nach oben

1989 | Buch

Kapitel lesen Erstes Kapitel lesen

Mathematical Problems from Combustion Theory

verfasst von: Jerrold Bebernes, David Eberly

Verlag: Springer New York

Buchreihe : Applied Mathematical Sciences

Enthalten in: Professional Book Archive

Einloggen, um Zugang zu erhalten

Über dieses Buch

This monograph evolved over the past five years. It had its origin as a set of lecture notes prepared for the Ninth Summer School of Mathematical Physics held at Ravello, Italy, in 1984 and was further refined in seminars and lectures given primarily at the University of Colorado. The material presented is the product of a single mathematical question raised by Dave Kassoy over ten years ago. This question and its partial resolution led to a successful, exciting, almost unique interdisciplinary col laborative scientific effort. The mathematical models described are often times deceptively simple in appearance. But they exhibit a mathematical richness and beauty that belies that simplicity and affirms their physical significance. The mathe matical tools required to resolve the various problems raised are diverse, and no systematic attempt is made to give the necessary mathematical background. The unifying theme of the monograph is the set of models themselves. This monograph would never have come to fruition without the enthu siasm and drive of Dave Eberly-a former student, now collaborator and coauthor-and without several significant breakthroughs in our understand ing of the phenomena of blowup or thermal runaway which certain models discussed possess. A collaborator and former student who has made significant contribu tions throughout is Alberto Bressan. There are many other collaborators William Troy, Watson Fulks, Andrew Lacey, Klaus Schmitt-and former students-Paul Talaga and Richard Ely-who must be acknowledged and thanked.

Inhaltsverzeichnis

Frontmatter

1. Introduction

Abstract

Extremely rapid exothermic chemical reactions can develop in combustible materials. For multicomponent reacting mixtures of N chemical species, the complete system of conservation equations can be expressed as

$$ \frac{{D\vec w}}{{Dt}} = \vec \nabla \, \bullet \,\left( {\vec f(\vec w)\, \bullet \,\vec \nabla \vec w} \right) + \vec g\,\left( {\vec w,\,\vec \nabla \vec w} \right)$$

(1.1)

where $ \vec w\left( {\vec x,\,t} \right) = \left( {\rho, \,\vec u,\,T,\,\vec y} \right) $ denotes the state of the system and where $ \frac{D}{{Dt}} = \frac{\partial }{{\partial t}} + \vec u\, \bullet \,\vec \nabla $ is the material derivative. The state $ \vec w $ includes the density ρ, the temperature T, the mass fractions $ \vec y = \left( {{y_1},...,{y_N}} \right) $ with $ \sum\nolimits_{i = 1}^N {{y_i} = 1} $ and the mass-average velocity $ \vec u = \sum\nolimits_{i = 1}^N {{y_i}{{\vec u}_i}} $ where $ {\vec u_i} $ is the velocity of species i. We also consider the pressure p, which is proportional to density and to temperature. The interaction of the chemistry of the species with the basic fluid flow is described by a highly nonlinear, extremely complex, degenerate, quasilinear parabolic system of partial differential equations. The problem of well-posedness for (1.1) has not been completely resolved [KZH2], [MAT].

Jerrold Bebernes, David Eberly

2. Steady-State Models

Abstract

The first section of this chapter deals with existence for the Dirichlet problem where the nonlinearity F(x, u) is a nonnegative function. The key result used is an existence theorem based on a priori knowledge of upper and lower solutions. We also analyze the spectrum of nonlinear eigenvalue problems and determine bounds on the critical eigenvalues.

