136
G. Biswas and K. C. Sahu
and experiments at low Reynolds numbers (Re < 0.01). They found that a bubble
with a small degree of nonsphericity, eventually becomes spherical, but a bubble with
a large degree of nonsphericity continues to deform with an increasing amplitude.
Later Refs. [78, 79] focussed on the decay rate of the oscillations for a viscous drop
in micro-gravity condition. Tsamopoulos and Brown [80] studied the effect of initial
amplitude on frequency for nonlinear inviscid oscillations of droplets. The shape
deformation of droplets in simple shear flows and in complex geometries with and
without external forcing has also been investigated by several researchers (see for
instance, Refs. [81, 82]).
6.4 Droplet Breakups
The phenomenon of liquid jet breakup is observed in many industrial applications,
such as ink-jet printing, particle sorting, atomization, mixing, combustion, separation
and spraying technologies [83]. Due to the Rayleigh–Plateau instability [84], a liquid
jet issued from an orifice becomes unstable and breaks up leading to formation of
satellite droplets due to the propagation of surface perturbations. The classical studies
of Plateau [84], Savart [85] and Rayleigh [47] provides a fundamental understanding
of the jet instabilities. Weber [86] investigated the influence of liquid viscosity and gas
density on jet breakup phenomenon. Goedde and Yuen [87] investigated the capillary
instability of water–glycerine interface. They showed that the nonlinear effects dominate the jet breakup process leading to formation of ligaments and satellite droplets.
The nozzle characteristics can also affect the breakup length of the jet occurring due
to the Rayleigh instability [88]. Unlike the aforementioned studies, which considered circular liquid jets, Kashyap et al. [89] and Farvardin and Dolatabadi [90] also
investigated the dynamics of elliptical-orifice liquid jets. It is found that for different
combinations of surface tension, inertia and aerodynamic interactions, four distinct
breakup regimes can be observed. They are (i) the Rayleigh breakup regime (varicose
perturbations), (ii) the first wind-induced breakup (sinuous perturbations), (iii) the
second wind-induced breakup and (iv) the atomization (spray) regime [91–94]. The
numerical simulations of liquid jets were conducted by Pan and Suga [95, 96] to
investigate the characteristics of breakup. The breakup length of a turbulent liquid
jet was studied by Lafrance [97].
In addition to the breakup of a liquid jet aligned with gravity (as discussed above),
Borthakur et al. [98, 99] investigated the dynamics of curved slender jets due to their
implications in the production of nanofibres from centrifugal spinning [100–102].
The variation of jet diameter along length for injections along the gravity (parallel
injection) and orthogonal to gravity (perpendicular injection) at t = 15 and t = 40
is shown in Fig. 12a and b, respectively. It can be seen that the pinch-up occurs early
in the case of perpendicular injection as compared to those in the parallel injection.
G. Biswas and K. C. Sahu
and experiments at low Reynolds numbers (Re < 0.01). They found that a bubble
with a small degree of nonsphericity, eventually becomes spherical, but a bubble with
a large degree of nonsphericity continues to deform with an increasing amplitude.
Later Refs. [78, 79] focussed on the decay rate of the oscillations for a viscous drop
in micro-gravity condition. Tsamopoulos and Brown [80] studied the effect of initial
amplitude on frequency for nonlinear inviscid oscillations of droplets. The shape
deformation of droplets in simple shear flows and in complex geometries with and
without external forcing has also been investigated by several researchers (see for
instance, Refs. [81, 82]).
6.4 Droplet Breakups
The phenomenon of liquid jet breakup is observed in many industrial applications,
such as ink-jet printing, particle sorting, atomization, mixing, combustion, separation
and spraying technologies [83]. Due to the Rayleigh–Plateau instability [84], a liquid
jet issued from an orifice becomes unstable and breaks up leading to formation of
satellite droplets due to the propagation of surface perturbations. The classical studies
of Plateau [84], Savart [85] and Rayleigh [47] provides a fundamental understanding
of the jet instabilities. Weber [86] investigated the influence of liquid viscosity and gas
density on jet breakup phenomenon. Goedde and Yuen [87] investigated the capillary
instability of water–glycerine interface. They showed that the nonlinear effects dominate the jet breakup process leading to formation of ligaments and satellite droplets.
The nozzle characteristics can also affect the breakup length of the jet occurring due
to the Rayleigh instability [88]. Unlike the aforementioned studies, which considered circular liquid jets, Kashyap et al. [89] and Farvardin and Dolatabadi [90] also
investigated the dynamics of elliptical-orifice liquid jets. It is found that for different
combinations of surface tension, inertia and aerodynamic interactions, four distinct
breakup regimes can be observed. They are (i) the Rayleigh breakup regime (varicose
perturbations), (ii) the first wind-induced breakup (sinuous perturbations), (iii) the
second wind-induced breakup and (iv) the atomization (spray) regime [91–94]. The
numerical simulations of liquid jets were conducted by Pan and Suga [95, 96] to
investigate the characteristics of breakup. The breakup length of a turbulent liquid
jet was studied by Lafrance [97].
In addition to the breakup of a liquid jet aligned with gravity (as discussed above),
Borthakur et al. [98, 99] investigated the dynamics of curved slender jets due to their
implications in the production of nanofibres from centrifugal spinning [100–102].
The variation of jet diameter along length for injections along the gravity (parallel
injection) and orthogonal to gravity (perpendicular injection) at t = 15 and t = 40
is shown in Fig. 12a and b, respectively. It can be seen that the pinch-up occurs early
in the case of perpendicular injection as compared to those in the parallel injection.