scholarly journals Longitudinal shock waves in a class of semi-infinite stretch-limited elastic strings

2021 ◽  
pp. 108128652110255
Author(s):  
Casey Rodriguez

In this paper, we initiate the study of wave propagation in a recently proposed mathematical model for stretch-limited elastic strings. We consider the longitudinal motion of a simple class of uniform, semi-infinite, stretch-limited strings under no external force with finite end held fixed and prescribed tension at the infinite end. We study a class of motions such that the string has one inextensible segment, where the local stretch is maximized, and one extensible segment. The equations of motion reduce to a simple and novel shock front problem in one spatial variable for which we prove existence and uniqueness of local-in-time solutions for appropriate initial data. We then prove the orbital asymptotic stability of an explicit two-parameter family of piece-wise constant stretched motions. If the prescribed tension at the infinite end is increasing in time, our results provide an open set of initial data launching motions resulting in the string becoming fully inextensible and tension blowing up in finite time.

Symmetry ◽  
2021 ◽  
Vol 13 (8) ◽  
pp. 1431
Author(s):  
Bilal Basti ◽  
Nacereddine Hammami ◽  
Imadeddine Berrabah ◽  
Farid Nouioua ◽  
Rabah Djemiat ◽  
...  

This paper discusses and provides some analytical studies for a modified fractional-order SIRD mathematical model of the COVID-19 epidemic in the sense of the Caputo–Katugampola fractional derivative that allows treating of the biological models of infectious diseases and unifies the Hadamard and Caputo fractional derivatives into a single form. By considering the vaccine parameter of the suspected population, we compute and derive several stability results based on some symmetrical parameters that satisfy some conditions that prevent the pandemic. The paper also investigates the problem of the existence and uniqueness of solutions for the modified SIRD model. It does so by applying the properties of Schauder’s and Banach’s fixed point theorems.


Author(s):  
Raphaël Danchin ◽  
Piotr Bogusław Mucha ◽  
Patrick Tolksdorf

AbstractWe are concerned with global-in-time existence and uniqueness results for models of pressureless gases that come up in the description of phenomena in astrophysics or collective behavior. The initial data are rough: in particular, the density is only bounded. Our results are based on interpolation and parabolic maximal regularity, where Lorentz spaces play a key role. We establish a novel maximal regularity estimate for parabolic systems in $$L_{q,r}(0,T;L_p(\Omega ))$$ L q , r ( 0 , T ; L p ( Ω ) ) spaces.


1998 ◽  
Vol 21 (2) ◽  
pp. 299-305 ◽  
Author(s):  
Fengxin Chen ◽  
Ping Wang ◽  
Chaoshun Qu

In this paper we study the system governing flows in the magnetic field within the earth. The system is similar to the magnetohydrodynamic (MHD) equations. For initial data in spaceLp, we obtained the local in time existence and uniqueness ofweak solutions of the system subject to appropriate initial and boundary conditions.


2020 ◽  
pp. 57-68
Author(s):  
V. A. Sidorov ◽  
E. V. Osowska

The article is devoted to the history of foundry production-casting of bells. This art was passed down from generation to generation, there were successful variants, but there were also defective bells. The authors analyzed in detail the different production of slotted bells. Bells are unique single products. They confirm the high level of skill of the master. The loss of sound properties of slotted bells depends on the ratio of the main dimensions, the size of the slits and their locations. The use of complementary methods for measuring mechanical and visualizing sound vibrations allows us to obtain initial data for the development of a mathematical model of sound vibrations of a bell. A continuation of the material about the art of manufacturing and the secrets of production will be published in the next issue of the magazine.


Author(s):  
Андрей Геннадьевич Деменков ◽  
Геннадий Георгиевич Черных

С применением математической модели, включающей осредненные уравнения движения и дифференциальные уравнения переноса нормальных рейнольдсовых напряжений и скорости диссипации, выполнено численное моделирование эволюции безымпульсного закрученного турбулентного следа с ненулевым моментом количества движения за телом вращения. Получено, что начиная с расстояний порядка 1000 диаметров от тела течение становится автомодельным. На основе анализа результатов численных экспериментов построены упрощенные математические модели дальнего следа. Swirling turbulent jet flows are of interest in connection with the design and development of various energy and chemical-technological devices as well as both study of flow around bodies and solving problems of environmental hydrodynamics, etc. An interesting example of such a flow is a swirling turbulent wake behind bodies of revolution. Analysis of the known works on the numerical simulation of swirling turbulent wakes behind bodies of revolution indicates lack of knowledge on the dynamics of the momentumless swirling turbulent wake. A special case of the motion of a body with a propulsor whose thrust compensates the swirl is studied, but there is a nonzero integral swirl in the flow. In previous works with the participation of the authors, a numerical simulation of the initial stage of the evolution of a swirling momentumless turbulent wake based on a hierarchy of second-order mathematical models was performed. It is shown that a satisfactory agreement of the results of calculations with the available experimental data is possible only with the use of a mathematical model that includes the averaged equations of motion and differential equations for the transfer of normal Reynolds stresses along the rate of dissipation. In the present work, based on the above mentioned mathematical model, a numerical simulation of the evolution of a far momentumless swirling turbulent wake with a nonzero angular momentum behind the body of revolution is performed. It is shown that starting from distances of the order of 1000 diameters from the body the flow becomes self-similar. Based on the analysis of the results of numerical experiments, simplified mathematical models of the far wake are constructed. The authors dedicate this work to the blessed memory of Vladimir Alekseevich Kostomakha.


