Unveiling the Spatiotemporal Evolution of Liquid-Lens Coalescence: Self-Similarity, Vortex Quadrupoles, and Turbulence in a Three-Phase Fluid System

We demonstrate that the three-phase Cahn-Hilliard-Navier-Stokes (CHNS3) system provides a natural theoretical framework for studying liquid-lens coalescence, which has been investigated in recent experiments. Our extensive direct numerical simulations (DNSs) of lens coalescence, in the two and three dimensional (2D and 3D) CHNS3, uncover the rich spatiotemporal evolution of the fluid velocity $\bf u$ and vorticity $\omega$, the concentration fields $c_1, \, c_2,$ and $c_3$ of the three liquids, and a generalized Laplace pressure $P^G_\mathcal{L}$, which we define in terms of these concentrations via a Poisson equation. We find, in agreement with experiments, that as the lenses coalesce, their neck height $h(t) \sim t^{\alpha_v}$, with $\alpha_v \simeq 1$ in the viscous regime, and $h(t) \sim t^{\alpha_i}$, with $\alpha_i \simeq 2/3$ in the inertial regime. We obtain the crossover from the viscous to the inertial regimes as a function of the Ohnesorge number $Oh$, a dimensionless combination of viscous stresses and inertial and surface tension forces. We show that a vortex quadrupole, which straddles the neck of the merging lenses, and $P^G_\mathcal{L}$ play crucial roles in distinguishing between the viscous- and inertial-regime growths of the merging lenses. In the inertial regime we find signatures of turbulence, which we quantify via kinetic-energy and concentration spectra. Finally, we examine the merger of asymmetric lenses, in which the initial stages of coalescence occur along the circular parts of the lens interfaces; in this case, we obtain power-law forms for the $h(t)$ with inertial-regime exponents that lie between their droplet-coalescence and lens-merger counterparts.Comment: 9 pages, 8 figure

An analytical and computational study of the incompressible Toner-Tu Equations

The incompressible Toner-Tu (ITT) partial differential equations (PDEs) are an important example of a set of active-fluid PDEs. While they share certain properties with the Navier-Stokes equations (NSEs), such as the same scaling invariance, there are also important differences. The NSEs are usually considered in either the decaying or the additively forced cases, whereas the ITT equations have no additive forcing. Instead, they include a linear, activity term \alpha \bu (\bu is the velocity field) which pumps energy into the system, but also a negative \bu|\bu|^{2}-term which provides a platform for either frozen or statistically steady states. Taken together, these differences make the ITT equations an intriguing candidate for study using a combination of PDE analysis and pseudo-spectral direct numerical simulations (DNSs). In the $d=2$ case, we have established global regularity of solutions, but we have also shown the existence of bounded hierarchies of weighted, time-averaged norms of both higher derivatives and higher moments of the velocity field. Similar bounded hierarchies for Leray-type weak solutions have also been established in the $d=3$ case. We present results for these norms from our DNSs in both $d=2$ and $d=3$, and contrast them with their Navier-Stokes counterparts