Jumping drops on the surface of the water

The paper proposes a theoretical model for the bouncing of a water drop on a free surface. The motion of a drop in air is described by the usual equations connecting the forces of inertia, gravity, and Stokes (viscosity resistance). The drop is considered spherical with a given surface tension. Numerical calculations were carried out using the same algorithm, but with different initial conditions. Some conditions are set for the droplet disintegration, others for the droplet reflection from the free surface. It is shown that the disintegration of a drop occurs periodically with a decrease in the drop size and an increase in the drop rise height. In the interval between droplet decays, periodic reflection from the free surface occurs with a decrease in the rise height.

Comparison of the numerical and self-similar solutions of Sedov’s problem on a point explosion in gas

Comparative analysis of solutions of Sedov’s problem of a point explosion in gas for the plane case, obtained by the analytical method and using the open software package of computational fluid dynamics OpenFOAM, is carried out. A brief analysis of methods of dimensionality and similarity theory used for the analytical self-similar solution of point explosion problem in a perfect gas (nitrogen) which determined by the density of uncompressed gas, magnitude of released energy, ratio of specific heat capacities and by the index of geometry of the explosion is given. The system of one-dimensional gas dynamics equations for a perfect gas includes the laws of conservation of mass, momentum, and energy is used. It is assumed that at the initial moment of time there is a point explosion with instantaneous release of energy. Analytical self-similar solutions for the Euler and Lagrangian coordinates, mass velocity, pressure, temperature, and density in the case of plane geometry are given. The numerical simulation of considered process in sonicFoam solver of OpenFOAM package built on the PISO algorithm was performed. For numerical modeling the system of differential equations of gas dynamics is used, including the equations of continuity, Navier-Stokes motion for a compressible medium and conservation of internal energy. Initial and boundary conditions were selected in accordance with the obtained analytical solution using the setFieldsDict, blockMeshDict, and uniformFixedValue utilities. The obtained analytical and numerical solutions have a satisfactory agreement.

Application of free software FreeFem++/Gmsh and FreeCAD/CalculiX for simulation of static elasticity problems

The paper discusses the stages of computer numerical simulation of engineering problems and ways to improve the accuracy of simulation; provides a brief overview of free software for simulation elasticity problems by the finite element method, as well as trends in the development of free CAD and CAE software. For a successful engineering study, it is necessary to choose a convenient tool that takes into account all the features of the problem being solved. Based on the solution of a test static problem of linear elasticity, two approaches to engineering modeling were demonstrated. The first approach requires programming skills - the full modeling cycle was written in the programming language of the FreeFem++ software. Additionally, the method mesh generating in the Gmsh program with subsequent use in the FreeFem++ program is shown. In the second approach, the full cycle of modeling is carried out through the interface of the FreeCAD program with the built-in CalculiX solver, which does not require programming skills. A way to parameterize the task using the Python interpreter built into FreeCAD is also proposed. The simulation results obtained using both approaches are compared for an object to which an external action is applied, determined by the Dirichlet or Neumann boundary conditions, and two types of object fastening are analyzed: rigid embedding and limitation by a plane with zero friction. The analysis of the use of computing resources by various direct and iterative methods is carried out. Within the framework of the considered test problem of static linear elasticity, the most optimal method in FreeFem++ is the iterative method of conjugate gradients CG both in terms of computation time and in terms of the memory used. The highest speed of calculations is provided by the Cholesky iterative method with conditioning by the incomplete Cholesky expansion in the CalculiX program.

