Biomedical Engineering Reference
In-Depth Information
(
(
d r s
2
d r s
2
)
__
)
___
___
μ
d z
u z
2
d z
4
μ u z , s
u z d 2 r s
μ __
____
_________
__________
1
____
r
=
E t σ
+
r s
×
d z 2
(11.19)
(
d r s
2
)
___
+
d z
11.2.5.2
Theory and Modeling of Jet Breakup
Under an electrical fi eld, highly charged aerosol droplets are generated from the apex of the liquid
cone jet. Hartman et al. [2,44] have developed an analytical modeling of jet breakup. Given a jet mov-
ing at constant velocity in a cylindrical coordinate system and harmonic perturbations only on the jet
surface, the following equation has been proposed to describe the surface properties of droplets:
2 π
___
r s
r jet
+
α 0 e ( ωt jmθ jkz ) ,
k
=
λ
(11.20)
where r s is the radius of the surface, r jet the radius of the unperturbed jet, r the radial component, z
the axial component, z the angular component, t the time, ω the growth rate of the perturbation, α 0
the amplitude of the perturbation at t
=
0, m a constant, k the wave number of the perturbation, and
λ is the wavelength of the perturbation.
The modes of the cone-jet breakup can fall into three categories depending on the value of m .
These three modes are shown in Figure 11.7. When m
=
0, the jet breakup is independent of the
1, the radius of the jet depends on angle θ (for half of the values of
θ , r s is larger than r jet , and for the other half, r s is smaller than r jet ). This mode will only occur when
the jet is slowed down by external forces, such as the drag force of the surrounding air or when the
jet is charged. When m
angular component θ . When m
=
2, the jet will be ramifi ed (occurring only when the jet is highly charged).
The electric stresses will transform the shape of the jet.
=
m =0
m =1
m =2
FIGURE 11.7 Three jet break-up modes: axisymmetric varicose, lateral kink, and ramifi ed jet. (Reprinted
from Hartman, R.P.A. et al., J. Aerosol Sci. , 31, 65, 2000. © Elsevier Science. With permission.)
 
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