Multiple solutions for a problem in buoyancy induced flow (Q1087096)

The authors study two related boundary-value problems \(P\pm:f'=\pm W(\phi,R),\quad \phi ''=f\phi '\quad ('=d/dn)\) with \(\phi (0)=1,\quad \phi (\infty)=0,\quad f(0)=0\), where R is the temperature excess ratio parameter and the function W represents buoyancy force with certain physically realistic properties. A derivation of these problems and their numerical solutions were given by \textit{J. M. Ramillson} and \textit{B. Gebhart} [Int. J. Heat Mass Transfer 23, 1521-1530 (1980; Zbl 0457.76080)]. These results have been extended considerably by \textit{B. Gebhart}, \textit{B. Hassard} and the authors [Numer. Heat Transfer 6, 337-352 (1983; Zbl 0557.76115)], who found several solutions of \(P\pm\) for some values of R. Here the authors analyze the problems of \(P\pm\) mathematically, and verify most of the previous numerical results, and in some respects they are also able to go beyond what was found numerically.