Interfaces as transport barriers in two-dimensional Cahn-Hilliard-Navier-Stokes turbulence

Abstract

We investigate the role of interfaces as transport barriers in binary-fluid turbulence by employing Lagrangian tracer particles. The Cahn-Hilliard-Navier-Stokes (CHNS) system of partial differential equations provides a natural theoretical framework for our investigations. For specificity, we utilize the two-dimensional (2D) CHNS system. We capture efficiently interfaces and their fluctuations in 2D binary-fluid turbulence by using extensive pseudospectral direct numerical simulations (DNSs) of the 2D CHNS equations. We begin with $n$ tracers within a droplet of one phase and examine their dispersal into the second phase. The tracers remain within the droplet for a long time before emerging from it, so interfaces act as transport barriers in binary-fluid turbulence. We show that the fraction of the number of particles inside the droplet decays exponentially and is characterized by a decay time $\tau_{\xi}\sim R_0^{3/2}$ that increases with $R_0$, the radius of the initially circular droplet. Furthermore, we demonstrate that the average first-passage time $\langle \tau \rangle$ for tracers inside a droplet is orders of magnitude larger than it is for transport out of a hypothetical circle with the same radius as the initially circular droplet. We examine the roles of the Okubo-Weiss parameter $\Lambda$, the fluctuations of the droplet perimeter, and the probability distribution function of $\cos(\theta)$, with $\theta$ the angle between the fluid velocity and the normal to a droplet interface, in trapping tracers inside droplets. We mention possible generalisations of our study.