Eulerian-Lagrangian Approach Research Articles

Three-dimensional simulation of binary alkane mixture flow in a ballistic cycle of a high-pressure direct fuel injector using a mass transfer cavitation model

A multi-species cavitation model is applied to a high-pressure injection cycle. Results of transient simulations with ballistic needle dynamics are presented. Species segregation in terms of local species demixing in the liquid phase is assessed. Therefore, the mass transfer cavitation model proposed by [101] is adopted and implemented in an incompressible large-eddy simulation flow solver. The model is based on the Rayleigh bubble dynamics equation, Raoult's and Dalton's law, vapor-liquid equilibrium, and mixing rules. It is initially tested on a hydrofoil test case and subsequently applied to a high-pressure injector. A heavier n-dodecane/n-heptane and a lighter n-octane/n-heptane mixture are considered. For both mixtures, essentially the same vortex and vapor structures are encountered. No distinct hysteresis between opening and closing phase of the cycle is present, which is related to a relatively low liquid density of both mixtures. Segregation correlates with the amount of vapor and is most pronounced for cloud cavitation at low needle lift levels and minor for string cavitation at high needle lift levels. It is considerably more pronounced for the heavier than the lighter mixture due to a wide spread of volatilities of n-dodecane and n-heptane. Even for the heavier mixture, segregation is much less pronounced than observed in the immediate environment of a single bubble. This observation is related to the homogeneous mixture approach, by which an averaged effect of segregation over the computational cell is assumed. An Euler-Lagrange approach is proposed for future studies, embedding details of mass transfer over the bubble interface into the CFD code.

Mathematical modelling of a semi-dry SO2 scrubber based on a Lagrangian-Eulerian approach

Semi-dry desulfurization is an efficient means of SO2 removal from the effluent gases from electrolysis cells in aluminum smelters. These gases are at low temperature and contain low concentrations of SO2, as opposed to thermal power plants. The removal is carried out by injecting powdered alkaline sorbent, hydrated lime (solid particles), into the SO2-containing gas (gas phase) in the presence of humidity. The reaction is controlled by the adsorption of SO2 onto the surface of lime. This study involves the mathematical modelling of a lab-scale scrubber using a Lagrangian-Eulerian approach in order to analyze the desulfurization efficiency. The model was validated based on experimental data. A parametric study was carried out to investigate the effects of particle size, sorbent amount, and relative humidity (RH) on the desulfurization efficiency. The results show that the particle size is the most important parameter; as the particle size decreases, the desulfurization efficiency increases. However, using finer particles may increase the process cost. The loss in SO2 capture efficiency due to the use of coarser particle size could be compensated by increasing the relative humidity (RH) of the gas, another key parameter of the process.

Parameters effect and optimization on calciner for treating magnesium salt based on computational fluid dynamics

The magnesium salt Mg(NO3)2·2H2O from laterite nickel ore nitric acid leaching process can be reused by thermal decomposition using calciner. To improve the performance of the calciner, a new numerical model integrating the gas-particle hydrodynamics and the particle pyrolysis reaction was established by Euler-Lagrange approach. In particular, the particle size variation caused by pyrolysis reaction was considered to accurately describe the gas-particle hydrodynamics. The particle pyrolysis model was developed to realize the interphase mass transfer. The P-1 model was modified to realize interphase radiation heat transfer. Then, the influence of different parameters on the thermal decomposition processes was discussed by using the established model. Finally, the optimal conditions for Mg(NO3)2·2H2O thermal decomposition were determined via response surface optimization analysis. The results show that the thermal decomposition of Mg(NO3)2·2H2O can be improved by increasing the initial hot gas temperature, reaction zone height, and decreasing treatment capacity of Mg(NO3)2·2H2O. The degree of influence of each parameter on thermal decomposition follows the order of initial hot gas temperature > treatment capacity of Mg(NO3)2·2H2O > reaction zone height. The decomposition rate of 99.99 % is obtained at optimized conditions of initial hot gas temperature 1111 K, Mg(NO3)2·2H2O treatment capacity 251 kg/h and reaction zone height 9 m.

