The motion of a viscous filament in a porous medium or Hele–Shaw cell: a physical realisation of the Cauchy–Riemann equations

We consider the motion of a thin filament of viscous fluid in a Hele–Shaw cell. The appropriate thin film analysis and use of Lagrangian variables leads to the Cauchy–Riemann system in a surprisingly direct way. We illustrate the inherent ill-posedness of these equations in various contexts.