I. Singular perturbation problems involving singular points and turning points. II. On the averaged Lagrangian technique for nonlinear dispersive waves

In Part I a class of linear boundary value problems is considered which is a simple model of boundary layer theory. The effect of zeros and singularities of the coefficients of the equations at the point where the boundary layer occurs is considered. The usual boundary layer techniques are still applicable in some cases and are used to derive uniform asymptotic expansions. In other cases it is shown that the inner and outer expansions do not overlap due to the presence of a turning point outside the boundary layer. The region near the turning point is described by a two-variable expansion. In these cases a related initial value problem is solved and then used to show formally that for the boundary value problem either a solution exists, except for a discrete set of eigenvalues, whose asymptotic behaviour is found, or the solution is non-unique. A proof is given of the validity of the two-variable expansion; in a special case this proof also demonstrates the validity of the inner and outer expansions.

Nonlinear dispersive wave equations which are governed by variational principles are considered in Part II. It is shown that the averaged Lagrangian variational principle is in fact exact. This result is used to construct perturbation schemes to enable higher order terms in the equations for the slowly varying quantities to be calculated. A simple scheme applicable to linear or near-linear equations is first derived. The specific form of the first order correction terms is derived for several examples. The stability of constant solutions to these equations is considered and it is shown that the correction terms lead to the instability cut-off found by Benjamin. A general stability criterion is given which explicitly demonstrates the conditions under which this cut-off occurs. The corrected set of equations are nonlinear dispersive equations and their stationary solutions are investigated. A more sophisticated scheme is developed for fully nonlinear equations by using an extension of the Hamiltonian formalism recently introduced by Whitham. Finally the averaged Lagrangian technique is extended to treat slowly varying multiply-periodic solutions. The adiabatic invariants for a separable mechanical system are derived by this method.

Forced convection film boiling of subcooled water around a sphere

An experimental investigation was made of forced convection film boiling of subcooled water around a sphere at atmospheric pressure. The water was sufficiently cool that the vapor condensed before leaving the film with the result that no vapor bubbles left the film. The experimental runs were made using inductively heated spheres at temperatures above 740°C. and using inlet water temperatures between 15°C. and 27°C. The spheres used had diameters of 1/2 inch, 9/16 inch, and 3/8 inch and were supported by the liquid flow. Reynolds numbers between 60 and 700 were used.

Analysis of the collected non-condensables indicated that oxygen and nitrogen dissolved in the water accumulated within the vapor film and that hetrogeneous chemical reactions occurred at the sphere surface. An iron-steam reaction resulted in more than 20% by volume hydrogen in the film at wall temperatures above 900°C. At temperatures near 1100°C. more than 80% by volume of the film was composed of hydrogen. It was found that gold plating of the sphere could eliminate this reaction.

Material and energy balances were used to derive equations which may be used to predict the overall average heat transfer coefficients for subcooled film boiling around a sphere. These equations include the effect of dissolved gases in the water. Equations also were derived which may be used to predict the composition of the film for cases in which an equilibrium exists between the dissolved gases and the gases in the film.

The derived equations were compared to the experimental results. It was found that a correlation existed between the Nusselt number for heat transfer from the vapor-liquid interface into the liquid and the Reynolds number, liquid Prandtl number product. In addition, it was found that the percentage of dissolved oxygen removed during the film boiling could be predicted to within 10%.

Dielectric recovery of short A-C arcs between low-boiling-point electrodes

The re-ignition characteristics (variation of re-ignition voltage with time after current zero) of short alternating current arcs between plane brass electrodes in air were studied by observing the average re-ignition voltages on the screen of a cathode-ray oscilloscope and controlling the rates of rise of voltage by varying the shunting capacitance and hence the natural period of oscillation of the reactors used to limit the current. The shape of these characteristics and the effects on them of varying the electrode separation, air pressure, and current strength were determined.

The results show that short arc spaces recover dielectric strength in two distinct stages. The first stage agrees in shape and magnitude with a previously developed theory that all voltage is concentrated across a partially deionized space charge layer which increases its breakdown voltage with diminishing density of ionization in the field-tree space. The second stage appears to follow complete deionization by the electric field due to displacement of the field-free region by the space charge layer, its magnitude and shape appearing to be due simply to increase in gas density due to cooling. Temperatures calculated from this second stage and ion densities determined from the first stage by means of the space charge equation and an extrapolation of the temperature curve are consistent with recent measurements of arc value by other methods. Analysis or the decrease with time of the apparent ion density shows that diffusion alone is adequate to explain the results and that volume recombination is not. The effects on the characteristics of variations in the parameters investigated are found to be in accord with previous results and with the theory if deionization mainly by diffusion be assumed.