Mixing in fluid flows is a fundamental mechanism that drives the redistribution of scalar quantities, such as temperature and chemical concentration. While diffusion alone acts slowly, advection by an incompressible velocity field can accelerate mixing through stretching, folding, and filamentation. In this talk, I will discuss how to quantify mixing in two-dimensional flows across two contrasting regimes: steady flows with integrable particle trajectories and time-dependent flows exhibiting Lagrangian chaos.
The first part focuses on passive scalar evolution in axisymmetric steady two-dimensional incompressible flows, where the advection–diffusion equation can be studied using analytical and spectral methods. I will show how even simple vortical kinematics can deform scalar distributions and enhance the effective action of diffusion.
In the second part, I will focus on chaotic advection, where repeated stretching and folding generate complex particle distributions. To quantify mixing from such Lagrangian data, I will introduce a mesh-free approach based on topological data analysis. By using the minimum spanning tree of Lagrangian particle clouds and Steele’s theorem, one obtains an estimator for the area of a chaotic mixing region directly from particle snapshots, without explicit boundary detection or spatial meshing. I will also discuss how the temporal growth of the minimum spanning tree length can serve as a coordinate-free estimator of the maximal Lyapunov exponent, without computing deformation gradients.