The double-layer Stefan problem with degeneration of one of the phases at the initial moment and a discontinuity in the boundary and initial conditions

2016 ◽  
pp. 238-272
Keyword(s):  
Double Layer ◽  
Stefan Problem ◽  
Initial Conditions ◽  
Initial Moment

scholarly journals Mathematical modeling of temperature changes impact on artificial ice islands

2021 ◽  
Vol 13 (1) ◽  
pp. 79-86
Author(s):  
Maxim V. Muratov ◽  
◽  
Vladimir A. Biryukov ◽  
Denis S. Konov ◽  
Igor B. Petrov ◽  
...  
Keyword(s):  
Numerical Solution ◽  
Stefan Problem ◽  
Initial Conditions ◽  
Economic Feasibility ◽  
The Arctic ◽  
Seasonal Temperature ◽  
Time Step ◽  
Temperature Changes ◽  
The Stability ◽  
The Impact

The article is devoted to the numerical solution of the Stefan problem for thermal effects on an artificial ice island. For modern tasks of the development of the Arctic, associated with the exploration and production of minerals, it is important to create artificial ice islands in the Arctic shelf, due to the speed of their construction, economic feasibility and other factors. The most important task for the exploitation of such islands is their stability, including against melting. This paper discusses the issue of the stability of ice islands to melting. For this, the Stefan problem on the change in the phase state of matter is formulated. An enthalpy solution method is constructed, and the applicability of this method is considered. For the numerical solution, the Peasman-Reckford scheme is used, which is unconditionally spectrally stable in the two-dimensional case, which allows to freely choose the time step. In addition, the developed approach takes into account the flow of water and the flow of melted water, which is important in the task at hand. The developed computational algorithms are parallelized for use on modern multiprocessor computing systems An approach is implemented for modeling thermal processes in the thickness of an arbitrary mass of substances, taking into account arbitrary initial conditions, environmental conditions, tidal currents of water, and solar radiation. This approach was used to calculate the temperature distribution in the thickness of the ice island, as well as to study the impact of seasonal temperature changes on the stability of the island.

On regularization of initial conditions of the modified Stefan problem

1995 ◽  
Vol 57 (5) ◽  
pp. 559-561
Author(s):  
G. A. Omel'yanov ◽  
V. G. Danilov ◽  
E. V. Radkevich
Keyword(s):  
Stefan Problem ◽  
Initial Conditions

scholarly journals УСТАНОВЛЕНИЕ ОГРАНИЧЕНИЙ ПО МАКСИМАЛЬНЫМ СКОРОСТЯМ ДЕСАНТИРОВАНИЯ С УЧЁТОМ ДЕГРАДАЦИИ ПРОЧНОСТНЫХ ХАРАКТЕРИСТИК МАТЕРИАЛОВ КУПОЛА ПАРАШЮТА. СООБЩЕНИЕ 1. ЗАВИСИМОСТЬ ПРОЧНОСТИ ПАРАШЮТОВ ОТ ДЕГРАДАЦИИ ПРОЧНОСТНЫХ ХАРАКТЕРИСТИК КОНСТРУКЦИОННЫХ МАТЕРИАЛОВ

2019 ◽  
pp. 65-71
Author(s):  
Петр Александрович Фомичев
Keyword(s):  
Initial Conditions ◽  
Free Fall ◽  
The Body ◽  
Design Stage ◽  
Initial Moment ◽  
Safety Factors ◽  
Simplified Approach ◽  
Term Operation ◽  
Parachute Canopy

It is introduced the concept of the degradation coefficients of the parachute canopy power structure materials strength characteristics. These coefficients are defined as the ratio of destructive loads after long-term operation or storage to the original, taken at the design stage. It is noted revealed the dependence of safety factors on degradation factors. It was determined the condition of safety factors equality after the long-term operation and during the design stage. It was shown that the ratio of maximum permissible loads is equal to the degradation coefficient. It was defined as the method for calculating the load on the parachute during deployment. It was applied to the simplified approach proposed by N. A. Lobanov. The following statement was determined according to this approach: a dynamic coefficient equal to two, a method for determining the dangerous section of the dome when assessing the fabric strength, the dependence of the speed at the moment of full filling of the parachute canopy from the generalized empirical coefficient. Characteristics of the standard atmosphere, depending on the height of throwing the aircraft are given by approximating functions. The movement of the body until the parachute opens is given in the form of differential equations with known initial conditions. The equation allows you to find the falling speed at the initial moment of the parachute opening, depending on the delay time. It is given the speed of steady free fall without the introduction of a parachute into the work and with a stabilizing parachute, the landing speed with the main parachute. The dependences of the maximum permissible loads on the dome at the opening moment on the strength degradation factors for the fabric of the dome, lines, and free ends of the suspension system are established. It was proposed the correlations for the maximum allowable speed at the time of the beginning of the parachute opening on the requirements of strength. This speed determines the maximum allowed landing speed for a particular type of parachute after long-term operation or storage.

