A cylindrical bucket of water has been spinning long enough that the water turns with…

PHYSICS · Subjective · Fermi University Physics · Mechanics & Dynamical Systems · Fluids & Continuum Physics · Kinematics, conservation laws and ideal flow

Fluids and Continuum Physics, Ch Kinematics of flow: fields, streamlines and the material derivative

Problem

A cylindrical bucket of water has been spinning long enough that the water turns with the bucket as a solid body. In coordinates fixed to the laboratory, with the origin on the axis of rotation and

z

vertically upward, the velocity field of the water is

\mathbf{u} = (-\Omega y,\; \Omega x,\; 0), \qquad \Omega = 1.5\ \mathrm{s^{-1}},

with

x

and

y

in metres. A velocity probe is clamped to the bench so that its sensing head stays at the fixed point

(0.10\ \mathrm{m},\,0)

; separately, a small neutrally buoyant bead is dropped in and its position filmed. Treat the water as a continuum and the field as exactly the one given. Which one of the following statements is correct? (a) The streamlines are circles centred on the axis; the probe record is Eulerian and the bead track is Lagrangian; the probe reads a velocity that does not change in time, of magnitude

0.15\ \mathrm{m\,s^{-1}}

in the

+y

direction. (b) The streamlines are straight lines radiating out from the axis; the probe record is Eulerian and the bead track is Lagrangian; the probe reads

0.15\ \mathrm{m\,s^{-1}}

in the

+x

direction. (c) The streamlines are circles centred on the axis; the probe record is Lagrangian, because the water it measures is moving, and the bead track is Eulerian. (d) The streamlines are circles centred on the axis; the probe record is Eulerian and the bead track is Lagrangian; but the probe reading varies sinusoidally in time, because the water goes round and round. Give the letter of the correct statement.

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