Polytropic solutions to the problem of spherically symmetric flow of an ideal gas

1980 ◽  
Vol 58 (8) ◽  
pp. 1085-1092 ◽  
Author(s):  
D. Summers
Keyword(s):  
Differential Equations ◽  
Fluid Dynamic ◽  
Mass Density ◽  
Ideal Gas ◽  
Radial Coordinate ◽  
Polytropic Index ◽  
Spherically Symmetric ◽  
Polytropic Model ◽  
Special Cases ◽  
Complete Set

An ideal, compressible gas is considered in steady, spherically symmetric, purely radial motion in the presence of a gravitating point mass situated at the origin of coordinates. The gas pressure p and mass density ρ are assumed to satisfy a simple polytrope law, [Formula: see text], where β is the polytropic index which is assumed to take on any real value; the self-gravitation of the gas is neglected. The model equations, which are expressed in a form appropriate to both the expansion and accretion cases, reduce to two non-linear ordinary differential equations, for which Bernoulli integrals are readily found. Singular points of the differential equations are analyzed, and the complete set of asymptotic solutions for the velocity, temperature and Mach number are given for λ → 0 and λ → ∞, where λ [Formula: see text] (radial coordinate)−1, as well as a special class of solutions valid for λ → constant (≠ 0). Families of velocity profiles are sketched which are representative of the complete range of β. The polytropic model, special cases of which have been used successfully in astrophysics in stellar wind and accretion problems, is here cast in general fluid-dynamic terms so that the complete set of solutions obtained may be applicable to a wide class of gas expansion and accretion problems.

Scattering by Spherically Symmetrical Objects

1972 ◽  
Vol 50 (8) ◽  
pp. 749-753 ◽  
Author(s):  
L. Shafai
Keyword(s):  
Differential Equations ◽  
Scattered Field ◽  
Initial Phase ◽  
Radial Coordinate ◽  
Spherically Symmetric ◽  
First Order ◽  
Auxiliary Functions ◽  
Vector Potentials ◽  
Phase Functions ◽  
First Order Differential Equations

The solutions of vector potentials in the presence of spherically symmetric objects are expressed in terms of two auxiliary functions, related respectively to the phase and amplitude of the resulting field. It is shown that these auxiliary functions satisfy first-order differential equations of the radial coordinate, and the scattered field is described by the phase functions alone. Furthermore, the differential equations satisfied by the phase functions are found to be independent of the amplitude functions and are solved numerically by using the well-known initial phase shifts readily obtained from the boundary conditions.

Three-Dimensional Free Vibration Analysis of a Homogeneous Transradially Isotropic Thermoelastic Sphere

2009 ◽  
Vol 77 (2) ◽  
Author(s):  
J. N. Sharma ◽  
N. Sharma
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Vibration Analysis ◽  
Spherical Wave ◽  
Three Dimensional ◽  
Coupled System ◽  
Solid Sphere ◽  
Radial Coordinate ◽  
Special Cases ◽  
Matlab Programming

In the present paper, an exact three-dimensional vibration analysis of a transradially isotropic, thermoelastic solid sphere subjected to stress-free, thermally insulated, or isothermal boundary conditions has been carried out. Nondimensional basic governing equations of motion and heat conduction for the considered thermoelastic sphere are uncoupled and simplified by using Helmholtz decomposition theorem. By using a spherical wave solution, a system of governing partial differential equations is further reduced to a coupled system of three ordinary differential equations in radial coordinate in addition to uncoupled equation for toroidal motion. Matrix Fröbenious method of extended power series is used to investigate motion along radial coordinate from the coupled system of equations. Secular equations for the existence of various types of possible modes of vibrations in the sphere are derived in the compact form by employing boundary conditions. Special cases of spheroidal and toroidal modes of vibrations of a solid sphere have also been deduced and discussed. It is observed that the toroidal motion remains independent of thermal variations as expected and spheroidal modes are in general affected by thermal variations. Finally, the numerical solution of the secular equation for spheroidal motion (S-modes) is carried out to compute lowest frequency and dissipation factor of different modes with MATLAB programming for zinc and cobalt materials. Computer simulated results have been presented graphically. The analyses may find applications in aerospace, navigation, and other industries where spherical structures are in frequent use.

scholarly journals SIMILARITY SOLUTIONS FOR STRONG SHOCKS IN A NON-IDEAL GAS

2012 ◽  
Vol 17 (3) ◽  
pp. 351-365 ◽  
Author(s):  
Rajan Arora ◽  
Amit Tomar ◽  
Ved Pal Singh
Keyword(s):  
Power Law ◽  
Initial Density ◽  
Similarity Solutions ◽  
Ideal Gas ◽  
Spherically Symmetric ◽  
State Dependent ◽  
Group Theoretic Method ◽  
Special Cases ◽  
Dependent Form ◽  
Flow Variables

