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


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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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


Constant defined by Eq. 16


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


Constant defined by Eq. 24


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)


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


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


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 :



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

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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 (2011).

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  • Void Fraction
  • Adjustment Factor
  • Bubble Radius
  • Bubble Wall
  • Compressible Liquid