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.
Modeling of the response of magnetic microgel particles of different structures to an applied magnetic field
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.
Copyright (c) 2026 Computational Continuum Mechanics https://creativecommons.org/licenses/by/4.0
Influence of viscosity on nonlinear resonance of forced oscillations of a bubble in a vibrating liquid
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.
Copyright (c) 2026 Computational Continuum Mechanics http://creativecommons.org/licenses/by-nc/4.0
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
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.
Implementation of С¹-continuous finite element method within strain gradient elasticity for crack problems
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.
Natural and forced vibrations of solid bodies with account for damping and prestressed state induced by piezoelectric elements
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.
Simulation of high-temperature oxide and sulfide-oxide corrosion of Fe76Cr24 alloy based on a coupled model of chemomechanics
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.
Numerical modeling of diurnal oscillation of insolation in the idealized model of general atmospheric circulation
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.
Fluid flow in a thick torus driven by distributed forces
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 performed in this study). In this case the curvature responsible for the occurrence of this maximum depends on the type of force. The obtained results are important for designing systems involving curved channels (e.g., heat exchangers, chemical reactors).
The influence of biomechanical parameters of an erythrocyte and viscosity of the surrounding fluid on the hydrodynamic interaction with a solid wall
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.
Interaction of mechanical, temperature and diffusion fields in a solid cylinder subjected to transient heating
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 different types of external thermal load is investigated.
Convection in a closed cavity with a large body of neutral buoyancy
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.
Modeling the loss of stability and failure of mine shaft tubing supports
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.
Numerical simulation of natural vibration of a partially fluid-filled truncated conical shell in a gravity field
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.
Copyright (c) 2025 Computational Continuum Mechanics http://creativecommons.org/licenses/by-nc/4.0
Numerical study of the motion of a solid particle in the field of ultrasonic standing wave
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 to describe the displacement of the particle in the direction perpendicular to the direction of wave propagation observed in experiments.
The onset and evolution of nonlinear convective regimes of binary mixtures in multilayer systems simulating synclinal geological folds
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).
Some coupled diffusion problems for ternary systems
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 coating.
On the development of an approximate theory of direct and inverse problem for an inhomogeneous rectangular region
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 inverse problems.
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
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.
Analysis of the bifurcation structure near the excitation threshold of convection during uniform fluid infiltration through a slightly asymmetric porous chip
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 process includes the Andronov--Hopf bifurcation, the birth and death of limiting cycles at the backward and forward tangential bifurcations of a pair of equilibria, respectively; and the formation of homoclinic trajectories. Stability maps are constructed on the planes of control parameters of the problem: the Rayleigh number for the porous medium, the thermal Péclet number, and the chip tilt angle, which acts as a symmetry imperfection parameter. Bifurcation diagrams of a dynamic system on its slow manifold are presented.
A magnetic field in the vicinity of a homogeneous magnet filling an axisymmetric, simply connected or multiply connected region
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 number of terms in a series increases, the error decreases, but the complexity of calculations increases. Moreover, it is impossible to verify the increase in accuracy experimentally. In the case of axisymmetric combined field sources, the derived expressions simplify the calculation of the magnetic field. The expression for the magnetic field distribution proposed by the authors can be used for the optimization of the design and manufacture of magnetofluid sensors.
Effect of a traveling magnetic field on heat and mass transfer during submerged heater crystal growth
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.
Simulation of convective polymerase chain reaction based on multiphase Lagrangian particle tracking model
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-tuning of the reaction kinetics. Due to this, the scope of applicability of the model and the range of related problems can be expanded.
Multi-harmonic balance method in structural element ratcheting problems
The ratcheting theory originated in the 1960s. In mechanics of deformable bodies, it is used to assess the strength of structural elements (gas turbines, high-temperature reactors, etc.) operating under intense cyclic thermal stresses. This study aims to investigate possible modes of elastic-plastic deformation in pressure vessels under cyclically varying temperature conditions, assuming an elastic-ideal plastic material behavior model. Due to material hardening, analytical methods are not sufficient to solve this problem, and numerical methods must be used. The number of calculation cycles significantly affects a process's result: a large number of these cycles can create modeling difficulties due to limited time and computational resources. Therefore, the Direct Cyclic Method (DCM) has become increasingly popular in engineering practice for solving this type of problems. This method is an application of the multi-harmonic balance technique to the problems of quasistatic cyclic loading of viscoelastic-plastic materials. The validity of the DCM method is evaluated by comparing it with the method of direct nonlinear analysis of loading history when assessing the adaptability of vessels under cyclic thermal stresses. The computational efficiency of the DCM was investigated for varying finite element mesh sizes and loading cycles. It is shown that the limitations of the DCM become apparent in cases where a stable periodic solution does not exist.
