Particle-image velocimetry of a steady

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

Particle-image velocimetry of a steady, planar, incompressible laboratory flow gives, close to the point

O

u = A y, \qquad v = -B x, \qquad w = 0,

with

A = 6.0\ \mathrm{s^{-1}}

B = 2.0\ \mathrm{s^{-1}}

and

x

y

in metres measured from

O

. Starting from the first-order relation

\delta u_i = (\partial u_i/\partial x_j)\,\delta x_j

for the relative velocity of two neighbouring fluid particles: ((a)) Write out the planar velocity gradient array

\partial u_i/\partial x_j

and confirm that the field is incompressible. ((b)) Derive the rate of strain tensor

e_{ij} = \tfrac{1}{2}(\partial u_i/\partial x_j + \partial u_j/\partial x_i)

and give

e_{xy}

(state the diagonal entries as well). ((c)) Derive the rotation tensor

\Omega_{ij} = \tfrac{1}{2}(\partial u_i/\partial x_j - \partial u_j/\partial x_i)

and give

\Omega_{xy}

. ((d)) Give the vorticity component

\omega_z

and the angular velocity at which a small floating cross of dye turns, stating its sense. ((e)) Add the two halves back together to recover

\partial u/\partial y

, as a check on the split.

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