Handbook of Solvents - George Wypych - ChemTech - Ventech!

366 Semyon Levitsky, Zinoviy Shulman
vr radial component of velocity in the liquid
The dynamic boundary condition that governs the forces balance at the interface, is:
−1
( , ) = ( , ) + 2 − ( , )
p R t p R t σR τ R t
[7.2.40]
g f rr
From [7.2.38] - [7.2.40] follows the equation of bubble dynamics:
−1
⎛ 3
J + pf( ∞) − pg + 2σR
= S, J = ρf
⎜RR
��
0 + R
�
⎝ 2
∞
3
−1
1 3 3
∫(
τrr τφφ )( ) , ( )
S = 2 − 3y
+ R dy y = r −R
3
0
2
⎞
⎟
⎠
[7.2.41]
where:
pf(∞) pressure in the liquid at infinity
pg pressure in the bubble
R, � R�� derivatives of the bubble radius with respect to time, t
The equation [7.2.41] was investigated 25 for growing and collapsing cavity in a liquid,
described by rheological model [7.2.26], at pf(∞) -pg= const. Similar analysis for another
rheological model of the solution (the so-called “yo-yo” model of the polymer dynamics)
was developed. 32 It was shown that viscoelastic properties of solution can be approximately
accounted for only in the close vicinity of the interface, that is through the boundary condition
[7.2.40]. In this case the integro-differential equation for R(t), following from [7.2.41],
[7.2.26], can be reduced to a simple differential equation. The latter was analyzed accounting
for the fact 7 that the effective viscosity ηl of a polymeric solution in elongational flow
around collapsing cavity can increase by the factor of 10 2 to 10 3 . If the corresponding
Reynolds number of the flow Re = (ηp/ηl)Rep (Rep =t0/tp,tp=4ηp(pf(∞)-pg) -1 ) is small, the inertial
terms in equation [7.2.41] can be neglected. For high-polymer solutions the inequalities
Re > 1.
Under these assumptions the equation for the relative velocity of the bubble surface takes
the form
2
( )( ) ( )
z� + 2 z −z z − z = 0, z = x� / x, x = R / R , z = − A/ 4± A / 16+ B / 2
1 2 0 12 ,
−1 −1 −1 −1
A = λ ( 1−β) ( 1− 2kλRe ) , B = ( 1−
β) kλ Re , λ = λ/
t , k = sign p − pf( ∞)
p p 0
g
12 /
( )
Here was adopted for simplicity that α = 1/2 and σ >1 mkm). Phase plot of this equation is presented in Figure 7.2.2. It is
seen that fork=-1(collapsing cavity) z →z1 as t →∞if z0 >z2. The stationary point z = z2 is unstable. The rate of the cavity collapsez=z1in the asymptotic regime satisfies inequality
zp ≤z1 ≤0, where zp = -Rep is equal to the collapse rate of the cavity in a pure viscous
fluid with viscosity of polymeric solution η. It means that the cavity closure in viscoelastic
solution of polymer at asymptotic stage is slower than in a viscous liquid with the same
equilibrium viscosity. On the contrary, the expansion under the same conditions is faster: at
k=1zp ≤z1 ≤ zx,
where zp=Repand zs=Res=(1 -β) -1 Rep is the asymptotic rate of the cavity
expansion in a pure solvent with the viscosity (1- β)η. This result is explained by different
behavior of the stress tensor component τrr, controlling the fluid rheology effect on the cav-