Modeling and CFD simulation of photocatalytic removal of tetracycline by ZnO/PbBiO2Cl nanocomposite catalyst

In this work, the photocatalytic degradation process under visible light in a batch photoreactor was investigated using computational fluid dynamics (CFD). The removal of tetracycline (TC) by the ZnO/PbBiO2Cl photocatalyst was modeled using the single organic species model based on Langmuir-Hinshelwood kinetics. In addition, the hydrodynamic behavior of the two-phase flow inside the photoreactor was simulated using the Eulerian-Lagrangian approach. In this study, the kinetic of the degradation reaction, velocity field of suspension, and concentration of TC were investigated. Based on the CFD results, it has been determined that a rotational velocity of 250 rpm in the reactor is the optimal speed that leads to the highest homogeneity (approximately 95 %) in suspension. Therefore, more light is absorbed by performing experiments in high-quality mixing, and maximum degradation can be achieved. The kinetic study showed that the degradation rate follows pseudo-first-order kinetics. The effect of different weight percentages of PbBiO2Cl and various loading of ZnO/PbBO2Cl on the degradation rate was also examined. The results showed that the 0.5 g/l of ZnO/PbBO2Cl (40 %) is the optimal amount. The degradation reaction kinetic constant was also calculated about 200 × 10−4 min−1. Finally, the obtained modeling results demonstrate excellent agreement with the experimental data.

Instabilities of a circular moderately dense particle cloud impacted by an incident shock

Shock-driven gas-particle flows exist in a wide variety of physical systems, covering the range from dilute to dense regime. There are a lot of fundamental questions remaining to be answered, like the evolutions of flow instability, two-phase interactions, and inter-particle collision effects. In this work, the Eulerian-Lagrangian approach with gas-particle four-way coupling is applied to model two-dimensional shock-driven gas-particle flow under the moderately dense regime. For the gas flow, the formation mechanism of primary and secondary vortexes is discussed. The primary vortex is caused by the roll-up of slip line, while the secondary vortexes are induced by the velocity and density difference between the penetrating flow of particle-laden zone and the outside mainstream. Parametric studies of different Mach numbers, particle diameters and initial volume fractions have been performed to investigate the effects on shock behaviours and slip line instabilities. For the particle phase, a high-volume-fraction compression region forms at the upstream cloud zone, which blocks the penetrating gas, and results in the formation of slip line. The cloud tail is mainly composed of the upstream edge particles. The inter-particle collisions have impact on the particle distributions, and greatly affect the slip line stability at the upstream cloud edge. This work provides valuable insights into the instability mechanism of compressible gas-solid flow and the particle dispersion behaviours. However, it is important to note that our analysis is based on a two-dimensional configuration. In a real three-dimensional context, the generation of vortices is likely to differ, and may affect the particle dynamics to a certain degree.

An interface tracking, finite volume code for modeling axisymmetric implosion of a rotating liquid metal liner with free surface

We present Integrated System Model-hydrodynamics (ISM-hydro)—an interface tracking, finite volume code for modeling a shaped implosion of a rotating, initially cylindrical, fluid shell (liner) with a free surface. The code is a novel implementation of the mixed Lagrangian–Eulerian approach, applied to a compressible fluid in an axisymmetric geometry described by cylindrical coordinates (r, ϕ, z). In ISM-hydro, a structured quadrilateral mesh follows fluid elements in the r-direction (radially Lagrangian) and is fixed in the z-direction (axially Eulerian). This approach accurately captures the motion of the liner's free surface, making it an interface tracking method. Using this mesh, we derive a finite volume discretization of the axisymmetric Euler equations for a rotating compressible fluid that has an exact balance of kinetic energy. An extensive comparison between ISM-hydro and the open-source software OpenFOAM is presented; results for different test cases show very good agreement in simulated implosion trajectories and flow fields. ISM-hydro is the purely hydrodynamic component of the Integrated System Model (ISM), a framework developed at General Fusion (GF) for comprehensive predictive modeling of GF's magnetized target fusion (MTF) scheme, where an imploding rotating liquid metal liner compresses a magnetized plasma target to fusion conditions. Among advantages of the code is its speed: a full implosion simulation with a coarse mesh takes on the order of one minute on a single core while preserving high accuracy. This makes ISM-hydro a valuable tool for the design optimization of GF's MTF machines.

