Environmental Engineering Reference
In-Depth Information
0.025
ψ
1
= -25 mV
ψ
2
= -25 mV
0.02
S
w
= 0.8
0.015
0.0
-25
-20
-15 -10
Potential (mV)
-5
0
Figure 5.8
Charge distribution in the capillary (After Ghazanfari, 2013)
Ψ
2
u
w
water
τ
ow
r
2
Ψ
1
r
1
τ
ow
u
o
Oil
Ψ
1
water
u
w
Ψ
2
Figure 5.9
Micro-scale representation of the interfacial forces
Where the shear stress at each phase is:
∂
∂
v
j
∂
∂
v
j
w
o
tm
=
and
tm
=
(5.80)
w
w
o
o
Where,
j
is the normal direction to the interface between the two fluids.
The velocity of each phase is found by solving equation 5.72 using the
charge density from equation 5.78, and the continuity conditions given in
equation 5.79 as:
(
)
−
(
)
r
2
Ikr Ikr
kI kr
ey
m
kE
∇
y
1
(
)
0
1
0
2
∫
v
=
w
2
[
y
+
1
fr
dr
(5.80)
(
)
(
)
o
2
2
I kr
o
0
2
0
1
r
2
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