DOCSLIB.ORG

On Coupled Lane-Emden Equations Arising in Dusty Fluid Models

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
On Coupled Lane-Emden Equations Arising in Dusty Fluid Models Home Search Collections Journals About Contact us My IOPscience On coupled Lane-Emden equations arising in dusty fluid models This article has been downloaded from IOPscience. Please scroll down to see the full text article. 2011 J. Phys.: Conf. Ser. 268 012006 (http://iopscience.iop.org/1742-6596/268/1/012006) View the table of contents for this issue, or go to the journal homepage for more Download details: IP Address: 193.175.53.21 The article was downloaded on 19/01/2012 at 16:06 Please note that terms and conditions apply. 5th International Workshop on Multi-Rate Processes and Hysteresis (MURPHYS 2010) IOP Publishing Journal of Physics: Conference Series 268 (2011) 012006 doi:10.1088/1742-6596/268/1/012006 On coupled Lane-Emden equations arising in dusty fluid models D Flockerzi1 and K Sundmacher 1,2 1 Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstr. 1, D-39106 Magdeburg, Germany 2 Otto-von-Guericke-University, Universit¨atsplatz 2, D-39106 Magdeburg, Germany E-mail: [email protected] Abstract. We investigate the existence of solutions for mixed boundary value problems in coupled Lane-Emden equations. Such problems arise e.g. in the study of multicomponent diffusion and reaction processes inside catalyst particles under spherical symmetry. In contrast to the frequently applied decomposition method of Adomian, we employ the geometric theory of ODEs to show that this boundary value problem can be transformed into a terminal value problem. To this end, we determine the integral manifold on which solutions of the boundary value problem necessarily lie. This will be done in suitable warped and blown-up coordinates. Moreover, we comment on the numerical implementation. 1. Introduction 1.1. Some history In astrophysics, the Lane-Emden equation is Poisson’s equation for the gravitational potential of a self-gravitating, spherically symmetric and polytropic fluid at hydrostatic equilibrium: d dρ dρ r2 + r2 ρn = 0 , ρ(0) = 1, (0+) = 0 . (1.1) dr dr dr n+1 n Here, the pressure P is assumed to be proportional to the power n of the ‘density’ δ = const·ρ via the polytropic relation P = const · δ1+1/n = const · ρn+1 of index n. So the solution of (1.1) provides the dynamics of a scaled density and hence of a scaled pressure. Jonathan Homer Lane (1819-1880), an American astrophysicist, was the first to perform a mathematical analysis of the sun as a gaseous body. With his work On the theoretical temperature of the sun under the hypothesis of a gaseous mass maintaining its volume by its internal heat and depending on the laws of gases known from terrestial experiments from 1870 (cf. [1]) he initiated the theory of stellar evolution. Jacob Robert Emden (1862-1940), a Swiss astrophysicist and meteorologist, provided a mathematical model as a basis of stellar structure by his work Gas balls: Applications of the mechanical heat theory to cosmological and meteorological problems in 1907 (cf. [2]). In a series of papers, Ralph Howard Fowler (1889-1944) considered a generalization of (1.1) of the form d dρ rα + rσ f(ρ) = 0 (1.2) dr dr Published under licence by IOP Publishing Ltd 1 5th International Workshop on Multi-Rate Processes and Hysteresis (MURPHYS 2010) IOP Publishing Journal of Physics: Conference Series 268 (2011) 012006 doi:10.1088/1742-6596/268/1/012006 in particular for f(ρ) equal to ρn or eρn (see e.g. [3], [4]). A simple coordinate change r 7→ θ γ entails uθθ +θ f(u) = 0 for some γ (cf. Chapter 7 of [5] or [6]). By now, there is a vast literature on the generalized Emden-Fowler equation d dρ p(r) + q(r) f(ρ) = 0 , (1.3) dr dr The review article [7] of J.S.W.Wong offers a comprehensive survey for the years up to 1975. For more recent theoretical and numerical work on generalized Emden-Fowler equations we refer to [8], [9], [10], [11], [12], [13], [14], [15] and [16], [17], [18], [19], [20] [21], [22] and the references therein. Often, numerical computations are based on Adomian’s decomposition method (see e.g. [16] or [20]). 