A vortex filament of circulation forms a closed square loop of side lying in a plane

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

Fluid Dynamics, Ch Vorticity: its sources, its field and the motion of vortices

Problem

A vortex filament of circulation Γ=0.5 m2 s−1\Gamma = 0.5\,\mathrm{m^2\,s^{-1}} forms a closed square loop of side L=0.4 mL=0.4\,\mathrm{m} lying in a plane, all four sides carrying the circulation in the same sense around the loop. Using the straight-segment Biot-Savart formula u=Γ4πd(cos⁡α1−cos⁡α2)u=\frac{\Gamma}{4\pi d}\left(\cos\alpha_1-\cos\alpha_2\right) for the field of one finite side at perpendicular distance dd, with α1,α2\alpha_1,\alpha_2 the angles from the filament to the lines joining the field point to its two ends, find the speed of the fluid at the centre of the square induced by the whole loop.

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