3 On the Spectrum of Relaxation Times in Coupled Diffusion …
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3.3.4 The Relaxation Time of Perturbations and Their
Asymptotics
The functions τ + (λ) and τ − (λ), depending on the dimensionless factors
M B /M A , V B /V A , kT /(GV m ), β/η (it is assumed that β A = β B = β), have been
obtained in a closed form and are not given here because of the cumbersome form
of their representation. The resulting expressions are analyzed using the computer
algebra tools. Figures 3.1 and 3.2 show the plots of τ − (λ) and τ + (λ) as a function of
dimensionless bulk viscosity β/η.
Both dependences τ − (λ), τ + (λ) have asymptotics at λ → 0 and λ → ∞ (Figs. 3.1
and 3.2).
For τ − (λ) at λ → ∞, the stress gradient does not affect the process of interdiffusion, which occurs by the Darken mechanism (Darken 1948; Mehrer 2007; Paul
et al. 2014) with the coefficient
Fig. 3.1 First branch (–) of the curve representing the relaxation time of perturbations as a function
of their characteristic length at different values of the bulk viscosity
Fig. 3.2 Second branch (+) of the curve representing the relaxation time of perturbations as a
function of their characteristic length at different values of the bulk viscosity