Numerical Study on the Rising Motion of Bubbles near the Wall

Based on the volume of fluid method (VOF), the rising characteristics of bubbles in near-wall static water are studied. In this study, the influence of the wall on the rising motion of the bubble was studied by changing the distance of the bubble wall, the diameter of the bubble, the arrangement of the bubble and the size ratio, etc. The influence is expressed as the average swing amplitude of the “Z”-shaped motion when the bubble rises. The study found that in the case of a single bubble, the wall surface has a certain influence on the rise of the bubble, and its degree is affected by the bubble wall distance and the bubble diameter. The influence of bubble wall distance is more obvious. The greater the bubble wall distance, the less the bubble is affected by the wall; in the case of double bubbles, the influence of the interaction force between the bubbles is significantly greater than the wall surface.

Symmetric double bubbles in the Grushin plane

We address the double bubble problem for the anisotropic Grushin perimeter Pα, α ≥ 0, and the Lebesgue measure in ℝ2, in the case of two equal volumes. We assume that the contact interface between the bubbles lies on either the vertical or the horizontal axis. We first prove existence of minimizers via the direct method by symmetrization arguments and then characterize them in terms of the given area by first variation techniques. Even though no regularity theory is available in this setting, we prove that angles at which minimal boundaries intersect satisfy the standard 120-degree rule up to a suitable change of coordinates. While for α = 0 the Grushin perimeter reduces to the Euclidean one and both minimizers coincide with the symmetric double bubble found in Foisy et al. [Pacific J. Math. 159 (1993) 47–59], for α = 1 vertical interface minimizers have Grushin perimeter strictly greater than horizontal interface minimizers. As the latter ones are obtained by translating and dilating the Grushin isoperimetric set found in Monti and Morbidelli [J. Geom. Anal. 14 (2004) 355–368], we conjecture that they solve the double bubble problem with no assumptions on the contact interface.

Double Bubbles on the Real Line with Log-Convex Density

The classic double bubble theorem says that the least-perimeter way to enclose and separate two prescribed volumes in ℝN is the standard double bubble. We seek the optimal double bubble in ℝN with density, which we assume to be strictly log-convex. For N = 1 we show that the solution is sometimes two contiguous intervals and sometimes three contiguous intervals. In higher dimensions we think that the solution is sometimes a standard double bubble and sometimes concentric spheres (e.g. for one volume small and the other large).