1.2. Lane-Emden equation in chemical engineering applications One of the important fields of application of the Lane-Emden equation is the analysis of the diffusive transport and chemical reaction of species inside a porous catalyst particle. Aris [23] and Metha & Aris [24] were among the first who solved this problem as boundary value problem, assuming power-law kinetics for a single chemical reaction taking place inside a spherical catalyst particle - the most relevant practical catalyst geometry: 1 d dc dc r2 = φ2 cn , c(0) = 1, (0+) = 0 (1.4a) r2 dr dr dr where the so called Thiele modulus φ is a constant parameter. The derivation of equation (1.4a) is based on the assumption that the diffusion of the considered species inside the porous catalyst obeys Fick’s law with a constant, i.e. concentration-independent, diffusivity. In multicomponent mixtures, the transport of species in porous catalysts should not be described by Fick’s law, but by the Maxwell-Stefan equations. The application of the latter equations to mass transport phenomena in porous solids leads to the so called Dusty Gas Model [25]. In this model, the solid catalyst is treated as ‘dust’, i.e. an ensemble of ‘huge’ motionless molecules which undergo exchange of momentum with the fluid species moving inside the porous catalyst structure. This approach accounts for three different kinds of transport contributions: bulk and Knudsen diffusion, surface diffusion and viscous Poiseuille-type flow. The Dusty Gas Model was originally formulated for ideal gas mixtures. Later it was generalized (named Dusty Fluid Model) to make it also applicable to nonideal multicomponent fluid mixtures, see e.g. [26], [27]. Combining the Dusty Fluid Model with the mass balances of species being transported inside a porous spherical catalyst body yields [28]: 1 d dx r2 D(x) = NR(x) , (1.4b) r2 dr dr where x stands for the vector of mole fractions of chemical species and D(x) for the concentration- dependent matrix of diffusion coefficients. Note that, based on the theory of irreversible thermodynamics, it can be shown that D is the product of two positive definite matrices (cf. [29]) and thus similar to a diagonal matrix with positive diagonal elements (cf. [30], Thm.7.6.3). On the right-hand side of eq. (1.4b), N represents the stoichiometry matrix of the considered reaction network, and R(x) represents the vector of reaction kinetic expressions. Assuming a constant D-matrix, one gets the following simplified catalyst diffusion model: 1 d dx r2 = f(x), f(x) = D−1 NR(x) . (1.4c) r2 dr dr 2 5th International Workshop on Multi-Rate Processes and Hysteresis (MURPHYS 2010) IOP Publishing Journal of Physics: Conference Series 268 (2011) 012006 doi:10.1088/1742-6596/268/1/012006 Among all possible rate expressions, linear, bilinear and quadratic functions R(x) are of highest relevance in chemical kinetics, due to the fact that chemical reactions are mostly based on mono-molecular and/or bi-molecular events. Focusing on this class of reaction kinetics, one can identify a sub-vector z containing the concentrations of all species which only appear in linear and bilinear rate expressions, and another sub-vector y containing the concentrations of the species which also appear in the quadratic terms. A practical example for this situation is the acid-catalysed dimerization of C4-olefines, a complex reaction accompanied by the formation of butane trimers, tetramers and isomers [31]. In this example the main reactant isobutene is the only species which appears in quadratic kinetic terms. The just discussed class of catalyst diffusion problems can be reformulated as quadratic 0 d Lane-Emden boundary value problem (with = dr ): 00 2 0 0 y + y = L1(y)[K11y + K12z] , y (0+) = 0, y(1) = β1 , r (1.5a) 2 z00 + z0 = L (y)[K y + K z] , z0(0+) = 0, z(1) = β . r 2 21 22 2 T T T for x = (y , z ) and r ∈ (0, 1] where the Lj(y) are matrices, linear in y, and the Kij are constant matrices, all of appropriate sizes. In a more compact notation, the problem can be given as follows: 2 x00 + x0 = L(y) Kx , x0(0+) = 0, x(1) = β, (1.5b) r for x = (yT, zT)T and r ∈ (0, 1]. 1.3. Outlook and summary In Section 2 we treat the scalar boundary value problem ( r2x0 )0 − k r2 xn = 0 , (1.6a) x0(0+) = 0 , x(1) = β > 0 (1.6b) 0 d of Lane-Emden-Fowler type (with = dr ). Based on the geometric theory of ordinary differential equations, we first determine the integral manifold S in (r, x, x0)-space of the (1.6a)-solutions which satisfy the boundary condition x0(0+) = 0 and thus reveal the underlying geometric structure. In the scalar case, the incoming manifold S will be shown to be, globally, the graph 0 of a smooth function x = ys(r, x) so that, in a second step, the boundary value problem 0 (1.6a)&(1.6b) can be converted to a terminal value problem (with x (1) = ys(1, β)). We establish in Section 3 that this geometric approach can also be taken for coupled Lane-Emden equations as (1.5a). The idea of converting boundary value into initial value problems has a long history. Here, we just refer to [6], where W.F. Ames and E. Adams have used group methods for the Emden-Fowler equations with boundary conditions that are different from ours.
