Skip to content
Publicly Available Published by De Gruyter March 7, 2020

Concentration profiles around and chemical composition within growing multicomponent bubble in presence of curvature and viscous effects

Anatoly E. Kuchma and Alexander K. Shchekin EMAIL logo

Abstract

The regularities of changing chemical composition and size of a ultra-small multicomponent gas bubble growing in a viscous solution have been analyzed. The full-scale effects of solution viscosity and bubble curvature at non-stationary diffusion of arbitrary number of dissolved gases with any value of gas supersaturations and solubilities in the surrounding liquid solution have been taken into account. The nonuniform concentration profiles of gas species in supersaturated solution around the growing bubble with changing composition have been found as a function of time and distance from the bubble center. Equations describing transition to stationary concentrations of gases in the bubble with increasing radius have been obtained. Analytic asymptotic solutions of these equations for a ternary system have been presented.

Introduction

There are two general approaches to theory of first-order phase transitions in a closed system with a limited availability of the nucleating species: the approach with the mean-field supersaturation and the approach with the excluded volume. The first approach implies that nucleation and growth of new-phase particles is governed by stationary diffusion of molecules and accompanied by a synchronous and uniform decrease in the mean supersaturation of the metastable phase [1], [2], [3], [4], [5]. The excluded volume approach is based on a self-similar solution for non-stationary diffusion onto the growing supercritical particles and takes into account that nucleation of new particles is strongly suppressed around growing ones [3], [4], [5], [6], [7], [8], [9], [10]. Only the theory of excluded volume describes strong swelling of liquid solutions with dissolved gas at fast decompressing [3], [5], [6].

In the case of liquid solution with several dissolved gases, both approaches assume stationary concentrations of gases in the bubbles growing with steady rate [3], [5]. However, strong influence of the Laplace pressure in a submicron gas bubble and viscosity of liquid solution on the growth rate of the bubble is able to postpone significantly establishing the stationary chemical composition in the bubble and its steady growth rate. As a result, application of the self-similar concentration profiles around the bubble in the theory of bubble nucleation is under the question. “Equations for bubble pressure and gas concentration profile around the bubble” of this paper answers how the dynamics of a multicomponent bubble growth in supersaturated solution with several dissolved gases should be reformulated to include the full-scale viscosity and capillarity (the Laplace pressure or curvature) effects, non-stationary diffusion of gas components at any value of their supersaturations and solubilities in the surrounding liquid solution and changing the bubble chemical composition. Equations describing transition to stationary concentrations of gases in the bubble with increasing radius are considered in “Joint evolution of the radius and chemical composition of a growing bubble”. Analytical asymptotic solutions of these equations for a ternary system are presented in “Asymptotics of reaching steady-state and self-similar diffusion growth regime”.

Equations for bubble pressure and gas concentration profile around the bubble

Let us consider a growing gas bubble in viscous liquid solution with arbitrary number k of dissolved gases. The dependence of total gas pressure Pg in a free supercritical spherical bubble growing in the liquid solution with supersaturated dissolved gases on the bubble radius R (and its derivatives R′ and R″ with respect to time) is described by the Rayleigh–Plesset equation [11], [12]

(1) ρ(RR+32R2)=PgP2σR4ηRR

where P is the pressure in the liquid solution, σ is the bubble surface tension, ρ and η are the liquid solution density and viscosity, respectively. We can neglect in most cases the contribution from inertial effects [given by the left-hand side of eq. (1)] [10] and to reduce eq. (1) to the form

(2) Pg=P+2σR+4ηRR.

Under assumption that all gases in the bubble are uniformly distributed and ideal, we have the following relation for the total gas concentration n(R)=i=1kni as a function of the bubble radius R with the full-scale account of the Laplace pressure 2σ/R and viscosity η:

(3) n(R)=PgkBT=n¯[1+RR(1+2ηRσ)]

where n¯P/kBT , R*≡2σ/P. If the bubble radius is larger ten nanometers, we can neglect adsorption of gas components at the bubble surface, then the total number N of gas molecules in the bubble equals

(4) N=4π3n(R)R3=4π3n¯R3[1+RR(1+2ηRσ)].

