic field in the vicinity of a homogeneous magnet filling an axisymmetric, simply connected or multiply connected region
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description: Designing inertial magnetorheological sensors requires calculating the spatial distribution of the magnetic field strength and its gradient, which characterize the magnetic system of their sensing elements. A sensing element includes a combined source of a constant magnetic field, consisting of a set of axially magnetized cylindrical, disc, or ring magnets with a finite aspect ratio of their main parameters and non-magnetic inserts. The assembly of magnets and non-magnetic elements is covered with a magnetic fluid that acts as a lubricant and is held in place by the field created by the assembly. The aim of the study is to obtain an expression that mathematically represents the magnetic field in the vicinity of an individual permanent magnet or a system of these magnets. The magnetostatic problem was solved in two stages using Ampere's method based on abstract magnetic poles. In the first stage, the north pole of a semi-infinite magnetized cylinder was considered. In the vicinity of this magnet an expression for the scalar magnetic potential was found. In the second stage, by applying the principle of superposition, an expression for the field near disc and ring magnets in a cylindrical coordinate system was obtained. The constructed expressions contain infinite series, which makes them difficult to use in practice. The number of terms in a series sufficient to describe the field with a predetermined accuracy was determined by comparing the data of analytical and numerical calculations. The corresponding two-dimensional axisymmetric magnetostatic problem was solved numerically using the Finite Element Method Magnetics package. For disc and ring magnets, the magnetic field strength magnitude at points in the surrounding space was calculated. It is shown that the first six terms in a series are sufficient for analytical and numerical solutions to coincide within 2%, which is an acceptable deviation equal to the typical instrument error of a modern teslameter. As the n…
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creator: Olga Sergeyevna Krylasova
creator: Mikhail Andreyevich Koskov
creator: Alexey Sergeyevich Ivanov
rights: Copyright (c) 2025 Computational Continuum Mechanics http://creativecommons.org/licenses/by-nc/4.0
date: 2025-12-14
publication date: 2025-12-14
starting page: 346
ending page: 353
doi: 10.7242/1999-6691/2025.18.3.25

Effect of a traveling magnetic field on heat and mass transfer during submerged heater crystal growth
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description: The effect of traveling magnetic field (TMF) on heat and mass transfer during the 4" Ga:Ge semiconductor single crystal growth by the Axial Heat Processing technique has been studied numerically. The heat and mass transfer processes in the composite multiphase system consisting of crystal, melt, crucible, and submerged heater were analyzed in an axisymmetric unsteady formulation, with the instantaneous position and shape of the crystallization front being unknown and determined during the simulation. The influence of the TMF is accounted for through an analytically defined Lorentz force applied for the case of electrically insulating boundaries. The flow patterns and dopant distribution in the melt and crystal were determined for various temperature profiles applied to the submerged heater, axial temperature gradients, and at different directions and induction values of the applied TMF. It was found that the temperature boundary conditions applied to the submerged heater significantly affect mass transfer in the melt and, consequently, dopant segregation in the grown crystal. The downward TMF improves the axial and radial dopant homogeneity in the grown crystal. Within a certain range of intensities, the TMF reduced the intensity of flow driven by the radial temperature gradient at the submerged heater and the curvature of the crystallization front. Moreover, the positive effect of the TMF increased with higher growth rates. In the investigated configuration, a fourfold reduction in radial dopant segregation was achieved under stringent growth conditions by expanding the convective dopant transfer zone toward the symmetry axis through the TMF, thereby eliminating the accumulation of dopant concentration in this region.
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creator: Oleg Anatol′yevich Khlybov
creator: Tatiana Petrovna Lyubimova
rights: Copyright (c) 2025 Computational Continuum Mechanics http://creativecommons.org/licenses/by-nc/4.0
date: 2025-12-14
publication date: 2025-12-14
starting page: 319
ending page: 331
doi: 10.7242/1999-6691/2025.18.3.23

Simulation of convective polymerase chain reaction based on multiphase Lagrangian particle tracking model
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description: The article considers the issues of modeling the convective polymerase chain reaction (PCR) under the assumption that the solution of DNA molecules is a dispersed medium. The Lagrangian model of multiphase flows serves as a basis for simulations. In the framework of this approach, we consider the liquid phase to be continuous, applying the Navier-Stokes equations to describe its dynamics. To model the dispersed phase, we integrate the equation of motion of individual particles (DNA molecules) along their trajectories. Since the volume fraction of DNA molecules during the PCR reaction is small enough, the interphase interaction is assumed to be one-sided: molecular phases do not affect the motion and heat exchange of the solvent. Therefore, the general problem is split into two subproblems: the hydrodynamic part and the transport-reaction problem for particles. The mathematical model incorporates the Navier-Stokes equations written in the Hele-Shaw approximation, the motion equations for DNA molecules, and the specific condition for the PCR reactions. To model the reaction, we develop an original technique in which the history of particle movement through the reaction zone determines the criterion for transformation of a particle from one type to another. A numerical simulation of the convective PCR in a Hele-Shaw cell was carried out. The distribution fields of DNA molecules and the hydrodynamic and thermal fields were obtained. Zones with ongoing reactions were visualized. The dependence of the number of DNA molecules on the reaction time and the doubling time of the number of molecules were established. The results are consistent with the data of other convective PCR models. The positive aspects of the model used are that it provides complete information about the behavior of individual particles, including their coordinates, velocities, residence times in reaction zones, and traveled distances. Additionally, the developed approach can be used to ensure fine-tunin…
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creator: Ramil′ Rifgatovich Siraev
rights: Copyright (c) 2025 Computational Continuum Mechanics http://creativecommons.org/licenses/by-nc/4.0
date: 2025-12-14
publication date: 2025-12-14
starting page: 307
ending page: 318
doi: 10.7242/1999-6691/2025.18.3.22