A spectral collocation and matrix eigenvalue method is used to study the linear stability of the trailing line vortex model of Batchelor. For both the inviscid and viscous stability problem, the entire unstable region in the swirl/axial wavenumber parameter space is mapped out for various azimuthal wavenumbers m. In the inviscid case, the non-axisymmetric perturbation with azimuthal wavenumber m = 1 has an unstable region of larger extent than any other, with an unusual two-lobed structure; also, the location and numerical value of the maximum disturbance growth rate previously reported for this case are shown to be incorrect. Exploiting the increasingly localized structure of perturbation eigenfunctions allows accurate results to be obtained up to values of m more than 3 orders of magnitude larger than previously, and the results for the most unstable mode are in excellent agreement with the asymptotic theory of Leibovich & Stewartson. A viscous analysis of these fundamentally inviscid modes reveals that the critical Reynolds number at which instability first occurs increases as O(m2) for m [Gt ] 1, and finds the critical values of swirl and wavenumber, which approach limiting values as m → ∞.In the viscous case, the instabilities for m = 0 and 1 recently reported by Khorrami are found via a simplified numerical approach and the entire unstable region for each of these modes is mapped out over a wide range of Reynolds numbers. The critical Reynolds numbers for these modes are found to be 322.42 and 17.527, respectively, the latter having been unreported previously. The instabilities persist in the limit of large Reynolds number, with corresponding disturbance growth rates decreasing roughly as 1/Re. In addition to the primary mode, a new family of long-wave viscous instabilities is found for the m = 1 case.

Abstract. We have shown by machine proof that F 24 = 2 2 24 + 1 is composite. The rigorous Pépin primality test was performed using independently developed programs running simultaneously on two different, physically separated processors. Each program employed a floating-point, FFT-based discrete weighted transform (DWT) to effect multiplication modulo F 24 . The final, respective Pépin residues obtained by these two machines were in complete agreement. Using intermediate residues stored periodically during one of the floating-point runs, a separate algorithm for pure-integer negacyclic convolution verified the result in a "wavefront" paradigm, by running simultaneously on numerous additional machines, to effect piecewise verification of a saturating set of deterministic links for the Pépin chain. We deposited a final Pépin residue for possible use by future investigators in the event that a proper factor of F 24 should be discovered; herein we report the more compact, traditional Selfridge-Hurwitz residues. For the sake of completeness, we also generated a Pépin residue for F 23 , and via the Suyama test determined that the known cofactor of this number is composite.

Evidence is presented for what appears to have been nonmodal disturbance growth in a pipe-flow experiment performed by Kaskel [Technical Report No. 32-138, Jet Propulsion Laboratory, California Institute of Technology, 1961], based on a comparison of the experimental traces of disturbance amplitude with predictions of transient amplitude growth calculated from the underlying linear stability operator. Owing to the geometry of the experimental disturbance generator, modes having azimuthal wavenumbers n=0, 6 and 12 are expected to contribute the bulk of the disturbance energy, and numerically predicted amplitude growth versus downstream distance is in good agreement with the experiment.

Results are presented for a class of self-similar solutions of the steady, axisymmetric Navier–Stokes equations, representing the flows in slender (quasi-cylindrical) vortices. Effects of vortex strength, axial gradients and compressibility are studied. The presence of viscosity is shown to couple the parameters describing the core growth rate and the external flow field, and numerical solutions show that the presence of an axial pressure gradient has a strong effect on the axial flow in the core. For the viscous compressible vortex, near-zero densities and pressures and low temperatures are seen on the vortex axis as the strength of the vortex increases. Compressibility is also shown to have a significant influence upon the distribution of vorticity in the vortex core.

Results are presented for a new class of selfsimilar solutions to the steady, axisymmetric Navier-Stokes equations, modelling the flows in vortices whose viscous cores grow proportionally to an arbitrary power of the axial coordinate. Effects of Reynolds number, growth rate, free-stream Mach number and vortex strength are investigated. Results for incompressible vortices are compared to experimental results for leading-edge vortices and to the inviscid limiting case. It is shown that the core of an incompressible vortex should, in absence of an axial pressure gradient, grow with the onehalf power of the axial coordinate. The total pressure level on the axis of the vortex is seen to 6e independent of Reynolds number and assumed growth rate, although the distribution of total pressure in the core is dependent on both. Large axial velocity surpluses are seen in the core as well as strong pressure gradients. Results for the viscous compressible vortex are compared to the inviscid potential and rotational model flows. Near-zero densities and pressures and very low temperatures are seen on the axis for reasonable freestream Mach numbers and vortex strengths. The presence of viscosity prevents the density from actually reaching zero, unlike with the inviscid models. Limiting velocity concepts are found to restrict the possible vortex strengths for a given freestream Mach number. Compressibility is shown to have a significant influence upon the distribution of vorticity in the core of the vortex.