1. The dry density of Ro remains constant,
3. The pressure P of water at n remains con-
R2
Frozen
stant at Pn ,
4. f2 vanishes in R1 and R2 unless ρ3 vanish-
n1
T1
es,
ξ
5. The flux f1 is negligibly small in R2,
R11
Frozen
6. Sensible heat terms are negligible in com-
Tσ
R1
R10
Fringe
parison with latent heat terms,
7. ρ1 is given in R1 and R2 as
x
ο
0C
n0
ρ = ρ ν(T ).
(19)
T
1
3
R0
Unfrozen
n
tions is known to be given by eq 19 where ν (T) ο
is an empirically determined and increasing
function of T. Hence, the assumption 7 implies
that ρ1 under dynamic conditions is also given
Figure 1. Quasi-steady freezing of soil.
by the same form as eq 19. We will assume that
ν (T) has a continuous first derivative.
We will seek a traveling wave solution to the problem in which the boundaries n (t), no
(t) and n1 (t) move with the same constant speed Vo, namely
Vo = - n = - no = - n1 .
˙
˙
˙
(20)
From a physical point of view, maintaining a constant pressure Pn is difficult at the moving
boundary n (t). However, a solution obtained under such an idealized condition is quite
useful for applications (Nakano and Primicerio 1995). If such a solution exists, it must sat-
isfy eq 13 15, and eq 17.
From eq 13, 14, and 15, we find that the flux of water f1 (ξ) is given in R1 (Nakano 1994a)
as
(
)
f1 = fo + s2 ρ
- ρ30 ν Vo - d2 (V - Vo ), 0 < ξ < δ
(21)
10
where ρ10 and ρ30 are the constant bulk densities of B1 and B3 in Ro, respectively, and
δ = n1 no, and fo is the constant flux of water in Ro. s2 is defined as
-1
s2 = 1 - d1 d2 .
(22)
Neglecting the gravitational effects and using Darcy's law, the flux of water fo in Ro is given
as
-
fo = Ko σo δo1
(23)
σo = Pn - Po , Po = P(0)
(24)
δo = no - n.
(25)
5
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