# 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.

## Keywords

Void Fraction Adjustment Factor Bubble Radius Bubble Wall Compressible Liquid## List of symbols

*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 (m

^{2}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]

## Greek symbols

*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

- φ
_{0} Initial void fraction defined by Eq. 31 (Dimensionless)

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

## Subscripts

- 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

## Notes

### Acknowledgments

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

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