Oxford Mathematician Dr Tom Crawford teaches Dr Trefor Bazett about Helmholtz’ Principle and how it can be applied to calculate the motion of a vortex in 2D potential flow.

Watch part 2 of the collaboration where Trefor teaches me a little algebraic topology, specifically the fundamental group here.

We being by looking at the setup for 2D potential flow which is both incompressible (zero divergence) and irrotational (zero curl). This gives rise to a velocity potential and a streamfunction which can be combined using the theory of Complex Analysis and the Cauchy-Riemann Equations to form a complex potential. The situation studied is for a vortex in the positive quadrant which is bounded below (along the x-axis) and to the left (along the y-axis). Using the Method of Images we derive the full complex potential and differentiate to get the velocity field. Helmholtz’ Principle is then applied to give the trajectory of the vortex as it moves in the fluid. Helmholtz’ Principle states that a vortex will move due to the velocity field of everything except itself.

For more information on 2D potential flow and the Method of Images see this video with 3Blue1Brown.