3 On the Spectrum of Relaxation Times in Coupled Diffusion …
49
Then, the components of the elastic strain energy can be written as Ge
e
: e
e
=
3/(8G)σ
2
m , K ε
e2
m = 1/(2K )σ
2
m , and the dissipation rate of the shear viscous flow is
η˙ e
v
: ˙
e
v
= 3/(8η)σ
2
m .
3.3.2 Field Equations
Equation (3.29) is written in terms of the velocity of the material point displacement
v(x, t) along the spatial coordinate x
∂v
∂ x
= −
3
4G
∂σ m
∂t
−
3
4η
σ m ,
(3.30)
where (and in the following) we set d/dt ≈ ∂/∂t on the assumption of small v.
The remaining field equations take the following form:
∂v
∂ x
= −V A
∂ j A
∂ x
− V B
∂ j B
∂ x
,
(3.31)
∂c B
∂t
+ c B
∂v
∂ x
= −
∂ j B
∂ x
,
(3.32)
j A = −c A M A
∂μ A
∂ x
, j B = −c B M B
∂μ B
∂ x
,
i A = V A
∂c A
∂t
+ c A V A
∂v
∂ x
, i B = V B
∂c B
∂t
+ c B V B
∂v
∂ x
.
(3.33)
They are closed by the constitutive Eq. (3.23) with regard to (3.20).
3.3.3 Perturbed System and Its Analysis
The system of coupled Eqs. (3.20), (3.23), (3.30)–(3.33) has a homogeneous
stationary solution
σ m (x, t) ≡ σ 0 , c A (x, t) ≡ c A0 , c B (x, t) ≡ c B0 ,
(3.34)
corresponding to some equilibrium state.
The spectrum of the relaxation times of the system is determined using the
perturbation method. To this end, this system is linearized in the vicinity of (3.34)
∂φ
∂t
− kT (ξ 0 M A + (1 − ξ 0 )M B )
∂
2
φ
∂ x 2 − φ 0 (1 − φ 0 )(M A V A β A + M B V B β B )
∂
3
φ
∂t∂ x 2