A three-phase hydrodynamic phase field model with stabilization of the Cahn-Hilliard equation
A numerical implementation of a three-phase hydrodynamic phase-field model is presented, which combines the Navier-Stokes equations for incompressible fluid and the Cahn-Hilliard equation. The phase field model is highly capable for modeling multiphase media, but the numerical implementation of the phase field method has a number of computational difficulties associated with the stability of numerical schemes and, as a consequence, the need to choose a small time step, which complicates the calculations. The paper proposes a method for stabilizing the numerical algorithm by adding a fictitious relaxation derivative of the chemical potential, which reduces the spectral radius of the iteration matrix for the Cahn-Hilliard equation and allows a significant increase in the time step without losing stability. The implementation uses semi-implicit discretization and the projection method to calculate the velocity and pressure. Verification calculations are performed in a square domain with solid walls for a model fluid consisting of three phases. In static tests for different values of surface tension coefficients, equilibrium configurations were obtained in accordance with the expected contact angles at the junction of the three phases. In dynamic tests, it was shown that the model correctly reproduces the fall of a drop through the boundary of a two-layer system, as well as the collision of two particles of different phases moving in the third phase.
Numerical simulation of flux expulsion in a plain channel MHD flow
Numerical simulation of forced magnetohydrodynamic (MHD) channel flow is performed under conditions where the magnetic Reynolds number is comparable to or significantly exceeds 1. The dimensionless parameters of the problem, used for verification, correspond to flow regimes characterized by the complex MHD interaction between the flow and the magnetic field. The aim of this work is to verify and analyze the performance and accuracy of numerical simulation software packages COMSOL Multiphysics, ANES, OpenFOAM, and Elmer in application to two different formulations of the considered problem: one written in terms of the induced magnetic field, and the other in terms of the magnetic vector potential. The results of the author's calculations are compared with the results from the approximate analytical solution for a one-dimensional formulation and with those from other authors. A comparative analysis indicates the key role of a nonlinear term in the equation of motion, which correctly models the phenomenon of hydraulic resistance crisis during the transition from Hartmann flow to Poiseuille flow. Neglecting this term leads to significant errors in the representation, especially, of unsteady Poiseuille flow. The computational experiments conducted by the authors, as well as those known from literature sources, show that the models of all software packages tested in this work are capable of reproducing the main flow patterns; however, the highest accuracy is achieved with strict adherence to the Courant criterion. The model implemented in OpenFOAM demonstrates values especially close to the published values in the Poiseuille regime. In terms of computational efficiency, the best performance is shown by the OpenFOAM system (open-source) and the freely distributed ANES, both implemented using the finite volume method. The finite element models of COMSOL and Elmer packages are characterized by longer computation time and lower parallelization efficiency. Based on the obtained results, one can select software packages that are best suited for numerical simulation of specific unsteady MHD flows in fusion-related problems.
Modeling the temperature field dynamics during the Czochralski single crystal growth in a non-stationary approximation
The transient process of growing axisymmetric crystals by the liquid-encapsulated Czochralski method is considered. The mathematical model accounts for heat transfer in the crucible-crystal-melt-encapsulant, formation of the melt/encapsulant meniscus, crystallization interface movement, and changes in crystal radius. A new algorithm was developed to determine the lateral surface shape of the crystal during the process. The proposed numerical approach utilizes a geometrically conservative difference scheme that guarantees the fulfillment of conservation laws of energy and mass. The special splitting technique is used to solve the corresponding set of finite difference equations. The proposed approach ensures the consistency of crystal shape evolution with conservation laws. The designed numerical procedure is used to evaluate the impact of external thermal regime on the shape of the growing crystal. When the heater temperature is maintained constant, the crystal radius gradually decreases over time. To study the influence of the external temperature field on the shape of the lateral surface, the mathematical model is supplemented with a proportional-integral temperature controller equation that links the change in heater temperature to the radius of the growing crystal. In a general case, the application of an integral temperature controller leads to fluctuations in the crystal radius around a set value, with both the frequency and amplitude of fluctuations increasing progressively. Based on the results of the numerical experiments, the parameter values for a proportional-integral temperature controller that ensure the growth of crystals with a nearly constant radius are determined. The results of transient numerical simulations are compared with the results obtained using the quasi-steady-state model of crystal growth.
MHD-stirring of a heavy impurity under a rotating magnetic field generated by a multi-section inductor
We study numerically the efficiency of stirring a two-phase dispersed fluid (liquid metal with impurity particles) in a cylindrical cell using a multiphase two-fluid model. The stirring is driven by an electromagnetic force generated by a sectional rotating magnetic field inductor. The inductor consists of six annular segments evenly distributed along the height of the cell. Each segment generates a rotating magnetic field independently. We consider three electromagnetic force configurations. The first configuration corresponds to a co-directional rotation of the magnetic fields of all six inductors, matching the case of a classical rotating magnetic field. In the second configuration, the three upper rings generate a magnetic field rotating in one direction, while the lower three rotate in the opposite direction. In the third configuration, the rotation directions of the magnetic fields alternate along the cell height. We introduce a parameter characterizing the inhomogeneity of the impurity distribution. It was shown that the most efficient mixing is achieved in the second configuration, which is attributed to the emergence of a large-scale oscillatory flow regime. The uniform rotating magnetic field configuration provides the least efficient mixing due to low poloidal velocity. In the third configuration, oscillations in the impurity distribution inhomogeneity parameter associated with the chaotic nature of the flow are observed. The optimal stirring time at which the degree of impurity distribution inhomogeneity is the least is determined.