It is follows from eq. (4) that the rate N′ of changing the total number N in time equals

(5) N=N(R)3RR(1+R3dlnn(R)dR)

where

(6) 1+R3dlnn(R)dR=[1+RR(1+2ηRσR)]1[1+2R3R(1+2ηRσ+ηRσdRdR)].

As seen from eqs. (4) and (5), the number Ni and the rate Ni of changing molecules of ith gas component in the bubble can be related to its volume concentrations ni in the similar to eq. (5) way

(7) Ni=Ni(R)3RR(1+R3dlnni(R)dR).

Because the surface of the growing bubble with radius R shifts at every point with the velocity of an incompressible liquid solvent, so we have for the rate Ni of ith gas component and the total rate N′ the following balance equations

(8) Ni=4πR2Dici(r,t)r|r=R(i=1,2,...,k),
(9) N=4πR2i=1kDici(r,t)r|r=R

where Di is the diffusivity of molecules of ith gas component (i=1, 2, …, k) in a solvent and ci(r, t) is the volume concentration of molecules of this component at time t in solution at distance r from the center of the bubble. The concentration ci(r, t) satisfies in the most general case the non-stationary diffusion equation [3], [5]

(10) ci(r,t)t=Dir2r[r2ci(r,t)r]R2(t)R(t)r2ci(r,t)r(i=1,2,...,k)

with boundary conditions

(11) ci(r,t)=ci0, ci(r=R(t),t)=ci(R).

Here ci(R) is the equilibrium concentration of ith gas component at the bubble surface. According to the Henry law, ci(R)=sini(R) where si is the solubility of the same component in the liquid solvent.

A self-similar solution of eq. (10) has been considered in [3], [5], [6], [9], [13] in the case of one dissolved gas in absence of the viscosity and the Laplace pressure effects and been extended in [3], [5] to the case of several dissolved gases. To include the viscous effects on bubble growth in the solution with one dissolved gas under conditions of nonstationary diffusion, it was recently proposed [14], [15] to use an approximate expression for the gas concentration profile around the bubble which is similar to that for the self-similar solution without viscosity. In this paper, we make a next step and propose an approximate anzatz for the concentration profile at nonstationary diffusion of ith (i=1, 2, …, k) gas component dissolved in multicomponent liquid solution in the most general case when the rates N′ and Ni of changing the number of molecules in the bubble is determined by eqs. (5)–(7). In this way, we will take into account the full-scale viscosity and capillarity effects.

In accordance with the previous results [3], [5], [13], [14], [15], the concentration profile around the growing multicomponent bubble can be approximately written in the form

(12) ci(ρ,R)=ci0(ci0ci(R))ρdzz2exp[RR2Dihi(R)(z2+2z3)]1dzz2exp[RR2Dihi(R)(z2+2z3)](i=1,2,...,k),ρ=rR(t).

Function hi(R) serves here as an important correction factor which can be determined from the requirement that the approximate solution (12) should provide a conservation of the total number of molecules of ith gas component in the bubble and solution. Note that such a correction factor has not previously been considered. Recognizing that eq. (8) for solution (12) can be rewritten as Ni=4πRDici(ρ,t)/ρ|ρ=1 , we see that an acceptable approximation (12) for the concentration profile of ith dissolved gas ensures the fulfillment of this relation if

(13) Ni=4πRDici0ci(R)1dzz2exp[RR2Dihi(R)(z2+2z3)](i=1,2,...,k).

For concentration profile (12), the decrease ΔNli(R) in the number of particles of ith gas component in the solution at time t is given by the expression

(14) ΔN1i(R)=4πR3(ci0ci(R))1dρρ2ρdzz2exp[RR2Dihi(R)(z2+2z3)]1dzz2exp[RR2Dihi(R)(z2+2z3)](i=1,2,...,k).

