Addendum to "A Variational Principle Associated with Surface Waves"

A second variational principle is given for the general linearinitial-boundary value problem associated with small amplitudecapillary-gravity waves on an incompressible viscous fluid.     

Curvilinear Co-ordinate Systems Associated with a Class of Boundary-Value Problems

A system of non-linear integrability equations is derived whichis associated with the differential form of a transformationfrom Cartesian co-ordinates to non-orthogonal curvilinear co-ordinates.A solution for this system is established when the curvilinearco-ordinate system contains two identical scaling factors andone right angle provided the unit normal to a smooth, finiteor infinite tube is prescribed. The general form of the transformationis obtained. It is shown that the transformation can also beobtained when the unit normal to the tube boundary section bythe plane normal to the given curve defining the orientationof the tube is prescribed. Moreover conditions are establishedunder which a completely orthogonal co-ordinate system can befound. An example is treated for a tube with circular cross-sectionand both the non-orthogonal and orthogonal co-ordinate systemsassociated with this are discussed.     

A Variational Principle Associated with Surface Waves

A variational principle is established for the general linearinitial-boundary value problem which is associated with small-amplitudewaves on the surface of an incompressible fluid. The effectsof surface tension and viscosity are included in the formulation.     

A Note on Some Variational Principles for a Class of Linear Initial-Boundary Value Problems

The method of time convolutions is used to derive variationalprinciples for some linear initial-boundary value problems inwhich time derivatives of the dependent variables appear explicitlyin the boundary conditions. The formulation includes the initialboundary value problems associated with the propagation of surfacewaves and the underwater transmission of sound.     

Interaction induced Raman light scattering as a probe of the local density structure of binary supercritical solutions

Interaction induced Raman light scattering is presented as a unique tool for the understanding of solvation processes from the solute's point of view in weakly interacting solute-solvent systems. A review of pertinent literature shows that this technique should be useful at least in single-phase binary mixtures such as supercritical solutions. Methane is used here as a probe molecule at 10mol% concentration (as the solute) and 90mol% CO and CO₂ are the solvents. The light scattering results, i.e., the dependence of the anisotropic intensities divided by density (I/d) on the density, are interpreted by use of the Duh-Haymet-Henderson closure (bridge) function of the Ornstein-Zernike integral equation. These data, together, are examined in the context of known supercritical solution thermodynamics and statistical mechanical results. It is shown that the light scattering I/d data versus density yield maxima in both attractive and repulsive solute-solvent systems. The local number density maxima were found near these same densities by the integral equation calculations for both methane + carbon monoxide or carbon dioxide using only Lennard-Jones single-centre parameters as input. The methane + carbon monoxide system is identified as weakly attractive (augmenting), whereas the methane + carbon dioxide system is identified as repulsive (avoidance).