A point source of strength sits at the origin

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Fluids & Continuum Physics · Kinematics, conservation laws and ideal flow

Fluid Dynamics, Ch Kinematics in three dimensions and Cauchy's equation of motion

Problem

A point source of strength Q=0.50 m3 s−1Q=0.50\,\mathrm{m^3\,s^{-1}} sits at the origin, with Stokes stream function Ψ=−Q4πcos⁡θ\Psi=-\dfrac{Q}{4\pi}\cos\theta. Someone wants the volume flux crossing the hemispherical cap between the cone θ1=90∘\theta_1=90^\circ and the cone θ2=180∘\theta_2=180^\circ (the whole lower half-space) and argues: “Ψ(θ1)=−Q4πcos⁡90∘=0\Psi(\theta_1)=-\dfrac{Q}{4\pi}\cos 90^\circ=0 and Ψ(θ2)=−Q4πcos⁡180∘=Q4π\Psi(\theta_2)=-\dfrac{Q}{4\pi}\cos 180^\circ=\dfrac{Q}{4\pi}, so the flux is ΔΨ=Ψ(θ2)−Ψ(θ1)=Q4π=0.0398 m3 s−1\Delta\Psi=\Psi(\theta_2)-\Psi(\theta_1)=\dfrac{Q}{4\pi}=0.0398\,\mathrm{m^3\,s^{-1}}.” Find the error in this reasoning and give the correct flux.

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