A shallow dish of golden syrup sits on a turntable that has been running for a long…

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

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

Problem

A shallow dish of golden syrup sits on a turntable that has been running for a long time at a constant

24

revolutions per minute, so the syrup turns with the dish as a solid body. In laboratory axes with origin on the axis of rotation,

x

and

y

horizontal and

z

vertically upward out of the plane towards the observer (vorticity positive anticlockwise seen this way), the velocity field in metres per second is

u = -\Omega y,\qquad v = \Omega x,

with

\Omega

the angular velocity of the turntable in radians per second. A small square of dye of side

10\ \mathrm{mm}

is injected at a radius of

60\ \mathrm{mm}

. One velocity probe is clamped to the laboratory bench above a fixed point of the dish's path; a second probe is bolted to the turntable and rides round with the same piece of syrup throughout. A student writes: "

\partial u/\partial y = -\Omega

is not zero, so this flow is a shear; the dye square must be sheared into a parallelogram." ((a)) Say which probe gives an Eulerian record and which a Lagrangian one. ((b)) Compute all components of

e_{ij}

and

\Omega_{ij}

for this field, and hence

\omega_z

and the rate at which a dye mark turns. ((c)) Say what actually happens to the dye square, and where the student's reasoning fails.

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