Using the identity

(15) 1dρρ2ρdzz2exp[b2(z2+2z3)]=13b,

which is valid for an arbitrary positive parameter b, we obtain from eq. (14)

(16) ΔN1i(R)=4πR33(ci0ci(R))DiRRhi(R)[1dzz2exp[RR2Dihi(R)(z2+2z3)]]1(i=1,2,...,k).

It follows from the condition of conservation of the number of particles of ith component of the dissolved gas in the form ΔNli(R)=Ni(R) and from eqs. (7), (13), (16) that function hi(R) equals

(17) hi(R)=1+R3dlnni(R)dR (i=1,2,...,k).

Thus the gas concentration profiles around the growing multicomponent bubble are completely determined by eqs. (12) and (17) if we know a relation between radius of the bubble and its chemical composition at any moment of time.

Joint evolution of the radius and chemical composition of a growing bubble

Taking into account definition N=i=1kNi , eqs. (4), (5) and (13), we obtain the nonlinear equation for the evolution of the bubble radius in the form

(18) RRn(R)(1+R3dlnn(R)dR)=i=1kDi(ci0ci(R))(1dzz2exp[RR2Dihi(R)(z2+2z3)])1.

In the case of one dissolved gas, in view of eqs. (6), (11) and (17), eq. (18) is sufficient for description of bubble growth. However, in the case of multicomponent solution with a number of dissolved gases, we need, in addition to evolution equation (18), to predict how the chemical composition of the bubble changes with time or with the bubble radius.

Let us introduce, together with volume concentrations ni of ith gas component in the bubble, the mole fraction xi of the same gas in the bubble as

(19) xi=Ni/N=ni/n, ni=xin, i=1kxi=1.

As follows from eq. (19), the rate xi of changing the mole fraction of ith gas component in the bubble with time is

(20) xi=NiNxiNN (i=1,2,...,k).

Let us also define the supersaturation ζi of ith gas component in the solution as

(21) ζici0cici (i=1,2,...,k)

where ci is the volume concentration of ith gas component at equilibrium with the bulk binary solution with the same solvent at presure P. Using eqs. (4), (11), (13), (19), equalities ci(R)=sini(R) and ci=si, eqs. (21) and (3) in eq. (20), we find

(22) xi=3R2(Disi((ζi+1)/[1+RR(1+2ηRσ)]xi)1dzz2exp[RR2Dihi(R)(z2+2z3)]xij=1kDjsj((ζj+1)/[1+RR(1+2ηRσ)]xj)1dzz2exp[RR2Djhj(R)(z2+2z3)]) (i=1,2,...,k).

With the help of eq. (19), eq. (17) for function hi(R) can be rewritten as

(23) hi(R)=1+R3(dlnndR+1xidxidR).

Substituting eq. (6) into the right-hand side of eq. (23) gives

(24) hi(R)=[1+RR(1+2ηRσ)]1[1+2R3R(1+2ηRσ+ηRσdRdR)]+R3xidxidR (i=1,2,...,k).

Using eqs. (3), (6), (11), (19) and (21) in eq. (18), we obtain

(25) RR(1+2R3R(1+2ηRσ+ηRσdRdR))=i=1kDisi(ζi+1xi(1+RR(1+2ηRσ)))1dzz2exp[RR2Dihi(R)(z2+2z3)].

The expression for dxi/dR=xi/R follows from eqs. (22):

(26) dxidR=3R2R(Disi((ζi+1)/[1+RR(1+2ηRσ)]xi)1dzz2exp[RR2Dihi(R)(z2+2z3)]xij=1kDjsj((ζj+1)/[1+RR(1+2ηRσ)]xj)1dzz2exp[RR2Djhj(R)(z2+2z3)]) (i=1,2,...,k).

It is obvious under the imposed conditions that the concentrations of all components of the solution are considered constant far away from the bubble, the requirement to preserve the total number of particles of each component in the solution together with the bubble is satisfied. However, the proposed approximate solution (12) [as well as the exact solution of the non-stationary diffusion equation (10)] provides the diffusion fluxes for each component consistent with the balance of these components at the bubble boundary precisely due to the factors hi. As can be seen from eqs. (25) and (26), the presence of these factors directly affects the dynamics of the size and chemical composition of the bubble with time.

