How strain and spin may make a star bi-polar

2014 ◽  
Vol 746 ◽  
pp. 332-367 ◽  
Author(s):  
Lawrence K. Forbes
Keyword(s):  
Point Source ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Nonlinear Effects ◽  
Solution Technique ◽  
Ambient Fluid ◽  
Apparent Contradiction ◽  
Spectral Solution ◽  
Linearized Theory ◽  
The One

AbstractA previous study by Forbes (ANZIAM J., vol. 53, 2011, pp. 87–121) has argued that, when a light fluid is injected from a point source into a heavier ambient fluid, the interface between them is most unstable to perturbations at the lowest spherical mode. This means that, regardless of initial conditions, the outflow from a point source eventually becomes a one-sided jet. However, two-sided (bi-polar) outflows are nevertheless often observed in astrophysics, in apparent contradiction to this prediction. While there are many possible explanations for this fact, the present paper considers the effect of a straining flow in the ambient fluid. In addition, solid-body rotation in the inner fluid is also accounted for, in a Boussinesq viscous model. It is shown analytically that there are circumstances under which straining flow alone is sufficient to convert the one-sided jet into a genuine bi-polar outflow, in linearized theory. This is confirmed in a numerical solution of a viscous model of the flow, based on a spectral solution technique that accounts for nonlinear effects. Rotation can also generate flows that are two-sided, and this is likewise revealed through an asymptotic analysis and numerical solutions of the nonlinear equations.

scholarly journals Numerical Solutions of Odd Order Linear and Nonlinear Initial Value Problems Using a Shifted Jacobi Spectral Approximations

2012 ◽  
Vol 2012 ◽  
pp. 1-25 ◽  
Author(s):  
A. H. Bhrawy ◽  
M. A. Alghamdi
Keyword(s):  
Galerkin Method ◽  
Initial Value Problems ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Variable Coefficients ◽  
Solution Technique ◽  
Initial Value ◽  
Spectral Approximations ◽  
Base Functions ◽  
And Performance

A shifted Jacobi Galerkin method is introduced to get a direct solution technique for solving the third- and fifth-order differential equations with constant coefficients subject to initial conditions. The key to the efficiency of these algorithms is to construct appropriate base functions, which lead to systems with specially structured matrices that can be efficiently inverted. A quadrature Galerkin method is introduced for the numerical solution of these problems with variable coefficients. A new shifted Jacobi collocation method based on basis functions satisfying the initial conditions is presented for solving nonlinear initial value problems. Through several numerical examples, we evaluate the accuracy and performance of the proposed algorithms. The algorithms are easy to implement and yield very accurate results.

On the formation of Moore curvature singularities in vortex sheets

1999 ◽  
Vol 378 ◽  
pp. 233-267 ◽  
Author(s):  
STEPHEN J. COWLEY ◽  
GREG R. BAKER ◽  
SALEH TANVEER
Keyword(s):  
Complex Plane ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Nonlinear Effects ◽  
Vortex Sheet ◽  
Finite Amplitude ◽  
Analytical Continuation ◽  
Curvature Singularity ◽  
Singularity Formation ◽  
Vortex Sheets

Moore (1979) demonstrated that the cumulative influence of small nonlinear effects on the evolution of a slightly perturbed vortex sheet is such that a curvature singularity can develop at a large, but finite, time. By means of an analytical continuation of the problem into the complex spatial plane, we find a consistent asymptotic solution to the problem posed by Moore. Our solution includes the shape of the vortex sheet as the curvature singularity forms. Analytic results are confirmed by comparison with numerical solutions. Further, for a wide class of initial conditions (including perturbations of finite amplitude), we demonstrate that 3/2-power singularities can spontaneously form at t=0+ in the complex plane. We show that these singularities propagate around the complex plane. If two singularities collide on the real axis, then a point of infinite curvature develops on the vortex sheet. For such an occurrence we give an asymptotic description of the vortex-sheet shape at times close to singularity formation.

scholarly journals Numerical Modelling on the Influence of Source in the Heat Transformation: An Application in the Metal Heating for Blacksmithing

2021 ◽  
Vol 7 (2) ◽  
pp. 97-101
Author(s):  
H. P. Kandel ◽  
J. Kafle ◽  
L. P. Bagale
Keyword(s):  
Numerical Methods ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Numerical Technique ◽  
Heat Equations ◽  
Science And Engineering ◽  
Physical Problems ◽  
Different Temperatures ◽  
The One ◽  
Heat Transformation

Many physical problems, such as heat transfer and wave transfer, are modeled in the real world using partial differential equations (PDEs). When the domain of such modeled problems is irregular in shape, computing analytic solution becomes difficult, if not impossible. In such a case, numerical methods can be used to compute the solution of such PDEs. The Finite difference method (FDM) is one of the numerical methods used to compute the solutions of PDEs by discretizing the domain into a finite number of regions. We used FDMs to compute the numerical solutions of the one dimensional heat equation with different position initial conditions and multiple initial conditions. Blacksmiths fashioned different metals into the desired shape by heating the objects with different temperatures and at different position. The numerical technique applied here can be used to solve heat equations observed in the field of science and engineering.

