DYNAMICS OF A SELF-GRAVITATING MAGNETIZED SOURCE

2007 ◽  
Vol 16 (02n03) ◽  
pp. 481-487 ◽  
Author(s):  
A. ULACIA REY ◽  
A. PÉREZ MARTÍNEZ ◽  
ROBERTO A. SUSSMAN
Keyword(s):  
Autonomous System ◽  
Initial Conditions ◽  
Numerical Study ◽  
Asymptotic State ◽  
Field Equations ◽  
Matter Source ◽  
Bianchi I ◽  
Stable Attractor ◽  
Linear Autonomous System ◽  
Free Parameters

We consider a magnetized degenerate gas of fermions as the matter source of a homogeneous but anisotropic Bianchi I space–time with a Kasner-like metric. We examine the dynamics of this system by means of a qualitative and numerical study of Einstein–Maxwell field equations which reduce to a non-linear autonomous system. For all initial conditions and combinations of free parameters, the gas evolves from an initial anisotropic singularity into an asymptotic state that is completely determined by a stable attractor. Depending on the initial conditions the anisotropic singularity is of the "cigar" or "plate" types.

Numerical Study of an Impact Oscillator

1995 ◽  
Author(s):  
Kannan Marudachalam ◽  
Faruk H. Bursal
Keyword(s):  
Dynamical Systems ◽  
Numerical Algorithm ◽  
Initial Conditions ◽  
Numerical Study ◽  
Harmonic Excitation ◽  
General Purpose ◽  
Constrained Systems ◽  
Global Dynamics ◽  
Impact Oscillator ◽  
Stick Slip

Abstract Systems with discontinuous dynamics can be found in diverse disciplines. Meshing gears with backlash, impact dampers, relative motion of components that exhibit stick-slip phenomena axe but a few examples from mechanical systems. These form a class of dynamical systems where the nonlinearity is so severe that analysis becomes formidable, especially when global behavior needs to be known. Only recently have researchers attempted to investigate such systems in terms of modern dynamical systems theory. In this work, an impact oscillator with two-sided rigid constraints is used as a paradigm for studying the characteristics of discontinuous dynamical systems. The oscillator has zero stiffness and is subjected to harmonic excitation. The system is linear without impacts. However, the impacts introduce nonlinearity and dissipation (assuming inelastic impacts). A numerical algorithm is developed for studying the global dynamics of the system. A peculiar type of solution in which the trajectories in phase space from a certain set of initial conditions merge in finite time, making the dynamics non-invertible, is investigated. Also, the effect of “grazing,” a behavior common to constrained systems, on the dynamics of the system is studied. Based on the experience gained in studying this system, the need for an efficient general-purpose numerical algorithm for solving discontinuous dynamical systems is motivated. Investigation of stress, vibration, wear, noise, etc. that are associated with impact phenomena can benefit greatly from such an algorithm.

GENERATING 3-D SCROLL GRID ATTRACTORS OF FRACTIONAL DIFFERENTIAL SYSTEMS VIA STAIR FUNCTION

2007 ◽  
Vol 17 (11) ◽  
pp. 3965-3983 ◽  
Author(s):  
WEIHUA DENG
Keyword(s):  
Differential System ◽  
Autonomous System ◽  
Function Approach ◽  
Differential Systems ◽  
Fractional Differential Systems ◽  
Fractional Differential System ◽  
Linear Autonomous System ◽  
Fractional Differential ◽  
The One ◽  
Number Of Equilibria

This paper discusses the stair function approach for the generation of scroll grid attractors of fractional differential systems. The one-directional (1-D) n-grid scroll, two-directional (2-D) (n × m)-grid scroll and three-directional (3-D) (n × m × l)-grid scroll attractors are created from a fractional linear autonomous system with a simple stair function controller. Being similar to the scroll grid attractors of classical differential systems, the scrolls of 1-D n-grid scroll, 2-D (n × m)-grid scroll and 3-D (n × m × l)-grid scroll attractors are located around the equilibria of fractional differential system on a line, on a plane or in 3D, respectively and the number of scrolls is equal to the corresponding number of equilibria.

scholarly journals Laminar Burning Velocities of Hydrogen-Blended Methane–Air and Natural Gas–Air Mixtures, Calculated from the Early Stage of p(t) Records in a Spherical Vessel

Energies ◽  
2021 ◽  
Vol 14 (22) ◽  
pp. 7556
Author(s):  
Maria Mitu ◽  
Domnina Razus ◽  
Volkmar Schroeder
Keyword(s):  
Natural Gas ◽  
Initial Conditions ◽  
Numerical Study ◽  
Early Stage ◽  
Linear Trend ◽  
Burning Velocity ◽  
Pressure Rise ◽  
Spherical Vessel ◽  
Practical Applications ◽  
Hydrogen Addition

