Oscillations, Waves, and Interactions - GWDG

188 R. Mettin
constant to M a = 2πρR 3 /3 where the bar indicates a time average [43]. This uncouples
the translational motion from the oscillation and introduces a certain error
which is not always negligible [44,45].
The bubble translational motion is counteracted by a viscous drag force. This
force can be found analytically for a spherical bubble and stationary low Reynolds
number flow (Re = 2|U|R/ν ≪ 1, ν being the kinematic viscosity). Depending on
the supposed boundary condition at the bubble surface it reads [6,46,47]
Here F (1)
D
F (1)
D
= −4πρνRU or F (2)
D
= −6πρνRU . (6)
is calculated with a zero shear stress boundary condition at the bubble
surface, and F (2)
D assumes a zero tangential velocity, i. e., no-slip boundary condition.
While the former case reflects an “ideal” gas-liquid interface, the latter formula is
valid for a “sticky” or “dirty” bubble, equivalent to a solid sphere. Indeed, surface active
contaminants in the liquid may gather at the bubble interface, and in many cases
experimental values from rising bubbles in non-clean liquids tend to yield F (1)
D [46].
If faster bubble translations are considered, the high Reynolds number limit may be
encountered, if the bubble is small enough to be still spherical and not deformed (in
water up to Re ≈ 800 [46]). In this situation the “clean” bubble boundary condition
yields [46,47]
F (3)
D
= −12πρνRU , (7)
and the “dirty” bubble case results in additional drag proportional to U 2 [46].
For oscillating bubbles, the drag becomes time dependent, and analysis gets more
complicated. Some limiting cases are derived in Ref. [48]. In particular, the limit of
high Re or high ( ˙ R/U)Re yields for an ideal (clean) bubble interface again F (3)
D , but
with the substitution of the constants R and U by their time-dependent counterparts
R(t) and U(t). Measurements of bubble motion in ultrasonic standing waves
at smaller Re, but higher ( ˙ R/U)Re support this result [10,49]. However, more experimental
investigations, in particular in strong fields, should be undertaken to verify
the theoretical approaches.
5 Bubble life cycles
If we combine all discussed aspects of single bubble dynamics, we can consider the
“life” of a bubble in a given liquid and sound field. We restrict our viewpoint to
translation and gas diffusion, both supposed to happen on a slower time scale than
the volume oscillation. In a standing wave, the fate of a bubble is determined by its
initial size R0 and position in space x, or, equivalently, the sound pressure amplitude
pa(x) at the initial position. Its life cycle can be represented by a trajectory in the
plane of R0 and pa. Main features can be seen if characteristic lines are drawn in that
plane: the threshold of spherical shape instability (SI), the inversion of the primary
Bjerknes force (BJ), and the rectified diffusion threshold (RD). Examples of such
diagrams are shown in Fig. 13.
The general direction of a bubble trajectory in this parameter space is towards the
right above the RD line, and towards the top below the BJ line. The space above