Weakly Nonlinear Lower-Branch Stability of Fully Developed and Developing Free-Surface Flows

We examine the weakly nonlinear stability of both fully developedand of developing liquid layers. The study of these free-surfaceflows is more complicated than that of many other flows owingto the fully nonlinear boundary conditions present. The scalingsused for the two problems follow from the work of J. S. B. Gajjar,who described their linear stability properties. We use thetechnique given by F. T. Smith to derive amplitude equationsof the type presented by J. T. Stuart and J. Watson. Both flowsare found to be supercritically stable in general and a varietyof asymptotic cases are considered.     

Upper-Branch Instability of Flow in Pipes of Large Aspect Ratio with Three-Dimensional Nonlinear Viscous Critical Layers

We consider the upper-branch neutral stability of flow in pipesof large aspect ratio, basically extending the work of F. T.Smith to the nonlinear regime. The inclusion of weak nonlinearityleads to an eigenproblem whose solution depends on the propertiesof three-dimensional nonlinear critical layers. Two specialcases are considered. The first is for very small amplitude perturbations, where R is a Reynolds numberbased on the height of the tube and which is assumed large.Then a fully analytical solution of the three-dimensional criticallayers is possible, from which the linear results of Smith maybe deduced. The second case studied is that of flow in a rectangularpipe, where a solution of the nonlinear critical layer problemcan be obtained. Further analysis of neutral modes in this lattercase suggests the possible existence, inter alia, of neutralmodes for finite aspect ratio tubes. These modes depend on thescaled amplitude and have O(1) wavespeeds.     

The effects of suction on the nonlinear stability of a three-dimensional compressible boundary layer

Permanent address (for correspondence), School of Mathematics and Statistics, University of Birmingham, Birmingham B15 2TT, UK. This study follows on from work by Bassom & Seddougui (1992,Proc. R. Soc. Lond. A 436, 405–15) on the effects of suctionon the nonlinear stability of the three-dimensional incompressibleboundary layer induced by a rotating disc. This flow has twotypes of stationary instability, one corresponds to the upper-branchinviscid mode and the other to the lower-branch viscous mode.This latter instability is characterized by an effective velocityprofile which has a zero shear stress at the wall and, as inBassom & Seddougui (1992), it is on this mode that interestis focused here. The effect of suction on the compressible flow,and its subsequent instability, is shown to be significant.