Recent Advances in Free Surface Flows
133
duct experiment on rising bubble in viscous liquids. The authors of Refs. [49, 52]
experimentally investigated the dynamics of an air bubble rising in aqueous sugar
solutions of differing concentrations. They identified regimes of spherical, oblate,
wobbling and skirted bubbles in the Reynolds and Eötvös numbers plane based on the
bubble shapes and motion. This phase plot was prepared using three dimensionless
parameters: the Reynolds number, the Eötvös number and the Morton number. The
Reynolds number was defined based on terminal velocity of the bubble in Refs. [49,
52]. The Gallilei number used by Sharaf et al. [51] and Tripathi et al. [19] is similar
to the Reynolds number, but uses
√
g R instead, as the velocity scale. Consequently,
the phase plots presented in Refs. [19, 51] also include unsteady bubbles, for which
there is no terminal velocity. The use of Gallilei and Eötvös numbers gives another
advantage as discussed by Landel et al. [54]. They showed that for the same volume
of air (i.e. constant Gallilei and Eötvös numbers) spherical cap bubbles with a range
of rise velocities (multiple Reynolds numbers) and volume of satellite bubbles can
be produced. Therefore, use of the Reynolds and Eötvös numbers may provide a
multivalued nature to the phase plot. The phase plot presented in Fig. 9 is a useful
extension to the classical region map of Bhaga and Weber [49] and Clift et al. [52].
6.3.2 Two Bubbles
The interactions and trajectories of a pair of air bubbles rising side-by-side in a liquid was also studied by many researchers in the Stokes and the potential flow limits
[55–57], and by performing simulations of the complete Navier–Stokes equations
[57–60] and also experimentally [61–63]. Kok [56, 61] experimentally found that
the two bubbles rising vertically in ultra-pure water tend to rotate to align themselves horizontally. Chen et al. [58] conducted two-dimensional simulations on two
bubbles rising side-by-side and showed that the bubbles coalesce and the resultant
bigger bubble exhibits shape oscillations [19, 51]. Duineveld [62] compared the rising dynamics of a single bubble and a pair of bubbles rising side-by-side and found
that the amplitude of oscillations is higher in case of pair of bubbles as compared
to that in case of single bubble. The coalescence and bouncing behaviours of two
bubbles for different values of Reynolds and Weber numbers were investigated by
Sanada et al. [63].
Recently, Tripathi et al. [64] conducted three-dimensional simulations and investigated the dynamics of a pair of air bubbles rising in water and investigated the
influence of inertia on the dynamics. They found that interaction between the wakes
of the bubbles, as shown in Fig. 10a, causes oscillatory motion/path instability as
shown in Fig. 10b. Chakraborty et al. [65, 66] studied two bubbles rising in an inline
configuration in stagnant liquid and studied deformation and coalescence dynamics
using a CLSVOF approach (see Fig. 11).
133
duct experiment on rising bubble in viscous liquids. The authors of Refs. [49, 52]
experimentally investigated the dynamics of an air bubble rising in aqueous sugar
solutions of differing concentrations. They identified regimes of spherical, oblate,
wobbling and skirted bubbles in the Reynolds and Eötvös numbers plane based on the
bubble shapes and motion. This phase plot was prepared using three dimensionless
parameters: the Reynolds number, the Eötvös number and the Morton number. The
Reynolds number was defined based on terminal velocity of the bubble in Refs. [49,
52]. The Gallilei number used by Sharaf et al. [51] and Tripathi et al. [19] is similar
to the Reynolds number, but uses
√
g R instead, as the velocity scale. Consequently,
the phase plots presented in Refs. [19, 51] also include unsteady bubbles, for which
there is no terminal velocity. The use of Gallilei and Eötvös numbers gives another
advantage as discussed by Landel et al. [54]. They showed that for the same volume
of air (i.e. constant Gallilei and Eötvös numbers) spherical cap bubbles with a range
of rise velocities (multiple Reynolds numbers) and volume of satellite bubbles can
be produced. Therefore, use of the Reynolds and Eötvös numbers may provide a
multivalued nature to the phase plot. The phase plot presented in Fig. 9 is a useful
extension to the classical region map of Bhaga and Weber [49] and Clift et al. [52].
6.3.2 Two Bubbles
The interactions and trajectories of a pair of air bubbles rising side-by-side in a liquid was also studied by many researchers in the Stokes and the potential flow limits
[55–57], and by performing simulations of the complete Navier–Stokes equations
[57–60] and also experimentally [61–63]. Kok [56, 61] experimentally found that
the two bubbles rising vertically in ultra-pure water tend to rotate to align themselves horizontally. Chen et al. [58] conducted two-dimensional simulations on two
bubbles rising side-by-side and showed that the bubbles coalesce and the resultant
bigger bubble exhibits shape oscillations [19, 51]. Duineveld [62] compared the rising dynamics of a single bubble and a pair of bubbles rising side-by-side and found
that the amplitude of oscillations is higher in case of pair of bubbles as compared
to that in case of single bubble. The coalescence and bouncing behaviours of two
bubbles for different values of Reynolds and Weber numbers were investigated by
Sanada et al. [63].
Recently, Tripathi et al. [64] conducted three-dimensional simulations and investigated the dynamics of a pair of air bubbles rising in water and investigated the
influence of inertia on the dynamics. They found that interaction between the wakes
of the bubbles, as shown in Fig. 10a, causes oscillatory motion/path instability as
shown in Fig. 10b. Chakraborty et al. [65, 66] studied two bubbles rising in an inline
configuration in stagnant liquid and studied deformation and coalescence dynamics
using a CLSVOF approach (see Fig. 11).