A wide channel carries the steady simple shear flow with shear rate

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 Vorticity, circulation and vortex motion

Problem

A wide channel carries the steady simple shear flow u=(Sy, 0, 0)\mathbf{u} = (Sy,\ 0,\ 0) with shear rate S=12 s−1S = 12\ \mathrm{s^{-1}}, where xx is horizontal along the channel and yy is measured across it. The streamlines are perfectly straight lines parallel to the xx axis. Consider the rectangular loop whose sides have length a=0.050 ma = 0.050\ \mathrm{m} along xx and b=0.020 mb = 0.020\ \mathrm{m} along yy, traversed anticlockwise when viewed with zz pointing out of the page. ((a)) Compute the circulation Γ\Gamma round the rectangle by adding the contributions of the four sides. ((b)) Compute ωz\omega_z for this flow and check that ωz\omega_z times the enclosed area gives the same Γ\Gamma.

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