A plane flow has velocity field u(x,y)=C , (-y

PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Vector calculus and field geometry · Topology and global structure

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

Problem

A plane flow has velocity field u(x,y)=C (−y, x)x2+y2+k (y, x),\mathbf u(x,y)=C\,\frac{(-y,\ x)}{x^2+y^2}+k\,(y,\ x), with C=0.30 m2 s−1C=0.30\,\mathrm{m^2\,s^{-1}} and k=0.50 s−1k=0.50\,\mathrm{s^{-1}}, defined for r>0r>0 with xx and yy in metres. ((a)) Show that u\mathbf u is curl-free wherever it is defined, and find its circulation round the unit circle. Does a single-valued potential exist on r>0r>0? ((b)) Cut the half-line θ=π\theta=\pi out of the domain. Construct ϕ(r)=−∫r0ru⋅dr′\phi(\mathbf r)=-\int_{\mathbf r_0}^{\mathbf r}\mathbf u\cdot\mathrm d\mathbf r' on the cut domain, with r0=(1,0) m\mathbf r_0=(1,0)\,\mathrm m, by integrating along the unit circle from r0\mathbf r_0 to the angle θ\theta and then out along the ray at that angle, and check that −∇ϕ=u-\nabla\phi=\mathbf u. ((c)) Evaluate ϕ\phi at (1.0, 1.0) m(1.0,\,1.0)\,\mathrm m, and find the jump ϕ(θ→π−)−ϕ(θ→−π+)\phi(\theta\to\pi^-)-\phi(\theta\to-\pi^+) across the cut at any radius.

Browse the problem library