A short, thin vortex tube of circulation and length lies along the stretching axis of…

PHYSICS · Multi-part · 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 short, thin vortex tube of circulation Γ0=0.20 m2 s−1\Gamma_0 = 0.20\,\mathrm{m^2\,s^{-1}} and length ℓ0=0.10 m\ell_0 = 0.10\,\mathrm{m} lies along the stretching axis of a steady pure strain u=(e1x,e2y,e3z)\mathbf{u} = (e_1x, e_2y, e_3z) with e1=0.50 s−1e_1 = 0.50\,\mathrm{s^{-1}} and e2=e3=−0.25 s−1e_2 = e_3 = -0.25\,\mathrm{s^{-1}} (incompressible, so e1+e2+e3=0e_1 + e_2 + e_3 = 0). The fluid is inviscid and barotropic and body forces are conservative. Treat the tube as thin enough to be replaced, for the purpose of the velocity it induces, by a straight vortex filament of the same length and circulation, centred on the origin and aligned with the xx-axis. ((a)) Using the stretching of a material line element under this strain (as in chapter 2), find the length of the tube at t=2 st = 2\,\mathrm{s}. ((b)) State, with a reason, the circulation of the tube at t=2 st = 2\,\mathrm{s}. ((c)) Find the speed induced at a point PP a perpendicular distance d=0.05 md = 0.05\,\mathrm{m} from the midpoint of the (now stretched) filament, using the Biot-Savart formula for a straight segment.

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