The flammable hydrogen-blended methane–air and natural gas–air mixtures raise specific safety and environmental issues in the industry and transportation; therefore, their explosion characteristics such as the explosion limits, explosion pressures, and rates of pressure rise have significant importance from a safety point of view. At the same time, the laminar burning velocities are the most useful parameters for practical applications and in basic studies for the validation of reaction mechanisms and modeling turbulent combustion. In the present study, an experimental and numerical study of the effect of hydrogen addition on the laminar burning velocity (LBV) of methane–air and natural gas–air mixtures was conducted, using mixtures with equivalence ratios within 0.90 and 1.30 and various hydrogen fractions rH within 0.0 and 0.5. The experiments were performed in a 14 L spherical vessel with central ignition at ambient initial conditions. The LBVs were calculated from p(t) data, determined in accordance with EN 15967, by using only the early stage of flame propagation. The results show that hydrogen addition determines an increase in LBV for all examined binary flammable mixtures. The LBV variation versus the fraction of added hydrogen, rH, follows a linear trend only at moderate hydrogen fractions. The further increase in rH results in a stronger variation in LBV, as shown by both experimental and computed LBVs. Hydrogen addition significantly changes the thermal diffusivity of flammable CH4–air or NG–air mixtures, the rate of heat release, and the concentration of active radical species in the flame front and contribute, thus, to LBV variation.

Evolution of anisotropic cosmic models in f(R,ϕ) gravity

2018 ◽  
Vol 27 (12) ◽  
pp. 1850115 ◽  
Author(s):  
M. Zubair ◽  
Farzana Kousar ◽  
Saira Waheed
Keyword(s):  
Scalar Field ◽  
Power Law ◽  
Bianchi Type ◽  
Scalar Potential ◽  
Anisotropic Fluid ◽  
Type I ◽  
Field Equations ◽  
Eos Parameter ◽  
Free Parameters ◽  
Physical Quantities

In this paper, we will discuss cosmological models using Bianchi type I for anisotropic fluid in [Formula: see text] theory of gravity which involves scalar potential. For this purpose, we consider power law assumptions of coupling function and scalar field along with the proportionality condition of expansion and shear scalars. We choose two [Formula: see text] models and obtain exact solutions of field equations in both cases. For these constructed models, the behavior of different physical quantities like EoS parameter, self-interacting potential as well as deceleration and skewness parameters are explored and illustrated graphically for the feasible ranges of free parameters. It is concluded that anisotropic fluid approaches isotropy in later cosmic times for both models.

scholarly journals New Results on the Motions of Asteroids in Resonances

1992 ◽  
Vol 152 ◽  
pp. 145-152 ◽  
Author(s):  
R. Dvorak
Keyword(s):  
Initial Conditions ◽  
Numerical Study ◽  
Equations Of Motion ◽  
Close Approach ◽  
Integration Time ◽  
Time Interval ◽  
Time Intervals ◽  
3 Dimensional ◽  
Special Cases ◽  
Good Agreement

In this article we present a numerical study of the motion of asteroids in the 2:1 and 3:1 resonance with Jupiter. We integrated the equations of motion of the elliptic restricted 3-body problem for a great number of initial conditions within this 2 resonances for a time interval of 104 periods and for special cases even longer (which corresponds in the the Sun-Jupiter system to time intervals up to 106 years). We present our results in the form of 3-dimensional diagrams (initial a versus initial e, and in the z-axes the highest value of the eccentricity during the whole integration time). In the 3:1 resonance an eccentricity higher than 0.3 can lead to a close approach to Mars and hence to an escape from the resonance. Asteroids in the 2:1 resonance with Jupiter with eccentricities higher than 0.5 suffer from possible close approaches to Jupiter itself and then again this leads in general to an escape from the resonance. In both resonances we found possible regions of escape (chaotic regions), but only for initial eccentricities e > 0.15. The comparison with recent results show quite a good agreement for the structure of the 3:1 resonance. For motions in the 2:1 resonance our numeric results are in contradiction to others: high eccentric orbits are also found which may lead to escapes and consequently to a depletion of this resonant regions.

A numerical study of the nonlinear fractional mathematical model of tumor cells in presence of chemotherapeutic treatment

2020 ◽  
Vol 13 (03) ◽  
pp. 2050021 ◽  
Author(s):  
Sachin Kumar ◽  
Abdon Atangana
Keyword(s):  
Tumor Cells ◽  
Immune Cells ◽  
Initial Conditions ◽  
Numerical Study ◽  
Operational Matrix ◽  
Medical Science ◽  
Algebraic Equations ◽  
Normal Cells ◽  
Operational Matrix Method ◽  
Chemotherapeutic Treatment

