A steady two-dimensional flow consists of a single line vortex of circulation on an axis

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

Conditions of Validity, Ch Gauss, Stokes and Green proved, and the first inversion of a differential operator

Problem

A steady two-dimensional flow consists of a single line vortex of circulation Γ=4.0 m2 s−1\Gamma = 4.0\,\mathrm{m^2\,s^{-1}} on an axis, with no line source (m=0m=0). Off the axis the velocity is u=uθθ^\mathbf u = u_\theta\hat{\boldsymbol\theta} with uθ=Γ/2πru_\theta = \Gamma/2\pi r. This field decays only as 1/r1/r, so the Helmholtz key result does not cover it; use the planar potentials of the spiral-flow example, with a reference length r0r_0, and check the split directly. ((a)) Write the Helmholtz decomposition u=−∇ϕ+∇×A\mathbf u = -\nabla\phi + \nabla\times\mathbf A for this flow, giving ϕ\phi and A\mathbf A in terms of rr, and explain in one sentence why ϕ=0\phi = 0 everywhere off the axis. ((b)) Find the speed of the flow at r=0.50 mr = 0.50\,\mathrm m.

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