Application of nonstationary self-similar variables for solving the three-dimensional problem of the decay of a special discontinuity

2021 ◽  
pp. 52-64
Author(s):  
Сергей Петрович Баутин ◽  
Сергей Львович Дерябин
Keyword(s):  
Cauchy Problem ◽  
Power Series ◽  
Initial Conditions ◽  
Three Dimensional ◽  
Physical Space ◽  
Initial Moment ◽  
Algebraic Equations ◽  
Gas Dynamics Equations ◽  
Self Similar ◽  
Isentropic Flows

Построение в физическом пространстве решения задачи о распаде специального разрыва, т.е. трехмерных изэнтропических течений политропного газа, возникающих после мгновенного разрушения в начальный момент времени непроницаемой стенки, отделяющей неоднородный движущийся газ от вакуума. В задаче учитывается действие силы тяжести и силы Кориолиса. В систему уравнений газовой динамики введена автомодельная особенность в переменную, которая выводит с поверхности раздела. Для полученной системы поставлена задача Коши с данными на звуковой характеристике. Решение задачи строилось в виде степенных рядов. Часть коэффициентов рядов определялась при решении алгебраических уравнений, а часть из решений - обыкновенных дифференциальных уравнений. Методом мажорант доказана сходимость построенных рядов. Построенное решение позволяет задавать начальные условия для разностной схемы при численном моделировании решений данной характеристической задачи Коши The aim of this study is to construct a solution to the problem of the decay of a special discontinuity in physical space. The problem reduces to finding of three-dimensional isentropic flows of a polytropic gas that occur after the instantaneous destruction of an impermeable wall separating an inhomogeneous moving gas from a vacuum at the initial moment of time. The problem takes into account the forces of gravity and Coriolis. Research methods. In the system of gas dynamics equations, a self-similar feature is introduced in a variable that outputs from the initial interface. For the resulting system, the Cauchy problem is formulated using conditions on the sound characteristic. The solution to this problem is constructed in the form of power series. The coefficients of the series are partly determined by solving algebraic equations, another part can be found as solutions of ordinary differential equations. The convergence of the constructed series is proved by the Majorant method The results obtained in the work. In the form of a convergent power series, solutions to the problem of the decay of a special discontinuity in physical space are constructed. Conclusions. The solution constructed in physical space allows setting the initial conditions for the numerical simulation of this characteristic Cauchy problem using a difference scheme.

scholarly journals Numerical Modeling of Material Points Evolution in a System with Gravity

2017 ◽  
Vol 21 (4) ◽  
pp. 1118-1140 ◽  
Author(s):  
A. V. Melkikh ◽  
E. A. Melkikh ◽  
V. A. Kozhevnikov
Keyword(s):  
Numerical Modeling ◽  
Gravitational Force ◽  
Initial Conditions ◽  
Velocity Distribution Function ◽  
Initial Moment ◽  
Maximum Lyapunov Exponent ◽  
3D Space ◽  
Material Points ◽  
Initial Distributions ◽  
Kinetic And Potential Energies

AbstractThe evolution of material points interacting via gravitational force in 3D space was investigated. At initial moment points with masses of 2.48 Sun masses are randomly distributed inside a cube with an edge of 5 light-years. The modeling was conducted at different initial distributions of velocities and different ratios between potential and kinetic energy of the points. As a result of modeling the time dependence of velocity distribution function of points was obtained. Dependence of particles fraction which had evaporated frominitial cluster on time for different initial conditions is obtained. In particular, it was obtained that the fraction of evaporated particles varies between 0,45 and 0,63.Mutual diffusion of two classes of particles at different initial conditions in the case when at initial moment of time both classes of particles occupy equal parts of cube was investigated.The maximum Lyapunov exponent of the system with different initial conditions was calculated. The obtained value weakly depends on the ratio between initial kinetic and potential energies and amounts approximately 10–5. Corresponding time of the particle trajectories divergence turned out to be 40-50 thousand years.

scholarly journals UNIFORMLY VALID ASYMPTOTICS FOR CARRIER’S MATHEMATICAL MODEL OF STRING OSCILLATIONS

2017 ◽  
Vol 22 (3) ◽  
pp. 337-351
Author(s):  
Paulius Miškinis ◽  
Aleksandras Krylovas ◽  
Olga Lavcel-Budko
Keyword(s):  
Mathematical Model ◽  
Small Parameter ◽  
Asymptotic Approximation ◽  
Wave Equations ◽  
Initial Conditions ◽  
Resonant Interaction ◽  
Nonlinear Wave Equations ◽  
Time Interval ◽  
Initial Moment ◽  
Weakly Nonlinear