Expressions (22), (24)–(26) make a set of equations describing the joint evolution of the radius of multicomponent bubble and its chemical composition {x}=(x1, x2,…, xk). They allow one to find radius of the bubble R and all molar fractions xi (i=1, 2, …, k) of gases in the bubble as a function of time or the bubble radius.

For a sufficiently large bubble, for which the influence of the curvature and viscosity becomes inessential, the self-similar regime of bubble growth is realized as an exact solution of nonstationary eq. (10). In this case we have [3], [5] xi=xis=const, hi(R)=1 (i=1, 2, …, k),

(27) ci(ρ)=ci0(ci0xisci)ρdzz2exp[(RR)s2Di(z2+2z3)]1dzz2exp[(RR)s2Di(z2+2z3)] (i=1,2,...,k),
(28) RR=(RR)s=Disi(ζi+1xis)xis1dzz2exp[(RR)s2Di(z2+2z3)]=i=1kDisi(ζi+1xis)1dzz2exp[(RR)s2Di(z2+2z3)].

The corresponding equations for stationary bubble composition {xs} and steady rate (RR′)s in the regime of self-similar growth look as

(29) xis=Disi(ζi+1)(RR)s1dzz2exp[(RR)s2Di(z2+2z3)]+Disi (i=1,2,...,k),
(30) i=1kDisi(ζi+1)(RR)s1dzz2exp[(RR)s2Di(z2+2z3)]+Disi=1.

In the limit of slow bubble growth at (RR′)s/Di≪1, one can find that

(31) 1dzz2exp[(RR)s2Di(z2+2z3)]1.

As a result, a steady-state diffusion of gases to the bubble is established, and eqs. (28)–(30) transform to

(32) (RR)s=Disi(ζi+1xis1)=i=1kDisi(ζi+1xis),
(33) xis=Disi(ζi+1)(RR)s+Disi (i=1,2,...,k),
(34) i=1kDisi(ζi+1)(RR)s+Disi=1.

Another limit is reached for fast bubble growth at (RR′)s/Di≫1 (i=1, 2, …, k). In this case,

(35) 1dzz2exp[(RR)s2Di(z2+2z3)](πDi6(RR)s)1/2,

and eqs. (28)–(30) transform as

(36) (RR)s1/2=(6Diπ)1/2si(ζi+1xis1)=(6π)1/2i=1kDi1/2si(ζi+1xis).
(37) xis=si(ζi+1)(π(RR)s/6Di)1/2+si (i=1,2,...,k),
(38) i=1ksi(ζi+1)(π(RR)s/6Di)1/2+si=1.

Let us note that in the case of a two-component bubble, the self-similar growth regime at stationary chemical composition of bubble was considered earlier in [16]. In particular, the differences in the stationary molar fractions of gases in the bubble for the Henry and Sieverts laws of gas dissolution have been analyzed.

Asymptotics of reaching steady-state and self-similar diffusion growth regime

As follows from the results of the previous section, if we consider bubble growth in the presence of viscosity and curvature effects, then the composition of the bubble changes while its size grows until these effects become negligible. It is important to establish the size interval for reaching the stationary composition in the bubble. In the case of ternary solution with two dissolved gases, we can describe asymptotic reaching the stationary composition {xs} in the growing bubble analytically.

Let us first consider the case of slow bubble growth at (RR′)s/Di≪1 (i=1, 2). Denote the molar fraction of first dissolved gas in solution as xx1. As follows from eqs. (25), (26) and (31) at x1= x, x2=1–x,

(39) RR[1+2R3R(1+2ηRσ+ηRσdRdR)]=D1s1[ζ1+1x(1+RR(1+2ηRσ))]+D2s2[ζ2+1(1x)(1+RR(1+2ηRσ))],
(40) dxdR=3R2R[D1s1(1x)(ζ1+11+RR(1+2ηRσ)x)xD2s2(ζ2+11+RR(1+2ηRσ)1+x)],

It is convenient to introduce new notation

(41) uRR(1+2ηRσ), v2R3R(1+2ηRσ+ηRσdRdR),Δx(R)xxs.