Combustion in a chamber furnace with a tangentially swirling vortex

Relevance. The need to analyze the furnace processes in boiler units with vertical vortex when arranging pulverized coal combustion with the enhancement of environmental parameters through the installation of tertiary blast nozzles. Despite the carbon-free policy in modern power engineering coal remains one of the main sources of energy, so the reconstruction of boiler units in order to reduce harmful emissions is very relevant. Aim. To study aerodynamics and combustion processes of pulverized coal fuel in the furnace chamber of a boiler unit with a tangential arrangement of burner devices using a combustion scheme with tertiary air blowing. Methods. The Eulerian–Lagrangian approach was applied to numerical study of flow characteristics, heat transfer and combustion in a furnace with a tangential burner arrangement. Also, k-ε model of gas turbulence, particle-turbulence interaction, diffusion model of particle dispersion, P-1 radiation modeling method and pulverized coal particle combustion model based on global particle kinetics and experimental data were used for numerical calculations. Results. Based on numerical calculations and analysis, the dependences of combustion product concentrations, temperature, hydrodynamics at combustion of Kuznetsky coal grade D in the furnace chamber with tangential arrangement of burners for brown coal, as well as at installation of tertiary blast nozzles have been obtained. Insufficient redistribution of air between burners and tertiary nozzles is revealed, as for maintenance of stable twist of vertical vortex and accordingly presence of high values of velocities at the outlet of burner devices it is not possible to increase the portion of tertiary air without reconstruction of burners. It is also established that the volume of the furnace chamber below the burners is poorly involved in heat exchange.

Airborne Transmission of SARS-CoV-2: The Contrast between Indoors and Outdoors

COVID-19 is an airborne disease, with the vast majority of infections occurring indoors. In comparison, little transmission occurs outdoors. Here, we investigate the airborne transmission pathways that differentiate the indoors from outdoors and conclude that profound differences exist, which help to explain why SARS-CoV-2 transmission is much more prevalent indoors. Near- and far-field transmission pathways are discussed along with factors that affect infection risk, with aerosol concentration, air entrainment, thermal plumes, and occupancy duration all identified as being influential. In particular, we present the fundamental equations that underpin the Wells–Riley model and show the mathematical relationship between inhaled virus particles and quanta of infection. A simple model is also presented for assessing infection risk in spaces with incomplete air mixing. Transmission risk is assessed in terms of aerosol concentration using simple 1D equations, followed by a description of thermal plume–ceiling interactions. With respect to this, we present new experimental results using Schlieren visualisation and computational fluid dynamics (CFD) based on the Eulerian–Lagrangian approach. Pathways of airborne infection are discussed, with the key differences identified between indoors and outdoors. In particular, the contribution of thermal and exhalation plumes is evaluated, and the presence of a near-field/far-field feedback loop is postulated, which is absent outdoors.

Dynamic Mesh Simulations in OpenFOAM: A Hybrid Eulerian–Lagrangian Approach

The past few decades have witnessed a growing popularity in Eulerian–Lagrangian solvers due to their significant potential for simulating aerodynamic flows, particularly in cases involving strong body–vortex interactions. In this hybrid approach, the two component solvers are mutually coupled in a two-way fashion. Initially, the Lagrangian solver can supply boundary conditions to the Eulerian solver, while the Eulerian solver functions as a corrector for the Lagrangian solution in regions where the latter cannot achieve high accuracy. To utilize such tools effectively, it is vital for them to be capable of handling dynamic mesh movements. This study builds upon the previous research conducted by our team and extends the capabilities of the hybrid solver to handle dynamic meshes. While OpenFOAM, the Eulerian component of this hybrid code, incorporates built-in dynamic mesh properties, certain modifications are necessary to ensure its compatibility with the Lagrangian solver. More specifically, the evolution algorithm of the pimpleFOAM solver needs to be divided into two discrete steps: first, updating the mesh, and later, evolving the solution. This division enables a proper coupling between pimpleFOAM and the Lagrangian solver as an intermediate step. Therefore, the primary objective of this specific paper is to adapt the OpenFOAM solver to meet the demands of the hybrid solver and subsequently validate that the hybrid solver can effectively address dynamic mesh challenges using this approach. This approach introduces a pioneering method for conducting dynamic mesh simulations within the OpenFOAM framework, showcasing its potential for broader applications. To validate the approach, various test cases involving dynamic mesh movements are employed. Specifically, all these cases employ the Lamb–Oseen diffusing vortex, but each case incorporates different types of mesh movements, including translational, rotational, oscillational, and combinations thereof. The results from these cases demonstrate the effectiveness of the proposed OpenFOAM algorithm, with the maximum relative errors —when compared to the analytical solution across all presented cases—capped at 2.0% for the worst-case scenario. This affirms the algorithm’s capability to successfully handle dynamic mesh simulations with the proposed solver.