Cancer belongs to the class of diseases which is symbolized by out of control cells growth. These cells affect DNAs and damage them. There exist many treatments available in medical science as radiation therapy, targeted therapy, surgery, palliative care and chemotherapy. Chemotherapy is one of the most popular treatments which depends on the type, location and grade of cancer. In this paper, we are working on modeling and prediction of the effect of chemotherapy on cancer cells using a fractional differential equation by using the differential operator in Caputo’s sense. The presented model depicts the interaction between tumor, normal and immune cells in a tumor by using a system of four coupled fractional partial differential equations (PDEs). For this system, initial conditions of tumor cells and dimensions are taken in such a way that tumor is spread out enough in size and can be detected easily with the clinical machines. An operational matrix method with Genocchi polynomials is applied to study this system of fractional PDEs (FPDEs). An operational matrix for fractional differentiation is derived. Applying the collocation method and using this matrix, the nonlinear system is reduced to a system of algebraic equations, which can be solved using Newton iteration method. The salient features of this paper are the pictorial presentations of the numerical solution of the concerned equation for different particular cases to show the effect of fractional exponent on diffusive nature of immune cells, tumor cells, normal cells and chemotherapeutic drug and depict the interaction among immune cells, normal cells and tumor cells in a tumor site.

An Investigation Into Nonlinear Growth Rate of Two-Dimensional and Three-Dimensional Single-Mode Richtmyer–Meshkov Instability Using an Arbitrary-Lagrangian–Eulerian Algorithm

2014 ◽  
Vol 136 (9) ◽  
Author(s):  
Mike Probyn ◽  
Ben Thornber ◽  
Dimitris Drikakis ◽  
David Youngs ◽  
Robin Williams
Keyword(s):  
Growth Rate ◽  
Numerical Scheme ◽  
Initial Conditions ◽  
Numerical Study ◽  
Three Dimensional ◽  
Single Mode ◽  
Computational Time ◽  
Initial Growth ◽  
Arbitrary Lagrangian Eulerian ◽  
Layer Width

This paper presents an investigation into the use of a moving mesh algorithm for solving unsteady turbulent mixing problems. The growth of a shock induced mixing zone following reshock, using an initial setup comparable to that of existing experimental work, is used to evaluate the behavior of the numerical scheme for single-mode Richtmyer–Meshkov instability (SM-RMI). Subsequently the code is used to evaluate the growth rate for a range of different initial conditions. The initial growth rate for three-dimensional (3D) SM Richtmyer–Meshkov is also presented for a number of different initial conditions. This numerical study details the development of the mixing layer width both prior to and after reshock. The numerical scheme used includes an arbitrary Lagrangian–Eulerian grid motion which is successfully used to reduce the mesh size and computational time while retaining the accuracy of the simulation results. Varying initial conditions shows that the growth rate after reshock is independent of the initial conditions for a SM provided that the initial growth remains in the linear regime.

scholarly journals Numerical study of pressure and velocity fluctuations in nearly isotropic turbulence

1978 ◽  
Vol 88 (4) ◽  
pp. 685-709 ◽  
Author(s):  
U. Schumann ◽  
G. S. Patterson
Keyword(s):  
Isotropic Turbulence ◽  
Initial Conditions ◽  
Numerical Study ◽  
Pressure Fluctuations ◽  
Reynolds Numbers ◽  
Companion Paper ◽  
Real Space ◽  
Pressure Gradients ◽  
Velocity Statistics ◽  
Decaying Turbulence

The spectral method of Orszag & Patterson has been extended to calculate the static pressure fluctuations in incompressible homogeneous decaying turbulence at Reynolds numbers Reλ [lsim ] 35. In real space 323 points are treated. Several cases starting from different isotropic initial conditions have been studied. Some departure from isotropy exists owing to the small number of modes at small wavenumbers. Root-mean-square pressure fluctuations, pressure gradients and integral length scales have been evaluated. The results agree rather well with predictions based on velocity statistics and on the assumption of normality. The normality assumption has been tested extensively for the simulated fields and found to be approximately valid as far as fourth-order velocity correlations are concerned. In addition, a model for the dissipation tensor has been proposed. The application of the present method to the study of the return of axisymmetric turbulence to isotropy is described in the companion paper.

Numerical Study of Superharmonic Resonances in a Mistuned Horizontal-Axis Wind-Turbine Blade-Rotor Set

2019 ◽  
Author(s):  
Ayse Sapmaz ◽  
Gizem D. Acar ◽  
Brian F. Feeny
Keyword(s):  
Wind Turbine ◽  
Initial Conditions ◽  
Numerical Study ◽  
Horizontal Axis ◽  
Rotor Speed ◽  
Horizontal Axis Wind Turbine ◽  
First Order ◽  
Direct Excitation ◽  
Rotor Angle ◽  
Perturbation Solutions

Abstract This paper is on a simplified model of an in-plane blade-hub dynamics of a horizontal-axis wind turbine with a mistuned blade. The model has cyclic parametric and direct excitation due to gravity and aerodynamics. This work follows up a previous perturbation study applied to the blade equations written in the rotor-angle domain and decoupled from the hub, in which superharmonic and primary resonances were analyzed. In this work, the effects of mistuning, damping, and forcing level are illustrated. The first-order perturbation solutions are verified with comparisons to numerical simulations at superharmonic resonance of order two. Additionally, the effect of rotor loading on the rotor speed and blade amplitudes is investigated for different initial conditions and mistuning cases.

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