scholarly journals Stability and solution of the time-dependent Bondi–Parker flow

2020 ◽  
Vol 493 (2) ◽  
pp. 2834-2840
Author(s):  
Eric Keto
Keyword(s):  
Steady State ◽  
Transonic Flow ◽  
Initial Conditions ◽  
Steady State Solution ◽  
State Solution ◽  
Time Dependent ◽  
Hamiltonian Description ◽  
Bernoulli’S Equation ◽  
Wide Range ◽  
Isothermal And Adiabatic Flows

ABSTRACT Bondi and Parker derived a steady-state solution for Bernoulli’s equation in spherical symmetry around a point mass for two cases, respectively, an inward accretion flow and an outward wind. Left unanswered were the stability of the steady-state solution, the solution itself of time-dependent flows, whether the time-dependent flows would evolve to the steady state, and under what conditions a transonic flow would develop. In a Hamiltonian description, we find that the steady-state solution is equivalent to the Lagrangian implying that time-dependent flows evolve to the steady state. We find that the second variation is definite in sign for isothermal and adiabatic flows, implying at least linear stability. We solve the partial differential equation for the time-dependent flow as an initial-value problem and find that a transonic flow develops under a wide range of realistic initial conditions. We present some examples of time-dependent solutions.

scholarly journals A second look at the Kurth solution in galactic dynamics

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Markus Kunze
Keyword(s):  
Steady State ◽  
Steady State Solution ◽  
Particle Energy ◽  
State Solution ◽  
Time Dependent ◽  
Galactic Dynamics ◽  
Poisson System ◽  
Second Look

<p style='text-indent:20px;'>The Kurth solution is a particular non-isotropic steady state solution to the gravitational Vlasov-Poisson system. It has the property that by means of a suitable time-dependent transformation it can be turned into a family of time-dependent solutions. Therefore, for a general steady state <inline-formula><tex-math id="M1">\begin{document}$ Q(x, v) = \tilde{Q}(e_Q, \beta) $\end{document}</tex-math></inline-formula>, depending upon the particle energy <inline-formula><tex-math id="M2">\begin{document}$ e_Q $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M3">\begin{document}$ \beta = \ell^2 = |x\wedge v|^2 $\end{document}</tex-math></inline-formula>, the question arises if solutions <inline-formula><tex-math id="M4">\begin{document}$ f $\end{document}</tex-math></inline-formula> could be generated that are of the form</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ f(t) = \tilde{Q}\Big(e_Q(R(t), P(t), B(t)), B(t)\Big) $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>for suitable functions <inline-formula><tex-math id="M5">\begin{document}$ R $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ P $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M7">\begin{document}$ B $\end{document}</tex-math></inline-formula>, all depending on <inline-formula><tex-math id="M8">\begin{document}$ (t, r, p_r, \beta) $\end{document}</tex-math></inline-formula> for <inline-formula><tex-math id="M9">\begin{document}$ r = |x| $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M10">\begin{document}$ p_r = \frac{x\cdot v}{|x|} $\end{document}</tex-math></inline-formula>. We are going to show that, under some mild assumptions, basically if <inline-formula><tex-math id="M11">\begin{document}$ R $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M12">\begin{document}$ P $\end{document}</tex-math></inline-formula> are independent of <inline-formula><tex-math id="M13">\begin{document}$ \beta $\end{document}</tex-math></inline-formula>, and if <inline-formula><tex-math id="M14">\begin{document}$ B = \beta $\end{document}</tex-math></inline-formula> is constant, then <inline-formula><tex-math id="M15">\begin{document}$ Q $\end{document}</tex-math></inline-formula> already has to be the Kurth solution.</p><p style='text-indent:20px;'>This paper is dedicated to the memory of Professor Robert Glassey.</p>

scholarly journals On Asymptotic Behavior of the Time-Dependent Solution of a Reliability Model

2014 ◽  
Vol 2 ◽  
pp. 1-11 ◽  
Author(s):  
Geni Gupur
Keyword(s):  
Asymptotic Behavior ◽  
Steady State ◽  
Steady State Solution ◽  
State Solution ◽  
Time Dependent ◽  
Reliability Model ◽  
Repair Rate ◽  
Lipschitz Continuous ◽  
Time Dependent Solution

On the basis of our previous work we study asymptotic behavior of the time-dependent solution of a reliability model of two identical units and a repairman and prove the following result: If the repair rate μ(x) is Lipschitz continuous and there exist two positive constants μ and μ such that 0 < μ ≤μ(x)≤ μ < ∞, then its time-dependent solution exponentially converges to its steady-state solution.

Compatibility and Uniqueness Analyses of Steady State Solution for Multi-variable Predictive Control Systems

2014 ◽  
Vol 39 (5) ◽  
pp. 519-529 ◽  
Author(s):  
Tao ZOU ◽  
Hai-Qiang LI ◽  
Bao-Cang DING ◽  
Ding-Ding WANG
Keyword(s):  
Steady State ◽  
Control Systems ◽  
Predictive Control ◽  
Steady State Solution ◽  
State Solution

Response of a Submerged Cylindrical Shell to an Axially Propagating Step Wave

1965 ◽  
Vol 32 (4) ◽  
pp. 788-792 ◽  
Author(s):  
M. J. Forrestal ◽  
G. Herrmann
Keyword(s):  
Cylindrical Shell ◽  
Steady State ◽  
Wave Front ◽  
Transverse Shear ◽  
Shell Theory ◽  
Steady State Solution ◽  
State Solution ◽  
Rotatory Inertia ◽  
Shear Deformations ◽  
Axial Coordinate

An infinitely long, circular, cylindrical shell is submerged in an acoustic medium and subjected to a plane, axially propagating step wave. The fluid-shell interaction is approximated by neglecting fluid motions in the axial direction, thereby assuming that cylindrical waves radiate away from the shell independently of the axial coordinate. Rotatory inertia and transverse shear deformations are included in the shell equations of motion, and a steady-state solution is obtained by combining the independent variables, time and the axial coordinate, through a transformation that measures the shell response from the advancing wave front. Results from the steady-state solution for the case of steel shells submerged in water are presented using both the Timoshenko-type shell theory and the bending shell theory. It is shown that previous solutions, which assumed plane waves radiated away from the vibrating shell, overestimated the dumping effect of the fluid, and that the inclusion of transverse shear deformations and rotatory inertia have an effect on the response ahead of the wave front.

scholarly journals Comparison of Solvers Performance for Load Flow Analysis

2019 ◽  
Vol 3 (1) ◽  
pp. 26 ◽  
Author(s):  
Vishnu Sidaarth Suresh
Keyword(s):  
Steady State ◽  
Power System ◽  
Reactive Power ◽  
Flow Analysis ◽  
Steady State Solution ◽  
Load Flow ◽  
State Solution ◽  
Computational Time ◽  
Load Flow Analysis ◽  
Active And Reactive Power

Load flow studies are carried out in order to find a steady state solution of a power system network. It is done to continuously monitor the system and decide upon future expansion of the system. The parameters of the system monitored are voltage magnitude, voltage angle, active and reactive power. This paper presents techniques used in order to obtain such parameters for a standard IEEE – 30 bus and IEEE-57 bus network and makes a comparison into the differences with regard to computational time and effectiveness of each solver

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