Universal phase and force diagrams for a microbubble or pendant drop in static fluid on a surface

Abstract

Dimensionless three-dimensional universal phase and lift force diagrams of a microbubble (or pendant drop) in static liquid on a solid surface (or orifice) are presented in this work. Microbubble dynamics has been found to play a vital role in mass, momentum, energy, and concentration transfer rates in contemporary micro- and nanosciences and technologies. In this study, dimensionless phase and force diagrams are introduced by utilizing the analytical solutions of the microbubble shape reported in the literature. It shows that phase and force diagrams can be universally specified by two dimensionless independent parameters, Bond number, and contact angle (or base radius). Based on the presence of an inflection point or neck on the microbubble surface, each diagram exhibits three regions. Growth, detachment, and entrapment of a microbubble can be described by path lines in three regions. The corresponding universal total lift forces include hydrostatic buoyancy, difference in gas, and hydrostatic pressures at the base, capillary pressure, as well as surface tension induced by the variation of circumference, which has not been treated in the literature so far. In the absence of viscous stress and Marangoni force, the total lift force equals surface tension induced by the variation of circumference. The latter can be an attaching or lifting force, depending on whether the state in the distinct regions and contact angle is less than or greater than a critical angle. The critical angle, which is slightly less than the inclination angle at the inflection point, is decreased with increasing Bond number.

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“…The smaller bubble radius creates higher internal pressure. As a result, the internal pressures of microbubbles are much higher than those of coarse bubbles and will squeeze the gas inside the bubble making it more likely to dissolve into water. − In addition, the gas–liquid mass transfer efficiency is proportional to the gas–liquid specific surface area, while the specific surface areas of the gas and liquid depend on the gas retention rate ( H 0 ) and the diameters of the bubbles ( d B ) as shown in eq where a is the gas–liquid specific surface, H 0 the gas retention rate, and d B the diameter of the bubble. − With the same gas retention rate H 0 , the mass transfer efficiencies are enhanced in microbubbles because of their smaller diameters. Besides, according to the Stokes’ law, microbubbles with smaller diameters have longer retention times in the liquid phase, as shown in eq where v is the bubble rising rate, μ the viscosity of water, and d the diameter of gas bubble, and ρ 1 and ρ g are the densities of liquid and gas .…”