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Thermal instability in pool boiling on wires at constant pressure. (English) Zbl 0543.35032

This paper gives a mathematical and numerical analysis of the steady- state of the equation \[ \partial \theta /\partial t-\partial^ 2\theta /\partial x^ 2+\sigma q(\theta)-a \lambda(1+\alpha \theta)=0,\quad 0<x<1, \] with \(\theta =0\) at \(x=0,1\), where \(q(\theta)\) is a highly nonlinear term expressing the heat flux density passing from an electric wire to a liquid vapour mixture. The parameter \(\lambda\), used as a eigenparameter for the nonlinear eigenvalue problem, is proportional to the \((current)^ 2\) in the electric wire. For various kinds of functions \(q(\theta)\), corresponding to whether or not radiation phenomena is taken into account, it is shown that if \(\lambda\) is large enough there are no steady-states (unless \(\alpha =0)\), and for \(\lambda\) small multiple steady-states are possible. The methods are those based on Leray-Schauder degree theory. Explicit computation of the solutions is given which verifies those given by the qualitative section and the shape of the bifurcation diagram.
Reviewer: G.C.Wake

MSC:

35J60 Nonlinear elliptic equations
35B32 Bifurcations in context of PDEs
82D15 Statistical mechanics of liquids
35B60 Continuation and prolongation of solutions to PDEs
35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs
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