Thermal Instability in Pool Boiling on Wires at Constant Pressure

This paper gives a mathematical and numerical analysis of the steady states of the equation $\frac{\partial\theta}{\partial t} - \frac{\partial^2\theta}{\partial x^2} + \sigma q(\theta) - a\lambda (1 + \alpha\theta) = 0,\quad 0 < x < 1.$ with Dirichlet boundary conditions, where q(θ) is a highly nonlinear term expressing the heat flux density passing from an electric wire to a liquid water vapor mixture. The varying parameter λ is proportional to i2, i being the current intensity along the electric wire. This analysis is performed for 2 kinds of functions q(θ) (corresponding to whether or not the radiation phenomenon is taken into account) and for α ≠ 0 or α = 0. With α ≠ 0 it is shown that no stable steady state exists for λ large enough, whereas for α = 0 and with radiation an S-shaped curve describes the solutions as λ varies. Numerical results, obtained by continuation methods, are in agreement with the mathematical predictions. It is shown that much care must be given to the approximating scheme in order to avoid "undulations" in the curve representing the approximate solutions as λ varies.