Dynamics of an Extending Contracting Spherical Bubble

In chemical technology, one often meets the problem about a spherically symmetric deformation (contraction or extension) of a gas bubble in an infinite viscous fluid. In the homobaric approximation (the pressure is homogeneous inside the bubble) [306, 312], only the motion of the outer fluid is of interest. The Navier-Stokes equations describing this motion in the spherical coordinates have the form 

If there is no mass transfer across the bubble boundary, the fluid velocity on the boundary is equal to the velocity of the boundary, 

The solution of Eq. (2.4.11) with regard to (2.4.12) and the condition that the fluid velocity is zero at infinity has the form 

By substituting (2.4.13) into (2.4.10) and then integrating with respect to R from a to oo, we arrive at the relations 

The fluid pressure P R=a at the bubble boundary can be determined from the condition on the jump of normal stresses on the discontinuity surface, that is, the bubble boundary [24, 430]. Under the homobaric assumptions, the gas in the bubble does not move, which implies that 

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