Biomedical Engineering Reference
In-Depth Information
Theoretically, the fl ow pattern inside a free jet of incompressible liquid can be described by the
Navier-Stokes equation as follows:
(
∂
u
j
∂
u
i
)
ρ
∂
u
1
___
__
___
___
∂
t
+
r
(
u
×
∇
)
u
=
−∇
p
+
∇
2
μ
e
ij
+
f
, e
ij
=
2
∂
x
j
+
∂
x
i
(11.21)
where
ρ
is the density,
p
the pressure,
u
the liquid velocity vector,
µ
the absolute viscosity, and
f
is
the summation of the external forces per unit volume.
When considering small velocity disturbances, the term
ρ
(
µ
·
∇
)
u
and the contribution of grav-
ity can be neglected. Also, when
i
0. Therefore, it can be concluded
that the radial integration of shear forces is equal to zero in the axial direction. The shear stress in
the axial direction can be assumed to be zero in the liquid jet. When the electrospraying in a cone-
jet mode for the varicose instabilities is considered, the value of
m
is equal to zero and all relations
become independent of
θ
. When
u
r
is assumed to depend on
r
linearly, and
u
z
is independent of
r
,
Equation 11.21 can be described as Equation 11.22:
≠
j
at the liquid surface, e
ij
=
(
(
ρ
∂
u
z
∂
2
u
z
ρ
∂
u
r
∂
p
∂
2
u
r
∂
p
)
)
___
___
____
___
___
____
∂
t
=
-
∂
r
+
μ
∂
z
2
,
∂
t
= -
∂
z
+
2
μ
∂
z
2
(11.22)
and the pressure disturbance and velocity disturbance inside the liquid cone can be described by
Equation 11.23:
α
0
ωr
_____
p
''e
(
wt
−
jkz
)
;
u
''
r
e
(
wt
−
jkz
)
;
u
''
r
u
''
z
e
(
wt
−
jkz
)
p
=
u
r
=
=
r
jet
;
u
z
=
(11.23)
When
u
r
depends linearly on
r
, the continuity equation
·
u
=
0, yielding Equation 11.24:
∇
2
∂
u
z
∂
u
r
−
j
2
ωα
0
1
___
__
___
_______
∂
r
+
∂
2
=
0,
u
''
=
kr
jet
(11.24)
In the cone-jet electrospraying mode, a grounded cylinder is assumed and the electric fi eld of an
undisturbed jet in the center of this cylinder can be described by Equation 11.25:
∆
V
___________
E
''(
r
)e
(
ωt
+
jmθ
−
jkz
)
E
r
=
r
jet
ln(
r
cyl
/
r
jet
)
E
=
(11.25)
where
r
cyl
is the radius of the cylinder,
E
the electric fi eld strength, and ∆
V
is the potential
difference.
An electric fi eld is always rotation-free and outside the jet there is no space charge. It is there-
fore concluded that
0. From these conditions, a further Equation 11.26 can
be yielded from Equation 11.25 and described as follows:
∆
×
E
=
0
and
∆
∙
E
=
r
2
d
2
E
''
r
d
E
''
____
____
(
m
2
k
2
r
2
)
E
''
d
r
2
+
d
r
−
+
=
0
(11.26)
The solution to Equation 11.26 can be approximated with
(
jk
υ
u
''
I
n
(
kr
)
−
I
−
n
(
kr
)
)
lim
n
_____
________________
E
''
=
ωr
jet
m
I
n
(
Kr
jet
)
(11.27)
→
I
−
n
(
kr
jet
)
−
The boundary conditions are
υ
___
E
'' (
r
=
r
cyl
)
=
0;
E
'' (
r
=
r
jet
)
=
n
''
z
r
jet
;
υ
=
∆
V
/ln(
r
cyl
/
r
jet
);
n
×
E
=
0;
r
cyl
→
∞
0 into the second relation of Equation 11.22 using Equations
11.23, 11.24, 11.26, and 11.27, a dimensionless parameter can be obtained as follows:
Substituting these results with
m
=
ηk
2
r
2
jet
k
2
r
jet
)
(24
+
________
___________
(8
2
ω
′
+
(
γρr
jet
)
1/2
×
k
2
r
jet
)
ω
′
+