Load more

Recommended publications
  • Smithsonian Miscellaneous Collections
    BULLETIN PHILOSOPHICAL SOCIETY WASHINGTON. VOL. IV. Containing the Minutes of the Society from the 185th Meeting, October 9, 1880, to the 2020! Meeting, June 11, 1881. PUBLISHED BY THE CO-OPERATION OF THE SMITHSONIAN INSTITUTION. WASHINGTON JUDD & DETWEILER, PRINTERS, WASHINGTON, D. C. CONTENTS. PAGE. Constitution of the Philosophical Society of Washington 5 Standing Rules of the Society 7 Standing Rules of the General Committee 11 Rules for the Publication of the Bulletin 13 List of Members of the Society 15 Minutes of the 185th Meeting, October 9th, 1880. —Cleveland Abbe on the Aurora Borealis , 21 Minutes of the 186th Meeting, October 25th, 1880. —Resolutions on the decease of Prof. Benj. Peirce, with remarks thereon by Messrs. Alvord, Elliott, Hilgard, Abbe, Goodfellow, and Newcomb. Lester F. Ward on the Animal Population of the Globe 23 Minutes of the 187th Meeting, November 6th, 1880. —Election of Officers of the Society. Tenth Annual Meeting 29 Minutes of the 188th Meeting, November 20th, 1880. —John Jay Knox on the Distribution of Loans in the Bank of France, the National Banks of the United States, and the Imperial of Bank Germany. J. J. Riddell's Woodward on Binocular Microscope. J. S. Billings on the Work carried on under the direction of the National Board of Health, 30 Minutes of the 189th Meeting, December 4th, 1880. —Annual Address of the retiring President, Simon Newcomb, on the Relation of Scientific Method to Social Progress. J. E. Hilgard on a Model of the Basin of the Gulf of Mexico 39 Minutes of the 190th Meeting, December iSth, 1880.
    [Show full text]
  • Homotopy Perturbation Method with Laplace Transform (LT-HPM) Is Given in “Homotopy Perturbation Method” Section
    Tripathi and Mishra SpringerPlus (2016) 5:1859 DOI 10.1186/s40064-016-3487-4 RESEARCH Open Access Homotopy perturbation method with Laplace Transform (LT‑HPM) for solving Lane–Emden type differential equations (LETDEs) Rajnee Tripathi and Hradyesh Kumar Mishra* *Correspondence: [email protected] Abstract Department In this communication, we describe the Homotopy Perturbation Method with Laplace of Mathematics, Jaypee University of Engineering Transform (LT-HPM), which is used to solve the Lane–Emden type differential equa- and Technology, Guna, MP tions. It’s very difficult to solve numerically the Lane–Emden types of the differential 473226, India equation. Here we implemented this method for two linear homogeneous, two linear nonhomogeneous, and four nonlinear homogeneous Lane–Emden type differential equations and use their appropriate comparisons with exact solutions. In the current study, some examples are better than other existing methods with their nearer results in the form of power series. The Laplace transform used to accelerate the convergence of power series and the results are shown in the tables and graphs which have good agreement with the other existing method in the literature. The results show that LT- HPM is very effective and easy to implement. Keywords: Homotopy Perturbation Method (HPM), Laplace Transform (LT), Singular Initial value problems (IVPs), Lane–Emden type equations Background Two astrophysicists, Jonathan Homer Lane and Robert had explained the Lane–Emden type differential equations. In this study, they had designed these types of differential equations, which is a dimensionless structure of Poisson’s equation for the gravitational potential of a self-gravitating, spherically symmetric, polytropic fluid and the thermal behavior of a spherical bunch of gas according to the laws of thermodynamics (Lane 1870; Richardson 1921).