Numerical simulation of spray atomization in crossflow using an empirical primary atomization model

The hybrid Eulerian-Lagrangian approach is used to simulate the atomization process of spray jet in subsonic crossflow. A coupled Volume of Fluid (VOF) - Lagrangian Particle Tracking (LPT) solver is implemented in the open-source platform OpenFOAM to solve the governing equations for the continuous and discrete phase respectively. An empirical model for primary atomization is proposed, in which the aerodynamic effect is taken into account, along with the turbulence effect. Thus, it is suitable for the flow at a low liquid-gas density ratio, i.e., less than 500. The performance of the presented model is evaluated and verified by comparisons of the simulated results with the experimental data. The flow characteristics of flow fields and distribution of droplet diameters are analyzed in detail. It is revealed that the vortex structures have a direct impact on the distribution of the droplets. The effects of the empirical constants in the atomization model are also examined. Finally, it is shown that the model performs better for the flow at a higher Weber number, in which the aerodynamic effect is more significant.

Finite element modeling in heat and mass transfer of potato slice dehydration, nonisotropic shrinkage kinetics using arbitrary Lagrangian–Eulerian algorithm and artificial neural network

AbstractThe present study optimized the drying temperature (50–80°C) for potato slices based on color, texture, and visual observations. At an optimized temperature (60°C), a 2D axisymmetric finite element method (FEM) was developed in COMSOL Multiphysics to predict the heat and mass transfer (HMT) in a disk‐shaped potato slice. The nonisotropic shrinkage was predicted for the potato slice by the arbitrary Lagrangian–Eulerian approach. The experimental dehydration results revealed that axial shrinkage (27.44%) was 2.5 times higher than radial shrinkage (67.39%). The simulated outcomes based on FEM revealed the realistic visualization of spatial heat transfer, moisture migration, and nonisotropic slice deformation. The predicted moisture content, surface temperature, and shrinkage properties were in good agreement with the experimental results. The shrinkage behavior was further validated using artificial neural network (ANN) to simulate the slice shrinkage. Results showed that both the COMSOL and ANN approaches can precisely predict the shrinkage‐dependent HMT model. The ANN model outperformed the COMSOL determined by mean absolute error, mean square error (MSE), root MSE, and Chi‐square (χ2) values. The successful application of the presented approach for determining dehydration characteristics may have potential for quality assessment and management of different fruits and vegetables.Practical applicationsThis journal article explores the practical industrial applications of combining finite element method (FEM)‐based heat and mass transfer model and artificial neural networks (ANNs) to improve the efficiency and quality of food drying processes. FEM is employed to simulate and predict the realistic visualization of heat and mass transfer phenomena along with non‐isotropic shrinkage, while ANN serves as a data‐driven modeling tool for process control and prediction. The integration of these two technologies offers significant advantages in the food industry, including quantification of precise temperature and moisture content, as well as to monitor the drying process of various food products, reduced energy consumption, and time.

Cavitation research with computational fluid dynamics: From Euler-Euler to Euler-Lagrange approach

Cavitation research with computational fluid dynamics: From Euler-Euler to Euler-Lagrange approach

A hybrid Eulerian–Lagrangian approach for non-self-similar expansion analysis of a cylindrical cavity in saturated and unsaturated critical state soils

AbstractThis paper proposes a powerful hybrid Eulerian–Lagrangian (HEL) approach for the analysis of cavity expansion problems. The new approach is applied to analysing the non-self-similar expansion process of a hollow cylinder of critical state soils, considering arbitrary saturation states of soils and both drained and undrained conditions. A closed-form solution for the stresses and displacements in the elastic zone is presented, taking the state-dependent soil moduli and outer boundary effect of the soil cylinder into account. Adopting large strain theory in the plastic zone, the non-self-similar cavity expansion process is formulated into a set of partial differential equations in terms of both Eulerian and Lagrangian descriptions, which is solved by a newly proposed algorithm. The HEL approach is compared with the conventional Eulerian and Lagrangian approaches for the cavity expansion analyses. It is found that the new approach can reduce to the Eulerian approach when the self-similar assumption is satisfied and to the Lagrangian approach when stress–total strain relationships are obtained analytically. Finally, the expansion process is proven to be non-self-similar by showing the stress and deformation paths, and the finite thickness of soil cylinders may greatly influence the cavity expansion behaviour, especially with a small thickness ratio. The HEL approach can provide useful tools for validating advanced numerical techniques for both saturated and unsaturated soils and interpreting pressuremeter tests in small-size calibration chambers.

