Dynamics and Stability of a Flexible Cylinder in a Narrow Coaxial Cylindrical Duct Subjected to Annular Flow

This paper considers analytically the dynamics of a flexible cylinder in a narrow coaxial cylindrical duct, subjected to annular flow. In the present analysis, in contrast to existing theory, the viscous forces are not derived by an adaptation of Taylor’s unconfined-flow relationships, but by a systematic, albeit approximate, solution of the Navier-Stokes equations, which accounts for the unsteady viscous effects much more fully than heretofore; it is found that, for very narrow annuli, the contribution of these unsteady viscous forces to the overall unsteady forces on the cylinder can be much larger than that of the steady skin friction and pressure-drop effects alone. The present analysis also differs from existing theory in that the in-viscid forces are not derived via the slender-body approximation, and hence the analysis is also applicable to bodies of relatively small length-to-radius ratio. The dynamics and stability of typical systems with fixed ends is investigated, concentrating mainly on viscous effects and comparing the results with those of previous work. It is found that, as the annular gap becomes narrower, the system loses stability by divergence at smaller flow velocities, provided the gap size is such that inviscid fluid effects are dominant. For very narrow annuli, however, where viscous forces predominate, this trend is reversed, and further narrowing of the annular gap has a stabilizing effect on the system; furthermore, in some cases the system loses stability by flutter rather than divergence.