2 Q Figure 2.5.1:The spherical coordinates The corresponding stream function at infinity follows by integration 2 in0. 2.5.6) Using the vector identity 7×(V×=7(7·)-7d (2.5.7) and (2.5.1),we get g=-V×(7×=-7×6 (2.5.8) Taking the curl of(2.5.2)and using (2.5.8)we get V×(7×G)=0 (2.5.9) After some straightforward algebra given in the Appendix,we can show that g-7× (2.5.10) r sin 0 and (=7×g=V×V× 山eo 包过 82b sin8 1 aψ X (2.5.11) r sin 0 rsin r288 sin0 80 Now from (2.5.9) vx×v×=vxxx(x】=02 z y x r o
Figure 2.5.1: The spherical coordinates The corresponding stream function at infinity follows by integration ψ = W 2 r2 sin2 θ, r ∼ ∞ (2.5.6) Using the vector identity ∇ × (∇ × ~q) = ∇(∇ · ~q) − ∇2 ~q (2.5.7) and (2.5.1), we get ∇2 ~q = −∇ × (∇ × ~q) = −∇ × ~ζ (2.5.8) Taking the curl of (2.5.2) and using (2.5.8) we get ∇ × (∇ × ~ζ) = 0 (2.5.9) After some straightforward algebra given in the Appendix, we can show that ~q = ∇ × Ã ψ~eφ r sin θ ! (2.5.10) and ~ζ = ∇ × ~q = ∇ × ∇ × Ã ψ~eφ r sin θ ! = − ~eφ r sin θ Ã ∂2ψ ∂r2 + sin θ r2 ∂ ∂θ Ã 1 sin θ ∂ψ ∂θ !! (2.5.11) Now from (2.5.9) ∇ × ∇ × (∇ × ~q) = ∇ × ∇ × " ∇ × Ã ∇ × ψ~eφ r sin θ !# = 0