Numerical simulation of gas-solid flow in a cyclone separator with additional inlet

In this study, an additional inlet was added to the gas-solid cyclone separator to enhance the separation efficiency. Four different heights were tested, including 0.95D, 1.4D, 1.5D, and 1.95D (D is the diameter of the cylindrical section). The investigation involved two inlet flow conditions: increasing and dividing the inlet flow rate. The finite volume method and Reynolds stress turbulence model were used to solve the averaged Navier–Stokes equations, whereas the Eulerian–Lagrangian approach and discrete phase model (DPM) were applied to track particles with a uniform diameter of 0.5–1.8 microns as the discrete phase. Owing to the low Stokes number and small and low-volume-fraction particles, a one-way coupling method was employed between air and the particles. The addition of an additional inlet reduced the static pressure in the center and downstream areas and increased the reverse flow velocity at the end of the cyclone. The installation of an additional inlet at 0.95D had the most positive effect on the separation efficiency, with an increase of 28.8% in the increasing flow rate case and 19.6% in the dividing flow rate case compared with the cyclone without an additional inlet. Furthermore, the increase in the separation efficiency of the submicron particles was greater than that of the larger particles in both flow distribution cases.

Developing a novel mucociliary clearance boundary condition (MCBC) to simulate micro-scale particle transfers inside the respiratory tract system without generating extra computational cells

The respiratory tract system is protected against pathogens and contaminants by the mucociliary clearance phenomenon. Simulating the effects of mucociliary clearance is an important numerical challenge besides the airflow inside the respiratory tract system. The open-source toolbox (OpenFOAM 7) is used to develop a novel method for modeling mucociliary clearance in the vicinity of airflow without extra computational cells. The new method is developed based on three components including inlet-outlet direction, cilia metachronal wave, and main flow core direction, and can be applied to any other respiratory geometries. The capability of the new method is assessed by applying it to the nasal cavity using the Eulerian-Lagrangian approach. The method is applied to the whole nasal surface except the vestibule region to model mucociliary clearance. The results show all deposited particles inside the nasal valve region are transferred alongside the nasal surface toward the main airway region within 12 min. Although mucociliary clearance does not affect the airflow and deposition location of particles due to its slight velocity, its effect on transferring deposited particles is significant. Particle tracking during 5 min inside a single nasal cavity shows 40 % of deposited particles inside the nasal valve are transferred to the main airway by mucociliary clearance. This novel numerical method can lead to the appearance of new insights into the discovery and development of mucosal-based therapies.

EFFECT OF NOZZLE CONFIGURATION ON PERFORMANCE OF A SPRAY DRYER

In this work, hydrodynamics and drying characteristics of spray dryer is numerically investigated using computational fluid dynamics (CFD) using Euler-Lagrangian (EL) approach. The gas phase is modelled as the continuous phase and solid particle as the dispersed phase. The turbulence in the gas phase is predicted using RNG version of k-ε model. As air flow pattern influences the time spent by particle in drying chamber, the spatial variation of air velocity and its circulation rate is quantified. Accordingly, optimum conditions for drying the feed slurry are determined. Further, five different outlet pipe locations are chosen and the optimum location is identified which supports the highest evaporation rate. To improve the product quality, conventional nozzle is modified and particle impact positions are analyzed. The particles impact positions on the dryer’s surface are found to be minimum for the proposed nozzle configuration and it improves the final product quality.

Численное моделирование осаждения твердых частиц в установке погружного горения

In this paper, the regularities of solid-phase sedimentation within a laboratory submerged combustion apparatus are considered. The study is conducted using the methods of computational fluid dynamics. A gas-liquid-solid three-phase flow is simulated. The gas-liquid and liquid-solid interactions are modeled using the Euler-Euler and Euler-Lagrange approaches, respectively. The thermal regime is considered without the vapor phase. The finite volume method is used to solve this problem. As a result, the trajectories of the solid particles are obtained and their correlation with the streamlines of the fluid is analyzed. The proportion of particles that settled down during the experiment is obtained for different numbers of particles. The conclusion is made on the nonlinear growth of the proportion of settled particles and their percentage on the right side. The velocity of the upward fluid flow is found to be higher than the deposition velocity for the entire considered range of solid particle diameters. It is concluded that the organized solid-phase withdrawal from the apparatus can be provided if the solid phase is extracted near the area of solid particle nucleation.

