PHYSICS · Multi-part · Fermi University Physics · Mathematical Methods for Physics · Fluids & Continuum Physics · Kinematics, conservation laws and ideal flow
Fluid Dynamics, Ch Vorticity: its sources, its field and the motion of vortices
Problem
Three point vortices, each of the same circulation Γ=1.5m2s−1 (positive, anticlockwise), sit at the vertices of an equilateral triangle of side d=0.40m in an unbounded, inviscid, incompressible fluid that is otherwise at rest. Each vortex moves with the velocity induced at its location by the other two, given by the planar Biot-Savart law for point vortices.
((a)) Without solving the equations of motion in full, argue from the symmetry of this initial arrangement and from the conserved quantities of point-vortex dynamics (the centroid and the angular impulse ∑iΓiri2 measured from it) that the triangle can only rotate rigidly about its centroid, keeping d fixed, and cannot expand, shrink or distort while the three circulations remain equal and the arrangement stays exactly equilateral.
((b)) Calculate the velocity induced at one vortex by the other two, and hence find the angular velocity Ω of the rigid rotation and the period T of one revolution.
((c)) Confirm explicitly that the motion found in (b) leaves the centroid fixed and ∑iΓiri2 unchanged, as required by (a).