    [Show full text]
  • A Chronological History of Electrical Development from 600 B.C
    From the collection of the n z m o PreTinger JJibrary San Francisco, California 2006 / A CHRONOLOGICAL HISTORY OF ELECTRICAL DEVELOPMENT FROM 600 B.C. PRICE $2.00 NATIONAL ELECTRICAL MANUFACTURERS ASSOCIATION 155 EAST 44th STREET NEW YORK 17, N. Y. Copyright 1946 National Electrical Manufacturers Association Printed in U. S. A. Excerpts from this book may be used without permission PREFACE presenting this Electrical Chronology, the National Elec- JNtrical Manufacturers Association, which has undertaken its compilation, has exercised all possible care in obtaining the data included. Basic sources of information have been search- ed; where possible, those in a position to know have been con- sulted; the works of others, who had a part in developments referred to in this Chronology, and who are now deceased, have been examined. There may be some discrepancies as to dates and data because it has been impossible to obtain unchallenged record of the per- son to whom should go the credit. In cases where there are several claimants every effort has been made to list all of them. The National Electrical Manufacturers Association accepts no responsibility as being a party to supporting the claims of any person, persons or organizations who may disagree with any of the dates, data or any other information forming a part of the Chronology, and leaves it to the reader to decide for him- self on those matters which may be controversial. No compilation of this kind is ever entirely complete or final and is always subject to revisions and additions. It should be understood that the Chronology consists only of basic data from which have stemmed many other electrical developments and uses.
    [Show full text]
  • Extended Harmonic Mapping Connects the Equations in Classical, Statistical, Fuid, Quantum Physics and General Relativity Xiaobo Zhai1,2, Changyu Huang1 & Gang Ren1*
    www.nature.com/scientificreports OPEN Extended harmonic mapping connects the equations in classical, statistical, fuid, quantum physics and general relativity Xiaobo Zhai1,2, Changyu Huang1 & Gang Ren1* One potential pathway to fnd an ultimate rule governing our universe is to hunt for a connection among the fundamental equations in physics. Recently, Ren et al. reported that the harmonic maps with potential introduced by Duan, named extended harmonic mapping (EHM), connect the equations of general relativity, chaos and quantum mechanics via a universal geodesic equation. The equation, expressed as Euler–Lagrange equations on the Riemannian manifold, was obtained from the principle of least action. Here, we further demonstrate that more than ten fundamental equations, including that of classical mechanics, fuid physics, statistical physics, astrophysics, quantum physics and general relativity, can be connected by the same universal geodesic equation. The connection sketches a family tree of the physics equations, and their intrinsic connections refect an alternative ultimate rule of our universe, i.e., the principle of least action on a Finsler manifold. One of the major unsolved problems in physics is a single unifed theory of everything1. Gauge feld theory has been introduced based on the assumption that forces are described as fermion interactions mediated by gauge bosons2. Grand unifcation theory, a special version of quantum feld theory, unifed three of the four forces, i.e., weak, strong, and electromagnetic forces. Te superstring theory3, as one of the candidates of the ultimate theory of the universe4, frst unifed the four fundamental forces of physics into a single fundamental force via particle interaction.
    [Show full text]
  • Extended Harmonic Mapping Connects the Equations in Classical, Statistical, Fluid, Quantum Physics and General Relativity
    Lawrence Berkeley National Laboratory Recent Work Title Extended harmonic mapping connects the equations in classical, statistical, fluid, quantum physics and general relativity. Permalink https://escholarship.org/uc/item/2pb137tw Journal Scientific reports, 10(1) ISSN 2045-2322 Authors Zhai, Xiaobo Huang, Changyu Ren, Gang Publication Date 2020-10-26 DOI 10.1038/s41598-020-75211-5 Peer reviewed eScholarship.org Powered by the California Digital Library University of California www.nature.com/scientificreports OPEN Extended harmonic mapping connects the equations in classical, statistical, fuid, quantum physics and general relativity Xiaobo Zhai1,2, Changyu Huang1 & Gang Ren1* One potential pathway to fnd an ultimate rule governing our universe is to hunt for a connection among the fundamental equations in physics. Recently, Ren et al. reported that the harmonic maps with potential introduced by Duan, named extended harmonic mapping (EHM), connect the equations of general relativity, chaos and quantum mechanics via a universal geodesic equation. The equation, expressed as Euler–Lagrange equations on the Riemannian manifold, was obtained from the principle of least action. Here, we further demonstrate that more than ten fundamental equations, including that of classical mechanics, fuid physics, statistical physics, astrophysics, quantum physics and general relativity, can be connected by the same universal geodesic equation. The connection sketches a family tree of the physics equations, and their intrinsic connections refect an alternative ultimate rule of our universe, i.e., the principle of least action on a Finsler manifold. One of the major unsolved problems in physics is a single unifed theory of everything1. Gauge feld theory has been introduced based on the assumption that forces are described as fermion interactions mediated by gauge bosons2.