Mathematical model of oil displacement by water in a plane channel

Unstable displacement of immiscible liquids in a plane channel is a topical research in both theoretical and practical applications. In this paper, we consider a plane channel filled with an incompressible fluid. Over time, another fluid is injected into the channel. The fluids are immiscible. The paper builds a mathematical model of the process of oil displacement by water in a plane channel, which allows further numerical studies and comparison of the results with the obtained experimental data using the example of the Hele-Show cell. The mathematical model for a multiphase, multicomponent flow consists of the Navier-Stokes equations, the equations of conservation of mass, momentum and energy. Modern methods for modeling the dynamics of "viscous fingers“ are based mainly on numerical methods for solving systems of differential equations using the pressure gradient, viscosity and capillary forces as parameters. The influence of these parameters must be determined experimentally. To solve the problem, a quasi-hydrodynamic approach is used, based on the addition of a certain small parameter and allowing one to describe stable schemes with central differences. The complexity of solving such problems lies in the size of the considered models, which in practice have a wide range of applications from micro-scale to orders of one centimeter. A comprehensive study will allow us to evaluate and analyze the entire process as a whole, as well as to establish flow parameters to improve the efficiency of displacement and increase oil recovery, since in the numerical modeling of the process it is easier to create many independent experiments with the same initial data, in contrast to the experimental study.

Terms number determination at the series truncation for the numerical solution of the problem of acoustic scattering from a sound-permeable spheres set

Acoustic scattering from small-sized obstacles under external influence is one of the most important problems in acoustics, primarily because of the practical applications of this phenomenon. The solution of this problem is reduced to solving the Helmholtz equation for a complex potential with certain boundary conditions. When using the calculation method based on the fast multipole method, the potentials are decomposed into series according to special spherical functions, the form of which depends on the region in which this potential is calculated. As a result, the numerical implementation of the resulting matrix system raises the question of the correct choice of the number of series members when truncating them, since with a small number of series members, the calculation accuracy will be low, and with a large one will be increase not only the accuracy, but also the calculation time. An analysis of the scientific literature has shown that there are two approaches to choosing the number of terms of a series when truncating for such problems. In the first approach, truncation of the series is based on comparing two consecutive values of the sum of the sought series until the required degree of accuracy is achieved. In the second approach, all series in each expansion are truncated for a fixed number of series terms determined using heuristic formulas. In this paper, using the example of three sound-permeable spheres of different radii in the case of their strong interaction, when numerical calculations become «sensitive» to the number of terms during truncation, we compared these approaches. The analysis of the obtained data showed that to determine the value of the desired function with the necessary accuracy, it is optimal to use a combination of the considered approaches.

Analytical review of the proceedings of the conference ”Multiphase Systems: Models, Experiment, Applications“

The presented analytical review includes a short description of the papers sent to the conference ”Multiphase Systems: Models, Experiments, Application“. The review consists of seven thematic sections corresponding to the research areas of the works presented by the authors. The list of sections traditionally present at conferences on multiphase systems has been supplemented with a section on ”Microhydrodynamics and Models of Biomedical Research“, which reflects one of the most rapidly developing areas of science and, at the same time, is closely related to the ideas and methods of mechanics of multiphase media. The works representing such new directions as gas hydrate and hydraulic fracturing research were included in the section ”Theory and practice of multiphase filtration“. The review considers more than sixty works in which the most striking results have been achieved, and most consistent with the spirit of the conference. However, we note that the total number of submitted papers exceeds one hundred and twenty.

Group classification of third order nonlinear wave equations chain’s

The paper presents the results of point symmetrys Lie algebras for third-order nonlinear wave equations calculating linked into a chain by Bäcklund transformations. Calculations are carried out by using Lie-Ovsyannikov method of group analysis. The basic algebras of point symmetries of the indicated equations are found, all possible cases of their extension are revealed, and the commutator tables of algebras found are calculated.

Dynamics of dislocations in the domain structure of the nematic liquid crystal

Dynamics and interaction of classical dislocations in the domain structure of π/2 nematic liquid crystal is studied. A feature of twisted nematics is that hydrodynamic flows in Williams domains, together with the tangential component of velocity, also have an axial component, the direction of which is opposite in neighboring domains. Dislocations can move both perpendicular (glide) to Williams domains, and along (climb) them. It was found that when dislocations collide with opposite topological charges S = ±1 at given voltage, their speed increases. It has been shown that dynamics and interaction of dislocations with topological charges S = ±1 are qualitatively well described by the perturbed sine-Gordon equation.