Handbook of Solvents - George Wypych - ChemTech - Ventech!

7.2 Bubbles dynamics and boiling 367
Figure 7.2.2. Phase plane for expanding and collapsing
cavity in polymeric solution. [By permission of
Nauka i Tekhnica Press, from the reference 25]
ity dynamics, in extensional and
compressional flows, respectively. 9 In the
former case, the τ rr value may be considerably
greater than in the latter one.
Heat transfer between phases is a
strong dissipative factor that in principle
can mask the rheological features in bubble
dynamics. Nevertheless, even with account
for heat dissipation the theoretical dependencies
of R(t) are sensitive to rheological
properties of solution. Typical results of air
bubble dynamics simulations at a sudden
pressure change in the solution withη p >>η s
are presented on the Figure 7.2.3,
where:
Figure 7.2.3. Heat transfer and rheodynamics at non-linear
oscillations of a bubble in polymeric liquid. [By permission
of Nauka i Tekhnica Press, from the reference
25]
R* dimensionless radius of the bubble, R* = R/R0 τ dimensionless time, τ = t/t0 τ* rr dimensionless radial component of the extra-stress tensor at the interface,
τ* rr = τrr(R,t)/pf0 0
θ1 Q*
0
dimensionless temperature at the center of the bubble, θ1 =Tg(0,t)/T0 dimensionless heat, transferred to liquid from the gas phase in a time τ,
Q* = Q/(R0T0kgt0) kg heat conductivity of gas
Δpf* dimensionless pressure change in the liquid at initial moment of time,
pf*(∞) =1+Δpf*h(t), Δpf*=Δpf/pf0 h(t) unit step function
Calculations have been done for the rheological model [7.2.25] with 20 relaxation elements
in the spectrum, distributed according to the law [7.2.29] withz=2.Toillustrate the
contribution of rheological non-linearity in equation [7.2.25] the numerical coefficient α
(α=1 or 0) was introduced in the term with λ k, containing material derivative. The value α=1
corresponds to non-linear model [7.2.25], while at α = 0 equation [7.2.25] is equivalent to
the linear hereditary model [7.2.10] with a discrete spectrum. Other parameters of the system
were chosen as follows: η p = 2 Pas, η s =10 -2 Pas, λ 1 =10 -5 s, R 0 = 50 mkm, Δp f* = 10,
p f0 =10 5 Pa, ρ f0 =10 3 kg/m 3 ,T 0= 293K, σ = 0.05 N/m. Thermodynamic parameters of the