A group theoretic method is used to obtain an entire class of similarity solutions to the problem of shocks propagating through a non-ideal gas and to characterize analytically the state dependent form of the medium ahead for which the problem is invariant and admits similarity solutions. Different cases of possible solutions, known in the literature, with a power law, exponential or logarithmic shock paths are recovered as special cases depending on the arbitrary constants occurring in the expression for the generators of the transformation. Particular case of collapse of imploding cylindrically and spherically symmetric shock in a medium in which initial density obeys power law is worked out in detail. Numerical calculations have been performed to obtain the similarity exponents and the profiles of the flow variables behind the shock, and comparison is made with the known results.

scholarly journals Simplified and improved criteria for oscillation of delay differential equations of fourth order

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
O. Moaaz ◽  
A. Muhib ◽  
D. Baleanu ◽  
W. Alharbi ◽  
E. E. Mahmoud
Keyword(s):  
Differential Equations ◽  
Fourth Order ◽  
Delay Differential Equations ◽  
Oscillatory Behavior ◽  
Interesting Point ◽  
All Solutions ◽  
Special Cases ◽  
Delay Differential ◽  
Noncanonical Operator

AbstractAn interesting point in studying the oscillatory behavior of solutions of delay differential equations is the abbreviation of the conditions that ensure the oscillation of all solutions, especially when studying the noncanonical case. Therefore, this study aims to reduce the oscillation conditions of the fourth-order delay differential equations with a noncanonical operator. Moreover, the approach used gives more accurate results when applied to some special cases, as we explained in the examples.

scholarly journals Ultrasonic Waves in Bubbly Liquids: An Analytic Approach

Mathematics ◽  
2021 ◽  
Vol 9 (11) ◽  
pp. 1309
Author(s):  
P. R. Gordoa ◽  
A. Pickering
Keyword(s):  
Mathematical Model ◽  
Partial Differential Equations ◽  
Differential Equations ◽  
Acoustic Waves ◽  
Elliptic Functions ◽  
Coupled System ◽  
Ultrasonic Waves ◽  
Partial Differential ◽  
Special Cases ◽  
Spherical Bubbles

We consider the problem of the propagation of high-intensity acoustic waves in a bubble layer consisting of spherical bubbles of identical size with a uniform distribution. The mathematical model is a coupled system of partial differential equations for the acoustic pressure and the instantaneous radius of the bubbles consisting of the wave equation coupled with the Rayleigh–Plesset equation. We perform an analytic analysis based on the study of Lie symmetries for this system of equations, concentrating our attention on the traveling wave case. We then consider mappings of the resulting reductions onto equations defining elliptic functions, and special cases thereof, for example, solvable in terms of hyperbolic functions. In this way, we construct exact solutions of the system of partial differential equations under consideration. We believe this to be the first analytic study of this particular mathematical model.

scholarly journals Coupled Systems of Sequential Caputo and Hadamard Fractional Differential Equations with Coupled Separated Boundary Conditions

Symmetry ◽  
2018 ◽  
Vol 10 (12) ◽  
pp. 701 ◽  
Author(s):  
Suphawat Asawasamrit ◽  
Sotiris Ntouyas ◽  
Jessada Tariboon ◽  
Woraphak Nithiarayaphaks
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Fractional Differential Equations ◽  
Coupled Systems ◽  
Uniqueness Of Solutions ◽  
Krasnoselskii’S Fixed Point Theorem ◽  
Hadamard Fractional Differential Equations ◽  
Special Cases ◽  
Separated Boundary Conditions ◽  
Fractional Differential

This paper studies the existence and uniqueness of solutions for a new coupled system of nonlinear sequential Caputo and Hadamard fractional differential equations with coupled separated boundary conditions, which include as special cases the well-known symmetric boundary conditions. Banach’s contraction principle, Leray–Schauder’s alternative, and Krasnoselskii’s fixed-point theorem were used to derive the desired results, which are well-illustrated with examples.

scholarly journals Anisotropic compact stars in higher-order curvature theory

2021 ◽  
Vol 81 (6) ◽  
Author(s):  
G. G. L. Nashed ◽  
S. D. Odintsov ◽  
V. K. Oikonomou
Keyword(s):  
General Relativity ◽  
Differential Equations ◽  
Higher Order ◽  
Specific Form ◽  
Ricci Scalar ◽  
Spherically Symmetric ◽  
Curvature Theory ◽  
Anisotropic Star ◽  
Symmetric Spacetime ◽  
Spherically Symmetric Spacetime