2008 ◽  
Vol 06 (04) ◽  
pp. 413-428 ◽  
Author(s):  
HARVEY SEGUR

It is known that an "explosive instability" can occur when nonlinear waves propagate in certain media that admit 3-wave mixing. In that context, three resonantly interacting wavetrains all gain energy from a background source, and all blow up together, in finite time. A recent paper [17] showed that explosive instabilities can occur even in media that admit no 3-wave mixing. Instead, the instability is caused by 4-wave mixing, and results in four resonantly interacting wavetrains all blowing up in finite time. In both cases, the instability occurs in systems with no dissipation. This paper reviews the earlier work, and shows that adding a common form of dissipation to the system, with either 3-wave or 4-wave mixing, provides an effective threshold for blow-up. Only initial data that exceed the respective thresholds blow up in finite time.


Author(s):  
Darina Hroncová

Urgency of the research. Computer models mean new quality in the knowledge process. Using a computer model, the properties of the subject under investigation can be tested under different operating conditions. By experimenting with a com-puter model, we learn about the modelled object. We can test different machine variants without having to produce and edit prototypes. Target setting. The development of computer technology has expanded the possibility of solving mathematical models and allowed to gradually automate the calculation of mathematical model equations. It is necessary to insert appropriate inputs of the mathematical model and monitor and evaluate the output results through the computer output device The target was to describe the mathematical apparatus required for mathematical modeling and subsequently to compile a model for computer modeling. Actual scientific researches and issues analysis. When formulating a mathematical model for a computer, the laws and the theory we use are always valid under more or less idealized conditions, and operate with fictitious concepts such as, material point, ideal gas, intangible spring, and the like. However, with these simplifications, we describe a realistic phenomenon where the initial assumptions are only met to a certain extent. In order for the results not to be different from the modeled reality, it is to be assumed that a good computer model arises gradually, by verifying and modifying it, which is one of the advantages of MSC Adams. Uninvestigated parts of general matters defining. The question of building a real manipulator model. Based on the above simulation, it is possible to build a real model. The research objective. Using MSC Adams to simulate multiple body systems and verify its suitability for simulating ma-nipulator and robot models. In various versions of the assembled model we can monitor its behavior under different operating conditions. The statement of basic materials. In computer simulation, MSC Adams-View is used to simulate mechanical systems. It has an interactive environment for automated dynamic analysis of parameterized mechanical systems with an arbitrary struc-ture of rigid and flexible bodies with geometric or force joints, in which act gravity, inertia, experimentally designed contact, friction, aerodynamic, hydrodynamic or electromechanical forces and have integrated control, hydraulic, pneumatic or elec-tromechanical circuits. Conclusions. Working with a mathematical model on a computer opens space for specific synthesis of empirical and ana-lytical method of scientific knowledge. Working with the computer model carries the characteristic features of classical experi-mentation. It represents a qualitatively new way of solving tasks that can not be experimented with on a real object. The result is the equivalence of the computer model and the object being investigated with the features and expressions chosen as essen-tial, with accuracy sufficient to the exact purpose.


2021 ◽  
Vol 7 ◽  
pp. 19-27
Author(s):  
Ю.Л. Николаев ◽  
П.Н. Шкатов ◽  
Э.Ф. Ахметшина ◽  
А.А. Саморуков

Theoretical and experimental researches of vibration-induction transducer (VIT) outlet signal formed during exposure to normal leakage magnetic field intensity component Hn over the defective area were carried out. Theoretical research is based on an assumption that VIT signal is a trigonometric series that is limited by first five harmonics. As initial data for mathematical model creation, well-known conformities for Hn distribution over the defective area were used. Based on acquired mathematical model conformities of VIT signal harmonical composition permutation during its movement over the defective area with varying amplitudes and vibration frequency were found. Theoretical research results were proven experimentally. Moreover, additional possibilities of this way of magnetic testing are shown in comparison with conventional ones.


2021 ◽  
Vol 28 (3) ◽  
pp. 171-185
Author(s):  
Oleg Baturin ◽  
Paul Nikolalde ◽  
Grigorii Popov ◽  
Anastssia Korneeva ◽  
Ivan Kudryashov

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