Multiple solutions in hollow geometries in the theory of thermal ignition

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.