A spherical air bubble of instantaneous radius sits in water and is expanding at rate

PHYSICS · Subjective · Fermi University Physics · Fluids & Continuum Physics · Fluid statics and the continuum picture · Fluid statics and the continuum picture

Fluid Dynamics, Ch Interfaces: capillary statics and the stress conditions at a surface

Problem

A spherical air bubble of instantaneous radius a=0.50 mma = 0.50\,\mathrm{mm} sits in water and is expanding at rate a˙=0.020 m s−1\dot a = 0.020\,\mathrm{m\,s^{-1}}. The water is incompressible and moves purely radially; the air inside expands uniformly, and its viscosity is negligible. Take water's viscosity as μ=1.00×10−3 Pa s\mu = 1.00\times10^{-3}\,\mathrm{Pa\,s}, and γ=0.0728 N m−1\gamma = 0.0728\,\mathrm{N\,m^{-1}}. Using a pillbox straddling the interface with n^\hat{\mathbf{n}} pointing from the air (fluid 1) into the water (fluid 2), derive the normal stress balance including the viscous normal stress in the water, and evaluate the pressure difference pair−pwater(a)p_{\text{air}}-p_{\text{water}}(a) at this instant.

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