The linear stability of steady flow in pipes with circular cross-section and sinusoidal axial variation in diameter is studied by finding global eigenmodes with axial wavelength commensurate with that of the wall corrugation, chosen to be equal to one pipe mean radius. The maximum peak-to-peak height of corrugation considered is approximately 8% of the mean diameter. At low corrugation amplitude and at low Reynolds numbers, the base flow remains attached to the wall, while at larger amplitudes and Reynolds numbers, an axisymmetric separation bubble forms within the corrugation. For all Reynolds numbers considered, flows remain stable to axisymmetric perturbations, but become unstable to standing-wave modes of low azimuthal wavenumber, with critical Reynolds number first falling, then increasing with increasing corrugation height. Both attached and separated flows exhibit similar types of instability modes, which in the case of separated flow are most energetic near the reattachment line of the base flow. The leading instability modes consist of counter-rotating vortices situated near the pipe wall.The stability of fully developed flow in straight-walled pipes of circular section has been extensively studied and it is generally accepted 1-3 that the flow is linearly stable to infinitesimal perturbations, gives rise to moderate linear transient energy growth whose magnitude scales with the square of Reynolds number, 4 supports equilibrium (but unstable) travelling-wave states of finite amplitude, and becomes turbulent in practice at a range of bulk-flow Reynolds numbers Re ¼ U bulk D/ ¼ 4Q/(pD) starting at approximately 2000. The size of flow perturbations which trigger transition falls with Reynolds number, again for Reynolds numbers larger than approximately 2000. 5 The travelling waves are associated with longitudinal rolls and axial streaks and it has recently been suggested that the equilibrium states can be obtained as solutions of a nonlinear eigenproblem involving a coupling of these basic flow elements. 6 All real pipes have some degree of geometric imperfection, which may be randomly distributed (e.g., "commercial roughness"), or more organized, either by manufacture (e.g., machining or deliberately introduced during forming for the case of flexible segmented pipelines and ducts) or from scale deposition in process equipment. It is notable that over a large range of surface roughness, typically observed transition Reynolds numbers remain in the range 2000-3000, 7,8 suggesting that the basic mechanism of instability remains the same in the presence of wall roughness.Compared to the situation for flow perturbation in straight pipes, the effect of wall shape variation on pipe flow stability has received little systematic attention. Arguably, the simplest case for study is presented by a sinusoidal axial variation in diameter where the corrugation amplitude is described by a ¼ 0.5(D max À D min) /D mean. Experimental studies have mainly concentrated on comparatively large corrugation amplitudes a ' 0:3 À 0:...

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Lower back pain is a very common presenting condition, with a large proportion resulting from discogenic causes, especially after strenuous activity. In patients with a history of exertion, lower back pain, and acute urinary retention, the obvious diagnosis to exclude would be cauda equina syndrome. We present a case of a 32-year-old man who presented with lower back pain, bilateral lower limb weakness, and acute urinary retention following a recent episode of heavy lifting. He was subsequently diagnosed with rhabdomyolysis. This case highlights that rarer conditions can masquerade as cauda equina syndrome, and even in seemingly straightforward presentations, alternative diagnoses should also be considered.