On interdependence of instabilities and average drop sizes in bag breakup

Abstract

A drop exposed to cross flow of air experiences sudden accelerations which deform it rapidly ultimately proceeding to disintegrate it into smaller fragments. In this work, we examine the breakup of a drop as a bag film with a bounding rim resulting from acceleration induced Rayleigh-Taylor instabilities and characterized through the Weber number, \textit{We}, representative of the competition between the disruptive aerodynamic force imparting acceleration and the restorative surface tension force. Our analysis reveals a previously overlooked parabolic dependence ($\sim We^2$) of the combination of dimensionless instability wavelengths $({\barλ}_{bag}^2/ {\barλ}_{rim}^4 {\barλ}_{film})$ developing on different segments of the deforming drop. Further, we extend these findings to deduce the dependence of the average dimensionless drop sizes for the rim, $\langle{\bar{D}}_{rim}\rangle$ and bag film, $\langle{\bar{D}}_{film}\rangle$ individually, on $We$ and see them to decrease linearly for the rim ($\sim We^{-1}$) and quadratically for the bag film ($\sim We^{-2}$). The reported work is expected to have far-reaching implications as it provides unique insights on destabilization and disintegration mechanisms based on theoretical scaling arguments involving the commonly encountered canonical geometries of a toroidal rim and a curved liquid film.

Figures (5)

• Figure 1: (a) (i) Initial undeformed drop (diameter, $D_0$) before it enters the air stream blowing from left to right and shown by hollow block arrows (ii) Flattening of the drop into a disc-like shape of cross stream dimension, $D_{bag}$ due to air impact and flattening in the radial (r) and streamwise (z) direction (iii) Incipience of Rayleigh-Taylor instability on $D_{bag}$ due streamwise acceleration, $\xi_{bag}$ and separation of drop into two distinct length scales, $h_{rim}$, and $h_{film}$ with $h_{rim}$ also accelerated radially, $\xi_{rim}$ (shown by blue arrows) (iv) Acceleration, $\xi_{film}$ of finite thickness ($h_{film}$) curved film (shown by blue arrows) (v) Instabilities on curved bag film due to radial acceleration, $\xi_{film}$ (vi) Bursting of curved liquid bag film (vii) Fragmentation of rim (b) 3D rendering of discoid (left) with the cut section (right) showing the different length scales, $h_{rim}$, $D_{bag}$ and $h_{film}$ (c) Illustration showing the approximation of the unstable curved film to one with flat interfaces since radius of the film, $\textcolor{black}{R_{film}} >> h_{rim}$, refer (a)-(v) (d) Schematic showing waves on rim, $\lambda_{rim}$ from z direction, refer (a)-(vi). Image sequence corresponds to $We \approx 16$ and 40% by weight glycerine-water solution. Dashed black line shows the trajectory of the deforming drop.
• Figure 2: Relation between the Rayleigh-Taylor instability dimensionless wavelengths on the drop (${\bar{\lambda}}_{bag}$), rim (${\bar{\lambda}}_{rim}$) and curved film (${\bar{\lambda}}_{film}$) with $We$ on log-log scale with an $R^2 \approx 0.97$. Inset shows $\forall_{rim}/ \forall_{film} \approx$ constant across all $We$ accompanied by a sketch to show accurate calculation of $\forall_{film}$ by considering difference between $\forall_{drop}$ and $\forall_{rim}$.
• Figure 3: (a) Initial bag (t = 12 ms) with formation of hole (t = 14 ms) triggering retraction of the curved film to form a "rolling rim" (t = 17 ms) which forms ligaments shedding droplets, zoomed view in dotted box (b) Schematic of curved bag film bursting leading to drop production (i) Initial curved film of radius, $R_{film}$ with a hole and rolling toroidal rim (ii) Approximation of the curved film as a flat sheet since $R_{film} >> h_{film}$ (iii) Formation of drops of size, $\langle D_{film}\rangle$ from ligaments of size, $d_{lig}$ produced from the rolling rim of diameter, $d_{roll}$ (c) Rim distortions of wavelength $\lambda_{rim}$ after bursting of the curved film (t = 24 ms) followed by emergence of bulbous regions due to capillary instability (t = 28 ms) finally resulting in a ring of spherical drops (t = 32 ms) (d) Schematic of rim breakup into drops (i) Approximation of toroidal ring as a flat interface since $h_{rim} << R_{max}$ (ii) Volume in one wavelength, $\lambda_{rim}$ equated to that of drops of average diameter, $\langle D_{rim}\rangle$. Test conditions, 50% glycerine-water, $We \approx 16$.
• Figure 4: Rim and bag drop sizes as a function $We$ derived theoretically in eqns \ref{['Eq5']} and \ref{['Eq6']} on semi-log scale. The average rim drops sizes, $\langle\overline{D}_{rim}\rangle$ decrease linearly whereas the average curved film sizes, $\langle\overline{D}_{film}\rangle$ decrease quadratically. Legend is same as Fig. \ref{['Fig2']}
• Figure :