Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Existence and multiplicity of solutions of an equation from pool boiling on wires

Author: Shin-Hwa Wang
Journal: Quart. Appl. Math. 58 (2000), 331-354
MSC: Primary 34B15; Secondary 80A20
DOI: https://doi.org/10.1090/qam/1753403
MathSciNet review: MR1753403
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Abstract: We investigate the existence and multiplicity of steady states of the equation

$\displaystyle \frac{{\partial \theta }}{{\partial t}} - \frac{{{\partial ^2}\th... ...theta \right) - a\lambda \left( 1 + \alpha \theta \right) = 0, \qquad 0 < x < 1$

with Dirichlet boundary conditions and initial conditions. This equation was derived and studied by Joly, Kernevez, and Llory [7] and Joly [8] in studying thermal effects from pool boiling, in which wires are heated by the Joule effect and are cooled in a bath of boiling water at constant pressure. They studied the steady-state problem for two kinds of heat flux density $ q\left( \theta \right)$ (corresponding to whether or not the radiation is taken into account) and for $ \alpha \ne 0$ or $ \alpha = 0$. (A) In the case with radiation and $ \alpha = 0$, for given specific function $ q\left( \theta \right)$ and constants $ \sigma > 0$, $ a > 0$, by numerical methods, they found an $ S$-shaped bifurcation diagram and three solutions for some parameter values. We prove this rigorously, for a specific range of parameters of physical interest. Specifically, we show that, for specific values of $ \alpha , \sigma $, and $ a$, there exist two positive numbers $ \mathop \lambda \limits_ - < \bar \lambda $ such that the steady-state problem has at least three solutions for $ \mathop \lambda \limits_ - < \lambda < \bar \lambda $, at least two solutions for $ \lambda = \mathop \lambda \limits_ -$ or $ \lambda = \bar \lambda $, and exactly one solution for $ 0 \le \lambda < \mathop \lambda \limits_ -$ or $ \lambda > \bar \lambda $. Moreover, we give lower and upper bounds for $ \mathop \lambda \limits_ -$ and $ \bar \lambda $. (B) In the case without radiation and $ \alpha \ne 0$, we show that there exist two positive numbers $ \mathop \lambda \limits_ - < \bar \lambda $ such that the steady-state problem has at least two solutions for $ \mathop \lambda \limits_ - < \lambda < \bar \lambda $, at least one solution for $ 0 \le \lambda \le \mathop \lambda \limits_ -$ or $ \lambda = \bar \lambda $, exactly one solution for $ 0 < \lambda < \mathop \lambda \limits_ -$ A and $ \lambda $ small enough, and no solution for $ \lambda > \bar \lambda $. Moreover, we give upper and lower bounds for $ \mathop \lambda \limits_ -$ and $ \bar \lambda $. Also, we find and correct two mistakes in [7, Proposition 2.6, (i), (ii)].

References [Enhancements On Off] (What's this?)

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DOI: https://doi.org/10.1090/qam/1753403
Article copyright: © Copyright 2000 American Mathematical Society

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