Scaling laws of top jet drop size and speed from bubble bursting including gravity and inviscid limit

Abstract

Jet droplets from bubble bursting are determined by a limited parametrical space: the liquid properties (surface tension, viscosity, and density), mother bubble size and acceleration of gravity. Thus, the two resulting parameters from dimensional analysis (usually, the Ohnesorge and Bond numbers, Oh and Bo) completely define this phenomenon when both the trapped gas in the bubble and the environment gas have negligible density. A detailed physical description of the ejection process to model both the ejected droplet radius and its initial launch speed is provided, leading to a scaling law including both Oh and Bo. Two critical values of Oh determine two limiting situations: one (Oh 1 =0.038) is the critical value for which the ejected droplet size is minimum and the ejection speed maximum, and the other (Oh 2 =0.0045) is a new critical value which signals when viscous effects vanish. Gravity effects (Bo) are consistently introduced from energy conservation principles. The proposed scaling laws produce a remarkable collapse of published experimental measurements collected for both the ejected droplet radius and ejection speed. * amgc@us.es 1 arXiv:1806.04199v2 [physics.flu-dyn]

Jetting in finite-amplitude, free, capillary-gravity waves

“…In figure 2(c) the formation of a slender jet with the ejection of a droplet from its tip is seen. This jetting seen here from free interfacial oscillations bears a qualitative resemblance to that observed during cavity collapse in bubble bursting (see Kroon 2012;Gañán-Calvo 2017;Lai et al 2018). We present a short review of the jetting literature below, focussing on analytical models and simulations.…”

“…This overshoot is a signature of jetting which the nonlinear theory manages to capture reasonably well. Another interesting feature seen in the upper insets of figures 6(b) and 6(c) is the appearance of a small dimple which is a known precursor to jet formation in other scenarios such as bubble bursting (Deike et al 2018;Gañán-Calvo 2017). Note that this dimple already starts appearing as a broader structure during the earlier downward motion of the cavity and at that instant is well described by the nonlinear theory (bottom insets in figures 6b and 6c).…”

“…Analytical solutions related to jetting have focussed on the surface singularity due to divergence of local curvature (Longuet-Higgins 1983;Zeff et al 2000) or the surface itself (Goodridge, Shi & Lathrop 1996;Shi, Goodridge & Lathrop 1997;Hogrefe et al 1998). Related studies include self-similar solutions close to the singularity (Zeff et al 2000;Duchemin et al 2002) and scaling analysis (Duchemin et al 2002;Gañán-Calvo 2017;Lai et al 2018). An early attempt at modelling these axisymmetric jets was by Longuet-Higgins (1983) who discovered a solution to the axisymmetric Laplace equation with a time-varying free surface.…”

“…The current consensus in the literature is that that the inception of these jets is due to focussing of capillary waves at the axis of symmetry (MacIntyre 1972;Duchemin et al 2002). The detailed physical mechanism of this focussing has been under active debate recently, in particular the role of viscosity in this focussing process as well as the dominant wavelength which gets focussed (Duchemin et al 2002;Gañán-Calvo 2017;Lai et al 2018;Gañán-Calvo 2018a;Gordillo & Rodríguez-Rodríguez 2018). A number of numerical simulations on jetting through bursting bubbles or Faraday waves have utilised the inviscid, irrotational framework employing the boundary integral technique (Boulton-Stone & Blake 1993;.…”

“…For example, these jets may be produced from collapsing bubbles near a solid wall, and being able to control their direction of ejection is a significant step in preventing erosion damage to turbomachinery surfaces (Blake & Gibson 1981). Bursting bubbles at the ocean surface (Veron 2015) or those in a glass of champagne (Ghabache et al 2014a) generate air cavities which collapse producing similar liquid jets which can release drops from their tip (Duchemin et al 2002;Gañán-Calvo 2017, 2018b. At the ocean surface, these ejected drops constitute an important source of sea-salt aerosols (Kientzler et al 1954;Veron 2015) while those in the champagne glass assist in the spread of aroma (Liger-Belair et al 2009;Ghabache et al 2014a;Ghabache & Séon 2016;Séon & Liger-Belair 2017).…”