A steady drain vortex sits in a large open tank of water ( )

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

Fluids and Continuum Physics, Ch Vorticity, circulation and vortex motion

Problem

A steady drain vortex sits in a large open tank of water (

\rho = 1000\ \text{kg m}^{-3}

). Viewed from above, with

z

pointing upwards, out of the page, the water circulates anticlockwise about a vertical axis. Outside a core of radius

0.10\ \text{m}

the flow is that of a line vortex,

u_\theta = \Gamma/(2\pi r)

, and it is steady, incompressible, inviscid and irrotational there; the radial and vertical velocity components are negligible. A Pitot-static tube is held in a horizontal plane at radius

r_1 = 0.30\ \text{m}

from the axis with its nose pointing into the local flow. It reads a difference (stagnation pressure minus static pressure) of

2.0\ \text{kPa}

. ((a)) Find the water speed at

r_1

. ((b)) Find the circulation round the horizontal circle of radius

r_1

centred on the axis, taken anticlockwise. ((c)) Find the speed at

r_2 = 0.75\ \text{m}

. ((d)) State, with a reason, the circulation round the horizontal circle of radius

r_2

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