On the nonlinear stability of the unsteady, viscous ow of an incompressible uid in a curved pipe

The stability of the ow of an incompressible, viscous uid through a pipe of circular cross-section, curved about a central axis is investigated in a weakly nonlinear regime. A sinusoidal pressure gradient with zero mean is imposed, acting along the pipe. A WKBJ perturbation solution is constructed, taking into account the need for an inner solution in the vicinity of the outer bend, which is obtained by identifying the saddle point of the Taylor number in the complex plane of the cross-sectional angle co-ordinate. The equation governing the nonlinear evolution of the leading order vortex amplitude is thus determined. The stability analysis of this ow to axially periodic disturbances leads to a partial di erential system dependent on three variables, and since the di erential operators in this system are periodic in time, Floquet theory may be applied to reduce this system to a coupled in nite system of ordinary di erential equations, together with homogeneous uncoupled boundary conditions. The eigenvalues of this system are calculated numerically to predict a critical Taylor number consistent with the analysis of Papageorgiou (1987). A discussion of how nonlinear e ects alter the linear stability analysis is also given, and the nature of the instability determined. This research was partially supported by the National Aeronautics and Space Administration under NASA Contract No. NAS1-19480 while the second author was in residence at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 236810001.