With increasing the bubble radius R and approaching stationary molar fraction xs in the bubble, quantities u, v, Δx(R) tend to zero and RR′→(RR′)s where the molar fraction xs and the rate (RR′)s of stationary bubble growth can be found from eqs. (33) and (34). Being interested in finding the asymptotic solutions of eqs. (39) and (40), we can set RR′=(RR′)s and, correspondingly, use R′=(RR′)s/R on the right-hand side of expressions for u and v:

(42) u=RR(1+2η(RR)sσR),v=2R3R(1+η(RR)sσR).

Keeping the terms of the first order with respect to small u, v and Δx(R), we obtain from eqs. (39)–(41) with the help of eq. (32)

(43) RR(RR)s(1v)(D1s1D2s2)Δx(D1s1xs+D2s2(1xs))u,
(44) dΔxdR3R(RR)s[((RR)s+D1s1(1xs)+D2s2xs)Δx+xs(1xs)(D1s1D2s2)u],

With use of eq. (42), eqs. (43) and (44) can be rewritten as

(45) RR(RR)s[1RR(23(1+η(RR)sσR)+D1s1xs+D2s2(1xs)(RR)s(1+2η(RR)sσR))](D1s1D2s2)Δx,
(46) dΔxdR3λRΔx3(RR)sxs(1xs)(D1s1D2s2)RR2(1+2η(RR)sσR)

where

(47) λ1+(1xs)D1s1+xsD2s2(RR)s=1+D1s1(1xs)+D2s2xsD1s1(ζ1+1xs)+D2s2(ζ2+xs)>1.

The general solution to eq. (47) with the initial condition for molar fraction deviation Δx(R)|R=R0=Δx0 at an arbitrary R=R0 has the form

(48) Δx(R)Δx0(R0R)3λ+3xs(1xs)(D1s1D2s2)(RR)sRR[13λ1(1(R0R)3λ1)+23λ2ησR(1(R0R)3λ2)].

Equations (48) and (45) give the desired asymptotic behavior in bubble radius R for molar fractions of gases in the bubble and the bubble growth rate at slow bubble growth and describe the transition to the steady-state diffusion. As we can see, the most long-lived are terms that decrease with increasing size like R−1 and R−2.

Let us now consider case of fast bubble growth at (RR′)s/Di≪1 (i=1, 2). In view of eq. (35) we have

(49) (1dxz2exp[RR2Dihi(R)(z2+2z3)])1(6πRRDihi(R))1/2.

As well as in the case of slow diffusion, with increasing the bubble radius R and approaching stationary molar fraction xs in the bubble, quantities u, v, Δx(R) tend to zero and RR′→→(RR′)s. However the molar fraction xs and the rate (RR′)s of stationary bubble growth should be sought now from eqs. (37) and (38). Keeping the terms of the first order with respect to small u, v and Δx(R), we obtain from eq. (25) at x1=x, x2=1–x in view of eqs. (41)

(50) (RR)1/2(1+v)=(6πD1h1(R))1/2s1(ζ1+1xsΔxxsu)+(6πD2h2(R))1/2s2(ζ2+xs+Δx(1xs)u).

Using eqs. (24) and (41), (42), we find with the same accuracy

(51) h11/2(R)1R6R(1+4ηRσ)+R6xsdΔxdR,
(52) h21/2(R)1R6R(1+4ηRσ)R6(1xs)dΔxdR,

Substituting h11/2(R) and h21/2(R) with the help of eqs. (51) and (52) into eq. (50) and using eqs. (36) and (42) allows us to rewrite expression for (RR′)1/2 in the form

(53) (RR)1/2=(RR)s1/2[1RR(56+4η(RR)s3σR+D11/2s1xs+D21/2s2(1xs)D11/2s1(ζ1+1xs)+D21/2s2(ζ2+xs)(1+2η(RR)sσR))](6π)1/2(D11/2s1D21/2s2)Δx(R).