Nonlinear wave packets in the Kelvin-Helmholtz instability

1979 ◽  
Vol 290 (1377) ◽  
pp. 639-681 ◽  
Keyword(s):  
Numerical Solutions ◽  
Initial Conditions ◽  
Density Ratio ◽  
Nonlinear Effects ◽  
Amplitude Equation ◽  
Two Dimensions ◽  
Spatial Modulation ◽  
Linear Solution ◽  
Nonlinear Stabilization ◽  
Nonlinear Solutions

The instability of two layers of immiscible inviscid and incompressible fluids in relative motion is studied with allowance for small, but finite, disturbances and for spatial as well as temporal development. By using the method of multiple scaling, a generalized formulation of the amplitude equation is obtained, applicable to both stable and marginally unstable regions of parameter space. Of principal concern is the neighbourhood of the critical point for instability, where weakly nonlinear solutions can be found for arbitrary initial conditions. Among the analytical results, it is shown that (1) the nonlinear effects can be stabilizing or destabilizing depending on the density ratio, (2) the existence of purely spatial instability depends upon the frame of reference, the density ratio, and whether the nonlinear effects are stabilizing, (3) exact nonlinear solutions of the amplitude equation exist representing modulations of permanent form travelling faster than the signal velocity of the linear equation (in particular, a solution is found that represents a solitary wave packet), and (4) the linear solution to the impulsive initial value problem has 'fronts’ which travel with the two (multiple) values of the group velocity (the packet as a whole moves with the mean of the two values). Numerical solutions of the amplitude equation (a nonlinear, unstable Klein-Gordon equation) are also presented for the case of nonlinear stabilization. These show that the development of a localized disturbance, in one or two dimensions, is highly dependent on the precise form of the initial conditions, even when the initial amplitude is very small. The exact solutions mentioned above play an important role in this development. The numerical experiments also show that the familiar uniform solution, an oscillatory function of time only, is unstable to spatial modulation if the amplitude of oscillation is large enough.

scholarly journals Simulations of autonomous fluid pulses between active elastic walls using the 1D-IRBFN method

2018 ◽  
Vol 13 (5) ◽  
pp. 47 ◽  
Author(s):  
Fatima Z. Ahmed ◽  
Mayada G. Mohammed ◽  
Dmitry V. Strunin ◽  
Duc Ngo-Cong
Keyword(s):  
Numerical Solutions ◽  
Initial Conditions ◽  
Radial Basis Function Network ◽  
Active Motion ◽  
One Dimensional ◽  
Pulse Trains ◽  
The One ◽  
Semi Empirical ◽  
Function Network ◽  
Elastic Walls

We present numerical solutions of the semi-empirical model of self-propagating fluid pulses (auto-pulses) through the channel simulating an artificial artery. The key mechanism behind the model is the active motion of the walls in line with the earlier model of Roberts. Our model is autonomous, nonlinear and is based on the partial differential equation describing the displacement of the wall in time and along the channel. A theoretical plane configuration is adopted for the walls at rest. For solving the equation we used the One-dimensional Integrated Radial Basis Function Network (1D-IRBFN) method. We demonstrated that different initial conditions always lead to the settling of pulse trains where an individual pulse has certain speed and amplitude controlled by the governing equation. A variety of pulse solutions is obtained using homogeneous and periodic boundary conditions. The dynamics of one, two, and three pulses per period are explored. The fluid mass flux due to the pulses is calculated.

scholarly journals Two Efficient Generalized Laguerre Spectral Algorithms for Fractional Initial Value Problems

2013 ◽  
Vol 2013 ◽  
pp. 1-10 ◽  
Author(s):  
D. Baleanu ◽  
A. H. Bhrawy ◽  
T. M. Taha
Keyword(s):  
Initial Conditions ◽  
Laguerre Polynomials ◽  
Operational Matrix ◽  
Infinite Interval ◽  
Solution Technique ◽  
Generalized Laguerre Polynomials ◽  
Spectral Solution ◽  
Spectral Algorithms ◽  
And Performance ◽  
Pseudo Spectral

