Shape optimisation for faster washout in recirculating flows

How to design an optimal biomedical device to minimise trapping of undesirable biological solutes/debris is a pertinent but complex question. We address this challenge, deriving particular motivation from \textit{ureteroscopy}, a minimally invasive surgical procedure for the removal of kidney stones by irrigating dust-like stone fragments with a saline solution. \\ We represent the renal pelvis as a 2D cavity and model fluid flow with the steady, incompressible Navier--Stokes equations. Within this modelling framework, the presence of vortices -- which arise as a result of flow symmetry breaking -- has previously been linked to long washout times of kidney stone dust; modelled via advection and diffusion of a passive tracer. \\ For a given flow field $\mathbf{u}$, vortices are characterised by regions where $\det\nabla\mathbf{u} > 0$. Thus, integrating a smooth form of $\max(0,\det\nabla\mathbf{u})$ over the domain provides an objective to minimise recirculation zones. We employ adjoint-based shape optimisation to identify ureteroscope tip geometries that reduce this objective. We show that a reduction in the vortex objective correlates with a reduction in washout time, and hence determine tip shapes which result in reduced washout times.