Stability analysis of a finite element approximation for the Navier-Stokes equation with free surface

In this work we study the numerical approximation of incompressible Navier-Stokes equations with free surface. The evolution of the free surface is driven by the kinematic boundary condition, and an Arbitrary Lagrangian Eulerian (ALE) approach is used to derive a (formal) weak formulation which involves three fields, namely, velocity, pressure, and the function describing the free surface. This formulation is discretised using finite elements in space and a time-advancing explicit finite difference scheme in time. In fact, the domain tracking algorithm is explicit: first, we solve the equation for the free surface, then move the mesh according to the sigma transform, and finally we compute the velocity and pressure in the updated domain. This explicit strategy is built in such a way that global conservation can be proven, which plays a pivotal role in the proof of stability of the discrete problem. The well-posedness and stability results are independent of the viscosity of the fluid, but while the proof of stability for the velocity is valid for all time steps, and all geometries, the stability for the free surface requires a CFL condition. The performance of the current approach is presented via numerical results and comparisons with the characteristics finite element method.

Fluid flow, slag entrainment, and composition evolution of slag inclusions during vacuum degassing refining

A multiphase mathematical model was established to study the fluid flow, the open-eye formation, and the residence time of slag inclusions in a bottom argon-blowing ladle. The fluid flow was solved using the Eulerian-Lagrangian multiphase approach, and the interface between different phases was described using the Volume of Fluid (VOF) model. The motion of argon bubbles and slag inclusions was tracked with the method of the discrete phase model (DPM). Moreover, the composition transformation of slag inclusions was predicted by employing a kinetic model based on the coupled reactions model. Numerical results show that the flow of the molten steel and the open-eye of the molten slag were sensitive to the argon blowing rate. The larger argon blowing rate expanded the volume of the plume and the area of the open-eye. Slag droplets were considered to move with the steel to the region away from the open-eye and then entrained into the steel to form slag inclusions as there was barely any downward velocity around the open-eye. Most slag inclusions were able to be removed immediately after the entrainment, while there still were some slag inclusions retaining in the steel for as long as thousands of seconds. The composition of slag inclusions had an apparent transformation due to reactions with the steel. The content of SiO2 and MgO in slag inclusions had a decrease while that of Al2O3 rose remarkably. The variation of SiO2 content occurred in the early one minute and the MgO content reduced gradually from 7.34wt% to 1.12wt% in 10 min. The composition transformation of slag inclusions was basically completed within 10 min in the current case.

Solitary deformation waves in two coaxial shells made of material with combined nonlinearity and forming the walls of annular and circular cross-section channels filled with viscous fluid

The aim of the paper is to obtain a system of nonlinear evolution equations for two coaxial cylindrical shells containing viscous fluid between them and in the inner shell, as well as numerical modeling of the propagation processes for nonlinear solitary longitudinal strain waves in these shells. The case when the stress-strain coupling law for the shell material has a hardening combined nonlinearity in the form of a function with fractional exponent and a quadratic function is considered. Methods. To formulate the problem of shell hydroelasticity, the Lagrangian–Eulerian approach for recording the equations of dynamics and boundary conditions is used. The multiscale perturbation method is applied to analyze the formulated problem. As a result of asymptotic analysis, a system of two evolution equations, which are generalized Schamel– Korteweg– de Vries equations, is obtained, and it is shown that, in general, the system requires numerical investigation. The new difference scheme obtained using the Grobner basis technique is proposed to discretize the system of evolution equations. Results. The exact solution of the system of evolution equations for the special case of no fluid in the inner shell is found. Numerical modeling has shown that in the absence of fluid in the inner shell, the solitary deformation waves have supersonic velocity. In addition, for the above case, it was found that the strain waves in the shells retain their velocity and amplitude after interaction, i.e., they are solitons. On the other hand, calculations have shown that in the presence of a viscous fluid in the inner shell, attenuation of strain solitons is observed, and their propagation velocity becomes subsonic.