We present a direct solution technique for approximating linear multiterm fractional differential equations (FDEs) on semi-infinite interval, using generalized Laguerre polynomials. We derive the operational matrix of Caputo fractional derivative of the generalized Laguerre polynomials which is applied together with generalized Laguerre tau approximation for implementing a spectral solution of linear multiterm FDEs on semi-infinite interval subject to initial conditions. The generalized Laguerre pseudo-spectral approximation based on the generalized Laguerre operational matrix is investigated to reduce the nonlinear multiterm FDEs and its initial conditions to nonlinear algebraic system, thus greatly simplifying the problem. Through several numerical examples, we confirm the accuracy and performance of the proposed spectral algorithms. Indeed, the methods yield accurate results, and the exact solutions are achieved for some tested problems.

scholarly journals Numerical Counterexamples of Lorenz System in Implicit Time Scheme

2018 ◽  
Author(s):  
Xuan Chen
Keyword(s):  
Lorenz System ◽  
Numerical Solutions ◽  
Initial Conditions ◽  
Explicit Scheme ◽  
Implicit Time ◽  
Lorenz Equations ◽  
Consistent System ◽  
Runge Kutta Method ◽  
Self Consistent ◽  
The One

In nonlinear self-consistent system, Lorenz system (Lorenz equations) is a classic case with chaos solutions which are sensitively dependent on the initial conditions. As it is difficult to get the analytical solution, the numerical methods and qualitative analytical methods are widely used in many studies. In these papers, Runge-Kutta method is the one most often used to solve these differential equations. However, this method is still a method based on explicit time scheme, which would be the main reason for the chaotic solutions to Lorenz system. In this work, numerical experiments based on implicit time scheme and explicit scheme are setup for comparison, the results show that: in implicit time scheme, the numerical solutions (counterexamples) are without chaos; for an original volume, the volume shrinks exponentially fast to 0 in common.

Sources of Nitrogen and Phosphorus in the Catchment Area of Neusiedlersee/Fertö, Austria/Hungary

1993 ◽  
Vol 28 (3-5) ◽  
pp. 101-110 ◽  
Author(s):  
W. v. d. Emde ◽  
H. Fleckseder ◽  
N. Matsché ◽  
F. Plahl-Wabnegg ◽  
G. Spatzierer ◽  
...  
Keyword(s):  
Point Source ◽  
Catchment Area ◽  
Management Practice ◽  
Water Erosion ◽  
Nonpoint Source ◽  
Point Sources ◽  
Source Control ◽  
Nitrogen And Phosphorus ◽  
Catchment Areas ◽  
The One

Neusiedlersee (in German) / Fertö tó (in Hungarian) is a shallow lake at the Austro-Hungarian border. In the late 1970s, the question arose what to do in order to protect the lake against eutrophication. A preliminary report established the need for point-source control as well as gave first estimates for non-point source inputs. The proposed point-source control was quickly implemented, non-point sources were - among other topics - studied in detail in the period 1982 - 1986. The preliminary work had shown, based on integrated sampling and data from literature, that the aeolic input outweighed the one via water erosion (work was for totP only). In contrast to this, the 1982 - 1986 study showed that (a) water erosion by far dominates over aeolic inputs and (b) the size of nonpoint-source inputs was assessed for the largest catchment area in pronounced detail, whereas additional estimates were undertaken for smaller additional catchment areas. The methods as well as the results are presented in the following. The paper concludes with some remarks on the present management practice of nonpoint-source inputs.

Nonlinear Sloshing in Fixed and Vertically Excited Containers

2002 ◽  
Author(s):  
Jannette B. Frandsen ◽  
Alistair G. L. Borthwick
Keyword(s):  
Perturbation Theory ◽  
Free Surface ◽  
Small Perturbation ◽  
Numerical Solutions ◽  
Vertical Direction ◽  
Nonlinear Effects ◽  
Breaking Waves ◽  
Surface Flow ◽  
Nonlinear Behaviour ◽  
Computational Domain

Nonlinear effects of standing wave motions in fixed and vertically excited tanks are numerically investigated. The present fully nonlinear model analyses two-dimensional waves in stable and unstable regions of the free-surface flow. Numerical solutions of the governing nonlinear potential flow equations are obtained using a finite-difference time-stepping scheme on adaptively mapped grids. A σ-transformation in the vertical direction that stretches directly between the free-surface and bed boundary is applied to map the moving free surface physical domain onto a fixed computational domain. A horizontal linear mapping is also applied, so that the resulting computational domain is rectangular, and consists of unit square cells. The small-amplitude free-surface predictions in the fixed and vertically excited tanks compare well with 2nd order small perturbation theory. For stable steep waves in the vertically excited tank, the free-surface exhibits nonlinear behaviour. Parametric resonance is evident in the instability zones, as the amplitudes grow exponentially, even for small forcing amplitudes. For steep initial amplitudes the predictions differ considerably from the small perturbation theory solution, demonstrating the importance of nonlinear effects. The present numerical model provides a simple way of simulating steep non-breaking waves. It is computationally quick and accurate, and there is no need for free surface smoothing because of the σ-transformation.

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