Page 63 - IJB-5-1

Gao D, et al.
The kinetic energy per unit volume of the liquid polarization force . Data for water and formamide tend
[26]
becomes of the order of the electrostatic stress once the to match with the IP scaling solution [26,31] .
jet is developed and the gradient of the kinetic energy in ·• VE-scaling: Dominance of viscous force
the axial direction is mostly balanced by the tangential and electrostatic suction in the limit of
electric stress resultant on the jet’s surface : 1 α µ
[66]
ρ
2 α << α µ 4 1 , << [26] :
ρQ ε EE zo ( ε −1) 4
0
no
~ (11) r µε 2 Q 3
. 05
4
RL R I = (γ KQ) , 0 ) / 18 (15)
D = (
o o o j 2
γ K
Here R = d ( Q ) 05. , L = d 0 Q , E no = ( 2γ ) . 05 The IP scaling law is applicable to high viscosity liquids
0
0
0
Q 0 Q 0 ε 00 with a sufficiently large electric conductivity. The trend of
d
, and E zo = E ( Q ) − . 05 are the characteristic values of this scaling law has been confirmed by experiments that
no
use glycerol with different electrical conductivities .
[26]
Q
the jet’s radius, axial length, outer normal electric field,
0
and tangential electric field. The characteristic flow rate 3.2.2.2 De la Mora’s Scaling Law
2 1
()
Q = γε 0 , characteristic distance D = γε 0 ] , and In the model of De la Mora and Loscertales, the conductivities
'
'
3
–4
0 ρ K ε 0 [ ρ K 2 of the tested liquids are all in the range of 10 S/m and above;
characteristic current I = γ ( ρ 0 05. are shown in the hence, these are considered as high conductivity liquids.
'
)
0
dimensionless analysis above . They concluded that the current and jet structure for liquids
[66]
with high conductivities are insensitive to changes of needle
Ganan-Calvo defined two dimensionless expressions voltage, diameter, the distance between two electrodes, the
for flow rate : meniscus shape, and the spray structure . The static Taylor
[26]
[31]
3
α = ρKQ , α = K 2 µ Q (12) equilibrium is maintained to the point where the flow time
becomes shorter than the electrical relaxation time. Beyond
µ
ρ
γε
23
0 εγ
this point, there is a rapid transition to a cone-jet. After the
0
same point, the ohmic bulk conduction current (I ) that is
When the inertia stress is large, the dimensionless ocd
flow rate is the ratio of flow rate, Q, and dimensional dominated in the cone also fast transfer to surface free charge
convection current (I ) by the accelerating liquid stream,
variable sets (γε /ρK). When viscous stress is large, the which is dominant close to jet’s end where is very small
scv
0
dimensionless flow rate is the ratio of flow rate, Q, and cross-section and large liquid velocity [31,32,55] . In the cone-jet
dimensional variable sets (γε 0 2 / µ K 2 ) . The dominance
3
3
of surface tension is a marginal situation . transition area between these two regions, both conduction
[26]
[55]
For six different situations, Ganan-Calvo et al. and convection are of the same order .
A non-dimensional factor, η , can be defined as the
2
identified different scaling laws for the jet diameter D and ratio of inertia pressure (ρQ /r ) where ρ is liquid density,
4
2
j
the electric current I . Three of them have been found Q is the flow rate, and r is local radius, and capillary
[26]
in published experimental data, and they are presented pressure is (γ/r) [31,32] .
below:
• IE-scaling: Dominance of inertia and electrostatic ρQ 2
2
1 α η =
ρ
suction in the limits of α µ 4 << α 1<<, ε − 1 Where, γ r 3
ρ
The formation of the jet may be determined either
r
D is the same as the characteristic value (R ) : by inertia or by charge relaxation, depending on which
[26]
0
j
ρε Q 3
I = (γ KQ) , 0 ) / 16 (13) of these two phenomena acts first as the cone apex is
. 05
D = (
j γ K approached . When η is smaller than unity, the diameter
[31]
The IE scaling law is the most common regime. It of jet scale as characteristic length r*. In the opposite limit,
has been widely verified in numerous experimental where η is much larger than one, the diameter of jet scales
works [26,66,69-71] , and data for octanol were accord with this as characteristic length R* . A characteristic distance λ
[31]
scaling law [26,31] . from the cone apex is closely related to the jet radius ,
[31]
• IP-scaling: Dominance of inertia and polarization and it also relates to the thickness of surface charge layer,
α
α
forces in the limit of µ << ρ << [26] : which is built up by the bulk conduction . When the jet
[72]
ε ( r − ) 1 4 ε −1 scaling length r*, which is the same magnitude of charge
r
ρ KQ 2 ρε Q 3
2
I = ( ), 0 ) / 16 (14) relaxation length, λ, where hydrodynamic time, t is of the
. 05
D = (
h
(ε − )ε j γ K order of the electrical relaxation time, t , the characteristic
1
r 0 e
The IP scaling law is used for polar liquids, and the distance, r*, is expressed as the following equation [31,32] .
scaling of jet diameter is the same as for IE scaling, no D ~ r = ( Q  ) /
r 0 13
*
matter whether the dominant factor is electrostatic or j K (17)
International Journal of Bioprinting (2019)–Volume 5, Issue 1 9