Handbook of Solvents - George Wypych - ChemTech - Ventech!

356 Semyon Levitsky, Zinoviy Shulman
7.2 BUBBLES DYNAMICS AND BOILING OF POLYMERIC SOLUTIONS
Semyon Levitsky
Negev Academic College of Engineering, Israel
Zinoviy Shulman
A.V. Luikov Heat and Mass Transfer Institute, Belarus
7.2.1 RHEOLOGY OF POLYMERIC SOLUTIONS AND BUBBLE DYNAMICS
7.2.1.1 Rheological characterization of solutions of polymers
Solutions of polymers exhibit a number of unusual effects in flows. 1 Complex mechanical
behavior of such liquids is governed by qualitatively different response of the medium to
applied forces than low-molecular fluids. In hydrodynamics of polymers this response is
described by rheological equation that relates the stress tensor, σ, to the velocity field. The
latter is described by the rate-of-strain tensor, e
e
ij
1⎛
v v
i
= ⎜
∂ ∂
+
2⎜
⎝∂xj∂x
j
i
⎞
⎟
⎠
[7.2.1]
where:
vi components of the velocity vector, � v
xi Cartesian coordinates (i, j=1,2,3)
The tensors σ and e include isotropic, p, ekk and deviatoric, τ, s contributions:
1 1
�
σ=− pI + τ, p =− σkk,
e = ekkI + s, ekk =∇⋅v
3 3
[7.2.2]
where:
I unit tensor
∇ Hamiltonian operator
For incompressible fluid ∇⋅v=0,e=sandrheological equation can be formulated in
the form of the τ- dependence from e. Compressibility of the liquid must be accounted for in
fast dynamic processes such as acoustic waves propagation, etc. For compressible medium
the dependence of pressure, p, on the density, ρ, and the temperature, T, should be specified
by equation, p=p(ρ,T), that is usually called the equation of state.
The simplest rheological equation corresponds to incompressible viscous Newtonian
liquid and has the form
τ=2η0e [7.2.3]
where:
η0 viscosity coefficient
Generalizations of the Newton’s flow law [7.2.3] for polymeric liquids are aimed to
describe in more or less details the features of their rheological behavior. The most important
among these features is the ability to accumulate elastic deformation during flow and
thus to exhibit the memory effects. At first we restrict ourselves to the case of small deformation
rates to discuss the basic principles of the general linear theory of viscoelasticity