Jerrold Bebernes, David Eberly

3. The Rigid Ignition Model

Abstract

We wish to analyze indepth the solid fuel ignition model (1.28-)(1.29)

$$ \begin{array}{*{20}{c}} {{\theta _t} - \Delta \theta = \delta {e^\theta },{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \left( {x,t} \right) \in \Omega {\mkern 1mu} {\mkern 1mu} \times {\mkern 1mu} {\mkern 1mu} \left( {0,T} \right)} \\ {\theta \left( {x,0} \right) = 0,{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} x \in \Omega } \\ {\theta \left( {x,t} \right) = 0,{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} \left( {x,t} \right) \in \partial \Omega {\mkern 1mu} {\mkern 1mu} \times {\mkern 1mu} {\mkern 1mu} \left( {0,T} \right)} \end{array} $$

and its relationship to the steady-state model (1.30)-(1.31)

$$ \begin{array}{*{20}{c}} { - \Delta \psi = \delta {e^\psi },{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} x \in \Omega } \\ {\psi (x) = 0,{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} x \in \partial \Omega .} \end{array} $$

Jerrold Bebernes, David Eberly

4. The Complete Model for Solid Fuel

Abstract

In this chapter we discuss comparison techniques, invariant sets, and existence results related to invariance. Our main application is the complete solid fuel model (1.24)-(1.25):

$$ \begin{array}{*{20}{c}} {\begin{array}{*{20}{c}} {{T_t} - \Delta T = \varepsilon \delta {y^m}\exp \left( {\frac{{T - 1}}{{\varepsilon T}}} \right)} \\ {{y_t} - \beta \Delta y = - \varepsilon \delta \Gamma {y^m}\exp \left( {\frac{{T - 1}}{{\varepsilon T}}} \right)} \end{array}}&{,\left( {x,t} \right) \in \Omega \times \left( {0,\infty } \right)} \end{array} $$

with initial-boundary conditions

$$ \begin{array}{*{20}{c}} {T\left( {x,0} \right) = 1,y\left( {x,0} \right) = 1,{\mkern 1mu} x \in \Omega } \\ {T\left( {x,t} \right) = 1,\left( {\frac{{\partial y\left( {x,t} \right)}}{{\partial \eta (x)}}} \right) = 0,{\mkern 1mu} \left( {x,t} \right) \in \partial \Omega \times \left( {0,\infty } \right).} \end{array} $$

where β ≥ 0, Γ > 0, and δ > 0. We prove there is a solution (T, y) for all (x, t) ∈ Ω × (0, ∞) such that y(x, t) → 0 as t → ∞.

Jerrold Bebernes, David Eberly

5. Gaseous Ignition Models

Abstract

We discuss in this chapter initial-boundary value problems of the form

$$ {u_t} - a\Delta u = f(u) + f(u) + g(t),{\mkern 1mu} {\mkern 1mu} a \geqslant 0,{\mkern 1mu} {\mkern 1mu} \left( {x,t} \right) \in \Omega \times \left( {0,T} \right) $$

with u(x, 0) = ϕ(x) for x ∈ Ω and u(x,t) = 0 for (x,t)∈ ∂Ω×(0,T). The reactive-diffusive gaseous model (1.39)-(1.40) and the nondiffusive model (1.41)-(1.40) are special cases.

Jerrold Bebernes, David Eberly

6. Conservation Systems for Reactive Gases

Abstract

In one space dimension, the conservation laws for reactive gases can be expressed as

$$ {u_t} + F{(u)_x} = B{u_{xx}} + G(u,{u_x}),(x,t) \in \Omega \times (0,T) \subset \mathbb{R} \times \mathbb{R} $$

(6.1)

where the solutions u are vector-valued functions of (x, t), and where B is a positive semidefinite matrix which will be referred to as the viscosity matrix.

Jerrold Bebernes, David Eberly

Backmatter

Titel: Mathematical Problems from Combustion Theory
verfasst von: Jerrold Bebernes
David Eberly
Verlag: Springer New York
Electronic ISBN: 978-1-4612-4546-9
Print ISBN: 978-1-4612-8872-5
DOI: https://doi.org/10.1007/978-1-4612-4546-9

Springer Professional

Über dieses Buch

Inhaltsverzeichnis

Frontmatter

1. Introduction

2. Steady-State Models

3. The Rigid Ignition Model

4. The Complete Model for Solid Fuel

5. Gaseous Ignition Models

6. Conservation Systems for Reactive Gases

Backmatter