The resistance to oscillatory motions of arbitrary wavelengths in an infinitely dilute lattice of identical spheres, immersed in a viscous fluid, is calculated from the linearized Navier–Stokes equation to lowest order in fluid inertia and sphere-volume fraction. The application we have in mind is to analyse the hydrodynamic modes in colloidal crystals (a lattice of Brownian particles repelling each other electrically), although other applications are possible. We find that the friction per particle for both compressional and transverse shear modes is close to the Stokes value at short wavelengths, whereas at long wavelengths fluid backflow within the lattice is important and causes the friction to increase for compressional modes. For shear modes, in which backflow is not present, the friction decreases from the Stokes value at short wavelengths to zero at long wavelengths. At sufficiently long wavelengths, when the shear-mode friction becomes small enough, propagating viscoelastic modes are possible in a lattice with elastic forces between spheres. Fluid inertia is most important for long-wavelength transverse motions, since a significant amount of fluid mass gets carried along by each particle. Explicit results for a bcc lattice are presented along with interpolation formulas, and the pertinence of these results to colloidal crystals is discussed. Finally, the effects of constraining walls are explored by considering a one-dimensional lattice near a wall. Backflow imposed by the wall increases the friction factors for the lattice modes, showing that propagating modes are unlikely in colloidal crystals that are confined to a cell thinner than a critical length.

An analysis is presented for electrophoretic motion of a charged non-conducting sphere in the proximity of rigid boundaries. An important assumption is that κa → ∞, where a is the particle radius and κ is the Debye screening parameter. Three boundary configurations are considered: single flat wall, two parallel walls (slit), and a long circular tube. The boundary is assumed a perfect electrical insulator except when the applied field is directed perpendicular to a single wall, in which case the wall is assumed to have a uniform potential (perfect conductor). There are three basic effects causing the particle velocity to deviate from the value given by Smoluchowski's classic equation: first, a charge on the boundary causes electro-osmotic flow of the suspending fluid; secondly, the boundary alters the interaction between the particle and applied electric field; and, thirdly, the boundary enhances viscous retardation of the particle as it tries to move in response to the applied field. Using a method of reflections, we determine the particle velocity for a constant applied field in increasing powers of λ up to O(λ6), where λ is the ratio of particle radius to distance from the boundary. Ignoring the O(λ0) electro-osmotic effect, the first effect attributable to proximity of the boundary is O(λ3) for all boundary configurations, and in cases when the applied field is parallel to the boundaries the electrophoretic velocity is proportional to ζp − ζw, the difference in zeta potential between the particle and boundary.

A screen of closely spaced, parallel, thin wires was placed downstream of a grid generating nearly isotropic turbulence. The screen was normal to the flow and was heated in one of two modes: (1) periodically in time, to generate a train of transversely uniform streamwise thermal ramps, each with a uniform streamwise gradient, and (2) steadily, with transverse non-uniformity, to generate a uniform transverse thermal ramp. The simple temperature and temperature-gradient fluctuation statistical properties in both cases were found to be comparable to those encountered in earlier works with a steadily heated grid producing a uniform transverse thermal ramp. In both modes of heating the temperature fluctuations decreased initially behind the screen and then increased monotonically. The turbulent-heat-transfer correlation coefficient attained an asymptotic magnitude between 0.7 and 0.8 for both modes of heating. The skewness of the temperature-fluctuation derivative in the direction of the mean gradient was founded to be non-zero despite the absence of mean shear.

I have used a novel approach based upon Hamiltonian mechanics to derive new equations for nearly geostrophic motion in a shallow homogeneous fluid. The equations have the same order accuracy as (say) the quasigeostrophic equations, but they allow order-one variations in the depth and Coriolis parameter. My equations exactly conserve proper analogues of the energy and potential vorticity, and they take a simple form in transformed coordinates.

Exact results are derived for the centroid and longitudinal variance of a passive contaminant distribution at large times after an instantaneous discharge in an oscillatory flow in a straight channel of constant cross-section. It is shown that the precise timing and cross-stream position of the discharge can have a substantial and persistent influence.