Approximate methods for calculating the impact of a massive body on a plate lying on the base

Normal impact of a massive body on a uniformly stretched plate lying on the base is investigated. A hinged round or rectangular plate on an elastic base, or an infinite plate on the surface of an ideal incompressible fluid is considered. The solution to the elastoplastic impact is in good agreement with numerical calculations and experimental data. With a small parameter of elastic collapse, that is, with the developed local plastic deformations, a solution to the problem of impact with rigid-plastic local collapse can be used. Approximate formulas for calculating the main characteristics of rigid-plastic impact are set up.

Mathematical modelling of the selective deep water intake process in a non-prismatic continuously stratified reservoir

Разработана математическая модель селективного водозаборного процесса в узко-глубоком непризматическом водоеме при наличии в нем прямой непрерывной плотностной стратификации. Модель представляет контактную краевую задачу потенциального движения воды в указанном водоеме. Движение воды обусловлено забором воды через два окна, устроенных одно над другим на напорной грани водоема. Учтена непризматическая конфигурация водоема в плане и по вертикали. Непризматические конфигурации описаны экспоненциальными функциями. В результате аналитического решения поставленной контактной краевой задачи получена совокупность расчетных формул, которая с привлечением конечноразностного метода Рунге-Кутты и компьютерных вычислительных систем позволила построить линии тока, приходящие к верхней кромке нижнего водозаборного окна. Вычислительные эксперименты показали, что по мере увеличения скорости потока воды через верхнее окно указанные линии тока опускаются вниз. При этом становится возможным управлять водозаборным процессом через нижнее окно с тем расчетом, чтобы в нижнее окно вода поступала из нижних холодных слоев водоема, что часто бывает необходимо для нужд теплоотвода от тепловыделяющих устройств предприятий, в том числе тепловых и атомных электростанций. Purpose. Mathematical simulation of the selective water intake process in a non-prismatic reservoir in the presence of continuous density stratification. Methodology. Water intake is carried out through two windows arranged one above the other on the pressure face of the reservoir. The non-prismatic configuration of the reservoir both in vertical and horizontal planes is taken into account. The contact initial-boundary value problem of the theory for surface and internal gravitational waves in an ideal incompressible fluid is used. Findings. As a result of the analytical solution of the mathematical model, a set of calculation formulas was obtained, which allows calculating the current lines coming to the upper edge of the lower water intake window. Originality/value. Authors obtain a set of formulas for the components of the water velocity vector. Using the set, a nonlinear boundary value problem is posed and solved for calculation of the current line coming to the upper edge of the lower water intake window by the finite-difference Runge-Kutta method. Based on the results of computational experiments, authors proved that the longitudinal and vertical non-prismatic configuration of a stratified reservoir significantly affects the process of selective water intake. It is proved that as the rate of water intake through the upper window increases, the thickness of the active layer from which water is taken through the lower window decreases to a certain minimum value. This hydrodynamic effect allows taking water from the deep cold layers of the reservoir

Bound Coherent Structures Propagating on the Free Surface of Deep Water

This article presents a study of bound periodically oscillating coherent structures arising on the free surface of deep water. Such structures resemble the well known bi-soliton solution of the nonlinear Schrödinger equation. The research was carried out in the super-compact Dyachenko-Zakharov equation model for unidirectional deep water waves and the full system of nonlinear equations for potential flows of an ideal incompressible fluid written in conformal variables. The special numerical algorithm that includes a damping procedure of radiation and velocity adjusting was used for obtaining such bound structures. The results showed that in both nonlinear models for deep water waves after the damping is turned off, a periodically oscillating bound structure remains on the fluid surface and propagates stably over hundreds of thousands of characteristic wave periods without losing energy.

