Chemistry Reference
In-Depth Information
Appendix B
Hydrodynamic Flow Fields with Axial Symmetry
In spherical coordinates, the velocity field is related to the stream function
ψ(
r
,θ)
[
1
]by
1
r
2
sin
∂ψ
∂θ
v
r
=−
(B.1)
θ
∂ψ
∂
1
r
sin
v
θ
=
(B.2)
θ
r
The stream function satistisfies the differential equation
E
4
ψ
=
0, where
∂
2
sin
θ
r
2
∂
∂θ
1
sin
∂
∂θ
E
2
ψ
=
r
2
+
ψ
(B.3)
∂
θ
Simple solutions to Eq.
B.3
are given by
A
1
r
4
A
4
r
sin
2
A
2
r
2
ψ
=
θ
+
+
+
A
3
r
(B.4)
where
A
1
, ....,
A
4
are constant parameters determined by boundary conditions. The
corresponding pressure field is given by
P
θ
20
A
1
r
r
2
+
P
∞
,
where
P
∞
is the pressure far from the spherical droplet. The body force acting
on the droplet which is balanced by forces exerted by the hydrohynamic flow is
f
P
=−
η
cos
+
2
A
3
/
A
3
[1].
The solution of the form given by Eq.
B.4
with boundary conditions
v
θ
(
=−
8
πη
r
=
a
)
=
v
s
(θ)
and
v
r
(
r
=
a
)
=
0 is determined in the frame of movingwith the droplet. Here,
v
s
(θ)
is the velocity on the surface of the droplet of radius
a
.Forlarge
r
, the Stokes'
flow requires motion at constant velocity
v
,
v
r
−
v
cos
(θ)
and
v
θ
v
sin
θ
.
From the latter conditions, it follows that
A
1
=
0, and 2
A
2
=−
v
. Because there
is no external force acting on the droplet,
f
P
=
0 and thus
A
3
=
0. The boundary
A
2
a
3
and 3
A
2
=
(κ) ∂
x
φ
conditions on the droplet surface imply that
A
4
=−
.The
corresponding flow field can be calculated using the relation between the stream