AbstractIn this paper we shall consider spherically symmetric spacetime solutions describing the interior of stellar compact objects, in the context of higher-order curvature theory of the $${{\mathrm {f(R)}}}$$ f ( R ) type. We shall derive the non-vacuum field equations of the higher-order curvature theory, without assuming any specific form of the $${{\mathrm {f(R)}}}$$ f ( R ) theory, specifying the analysis for a spherically symmetric spacetime with two unknown functions. We obtain a system of highly non-linear differential equations, which consists of four differential equations with six unknown functions. To solve such a system, we assume a specific form of metric potentials, using the Krori–Barua ansatz. We successfully solve the system of differential equations, and we derive all the components of the energy–momentum tensor. Moreover, we derive the non-trivial general form of $${{\mathrm {f(R)}}}$$ f ( R ) that may generate such solutions and calculate the dynamic Ricci scalar of the anisotropic star. Accordingly, we calculate the asymptotic form of the function $${\mathrm {f(R)}}$$ f ( R ) , which is a polynomial function. We match the derived interior solution with the exterior one, which was derived in [1], with the latter also resulting to a non-trivial form of the Ricci scalar. Notably but rather expected, the exterior solution differs from the Schwarzschild one in the context of general relativity. The matching procedure will eventually relate two constants with the mass and radius of the compact stellar object. We list the necessary conditions that any compact anisotropic star must satisfy and explain in detail that our model bypasses all of these conditions for a special compact star $$\textit{Her X--1}$$ Her X - - 1 , which has an estimated mass and radius $$(mass = 0.85 \pm 0.15M_{\circledcirc }\ and\ radius = 8.1 \pm 0.41~\text {km}$$ ( m a s s = 0.85 ± 0.15 M ⊚ a n d r a d i u s = 8.1 ± 0.41 km ). Moreover, we study the stability of this model by using the Tolman–Oppenheimer–Volkoff equation and adiabatic index, and we show that the considered model is different and more stable compared to the corresponding models in the context of general relativity.

scholarly journals Узагальнення моделі Голанда і Рейсснера на випадок осьової симетрії

2021 ◽  
pp. 12-19
Author(s):  
Костянтин Петрович Барахов
Keyword(s):  
Stress State ◽  
Analytical Solution ◽  
Differential Equations ◽  
Boundary Conditions ◽  
Bessel Functions ◽  
Classical Model ◽  
Adhesive Bond ◽  
Radial Coordinate ◽  
Round Hole ◽  
Normal Stresses

The purpose of this work is to create a mathematical model of the stress state of overlapped circular axisymmetric adhesive joints and to build an appropriate analytical solution to the problem. To solve the problem, a simplified model of the adhesive bond of two overlapped plates is proposed. The simplification is that the movement of the layers depends only on the radial coordinate and does not depend on the angular one. The model is a generalization of the classical model of the connection of Holland and Reissner in the case of axial symmetry. The stresses are considered to be evenly distributed over the thickness of the layers, and the adhesive layer works only on the shift. These simplifications allowed us to obtain an analytical solution to the studied problem. The problem of the stress state of the adhesive bond of two plates is solved, one of which is weakened by a round hole, and the other is a round plate concentric with the hole. A load is applied to the plate weakened by a round hole. The discussed area is divided into three parts: the area of bonding, as well as areas inside and outside the bonding. In the field of bonding, the problem is reduced to third- and fourth-order differential equations concerning tangent and normal stresses, respectively, the solutions of which are constructed as linear combinations of Bessel functions of the first and second genera and modified Bessel functions of the first and second genera. Using the found tangential and normal stresses, we obtain linear inhomogeneous Euler differential equations concerning longitudinal and transverse displacements. The solution of the obtained equations is also constructed using Bessel functions. Outside the area of bonding, displacements are described by the equations of bending of round plates in the absence of shear forces. Boundary conditions are met exactly. The satisfaction of marginal conditions, as well as boundary conditions, leads to a system of linear equations concerning the unknown coefficients of the obtained solutions. The model problem is solved and the numerical results are compared with the results of calculations performed by using the finite element method. It is shown that the proposed model has sufficient accuracy for engineering problems and can be used to solve problems of the design of aerospace structures.

scholarly journals The diagrammatic coaction beyond one loop

2021 ◽  
Vol 2021 (10) ◽  
Author(s):  
Samuel Abreu ◽  
Ruth Britto ◽  
Claude Duhr ◽  
Einan Gardi ◽  
James Matthew
Keyword(s):  
Differential Equations ◽  
Hypergeometric Functions ◽  
Differential Forms ◽  
Feynman Graph ◽  
Loop Order ◽  
Feynman Integrals ◽  
Mathematical Properties ◽  
Multiple Polylogarithms ◽  
Complete Set ◽  
General Tool

Abstract The diagrammatic coaction maps any given Feynman graph into pairs of graphs and cut graphs such that, conjecturally, when these graphs are replaced by the corresponding Feynman integrals one obtains a coaction on the respective functions. The coaction on the functions is constructed by pairing a basis of differential forms, corresponding to master integrals, with a basis of integration contours, corresponding to independent cut integrals. At one loop, a general diagrammatic coaction was established using dimensional regularisation, which may be realised in terms of a global coaction on hypergeometric functions, or equivalently, order by order in the ϵ expansion, via a local coaction on multiple polylogarithms. The present paper takes the first steps in generalising the diagrammatic coaction beyond one loop. We first establish general properties that govern the diagrammatic coaction at any loop order. We then focus on examples of two-loop topologies for which all integrals expand into polylogarithms. In each case we determine bases of master integrals and cuts in terms of hypergeometric functions, and then use the global coaction to establish the diagrammatic coaction of all master integrals in the topology. The diagrammatic coaction encodes the complete set of discontinuities of Feynman integrals, as well as the differential equations they satisfy, providing a general tool to understand their physical and mathematical properties.

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