In the paper, an asymptotic analysis of G.F. Carrier’s mathematical model of string oscillation is presented. The model consists of a system of two nonlinear second order partial differential equations and periodic initial conditions. The longitudinal and transversal string oscillations are analyzed together when at the initial moment of time the system’s solutions have amplitudes proportional to a small parameter. The problem is reduced to a system of two weakly nonlinear wave equations. The resonant interaction of periodic waves is analyzed. An uniformly valid asymptotic approximation in the long time interval, which is inversely proportional to the small parameter, is constructed. This asymptotic approximation is a solution of averaged along characteristics integro-differential system. Conditions of appearance of combinatoric resonances in the system have been established. The results of numerical experiments are presented.

Rotating Incoherent Matter in General Relativity

1973 ◽  
Vol 51 (4) ◽  
pp. 477-490 ◽  
Author(s):  
J. Pachner
Keyword(s):  
External Field ◽  
Initial Conditions ◽  
Rotational Symmetry ◽  
Einstein Equations ◽  
Initial Moment ◽  
Physical Situation ◽  
Field Equations ◽  
Axial Acceleration ◽  
Indeterminate Form ◽  
Analytical Proof

Under the assumption of rotational symmetry a method is developed for the numerical integration of exact Einstein equations describing the time evolution of a rotating incoherent matter. The complete system of equations deduced earlier for the case of an ideal fluid reduces here to six Einstein field equations and to the equation of continuity. The choice of the system of comoving cylindrical coordinates (z, r, [Formula: see text], t) is restricted by further conditions so that the only allowed coordinate transformations are the translations of the origin of the coordinates z, [Formula: see text], and t, and the reflections in these three axes. To any set of the permissible initial values of the field corresponds then only one physical situation. The exact field equations are deduced. With the exception of a few terms they are invariant with respect to certain permutations. The requirements upon the field to be regular at the initial moment and its infinitesimal past and future imposes restrictions on the functions appearing in the components of the metric field. Even then the field equations contain terms that have an indeterminate form 0/0, but are finite, at the z axis. Therefore two sets of field equations must be used : one set for the z axis in which the indeterminacy is analytically evaluated, and the other set for the space with r > 0. The junction conditions for the external field are formulated and proved that the corresponding external field does exist. The problem of the Lichnerowicz initial conditions is solved. An analytical proof is given that the rotation can stop the contraction of incoherent matter and revert it to a new expansion even along the axis of rotation. In terrestrial physics the axial acceleration is negligibly small, but it may play an essential role in cosmology.

scholarly journals The Peculiarities of Charge Motion in the Molecular Polynucleotide Chains of Finite Length. The Rapid Formation of a Moving Polaron State

2018 ◽  
Vol 13 (2) ◽  
pp. 534-550 ◽  
Author(s):  
A.N. Korshunova ◽  
V.D. Lakhno
Keyword(s):  
Charge Transfer ◽  
Electric Field ◽  
Finite Length ◽  
Numerical Experiments ◽  
Initial Conditions ◽  
Initial Time ◽  
Initial Moment ◽  
Polaron State ◽  
Rapid Formation ◽  
Dna Chain

The numerical experiments which demonstrate the possibility of charge transfer in a homogeneous G/C DNA chain in the absence of an electric field have been carried out. As a model, which describes the dynamics of a DNA molecule, was considered the nonlinear Peyrard-Bishop-Dauxois-Holstein model. It is commonly supposed that the main electric current carrier in homogeneous synthetic polynucleotide chains is the polaron. We have previously studied the peculiarities of polaron motion in molecular polynucleotide chains of finite length. It was shown that a polaron placed at the initial moment of time not in the center of the chain acquires the ability to move in the absence of an electric field and in the absence of any additional excitations in the chain. The numerical experiments which demonstrate the possibility of polaron charge transfer in a homogeneous finite unclosed G/C DNA chain due to the interaction with localized excitations have been carried out in the absence of an electric field. In this study, at the initial moment of time, a polaron is not added to the chain, but a charge localized in the region of a certain number of neighboring sites displaced from the equilibrium positions. The motion of the charge in the chain is caused by choice of these specified initial conditions, which ensure the rapid formation of the polaron state and, as a consequence, charge transfer along the chain. For the assignment of the external nonlinear excitations, we used nonzero values of the displacements of particles and/or their velocities at the initial instant of time. Non-zero values of chain sites velocities at the initial time were used to stimulate the motion of the charge. It is shown that for the rapid formation of the polaron state, the initial conditions must correspond to the parameters of the polaron, which is formed in the chain under the chosen parameters. It is shown that, depending on the parameters of the chain and on the parameters of the selected initial conditions, the charge can be transferred along the chain over long distances.

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