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## Immersed boundary simulation of drop stability

The stability of the quasi-two-dimensional droplet flow is of great importance in microfluidic devices. We check the drop's stability in the square box using the immersed boundary and lattice Boltzmann methods. We implement two-dimensional equations within the immersed boundary approach in the Palabos programming platform. We check the influence

of the boundaries on the drop movement. We estimate fluctuations in the quantities while applying different initial conditions of the linear and angular velocities. We found that the level of fluctuations depends on the symmetrical displacement of drop at the initial state. The effect is connected with the hydrodynamic interaction of drop with the walls.

Lattice Boltzmann method is a mesoscopic method used for solving hydrodynamics problems of both incompressible and compressible fluids. Although the method is widely used, reliability of the results is unclear. Therefore, we use the method to solve a fundamental problem with a known analytical solution, the Couette flow. We estimate the accuracy of the simulation results obtained by setting different types of spatial grids, boundary conditions, and equilibrium distribution functions. However, the method imposes restrictions on a large number of simulation parameters such as Reynolds and Mach numbers. During simulation we discovered an unexpected behavior of the solution using classical lattice Boltzmann method. In these simulations we find that the conservation law is violated due to an unexpected inflow in the upper corners of the computational domain.

We discuss the applicability of multiphase lattice Boltzmann method for the simulation of the drop oscillation. We demonstrate that the simulation of the single drop excited to the first eigenmode does follow Rayleigh formula. Simulations show no sensitivity to the number of the discrete velocities with D3Q19 and D3Q27 representations of the distribution function in three dimensions. The boundaries do influent the motion of the drop—division of the computational area by the even and the odd number of cells comes out important and leads to symmetry violence. The second part of the chapter describes the oscillations of the ensemble of three drops due to the excitation of the central drop in the first eigenmode. The motion of the backdrops does strongly depend on the viscosity of the fluid. We provide future details of simulations.

We simulate the oscillation of the viscous drop in the viscous liquid. We combine methods of chromodynamics model and Shan-Chen pseudo-potential for the immiscible fluids. We measure the frequency of the first nontrivial eigenmode using the initial ellipsoid form of the drop. Drop oscillates about the equilibrium spherical form of radius $R$. Computed frequency as a function of the radius $R$ follows to the well known Rayleigh formula. We discuss the simulation setup in the framework of the Lattice Boltzmann method.

The classical problem of oscillations of liquid droplets is a good test for the applicability of computer simulation. We discuss the details of our approach to a simulation scheme based on the Boltzmann lattice equation. We show the results of modeling induced vibrations in a chain of three drops in a closed tube. In the initial position, the central drop has formed as an ellipsoid, out of the spherical equilibrium form. The excitation of vibrations in the left and right droplets depends on the viscosity of the surrounding fluid and the surface tension. Droplets are moving out of the initial position as well. We discuss the limits of the applicability of our model for the study of such a problem. We will also show the dynamics of the simulated process.

The classical problem of oscillations of liquid droplets is a good test for the applicability of computer simulation. We discuss the details of our approach to a simulation scheme based on the Boltzmann lattice equation. We show the results of modeling induced vibrations in a chain of three drops in a closed tube. In the initial position, the central drop has formed as an ellipsoid, out of the spherical equilibrium form. The excitation of vibrations in the left and right droplets depends on the viscosity of the surrounding fluid and the surface tension. Droplets are moving out of the initial position as well. We discuss the limits of the applicability of our model for the study of such a problem. We will also show the dynamics of the simulated process.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

Event logs collected by modern information and technical systems usually contain enough data for automated process models discovery. A variety of algorithms was developed for process models discovery, conformance checking, log to model alignment, comparison of process models, etc., nevertheless a quick analysis of ad-hoc selected parts of a journal still have not get a full-fledged implementation. This paper describes an ROLAP-based method of multidimensional event logs storage for process mining. The result of the analysis of the journal is visualized as directed graph representing the union of all possible event sequences, ranked by their occurrence probability. Our implementation allows the analyst to discover process models for sublogs defined by ad-hoc selection of criteria and value of occurrence probability

The dynamics of a two-component Davydov-Scott (DS) soliton with a small mismatch of the initial location or velocity of the high-frequency (HF) component was investigated within the framework of the Zakharov-type system of two coupled equations for the HF and low-frequency (LF) fields. In this system, the HF field is described by the linear Schrödinger equation with the potential generated by the LF component varying in time and space. The LF component in this system is described by the Korteweg-de Vries equation with a term of quadratic influence of the HF field on the LF field. The frequency of the DS soliton`s component oscillation was found analytically using the balance equation. The perturbed DS soliton was shown to be stable. The analytical results were confirmed by numerical simulations.