On a horizontal water table used for flow visualisation

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

On a horizontal water table used for flow visualisation, the steady velocity of the thin water layer in the gap between two diagonal guide vanes is fitted, in metres per second, by

u = S y,\qquad v = S x,\qquad S = 2.5\ \mathrm{s^{-1}},

with

x

y

in metres in the horizontal plane and

z

vertically upward out of that plane towards the observer; vorticity is positive anticlockwise seen this way. A particle-image velocimetry (PIV) system, which photographs seed particles in the water, reports the velocity at each point of a fixed grid. A small light cross of two straws, which drifts with the water and whose arms follow the local fluid lines, is laid down at the point

(0.10\ \mathrm{m},\,0.05\ \mathrm{m})

and filmed. A student writes: "The streamlines through this region are curved, so the fluid is going round a bend; a floating cross there must spin." ((a)) Say which of the two records, the fixed PIV grid and the film of the cross, is Eulerian and which is Lagrangian. ((b)) Using the tangency condition

\mathrm{d}x/u = \mathrm{d}y/v

, show that the streamlines are the curves

x^{2} - y^{2} = \text{constant}

. ((c)) Compute

e_{ij}

and

\omega_z

for this field, and state the rate at which the fluid is stretched along the line

y = x

. ((d)) Say whether the cross spins, and where the student's reasoning fails.

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