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
Water (ρ=998kgm−3) drains from a wide basin, forming a steady vertical vortex whose core has radius a0=0.05m. Convergent flow toward the drain stretches the vortex tube axially, and its core narrows to a=0.02m; the fluid outside the core is irrotational. The flow is inviscid, incompressible and steady, and the free surface is treated as flat (no change of elevation is considered).
((a)) Given that the tube's circulation was Γ0=0.50m2s−1 before it narrowed, state its circulation once narrowed to a=0.02m, giving the property of an inviscid barotropic fluid that justifies this.
((b)) Treating the narrowed core as an infinite straight vortex filament of that circulation, find the speed of the water at a point r=0.10m from the axis (outside the core), using the Biot-Savart law.
((c)) Far from the vortex the pressure is the ambient value p∞ and the velocity is negligible. Using Bernoulli's theorem for steady, incompressible, irrotational flow (valid throughout the flow, not only along one streamline, because the flow outside the core is irrotational), find the pressure deficit p∞−p(r) at r=0.10m.