Water swirls round a plughole with velocity

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

Mathematical Methods for Physics, Ch Gradient, divergence and curl

Problem

Water swirls round a plughole with velocity v=k(−y, x, 0)/(x2+y2)\mathbf v = k(-y,\ x,\ 0)/(x^2+y^2), the plughole swirl, where k=0.020 m2 s−1k = 0.020\ \mathrm{m^2\,s^{-1}}. (a) In cylindrical coordinates (ρ,ϕ,z)(\rho,\phi,z), show that φ=−kϕ\varphi = -k\phi satisfies v=−∇φ\mathbf v = -\nabla\varphi wherever ϕ\phi is defined. (b) On the half-space y>0y>0, take ϕ\phi between 00 and π\pi. Find φ(B)−φ(A)\varphi(B) - \varphi(A) for A=(1.0, 1.0, 0) mA = (1.0,\ 1.0,\ 0)\ \mathrm m and B=(−1.0, 1.0, 0) mB = (-1.0,\ 1.0,\ 0)\ \mathrm m. (c) Which identity guarantees ∇×v=0\nabla\times\mathbf v = \mathbf 0 wherever φ\varphi is a smooth potential, and why must it hold whatever φ\varphi is? The circulation round a circle about the axis is nevertheless 2πk2\pi k. Find its value and explain why this does not contradict the identity.

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