A steady two-dimensional flow round an axis is the sum of a line source of strength…

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 round an axis is the sum of a line source of strength

m = 3.0\,\mathrm{m^2\,s^{-1}}

and a line vortex of circulation

\Gamma = 1.5\,\mathrm{m^2\,s^{-1}}

, both on the same axis. Off the axis,

u_r = m/2\pi r

and

u_\theta = \Gamma/2\pi r

. This field decays only as

1/r

, so the Helmholtz key result does not cover it; use the planar potentials of the spiral-flow example, with a reference length

r_0

, and check the split directly. ((a)) Write the Helmholtz decomposition

\mathbf u = -\nabla\phi + \nabla\times\mathbf A

for this flow, giving

\phi

and

\mathbf A

explicitly and stating which axis quantity generates each. ((b)) Find the speed and the angle to the radius of the flow at

r = 2.0\,\mathrm m

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