The flow of a viscous fluid through axially symmetric pipes and symmetrical channels is investigated under the assumption that the Reynolds number is small enough for the Stokes flow approximations to be made. It is assumed that the cross-section of the pipe or channel varies sinusoidally along the length. The flow is produced by a prescribed pressure gradient and by the variation in cross-section that occurs during the passage of a prescribed sinusoidal peristaltic wave along the walls. The theory is applied in particular to two extreme cases, peristaltic motion with no pressure gradient and flow under pressure along a pipe or channel with fixed walls and sinusoidally varying cross-section. Perturbation solutions are found for the stream function in powers of the ratio of the amplitude of the variation in the pipe radius or channel breadth to the mean radius or breadth respectively. These solutions are used to calculate, in particular, the flux through the pipe or channel for a given wave velocity in the first case and for a given pressure gradient in the second case. With a suitable notation it is possible to combine the analysis required for the two cases of pipe and channel flow.

Classical shallow-water theory for the propagation of long waves in running water is modified by the inclusion of the effects of the vorticity present in the main stream as the result of the action of viscosity. When this vorticity is assumed constant, a non-linear theory can be used, but for more general velocity distributions in the main stream it is necessary to linearize the problem.In the linearized theory, a general equation is obtained connecting the wave velocity with the velocity in the undisturbed stream and this is solved in several special cases. It is shown generally that the wave velocity relative to the mean flow is always greater than the value given by the classical theory. The wave velocity relative to the bottom of the stream has two values, one less than the minimum stream velocity and the other greater than the maximum stream velocity.

Some genera of ciliates, such as Oxytricha and Stylonychia, undergo massive genome reorganization during development and provide model organisms to study DNA rearrangement. A common feature of these ciliates is the presence of two types of nuclei: a germline micronucleus and a transcriptionally-active somatic macronucleus containing over 16,000 gene sized “nanochromosomes”. During conjugation the old parental macronucleus disintegrates and a new macronucleus forms from a copy of the zygotic micronucleus. During this process, macronuclear chromosomes assemble through DNA processing events that delete 90-98% of the DNA content of the micronucleus. This includes the deletion of noncoding DNA segments that interrupt precursor DNA regions in the micronucleus, as well as transposons and other germline-limited DNA. Each macronuclear locus may be present in the micronucleus as several nonconsecutive, permuted, and/or inverted DNA segments. Here we investigate the genome-wide range of scrambled gene architectures that describe all precursor-product relationships in Oxytricha trifallax, the first completely sequenced scrambled genome. We find that five general, recurrent patterns in the sets of scrambled micronuclear precursor pieces can describe over 80% of Oxytricha's scrambled genes. These include instances of translocations and inversions, and other specific patterns characterized by alternating stretches of consecutive odd and even DNA segments. Moreover, we find that iterating patterns of alternating odd-even segments up to four times can describe over 96% of the scrambled precursor loci. Recurrence of these highly structured genetic architectures within scrambled genes presumably reflects recurrent evolutionary events that gave rise to over 3,000 of scrambled loci in the germline genome.