As follows from eqs. (45) and (53), the ratio between the contributions of viscous forces and Laplace forces during the evolution of the bubble substantially depends on the degree of non-stationarity of diffusion flows of gas molecules in solution. Quantitatively, this ratio can be characterized by the value of a dimensionless quantity η(RR′)sR. Thus, with an increase in the degree of nonstationary diffusion and a corresponding increase in the magnitude (RR′)s, the relative role of the viscosity forces, ceteris paribus, increases monotonously.

As follows from eqs. (26) at x1=x, x2=1–x in view of eqs. (41), (49) and smallness of u, v, RR′–(RR′)s and Δx,

(54) dΔxdR3R(6π1(RR)s)1/2[(D1h1(R))1/2s1(1xsΔx)(ζ1+1xsΔxxsu) (xs+Δx)(D2h2(R))1/2s2(ζ2+xs+Δx(1xs)u)].

Substituting h11/2(R) and h21/2(R) with the help of eqs. (51) and (52) into eq. (54) and using eqs. (36) and (42) allows us to rewrite expression for Δx in the form

(55) dΔxdR6μRΔx6D11/2s1D21/2s2D11/2s1(ζ1+1xs)+D21/2s2(ζ2+xs)xs(1xs)RR2(1+2η(RR)sσR)

where

(56) μ1+(6π1(RR)s)1/2(D11/2s1(1xs)+D21/2s2xs)=1+D11/2s1(1xs)+D21/2s2xsD11/2s1(ζ1+1xs)+D21/2s2(ζ2+xs)>1.

The solution to eq. (55) can be constructed in the same way as for eq. (43) (the equations differ only in the coefficients in terms on the right-hand side). Integration of eq. (55) with initial condition Δx(R)|R=R0=Δx0 at an arbitrary R=R0 gives

(57) Δx(R)Δx0(R0R)6μ+6(D11/2s1D21/2s2)xs(1xs)D11/2s1(ζ1+1xs)+D21/2s2(ζ2+xs)RR×[16μ1(1(R0R)6μ1)+16μ22η(RR)sσR(1(R0R)6μ2)].

Equations (57) and (55) give the desired asymptotic behavior in bubble radius R for molar fractions of gases in the bubble and the bubble growth rate at fast bubble growth and describe the transition to the self-similar diffusion of gas components. Finally, we see from eqs. (57) and (53) that, as in the case of slow growth, the most long-lived terms are those that decrease with increasing size as R−1 and R−2.

Conclusions

The results of this research can be directly applied to kinetics of distribution of bubbles in radii and composition at degassing of a multicomponent decompressed solution. Using an approximate solution (12) of the non-stationary diffusion equation (10) allows one to completely take into account the capillary and viscosity effects in the evolution of the bubble distribution in time in the entire range of bubble radii above the unstable equilibrium radius. Thus, the general approach to an adequate description of the kinetics of the nucleation stage of the bubbles arises, not only under the assumption of their self-similar growth, but also in a such situation when, by the end of this stage, the self-similar growth mode and stationary composition of the bubbles have not yet been established. A significant difference between expression (12) and that proposed in [14], [15] in the case of one-component bubble growth in a viscous solution is the presence of factor h [hi in the one-component case determined by eq. (17)], the choice of which, as shown, provides the requirement to preserve the total number of gas molecules in the system. Setting h=1 generally violates the condition of such a balance.

The role of the capillary effects jointly with the viscosity effects, as well as the multicomponent case with arbitrary number of dissolved gases, has not been considered previously. As said above, we were focused in this study on possible application to nucleation theory, in particular from the point of reaching the stationary growth and steady composition of supercritical bubbles at the nucleation stage. However, the obtained results should be of interest for planning experiments with single bubbles which were intensively studied in [17], [18].


Article note

A collection of invited papers based on presentations at 21st Mendeleev Congress on General and Applied Chemistry (Mendeleev-21), held in Saint Petersburg, Russian Federation, 9–13 September 2019.


