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.