A spherical gas bubble of radius sits in water (

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Solid continuum mechanics and complex fluids · Solid continuum mechanics and complex fluids

Bending, Ringing, Breaking and Flowing, Ch Bubbles, drops and cavitation

Problem

A spherical gas bubble of radius R=0.20 mmR = 0.20\,\mathrm{mm} sits in water (ρ=1000 kg m−3\rho = 1000\,\mathrm{kg\,m^{-3}}, surface tension γ=0.072 N m−1\gamma = 0.072\,\mathrm{N\,m^{-1}}, viscosity η=1.0×10−3 Pa s\eta = 1.0\times10^{-3}\,\mathrm{Pa\,s}) and is expanding with wall speed R˙=0.50 m s−1\dot R = 0.50\,\mathrm{m\,s^{-1}}. The gas inside is at pressure pB=150 kPap_B = 150\,\mathrm{kPa}. Treat the liquid as incompressible and Newtonian, the flow as purely radial, and the bubble as a sphere throughout. ((a)) Starting from the radial flow field vr(r)=R2R˙/r2v_r(r) = R^2\dot R/r^2 fixed by incompressibility, find the viscous normal stress the liquid exerts at the wall, 2η ∂vr/∂r2\eta\,\partial v_r/\partial r evaluated at r=Rr=R, and hence derive the normal-stress balance at the wall in the form p(R+)=pB−2γR−4ηR˙Rp(R^+) = p_B - \dfrac{2\gamma}{R} - \dfrac{4\eta\dot R}{R}. ((b)) Evaluate the surface-tension term and the viscous term numerically, and find the liquid pressure p(R+)p(R^+) just outside the wall.

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