A Note on Dual Extremum Principles in Magnetohydrodynamic Pipe Flow

A direct application of dual extremum principles to the linearequations for magneto-hydrodynamic pipe flow is described. AHamiltonian for the system is constructed and it is shown thatit is a saddle functional with respect to two particular vectorsone consisting of the fluid velocity and the gradients of theinduced magnetic field and the other the induced field and thegradients of the velocity. The generalization of the techniqueto certain linear boundary-value problems is also included.     

The Reflection of Surface Waves by Two Parallel Vertical Barriers

A wave on the free surface of a liquid is incident on two parallelbarriers, each of infinite length and completely submerged tothe same depth. Here we calculate the coefficients of reflectionand transmission for this configuration to the second orderwhen the distance between the plates is large.     

Amplitude-dependent Stability of Boundary-layer Flow with a Strongly Non-linear Critical Layer

The amplitude-dependent neutral stability properties, mainlyof an accelerating boundary-layer flow, are studied theoreticallyfor large Reynolds numbers when the disturbance size is sufficientlylarge to provoke a strongly non-linear critical layer withinthe flow field. The theory has a rational basis aimed at a detailedunderstanding of the delicate physical balances controllingstability. It shows that when the fundamental disturbance size rises to O(R^-1/3, where R is the Reynolds number based on theboundary-layer thickness, the neutral wavelength shortens andthe wavespeed increases in such a way that they become comparablewith the typical thickness and speed, respectively, of the basicflow. In this Rayleigh-like situation a new (previously negligible)feature emerges, that of a substantial pressure variation acrossthe critical layer, which strongly affects the jump conditionson the Rayleigh solutions holding outside the critical layer.As a result of the strong non-linearity the total velocity jumpis affected non-linearly by the critical layer vorticity, whilein contrast the phase shift remains linearly dependent on thevorticity. Furthermore, it is shown that the phase shift, notthe total velocity jump, dictates the neutral stability criteria. Also, flow reversal occurs near the wall where the disturbanceis greater than the basic flow. The link between the viscouseffects in the wall layers and in the critical layer fixes theamplitude-dependence of the neutral modes throughout. As thedisturbance amplitude increases the critical layer with vorticitytrapped within it moves toward the edge of the boundary layerand is forced to leave the boundary layer when exceeds O(R^-1/3,if neutral stability is to be maintained. This departure israther abrupt, involving a dependence on (scaled amplitude)^–12.A study of the more practical application to temporally growingdisturbances should be interesting.     

PHOTOSENSITIZATION WITH DERIVATIVES OF CHLOROPHYLL

The properties of several chlorophyll derivatives were examined: the methyl esters of pheophorbide A, pheophorbide B and pheophytin. In spite of structural differences, all products were equally effective sensitizers in vitro and were localized equally well by murine tumors in vivo after 1 h. But only the pheophytins persisted at neoplastic loci for 24 h. There was no evidence of hydrolysis of the methyl esters, but the phytyl ester linkage was labile in vivo.     

High-resolution Fourier-transform infrared spectroscopy of the ν1 and ν8 fundamental bands of sulphur tetrafluoride (SF4)

The vibration-rotation spectra of the ν₁ and ν₈ fundamental bands of ³²SF₄ have been observed using Fourier-transform infrared spectroscopy. The band centre of the c-type ν₁ symmetric sulphur-equatorial-fluorine stretching vibration was observed at 891.6 cm^?1 and that for the b-type ν₈ asymmetric sulphur-equatorial-fluorine stretching vibration at 864.6 cm^?1. In total, 2044 rovibrational transitions have been assigned. Analysis of the spectra showed that the rotational states of the ν₁ = 1 and ν₈ = 1 upper vibrational levels are coupled by an a-type Coriolis interaction. This coupling has been treated both using perturbation theory and by the explicit inclusion of an appropriate Hamiltonian matrix element in a combined fit of the data for both bands. Spectroscopic parameters have been determined for the ground, ν₁ = 1 and ν₈ = 1 vibrational levels. Weaker transitions resulting from difference bands and the fundamental bands of the ³⁴SF₄ isotopomer have been identified but could not be assigned, because of the density of lines in the room-temperature spectrum. The possibility that discrepancies between the observed and predicted spectra of the ν₁ fundamental may result from either a Coriolis interaction with the states of another vibrational level, or the effects of intramolecular exchange of axial and equatorial fluorine atoms is considered. The discussion is supported by theoretical calculations which show that the likely path for intramolecular exchange is via a C _4v transition state.     

Optimum use of holding tanks to avoid pollution surges in narrow estuaries

It is common practice to discharge sewage and industrial effluentsinto estuaries. The absence of an advective current at the turnof the tide may allow the immediate pollution level close tothe source (or the subsequent dissolved oxygen sag) to buildup to unacceptably high levels. One way of alleviating the worstexcesses is to use holding tanks and to vary the rate of discharge.This paper gives explicit formulae for the optimal rate of dischargein narrow estuaries to match the dilution capacity of the tidalflow and thereby to minimize the maximum pollution level. Itis also shown how oxygenation of the discharge can be optimizedto satisfy a specified limit upon the permissible maximum oxygensag.