    [Show full text]
  • Numerical Study of Fractional Differential Equations of Lane-Emden Type by Method of Collocation
    Applied Mathematics, 2012, 3, 851-856 http://dx.doi.org/10.4236/am.2012.38126 Published Online August 2012 (http://www.SciRP.org/journal/am) Numerical Study of Fractional Differential Equations of Lane-Emden Type by Method of Collocation Mohammed S. Mechee1,2, Norazak Senu3 1Institute of Mathematical Sciences, University of Malaya, Kuala Lumpur, Malaysia 2Department of Mathematics, College of Mathematics and Computer Sciences, University of Kufa, Najaf, Iraq 3Department of Mathematics, Institute for Mathematical Research, Universiti Putra Malaysia, Selangor, Malaysia Email: [email protected], [email protected] Received December 29, 2011; revised July 12, 2012; accepted July 19, 2012 ABSTRACT Lane-Emden differential equations of order fractional has been studied. Numerical solution of this type is considered by collocation method. Some of examples are illustrated. The comparison between numerical and analytic methods has been introduced. Keywords: Fractional Calculus; Fractional Differential Equation; Lane-Emden Equation; Numerical Collection Method 1. Introduction further explored in detail by Emden [7], represents such phenomena and having significant applications, is a Lane-Emden Differential Equation has the following second-order ordinary differential equation with an arbi- form: trary index, known as the polytropic index, involved in k yt yt fty ,= gt ,0<1,0 t k (1) one of its terms. The Lane-Emden equation describes a t variety of phenomena in physics and astrophysics, in- with the initial condition cluding aspects of stellar structure, the thermal history of a spherical cloud of gas, isothermal gas spheres,and y 0=Ay , 0= B , thermionic currents [8]. where A, B are constants, f ty, is a continuous real The solution of the Lane-Emden problem, as well as valued function and gtC0,1 (see [1]).
    [Show full text]
  • Solution of New Nonlinear Second Order Singular Perturbed Lane-Emden Equation by the Numerical Spectral Collocation Method
    Num. Com. Meth. Sci. Eng. 2, No. 1, 11-19 (2020) 11 Numerical and Computational Methods in Sciences & Engineering An International Journal http://dx.doi.org/10.18576/ncmse/020102 Solution of new nonlinear second order singular perturbed Lane-Emden equation by the numerical spectral collocation method M. A. Abdelkawy1,2,∗ and Zulqurnain Sabir3 1Department of Mathematics, Faculty of Science, Beni-Suef University, Beni-Suef, Egypt 2Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University, Riyadh, Saudi Arabia 3Department of Mathematics and Statistics, Hazara University Mansehra, Pakistan Received: 12 Dec. 2019, Revised: 23 Feb. 2020, Accepted: 27 Feb. 2020 Published online: 1 Mar. 2020 Abstract: The aim of the present study is to design a new nonlinear second order singularly perturbed Lane-Emden equation and report numerical solutions by using a well-known spectral collocation technique. The idea of the present study has been taken from the standard Lane-Emden equation. For the model validation, three different examples based on the singular perturbed Lane-Emden equation along with three cases have been presented and the solutions are numerically investigated by using a spectral collocation technique. Comparison of the present outcomes with the exact solutions shows the exactness, correctness and stability of the designed model as well as the present scheme. Moreover, absolute error and convergence are derived in the form of plots as well as tables. Keywords: Nonlinear singular perturbed Lane-Emden; spectral collocation technique; shape factor. 1 Introduction Singular Lane-Emden equation was introduced first time by famous astrophysicist Jonathan Homer Lane [1] and Robert Emden [2] working on the thermal execution of a spherical cloud of gas and thermodynamics classical laws [3].
