(21)
where Rb (m) is the bubble radius. The rate of energy dissipation per unit mass for a
surface tension of 64 mN·m−^2 , a hemispherical bubble of 3.5 mm diameter and a film
thickness of 15 μm is 7000 m^2 .s−^3. This value corresponds to a turbulent eddy size of 3.5
μm and a laminar shear stress of 95 N·m−^2 , which both can cause cell death. However, it
is unknown how the energy released is dissipated in the rim. Kowalski and Thomas
(1995) studied the breaking of a bubble in solution of SDS in water with and without
Pluronic F68. Without Pluronic, rapid expansion of the hole was observed together with
the tangential ejection of threads (20–60 in number) breaking up in small droplets (30–
300 μm) at velocities of 10 m·s−^1. Assuming a distance of acceleration of 1 mm, this
results in a local energy dissipation of 10^5 m^2 .s−^3 , which is of the same order as the value
calculated by Cherry and Hulle (1992). In the presence of Pluronic formation of the
threads was suppressed, which was attributed to polymeric inhibition of the ultra-micro-
scale motions.
In conclusion, in the absence of additives almost all cells present in the bubble film are
killed upon rupture. This is most likely due to intense energy dissipation in the retracting
rim, which results in the tangential ejection of liquid drops. This cell death can be
prohibited by the addition of protective additives, primarily because the rupture process
itself is less severe and cells have become less fragile. In addition, Pluronic, Methocel
and PVA cause rapid draining of the cells out of the film and thus out of the danger zone.
Bubble cavityChalmers and Bavarian (1991) calculated that the shear stresses associated with the
collapse of the bubble cavity are of the order of 200 N·m−^2 , which is a magnitude larger
than the shear stresses reported to cause cell damage. This is an average shear stress
calculated on the assumption of simple boundary-layer flow. Garcia-Briones and
Chalmers (1994) stated that for the more complex flows the shear stress does not
represent the local state of stress. They suggested using two flow parameters of general
nature to study hydrodynamic-related cell injury. These are the state of stress
(characterised by the second invariant of the stress tensor) and the flow classification
parameter RD, which is related to the possibility of stress relaxation by rotation. RD ranges
from infinity for rigid-body rotation to unity for viscometric flows to zero for
elongational flows. Flows with an RD above unity are weaker than the viscometric flows
and were assumed not to be able to damage cells (Garcia-Briones and Chalmers 1994).
By numerically simulating the collapse of the bubble cavity during the bubble breakage
process, Boulton-Stone and Blake (1993) and Garcia-Briones and Chalmers (1994)
showed that the regions with the highest state of stress develop from the cavity edge,
spread rapidly along the cavity interface and reach the highest value upon convergence at
the bottom of the cavity, where two opposing jets are formed. Figure 15.5 schematically
shows the regions of high state of stress during cavity collapse. Garcia-Briones and
Chalmers (1994) showed that in particular for smaller bubbles (0.77 and 1.7 mm), a
strong downward and upward jet are formed, which do not occur for a bubble of 6.3 mm
in diameter. Furthermore, they showed that the magnitude of the stresses decreases
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