Acknowledgments

This work was supported by Russian Foundation for Basic Research (Funder Id: http://dx.doi.org/10.13039/501100002261, grant 19-03-00997).

References

[1] F. M. Kuni, A. P. Grinin. Colloid J. 46, 412 (1984).Search in Google Scholar

[2] V. V. Slezov. Kinetics of First-Order Phase Transitions. Wiley-VCH, Berlin (2009).10.1002/9783527627769Search in Google Scholar

[3] A. E. Kuchma, A. K. Shchekin, D. S. Martyukova. J. Chem. Phys. 148, 234103 (2018).10.1063/1.5026399Search in Google Scholar PubMed

[4] A. E. Kuchma, A. K. Shchekin. J. Chem. Phys. 150, 054104 (2019).10.1063/1.5077006Search in Google Scholar PubMed

[5] A. K. Shchekin, A. E. Kuchma. Colloid J. 82, (2020) (to be published).10.1134/S1061933X20060198Search in Google Scholar

[6] A. E. Kuchma, F. M. Kuni, A. K. Shchekin. Phys. Rev. E 80, 061125 (2009).10.1103/PhysRevE.80.061125Search in Google Scholar PubMed

[7] A. E. Kuchma, M. Markov, A. K. Shchekin. Physica A 402, 255 (2014).10.1016/j.physa.2014.02.005Search in Google Scholar

[8] A. E. Kuchma, A. K. Shchekin, M. N. Markov. Colloids Surf. A 483, 307 (2015).10.1016/j.colsurfa.2015.04.020Search in Google Scholar

[9] A. E. Kuchma, A. K. Shchekin, M. Y. Bulgakov. Physica A 468, 228 (2017).10.1016/j.physa.2016.11.007Search in Google Scholar

[10] A. E. Kuchma, A. K. Shchekin, D. S. Martyukova, A. V. Savin. Fluid Phase Equil. 455, 63 (2018).10.1016/j.fluid.2017.09.022Search in Google Scholar

[11] C. E. Brennen. Cavitation and Bubble Dynamics. University Press, Oxford (1995).Search in Google Scholar

[12] M. S. Plesset, A. Prosperetti. Ann. Rev. Fluid Mech. 9, 14 (1977).10.1146/annurev.fl.09.010177.001045Search in Google Scholar

[13] L. E. Scriven, Chem. Eng. Sci. 10, 1 (1959).10.1016/0009-2509(59)80019-1Search in Google Scholar

[14] A. A. Chernov, A. A. Pil’nik, M. N. Davydov, E. V. Ermanyuk, M. A. Pakhomov. Int. J. Heat Mass Transf. 123, 1101 (2018).10.1016/j.ijheatmasstransfer.2018.03.045Search in Google Scholar

[15] A. A. Chernov, A. A. Pil’nik, M. N. Davydov. J. Phys. Conf. Ser. 1382, 012107 (2019).10.1088/1742-6596/1382/1/012107Search in Google Scholar

[16] G. Y. Gor, A. E. Kuchma. J. Chem. Phys. 131, 234705 (2009).10.1063/1.3276708Search in Google Scholar PubMed

[17] V. P. Skripov, M. Z. Faizullin. Crystal-Liquid-Gas Phase Transitions and Thermodynamic Similarity. Wiley-VCH, Berlin, Weinheim (2006).10.1002/3527608052Search in Google Scholar

[18] V. G. Baidakov. Explosive Boing of Superheated Cryogenic Liquids. Wiley-VCH, Berlin, Weinheim (2007).10.1002/9783527610662Search in Google Scholar

Published Online: 2020-03-07
Published in Print: 2020-07-28

©2020 IUPAC & De Gruyter. This work is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License. For more information, please visit: http://creativecommons.org/licenses/by-nc-nd/4.0/

Downloaded on 8.12.2022 from https://www.degruyter.com/document/doi/10.1515/pac-2020-0101/html
Scroll Up Arrow