    [Show full text]
  • A Reliable Iterative Method for Solving the Time-Dependent Singular Emden-Fowler Equations
    Cent. Eur. J. Eng. • 3(1) • 2013 • 99-105 DOI: 10.2478/s13531-012-0028-y Central European Journal of Engineering A reliable iterative method for solving the time-dependent singular Emden-Fowler equations Research Article Abdul-Majid Wazwaz Department of Mathematics Saint Xavier University, Chicago, IL 60655 Received 12 April 2012; accepted 18 June 2012 Abstract: Many problems of physical sciences and engineering are modelled by singular boundary value problems. In this paper, the variational iteration method (VIM) is used to study the time-dependent singular Emden-Fowler equations. This method overcomes the difficulties of singularity behavior. The VIM reveals quite a number of features over numerical methods that make it helpful for singular and nonsingular equations. The work is supported by analyzing few examples where the convergence of the results is observed. Keywords: Emden-Fowler equation • Variational iteration method • Lagrange multiplier © Versita sp. z o.o. 1. Introduction n Singular boundary value problems constitute an important For f( x) =1 and g(y) = y , Equation (1) is the standard class of boundary value problems and are frequently en- Lane-Emden equation that has been used to model several countered in the modeling of many problems of physical phenomena in mathematical physics and astrophysics and engineering sciences [1–8], such as reaction engineer- such as the theory of stellar structure and the thermal ing, heat transfer, fluid mechanics, quantum mechanics behavior of a spherical cloud of gas [11–14]. The Lane- and astrophysics [8]. In chemical engineering too, several Emden equation was first studied by the astrophysicists processes, such as isothermal and non-isothermal reac- Jonathan Homer Lane and Robert Emden, where they tion diffusion process and heat transfer are represented by considered the thermal behavior of a spherical cloud of singular boundary value problems [8].
    [Show full text]
  • Numerical Study of Singular Fractional Lane–Emden Type Equations Arising in Astrophysics
    J. Astrophys. Astr. (2019) 40:27 © Indian Academy of Sciences https://doi.org/10.1007/s12036-019-9587-0 Numerical study of singular fractional Lane–Emden type equations arising in astrophysics ABBAS SAADATMANDI∗ , AZAM GHASEMI-NASRABADY and ALI EFTEKHARI Department of Applied Mathematics, University Kashan, Kashan 87317-53153, Iran. ∗Corresponding author. E-mail: [email protected] MS received 20 December 2018; accepted 13 April 2019; published online 17 June 2019 Abstract. The well-known Lane–Emden equation plays an important role in describing some phenomena in mathematical physics and astrophysics. Recently, a new type of this equation with fractional order derivative in the Caputo sense has been introduced. In this paper, two computational schemes based on collocation method with operational matrices of orthonormal Bernstein polynomials are presented to obtain numerical approximate solutions of singular Lane–Emden equations of fractional order. Four illustrative examples are implemented in order to verify the efficiency and demonstrate solution accuracy. Keywords. Fractional Lane–Emden type equations—orthonormal Bernstein polynomials—operational matrices—collocation method—Caputo derivative. 1. Introduction hydraulic pressure at a certain radius r. Due to the contribution of photons emitted from a cloud in radi- The Lane–Emden equation has been used to model ation, the total pressure due to the usual gas pressure many phenomena in mathematical physics, astrophysics and photon radiation is formulated as follows: and celestial mechanics such as the thermal behavior of 1 RT a spherical cloud of gas under mutual attraction of its P = ξ T 4 + , 3 V molecules, the theory of stellar structure, and the theory ξ, , of thermionic currents (Chandrasekhar 1967).
    [Show full text]
  • The Sun Liquid Or Gaseous?
