Handbook of Solvents - George Wypych - ChemTech - Ventech!

7.2 Bubbles dynamics and boiling 369
Typical data for Λ and μ, calculated from the solution of linearized equation [7.2.42]
for air bubble in the case of discrete spectrum with n1 relaxation times, distributed by the
Rouse law, are presented in Figures 7.2.4, 7.2.5. The maximum time, λ1, was evaluated from
equations [7.2.27], [7.2.32]. For curves 1- 6 and 1�-6�� �1 = 1, 5, 10, 20, 50 and 100, correspondingly.
Curve 7 refers to Newtonian fluid with η = ηp. For curves 1- 6, A = 200, for
curves 1�-6�, A = 1000 with dimensionless parameterA=pf0M[η]/(RGT0). Other parameters
were adopted as follows: kM= 0.4,ηs*=10 -2 , γ = 1.36. For atmospheric pressure andρf0=10 3
kg/m 3 these values correspond approximately to ηs = 0.1 Pas, R0 =10 -3 m. At these parameters,
the relative effect of acoustic dissipation on damping of bubble pulsations is small. 31
The value of heat decrement is denoted in Figure 7.2.4 by Λ1, the value of the rheological
one-byΛ3. It is seen from the plots that viscoelasticity of the solution is responsible for reduction
of rheological dissipative losses. The effect increases with reduced concentration of the
polymer, ~ c. When the number of relaxation times in the spectrum, accounted for in simulations,
grows, the magnitude of Λ3 also grows. Nevertheless, for all considered values the
weakening of the damping, as compared with Newtonian fluid of the same viscosity, is essential.
It follows from the analysis that in polymeric solution, pulsations of the bubble are
possible even at relatively large c ~
values (to the right of the dashed line in Figure 7.2.4). In
this region of reduced concentrations the Newtonian viscosity of the solution is so high that
only aperiodic solution for ΔR exists when the liquid follows the Newtonian model. The increase
in molecular mass of the dissolved polymer at ~ c = const causes decrease in rheological
losses.
The natural frequency of the bubble
raises with ~ c and this behavior is qualita-
Figure 7.2.4. Rheological dissipative losses versus reduced
concentration of a polymer. [By permission of
Nauka i Tekhnica Press, from the reference 25]
Figure 7.2.5. The effect of reduced polymer concentration
on the natural frequency of bubbles. [By permission
of Nauka i Tekhnica Press, from the reference 25]