Numerical simulation of the aerodynamics of a cycloidal rotor in flight
The paper presents numerical simulation of the aerodynamics of a cycloidal rotor in various modes. The cycloidal rotor is a rotor consisting of several blades parallel to its axis and capable of changing their pitch angle as it rotates. The rotor’s aerodynamics is modeled in the formulation of a three-dimensional turbulent incompressible air flow. The numerical model is based on the computational fluid dynamics methods, in particular, the computational domain is discretized by the control volume method, and the motion of blades is modeled using the sliding mesh method. The results of test calculations are in good agreement with the experimental data. It is shown that, during rotation, the blade passes two parts of rotor thrust generation, most of which falls on the lower half of the cycle. To find the rotor characteristics under flight conditions, calculations were performed for the flow around a single rotor by the incident flow with varying velocity and direction. It was found that the vertical component of the force increases with increasing rotor descent speed. Under a horizontal flow, the power and, accordingly, the rotor efficiency change greatly. The calculations performed for two rotors in the incident flow show that their parameters change significantly and in opposite directions with increasing flow velocity. In this case, the torque on the front rotor decreases with increasing flow velocity, and on the rear rotor it increases. At a moderate incident flow velocity, the rotors have almost no effect on each other. As the incident flow velocity increases, rotor control requires deflection of the air jet thrown off by the rotors, as a result of which the jet created by the front rotor falls on the rear rotor, which leads to deterioration of its characteristics.
Numerical modeling of fatigue fracture in the light alloys produced by additive technology
A new approach to optimization of process parameters of selective laser melting is proposed in this paper. The required fatigue life of the final SLM product is used as an optimization parameter. A two-stage algorithm is proposed to implement this approach. At the first stage, a three-dimensional non-stationary nonlinear problem of heat conduction for a multiphase system is solved. As a result of modeling, geometric parameters of single-layer and multilayer systems of overlapping tracks are determined. The influence of laser melting process parameters (power, speed, laser beam pitch) on the topological features of the formed microstructure is studied. Characteristic types and geometric parameters of quasi-regular defects of the material produced by selective laser melting in the form of "unmelted" and multiple "overmelted" areas are obtained. At the second stage, the influence of single and multiple defects on the fatigue strength of printed samples under high-frequency loading is studied using the previously proposed multi-mode fatigue failure model. It is shown that internal heterogeneity of the microstructure of materials printed by selective laser melting can lead to earlier subsurface initiation of fatigue cracks and significantly reduce the fatigue strength and durability of the product. This effect is more pronounced for systems of multiple defects. The proposed models and calculation algorithms allow calculating the fatigue strength and durability of samples or products obtained by selective laser melting, depending on the selected process parameters. Mathematical modeling and comparison of the fatigue life of corset samples obtained by selective laser melting with experimental data are carried out. Qualitative and quantitative correspondence between the modeling and experimental results is noted.
Stability of a liquid layer in a rotating Hele–Shaw reactor under competition of buoyancy effects generated by inertial forces
Technologies that involve studying heat and mass transfer processes in a rotating Hele-Shaw cell can be effectively used in particular in designing microfluidic devices and small-scale flow-type chemical reactors. A quasi-two-dimensional model allows recording density fields by optical methods, and rotation makes it possible to control them through spatially distributed inertial forces. As is known, in the limit of an infinitely thin layer, the Coriolis force disappears in a standard mathematical model. However, the experimental observations of a fluid flow in a rotating Hele-Shaw cell indicate that the Coriolis effect fully manifests itself. The correct derivation of the equation of motion in the Hele-Shaw and Boussinesq approximation leads to the appearance of a term responsible for the buoyancy of the medium element caused by the Coriolis force. In this paper, to study the new effect, we consider the problem of convective stability of a fluid with internal generation of a transport component, which can be either the concentration of a dissolved substance or the temperature of the medium. The study of the system includes finding its basic state and linear analysis of its stability, analysis of weakly nonlinear solutions near the first bifurcation point, direct numerical modeling of nonlinear convection regimes, and evaluation of the general properties of a disturbance spectrum and the branching of solutions near the equilibrium bifurcation. The character of the branching of solutions is determined by applying the method of multiple time scales. The weakly nonlinear analysis indicates that, when the Rayleigh number reaches a critical value, the steady-state equilibrium of the fluid is replaced by oscillatory convection. The finite difference method is used to obtain a complete picture of the nature of nonlinear dynamics. It is shown that the Coriolis force has a stabilizing effect on the basic state of the system. In the strongly nonlinear regimes of convection, the Coriolis buoyancy leads to a complication of the scenario of transition to chaotic convection. It is demonstrated that the transition is accompanied by a series of bifurcations of limit cycles and tori, the final destruction of 2-D or 3-D tori, the appearance of strange attractors of the toroidal type. A stability map is constructed in the Rayleigh number-Ekman number parameter space.