Multiple solutions in hollow geometries in the theory of thermal ignition

E. Balakrishnan, A. Swift, G. C. Wake

Research output: Contribution to journalArticle

2 Citations (Scopus)

Abstract

Following Burnell et al., the dimensionless form of the steady state heat balance equation for material undergoing an exothermic reaction over the symmetrical N-dimensional unit sphere is u″(r) + (N - 1)r-1u′(r) + λe-(1/u) = 0. The parameter λ is held constant so that the solution structure is dependent on the bifurcation parameter U appearing in the boundary condition u(1) = U. In previous papers we discussed the existence of multiple solutions for both class A (slab, infinite cylinder, and sphere) and nonclass A geometries and showed that multiple solutions (of multiplicity greater that three) occur for 2 <N <12. In this paper, we present numerical results for some hollow geometries which indicate that similar solution structure to that of previous cases is preserved and that the multiplicity of solutions does not always depend on the size of the hollow region.

Original languageEnglish
Pages (from-to)41-46
Number of pages6
JournalApplied Mathematics Letters
Volume10
Issue number5
Publication statusPublished - Sep 12 1997

Fingerprint

Ignition
Multiple Solutions
Exothermic reactions
Multiplicity of Solutions
Bifurcation (mathematics)
Geometry
Balance Equations
Unit Sphere
Dimensionless
Heat Equation
Multiplicity
Bifurcation
Boundary conditions
Numerical Results
Dependent
Hot Temperature
Class
Form

Keywords

  • Boundary value problems
  • Combustion
  • Nonlinear systems

ASJC Scopus subject areas

  • Computational Mechanics
  • Control and Systems Engineering
  • Applied Mathematics
  • Numerical Analysis

Cite this

Multiple solutions in hollow geometries in the theory of thermal ignition. / Balakrishnan, E.; Swift, A.; Wake, G. C.

In: Applied Mathematics Letters, Vol. 10, No. 5, 12.09.1997, p. 41-46.

Research output: Contribution to journalArticle

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