Bound Coherent Structures Propagating on the Free Surface of Deep Water

<p><span>               The work presents the results of studying the bound coherent structures propagating on the free surface of ideal incompressible fluid of infinite depth. Examples of such structures are bi-solitons which are exact solutions of the known approximate model for deep water waves — the nonlinear Schrödinger equation (NLSE). Recently, when studying multiple breathers collisions, the occurrence of such objects was found in a more accurate model of the supercompact equation for unidirectional water waves [1]. The aim of this work is obtaining and further studying such structures with different parameters in the supercompact equation and in the full system of nonlinear equations for potential flows of an ideal incompressible fluid written in conformal variables. </span><span>The algorithm used for finding the bound coherent objects was similar to the one described in [2]. As the initial conditions for obtaining such structures in the framework of the above models, the NLSE bi-soliton solutions were used, as well as two single breathers numerically found by the Petviashvili method and placed in a same point of the computational domain. During the evolution calculation the initial structures emitted incoherent waves which were filtered at the boundaries of the domain using the damping procedure. It is shown that after switching off the filtering of radiation, periodically oscillating coherent objects remain on the surface of the liquid, propagate stably during one hundred thousand characteristic wave periods and do not lose energy. The profiles of such structures at different parameters are compared.</span></p><p><span>This work was supported by RSF grant </span><span>19-72-30028</span><span> and </span><span>RFBR grant </span><span>20-31-90093</span><span>.</span></p><p><span>[1] Kachulin D., Dyachenko A., Dremov S. Multiple Soliton Interactions on the Surface of Deep Water //Fluids. – 2020. – Т. 5. – №. 2. – С. 65.</span></p><p><span>[2] Dyachenko A. I., Zakharov V. E. On the formation of freak waves on the surface of deep water //JETP letters. – 2008. – Т. 88. – №. 5. – С. 307.</span></p>

A New Solving Procedure for the Kelvin–Kirchhoff Equations in Case of a Falling Rotating Torus

We present a new solving procedure in this paper for Kelvin–Kirchhoff equations, considering the dynamics of a falling rigid rotating torus in an ideal incompressible fluid, assuming additionally the dynamical symmetry of rotation for the rotating body, [Formula: see text]. The fundamental law of angular momentum conservation is used for the aforementioned solving procedure. The system of Euler equations for the dynamics of torus rotation is explored for an analytic way of presentation of the approximated solution (where we consider the case of laminar flow at slow regime of torus rotation). The second finding is that the Stokes boundary layer phenomenon on the boundaries of the torus also assumed additionally at the formulation of basic Kelvin–Kirchhoff equations (for which the analytical expressions for the components of fluid’s torque vector [Formula: see text] were obtained earlier). The results for calculating the components of angular velocity [Formula: see text] should then be used for full solving the momentum equation of Kelvin–Kirchhoff system. The trajectories of motion can be divided into, preferably, three classes: zigzagging, helical spiral motion, and the chaotic regime of oscillations.

Влияние дискообразных непроводящих включений на электропроводность материала на постоянном токе

The influence of non-conductive disk-like inclusions (fractures) on the conductivity of a conducting medium in direct current is considered. To find the additional specific electrical resistance due to cracks, a self-consistent problem for the current density has been solved. In this case, a hydrodynamic analogy was used between the motion of an ideal incompressible fluid during potential flow around solids and electric current flowing around fractures. The functional dependences of the resulting relative additional specific electrical resistance of the material on the fracturing coefficient for thick and thin samples are found. A graphical visualization of these dependences shows that for a thick specimen, for all real values of the coefficient of fracturing, it is linear and is determined by the shape of the fractures. For a thin sample, the dependence of the relative additional the specific electrical resistance of the material on the fracturing coefficient is proportional to the thickness of the sample and is nonlinear. The obtained dependences of the additional electrical resistivity can be used, in particular, as a non-invasive method for determining the fracturing coefficient of rock samples conducting electric current.

Lagrangian and Dirac constraints for the ideal incompressible fluid and magnetohydrodynamics

The incompressibility constraint for fluid flow was imposed by Lagrange in the so-called Lagrangian variable description using his method of multipliers in the Lagrangian (variational) formulation. An alternative is the imposition of incompressibility in the Eulerian variable description by a generalization of Dirac’s constraint method using noncanonical Poisson brackets. Here it is shown how to impose the incompressibility constraint using Dirac’s method in terms of both the canonical Poisson brackets in the Lagrangian variable description and the noncanonical Poisson brackets in the Eulerian description, allowing for the advection of density. Both cases give the dynamics of infinite-dimensional geodesic flow on the group of volume preserving diffeomorphisms and explicit expressions for this dynamics in terms of the constraints and original variables is given. Because Lagrangian and Eulerian conservation laws are not identical, comparison of the various methods is made.

INVESTIGATION OF THE PROBLEMS OF LATERAL OUT LOAD OF LIQUIDS

The paper studies the problem of the flow of an ideal incompressible fluid in the presence of outflow to the side channels

The Elastoplastic Impact Process Study of the Indenter and the Plate Lying on the Base

The massive body impact on a uniformly stretched pivotally supported rectangular plate on an elastic Winkler base and on an endless plate lying on the surface of an ideal incompressible fluid is investigated. The asymptotic behavior of the Green influence function is used. The elastoplastic impact solution is in good agreement with numerical calculations and experimental data. The results to study the material properties are used when an indenter impacts ice.