ON THE STABILITY OF STEADY SURFACE-TENSION DRIVEN FLOWS

Abstract

In this paper, we study the stability of similarity solutions for the problem $f'''+Q(Aff''-(f')^{2})=\beta$, with $Q\gt 0$, $A\geq 1$ and $\beta \in \Bbb R$. The given problem was derived from the symmetric reduction of similarity transformations from the Navier-Stokes equation for the planar flows. By imposing additional eigenvalue problems, our numerical studies show that the resultant steady flows are unstable as $Q$ becomes large for various $A\lt 2$. Furthermore, our analytical result gives that the steady flows are stable for small $Q$, when $1\leq A\lt 2$, or for any $Q\gt 0$ when $A\geq 2$. Moreover, the existence of asymmetric flows for various $A\lt 2$ is also found numerically.