energy of the ab interface per unit mass of interface while ˆ
E
ab
¼ r
ab
a
ab
E
ab
is the excess energy per unit volume of the system. Finally, the common line energy per unit mass of
common line is denoted as E
wns
while the common line internal energy per unit volume of the system is
ˆ E
wns
¼ r
wns
l
wns
E
wns
. Note that in the case of massless inter- faces and common lines, the energies per unit volume are
still directly meaningful functions, while the product of mass density times energy per mass must be evaluated in
the limit as the density approaches zero.
The objective of this section is to provide the needed balance equations rather than reproduce their derivation
from the earlier general work.
20
2.1 Phase conservation equations
The balance equations for the three-phase system are essen- tially unchanged from the general case with more phases.
For the current study, each phase may have two different kinds of interfaces at its boundary. For example, the phase is
bounded by some combination of wn and ws interfaces. The balance equations for the phases are as follows:
Macroscale mass conservation for the a-phase D
a
e
a
r
a
Dt þ
e
a
r
a
=·v
a
¼ X
bÞa
ˆe
a ab
a ¼ w ,
n ,
s 1
Macroscale momentum conservation for the a-phase e
a
r
a
D
a
v
a
Dt ¹ =
· e
a
t
a
¹ e
a
r
a
g
a
¼ X
bÞa
ˆ T
a ab
a ¼ w ,
n ,
s 2
Macroscale energy conservation for the a-phase D
a
ˆ E
a
Dt ¹ =
· e
a
q
a
¹ e
a
t
a
¹ ˆ E
a
I : =v
a
¹ e
a
r
a
h
a
¼ E
a
X
bÞa
ˆe
a ab
þ X
bÞa
ˆ Q
a ab
a ¼ w ,
n ,
s ð
3Þ The terms on the right side of the equations account for
exchanges with the bounding interfaces. The complete notation used is provided at the beginning of the text.
2.2 Interface conservation equations
These equations express conservation of mass, momentum, and energy of the interface. The interfaces may exchange
properties with adjacent phases and with the common line. The balance equations are as follows:
Macroscale mass conservation for the ab-interface D
ab
a
ab
r
ab
Dt þ
a
ab
r
ab
=·v
ab
¼ ¹ ˆe
a ab
þ ˆe
b ab
þ ˆe
ab wns
ab ¼ wn ,
ws ,
ns ð
4Þ Macroscale momentum conservation for the ab-interface
a
ab
r
ab
D
ab
v
ab
Dt ¹ =
· a
ab
t
ab
¹ a
ab
r
ab
g
ab
¼ ¹ X
i ¼ a ,
b
ˆe
i ab
v
i ,
ab
þ ˆ T
i ab
þ ˆ T
ab wns
ab ¼ wn ,
ws ,
ns ð
5Þ Macroscale energy conservation for the ab-interface
D
ab
ˆ E
ab
Dt ¹=
· a
ab
q
ab
¹ a
ab
t
ab
¹ ˆ E
ab
I :=v
ab
¹ a
ab
r
ab
h
ab
¼ ¹ X
i ¼ a ,
b
{ˆe
i ab
[ E
i
þ v
i ,
ab 2
= 2
]
þ ˆ T
i ab
·v
i ,
ab
þ ˆ Q
i ab
} þ
E
ab
ˆe
ab wns
þ ˆ Q
ab wns
ð 6Þ
2.3 Common line conservation equations
The balance equations for the common line account for the properties of the common line and the exchange of those
properties with the interfaces that meet to form the common line. The appropriate equations for the case where there are
three phases, and thus only one common line and no com- mon points, are as follows.
Macroscale mass conservation for the wns-common line D
wns
l
wns
r
wns
Dt þ
l
wns
r
wns
=·v
wns
¼ ¹ ˆe
wn wns
þ ˆe
ws wns
þ ˆe
ns wns
7
Fig. 1. Depiction of a three-phase system at a macroscale point top and from the microscale perspective bottom with notation
employed to identify phases, interfaces, and the common line.
526 W. G. Gray
Macroscale momentum balance for the wns-common line l
wns
r
wns
D
wns
v
wns
Dt ¹ =
· l
wns
t
wns
¹ l
wns
r
wns
g
wns
¼ ¹ X
ij ¼ wn ,
ws ,
ns
ˆe
ij wns
v
ij ,
wns
þ ˆ T
ij wns
ð 8Þ
Macroscale energy conservation for the wns-common line D
wns
ˆ E
wns
Dt ¹ =
· l
wns
q
wns
¹ l
wns
t
wns
¹ ˆ E
wns
I :=v
wns
¹ l
wns
r
wns
h
wns
¼ ¹ X
ij ¼ wn ,
ws ,
ns
{ˆe
ij wns
[ E
ij
þ v
ij ,
wns 2
= 2ÿ þ ˆ
T
ij wns
·v
ij ,
wns
þ ˆ Q
ij wns
} ð
9Þ
2.4 Entropy inequality
