A spherical cavitation bubble of radius grows in water (viscosity )

PHYSICS · Multi-part · Fermi University Physics · Fluids & Continuum Physics · Fluid statics and the continuum picture · Kinematics, conservation laws and ideal flow

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

Problem

A spherical cavitation bubble of radius R(t)R(t) grows in water (viscosity μ=1.00×10−3 Pa s\mu = 1.00\times10^{-3}\,\mathrm{Pa\,s}). Far-field effects aside, the surrounding water moves purely radially, ur(r,t)=A(t)r2(r≥R(t)),u_r(r,t)=\frac{A(t)}{r^2}\qquad (r\ge R(t)), which satisfies ∇⋅u=0\nabla\cdot\mathbf{u}=0 for r>0r>0 (this is the incompressible condition of Chapter 1). At the instant considered, A=5.0×10−7 m3 s−1A=5.0\times10^{-7}\,\mathrm{m^3\,s^{-1}} and R=0.50 mmR=0.50\,\mathrm{mm}. There is no evaporation or condensation at the bubble wall, so the wall is a material surface of the liquid, F(r,t)=r−R(t)=0F(r,t)=r-R(t)=0. ((a)) Using the fact that particles labelled by the Lagrangian map of Chapter 2 remain on a material surface for all time, derive the radial kinematic condition relating uru_r at r=Rr=R to R˙=dR/dt\dot R=\mathrm{d}R/\mathrm{d}t, and hence find R˙\dot R at the instant given. ((b)) Using the two-viscosity Newtonian stress law of Chapter 1 (with the bulk-viscosity term dropped, since the flow is incompressible), compute the viscous part of the radial normal stress in the water, σrr′=2μ ∂ur/∂r\sigma_{rr}'=2\mu\,\partial u_r/\partial r, evaluated at the bubble wall.

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