    2011 Volume 3 The Journal on Advanced Studies in Theoretical and Experimental Physics, including Related Themes from Mathematics PROGRESS IN PHYSICS SPECIAL ISSUE The Sun — Liquid or Gaseous? A Thermodynamic Analysis “All scientists shall have the right to present their scientific research results, in whole or in part, at relevant scientific conferences, and to publish the same in printed scientific journals, electronic archives, and any other media.” — Declaration of Academic Freedom, Article 8 ISSN 1555-5534 The Journal on Advanced Studies in Theoretical and Experimental Physics, including Related Themes from Mathematics PROGRESS IN PHYSICS A quarterly issue scientific journal, registered with the Library of Congress (DC, USA). This journal is peer reviewed and included in the abs- tracting and indexing coverage of: Mathematical Reviews and MathSciNet (AMS, USA), DOAJ of Lund University (Sweden), Zentralblatt MATH (Germany), Scientific Commons of the University of St. Gallen (Switzerland), Open-J-Gate (India), Referativnyi Zhurnal VINITI (Russia), etc. Electronic version of this journal: JULY 2011 VOLUME 3 http://www.ptep-online.com Editorial Board Dmitri Rabounski, Editor-in-Chief SPECIAL ISSUE [email protected] Florentin Smarandache, Assoc. Editor [email protected] The Sun — Gaseous or Liquid? Larissa Borissova, Assoc. Editor [email protected] A Thermodynamic Analysis Editorial Team Gunn Quznetsov [email protected] Andreas Ries CONTENTS [email protected] Chifu Ebenezer Ndikilar [email protected] Robitaille P.M. A Thermodynamic History of the Solar Constitution — I: The Journey Felix Scholkmann toaGaseousSun.............................................................3 [email protected] Secchi A. On the Theory of Solar Spots Proposed by Signor Kirchoff ..................26 Postal Address Secchi A.
    [Show full text]
  • Rational Chebyshev of Second Kind Collocation Method for Solving a Class of Astrophysics Problems
    The Rational Chebyshev of Second Kind Collocation Method for Solving a Class of Astrophysics Problems K. Paranda∗ , S. Khaleqiay aDepartment of Mathematics, Faculty of Mathematical Sciences, Shahid Beheshti University, Evin,Tehran 19839,Iran September 21, 2018 Abstract The Lane-Emden equation has been used to model several phenomenas in theoretical physics, math- ematical physics and astrophysics such as the theory of stellar structure. This study is an attempt to utilize the collocation method with the Rational Chebyshev of Second Kind function (RSC) to solve the Lane-Emden equation over the semi-infinit interval [0; +1). According to well-known results and comparing with previous methods, it can be said that this method is efficient and applicable. Keywords| Lane-Emden,Collocation method,Rational Chebyshev of Second Kind,Nonlinear ODE,Astrophysics arXiv:1508.07240v1 [cs.NA] 28 Aug 2015 ∗E-mail address: k [email protected], Corresponding author, (K. Parand) yE-mail address: [email protected], (S. Khaleghi) 1 1 introduction Solving science and engineering problems appeared in unbounded domains, scientists have studied about it in the last decade. Spectral method is one of the famous solution. Some researchers have proposed several spectral method to solve these kinds of problems. Using spectral method related to orthogonal systems in unbounded domains such as the Hermite spectral method and the Laguerre Method is called direct approx- imation [1{8]. In an indirect approximation, the original problem mapping in an unbounded domain is changed to a problem in a bounded domain. Gua has used this method by choosing a suitable Jacobi polynomials to approximate the results [9{11].
    [Show full text]
  • The Source of Solar Energy, Ca. 1840-1910: from Meteoric Hypothesis to Radioactive Speculations
    1 The Source of Solar Energy, ca. 1840-1910: From Meteoric Hypothesis to Radioactive Speculations Helge Kragh Abstract: Why does the Sun shine? Today we know the answer to the question and we also know that earlier answers were quite wrong. The problem of the source of solar energy became an important part of physics and astronomy only with the emergence of the law of energy conservation in the 1840s. The first theory of solar heat based on the new law, due to J. R. Mayer, assumed the heat to be the result of meteors or asteroids falling into the Sun. A different and more successful version of gravitation-to-heat energy conversion was proposed by H. Helmholtz in 1854 and further developed by W. Thomson. For more than forty years the once so celebrated Helmholtz-Thomson contraction theory was accepted as the standard theory of solar heat despite its prediction of an age of the Sun of only 20 million years. In between the gradual demise of this theory and the radically different one based on nuclear processes there was a period in which radioactivity was considered a possible alternative to gravitational contraction. The essay discusses various pre- nuclear ideas of solar energy production, including the broader relevance of the question as it was conceived in the Victorian era. 1 Introduction When Hans Bethe belatedly was awarded the Nobel Prize in 1967, the presentation speech was given by Oskar Klein, the eminent Swedish theoretical physicist. Klein pointed out that Bethe’s celebrated theory of 1939 of stellar energy production had finally solved an age-old riddle, namely “how it has been possible for the sun to emit light and heat without exhausting its source.”1 Bethe’s theory marked indeed a watershed in theoretical astrophysics in general and in solar physics in particular.
    [Show full text]