Handbook of Solvents - George Wypych - ChemTech - Ventech!

360 Semyon Levitsky, Zinoviy Shulman
where:
k heat conductivity of the liquid
cv equilibrium isochoric specific heat capacity
Equations [7.2.10], [7.2.11], [7.2.21] - [7.2.23] constitute complete set of equations
for linear thermohydrodynamics of polymeric solutions.
The non-linear generalization of equation [7.2.4] is not single. According to the
Kohlemann-Noll theory of a “simple” liquid, 5 a general nonlinear rheological equation for a
compressible material with memory may be represented in terms of the tensor functional
that defines the relationship between the stress tensor, σ, and the deformation history. The
form of this functional determines specific non-linear rheological model for a hereditary
medium. Such models are numerous, the basic part of encountered ones can be found elsewhere.
1,6-8 It is important, nevertheless, that in most cases the integral rheological relationships
of such a type for an incompressible liquid may be brought to the set of the first-order
differential equations. 9 Important special case of this model represents the generalized
Maxwell’s model that includes the most general time-derivative of the symmetrical tensor
where:
∑
( k)
( k)
( k )
τ= τ , τ + λ F τ = 2 η e
[7.2.24]
F
abc τ
k
k abc
k
D
D d
a( e e ) bI ( e) ce ( )
Dt
Dt dt w
τ
τ τ
= + τ ⋅ + ⋅ τ + τ ⋅ + τ = − ⋅ τ + τ ⋅ w w = ∇v − ∇v
T � �
, 1
2
tr tr , ( )
D/Dt Jaumann’s derivative 1
d/dt ordinary total derivative
tr trace of the tensor, trτ = τkk w
∇
vorticity tensor
� v T
transpose of the tensor ∇ � v
λk, ηk parameters, corresponding to the Maxwell-type element with the number k
Ata=-1,b=c=0,equations [7.2.24] correspond to the Maxwell liquid with a discrete
spectrum of relaxation times and the upper convective time derivative. 3 For solution of
polymer in a pure viscous liquid, it is convenient to represent this model in such a form that
the solvent contribution into total stress tensor will be explicit:
⎡ τ
⎤
τ= τ + η τ + λ ⎢ −( τ ⋅ + ⋅τ
) ⎥
⎣
⎦
=
( k)
( k)
( k)
D
( k) ( k)
∑ 2 se,
k
e e 2ηke [7.2.25]
k
Dt
To select a particular nonlinear rheological model for hydrodynamic description of the
fluid flow, it is necessary to account for kinematic type of the latter. 7 For example, the radial
flows arising from the bubble growth, collapse or pulsations in liquid belong to the
elongational type. 3 Therefore, the agreement between the experimental and theoretically
predicted dependencies of elongational viscosity on the elongational deformation rate
should be a basic guideline in choosing the model. According to data 10-13 the features of
elongational viscosity in a number of cases can be described by equations [7.2.25]. More
simple version of equation [7.2.25] includes single relaxation time and additional parameter
1/2 ≤α ≤1,
controlling the input of nonlinear terms: 7