Nonexistence of the reversed flow solutions of the Falkner-Skan equations
The authors study the third order boundary value problem on the half-line \[ f'''+ff''+\lambda(1-{f'}^2)=0,\;\;\; f(0)=f'(0)=0,\;\;f'(\infty)=1,\tag{1} \] where \(\lambda\) is a negative parameter. A reversed flow solution of (1) is a solution \(f\) with \(f''(0)<0\). Numerical experiments show that the number \(\lambda_0=-0.1988\) appears to have the property that (1) has no reversed-flow solution for \(\lambda<\lambda_0\) and has at least one reversed flow solution for \(\lambda_0<\lambda<0\). See, e.g., [\textit{S. N. Brown} and \textit{K. Stewartson}, ``On the reversed flow solutions of the Falkner-Skan equation'', Mathematika, Lond. 13, 1--6 (1966; Zbl 0148.22002)]. In this paper, the conditions \(f''(0)<0\) and \(|f'|<1\) are added to (1) and it is shown analytically that there exists a number \(\lambda^*<0\) such that, in fact, no such solutions appear for \(\lambda\leq\lambda^*\). An estimate of \(\lambda^*\), and, hence, a lower bound of \(\lambda_0\), is obtained. Moreover, a new estimate of \(f''(0)\) in terms of \(\lambda \) is given. The proof proceeds as follows: for solutions of (1), it is shown that there is a number \(\eta_\lambda\) such that \(f'\) is strictly decreasing in \((0,\eta_\lambda)\) and strictly increasing in \((\eta_\lambda, \infty)\); this property allows to take \(f'\) as a new independent variable in two different domains and the problem is written in the form of a system of two singular integral equations.
scientific article; zbMATH DE number 5930928
scientific article