Handbook of Solvents - George Wypych - ChemTech - Ventech!

312 E. Ya. Denisyuk, V. V. Tereshatov
( , ) () ( / () )
uxt =ψ tθ x ϕ t
[6.1.37]
where the function θξ ( )satisfies conditions of Eq. [6.1.32] and defines the profile of the diffusion
wave, ψ(t) describes boundary conditions and ϕ(t) the function specifies the penetration
depth of diffusion wave.
Eq. [6.1.37] satisfies the boundary condition of Eq. [6.1.34] and Eq. [6.1.23] on the
half-line with the diffusion coefficient of Eq. [6.1.35] in the following cases: 1) ψ(t), ϕ(t) are
the functions of power type (power swelling mode); 2) ψ(t), ϕ(t) are the function exponentially
depending on time (exponential swelling mode); 3) ψ(t)~(t0-t) m ,ϕ(t)~(t0 -t) n , where
m,ns c, the swelling
mode is of a blow-up nature. The solutions describing these modes are given below.
I. Power mode (s < s c):
2 ( ) ( )
m m ms n
uxt , = M t θξ, ξ = x/ M t
[6.1.39]
2
2 −1
() ()
2
q q 2r
r
g t = M M t , g t = M t
[6.1.40]
1 1 2
2 2
1 1/ d −2
1/ d −1
1
m = , n = , q = , r =
[6.1.41]
s −s
s −s
s −s
ds s
c c c c
II. Exponential mode (s = s c):
2 ( ) ( )
( − )
2
{ 2 0 } θξ () ξ 2 { c 2(
0)
2}
u x, t = exp M t − t , = xM /exp s M t −t
/
{ c
0 }
2
() = ( / ) exp ( / 2+ 1)
( − )
g t M M s M t t
1 1 2 2
{ c
0 }
2
() = exp ( / 2+ 1)
( − )
g t s M t t
2 2
III. Blow-up mode (s > s c):
2 ( , ) = ( − ) θξ ( ) , ξ=
/ ( − )
u x t M t t x M t t
m m ms n
2 0 2 0
2 −1
2
() = ( − ) , () = ( − )
g t M M t t g t M t t
q q r r
1 1 2 0 2 2 0
where the exponents are defined by Eqs. [6.1.41] but if s>sc, then m, n, q, r