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 (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]
 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
 0:

Initialvalue 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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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 (2011). https://doi.org/10.1007/s0023101108139
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Keywords
 Void Fraction
 Adjustment Factor
 Bubble Radius
 Bubble Wall
 Compressible Liquid