Two-sphere System Research Articles

Hydrodynamic interaction of two unequal-sized spheres in a slightly rarefied gas: resistance and mobility functions

The problem of the hydrodynamic interaction of two unequal-sized spheres in a slightly rarefied gas is treated following the singular perturbation scheme of Sone & Onishi (1978), valid at small, but finite, particle Knudsen numbers. In this method the solution to the linearized BGKW transport equation governing the gas molecular motion consists of two parts: one describing a Knudsen layer where the actual microscopic boundary conditions are applied and the other describing a Hilbert region where the Stokes equations of continuum hydrodynamics hold. The Knudsen-layer solution establishes the ‘slip’ boundary conditions for the Stokes equations. Here we clearly distinguish between particle ‘slip’ due to the type of boundary conditions and particle ‘slip’ due to lengthscale effects as measured by the Knudsen number. The present analysis has been carried out to first order in particle Knudsen number for the case of diffuse reflective molecular boundary conditions. General relationships between the first- and zero-order velocity fields, both of which are written in the form of Lamb's (1932) solution to the Stokes equation, are established. It is illustrated how these general relationships can be used to determine the force and torque acting on a single sphere translating and rotating in a slightly rarefied gas. Finally, we have treated the two-sphere problem in a slightly rarefied gas using the twin multipole expansion method of Jeffrey & Onishi (1984). Here again, general relationships are established between the solutions of the first-order fluid velocity field and the zero-order velocity field, the latter being shown to recover Jeffrey & Onishi's results for stick boundary conditions. These general relationships are subsequently used to determine the complete resistance and mobility matrices of the two-sphere system. The symmetric properties of the resistance and mobility matrices are demonstrated for slip boundary conditions, in agreement with the general proof of Landau & Lifshitz (1980) and Bedeaux, Albano & Mazur (1977).

Double-layer interaction energy for two unequal spheres

The potential distribution surrounding two unequal spheres of any surface potential has been obtained by a finite element solution of the Poisson–Boltzmann equation in bispherical coordinates using constant potential boundary conditions. The electrostatic energy of the two-sphere system was obtained from the potential distribution. This method is completely general. It is not restricted to “plate-like” particles or to small surface potentials. A comparison is made between this method of calculating the double-layer interaction energy and other methods which utilize the parallel-plate solution to the Poisson–Boltzmann equation. Both methods give linear results when log(energy) is plotted against separation. However, this method gives a more negative slope than the one-dimensional methods.

Experimental results of dependent light scattering by two spheres

Microwave analog measurements of the forward scattering produced by two spheres yield the extinction as a function of sphere separation for the case of one sphere's being shadowed by the other. The effects of dependent scattering are obvious up to a separation distance of about ten sphere diameters. Side-scattering measurements show a resonance when the axis of the two-sphere system is in the scattering plane and bisects the scattering angle. The magnitude of at least one measured resonance is a factor of 44 larger than the scattering that is due to a single sphere.

Influence of Electrode Curvature on Electrical Breakdown in Vacuum

It has been observed by others that a decrease in curvature of one (or both) electrode(s) of a vacuum gap can lead to either an increase or decrease, depending upon the gap parameters, in the breakdown voltage. This electrode curvature effect is investigated theoretically for a system consisting of two isolated spheres. For each sphere the maximum surface field intensity and the area over which the field magnitude is no less than a given fraction of this maximum are found. These results are considered in conjunction with several theories of vacuum breakdown in order to predict the expected dependence of breakdown voltage upon electrode curvature at varying gap lengths. Comparisons with applicable published experimental results are made. The following conclusions are drawn: (1) The cathode crossover effect, that is, the observation that a decrease in curvature of the cathode can produce a decrease in breakdown voltage at short gap lengths and an increase in breakdown voltage at long gap lengths, can be predicted explicitly for a two-sphere system using any major vacuum breakdown theory. This crossover is confirmed experimentally. (2) The anode crossover effect is experimentally supported, but not as firmly as the cathode crossover. The theoretical analysis suggests that this anode crossover effect results from a transition, at a critical gap length, in the dominant mechanism in electrical breakdown in vacuum. (3) The increase in breakdown voltage accompanying an increase in curvature of one (or both) electrode(s) at short gap lengths is explained on the basis of a decrease in breakdown area (sites).