A Sphere Theorem on the Stokes Equation for Axisymmetric Viscous Flow

概要

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A theorem is presented which gives the perturbation stream function ψ(γ,θ) when a sphere (γ=a) is introduced into an unlimited viscous fluid in axisymmetric motion obeying the Stokes equation, of which the stream function is ψ(γ,θ), where(γ,θ,ψ) are the polar coordinates. ψ(γ,θ) is given by ψ(γ,θ)=ψ_1(γ,θ)+a^2ψ_2(γ,θ), ψ_1(γ,θ)=(a-γ^2/a)∂/<∂R>ψ_1(R,θ)-(<3γ>/<2a>-<1γ^3>/<2a^3>)ψ_1(R,θ),ψ_2(γ,θ)=-<sinθ>/<4√<γ>∫^γ_0√<γ>ψ(γ,θ)dγ, ψ_1(γ,θ)=ψ(γ,θ)-γ^2,ψ_2(γ,θ) where ω is the vorticity, and R=a^2/r. This gives the stream sheets directly without recourse to manipulation of spherical harmonics, and is very simple when ψ(γ,θ) is irrotational, i.e. ω=ψ_2=0. As examples of the applications, of this theorem, the stream functions for uniform flow, source flow and parabolic flow past a sphere are derived.
社団法人日本物理学会の論文
1956-07-05