Consider steady flow at velocity U relative to a spherical bubble of radius R, as in Fig.

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Consider steady flow at velocity U relative to a spherical bubble of radius R, as in Fig. P5.1. The flow is axisymmetric (independent of ϕ and with vϕ = 0) and UR/ν → 0 in both fluids. The Hadamard–Rybczyński analysis predicts that

inside the bubble. Determine vr(r, θ) inside (0 ≤ r ≤ R). Note that vr(R, θ) = 0 for a bubble of fixed size.

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