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Growth of a gas bubble in a supersaturated and slightly compressible liquid at low Mach number

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Abstract

In this paper, the growth of a gas bubble in a supersaturated and slightly compressible liquid is discussed. The mathematical model is solved analytically by using the modified Plesset and Zwick method. The growth process is affected by: sonic speed in the liquid, polytropic exponent, diffusion coefficient, initial concentration difference, surface tension, viscosity, adjustment factor and void fraction. The famous formula of Plesset and Zwick is produced as a special case of the result at some values of the adjustment factor. Moreover, the resultant formula is implemented to the case of the growth of underwater gas bubble.

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Abbreviations

A :

Constant defined by Eq. 11

b :

Adjustment factors (dimensionless)

C :

Concentration of the gas in liquid (kg m−3)

C 1 :

Velocity of the sound in the liquid (m s−1)

C R :

Instantaneous gas concentration at the bubble boundary (kg m−3)

D :

Diffusivity constant (m2 s−1)

\( \Updelta C_{0} \) :

The concentration difference defined by Eq. 12 (kg m−3)

\( \Updelta C_{R}^{*} \) :

Instantaneous concentration difference, defined by Eq. 14 (kg m−3)

\( J_{a} \) :

Jacob number for the case of mass diffusion [26], given by the Eq. 31

M :

Mach number (The ratio of the bubble growth velocity to the sonic speed in the liquid)

P g :

Pressure of the bubble wall (N m−2)

r :

The distance from the origin of the bubble (m)

R 0 :

Initial bubble wall radius (m)

R :

Instantaneous bubble wall radius (m)

\( \dot{R} \) :

Instantaneous bubble wall velocity (m s−1)

\( \ddot{R} \) :

Instantaneous bubble wall acceleration (m s−2)

t :

Time elapsed [s]

α :

Constant defined by Eq. 26

\( \beta \) :

Constant defined by Eq. 12

\( \hat{\beta } \) :

Constant defined by Eq. 25

γ:

Constant defined by Eq. 16

κ:

Polytropic exponent \( \left\{ {\begin{array}{*{20}c} {\kappa = 0:{\text{Isoparic}}\;{\text{system}}\;({\text{Const}} .\;{\text{pressure}})} \\ {\kappa = 1:{\text{Isothermal}}\;{\text{system}}\;({\text{Const}} .\;{\text{temperature}})} \\ {\kappa = C_{p} /C_{v} :{\text{Adiabatic}}\;{\text{system}}\;({\text{No heat transfer}})} \\ \end{array} } \right. \)

λ:

Constant defined by Eq. 24

μ:

Viscosity [Pa s]

ρ g :

Density of the gas inside the bubble (kg m−3)

ρ l :

Density of the liquid surrounding the bubble (kg m−3)

σ:

The surface tension of liquid surrounding the bubble (N m−1)

τ:

Dimensionless variable defined by Eq. 16

φ0 :

Initial void fraction defined by Eq. 31 (Dimensionless)

Ψ :

Dimensionless volume variable (instantaneous volume to initial bubble volume) defined by Eq. 15

0:

Initial-value quantities

g :

Variables corresponding to the gas bubble

l :

Variables corresponding to the liquid in which the bubble growing in

m :

Maximum value

R :

Bubble boundary

sat :

Saturation

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Acknowledgments

The authors would like to thank the reviewers for their valuable comments and suggestions for improving the original manuscript.

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Authors and Affiliations

  1. Department of Mathematics, Faculty of Science, Tanta University, Tanta, Egypt

    S. A. Mohammadein

  2. Department of Mathematics, Faculty of Science, Benha University, Benha, Egypt

    K. G. Mohamed

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  1. S. A. Mohammadein
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  2. K. G. Mohamed
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Correspondence to S. A. Mohammadein.

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Mohammadein, S.A., Mohamed, K.G. Growth of a gas bubble in a supersaturated and slightly compressible liquid at low Mach number. Heat Mass Transfer 47, 1621–1628 (2011). https://doi.org/10.1007/s00231-011-0813-9

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