Abstract: Fluid mixing that is induced by hydrodynamic instability is ubiquitous in nature; the material interface between two fluids when perturbed even slightly, changes shape under the influence of hydrodynamic forces, and an additional zone called the mixing layer where the two fluids mix, develops and grows in size. Examples include a drop of ink mixing in water, and water vapor at the edge of clouds mixing with atmospheric air. At finite temperature, motion of the microscopic constituents of a fluid leads to fluctuations in its thermodynamic quantities such as density and pressure; these are called thermal fluctuations, and are thought to be significant in small systems such as nanochannels. Here, we study the role of thermal fluctuations in fluid mixing at the interface separating two miscible fluids under the influence of two instabilities; the Rayleigh-Taylor (RTI) and Richtmyer-Meshkov (RMI) instabilities. The study was conducted using numerical simulations after verification of the simulation methodology. Specifically, fluctuating hydrodynamics simulations were used; the fluctuating compressible Navier-Stokes equations comprise the physical model of the system, and they are solved using numerical methods that were developed and implemented in-house.
Our results indicate that thermal fluctuations can trigger the onset of RTI at an initially unperturbed interface, which subsequently leads to mixing of multi-mode character. In addition we find that for both RMI and RTI, whether or not thermal fluctuations quantitatively affect the mixing behavior, depends on the magnitude of the dimensionless Boltzmann number of the hydrodynamic system in question, and not solely on its size. When the Boltzmann number is much smaller than unity, the quantitative effect of thermal fluctuations on the mixing behavior is negligible. Under this circumstance, we also show that mixing behavior observed in the hydrodynamic system is the average of the outcome from several stochastic instances, with the ensemble of stochastic instances providing the bounds. Most macroscopic hydrodynamic systems fall in this category. However, when the system is such that the Boltzmann number is of order unity, we show that thermal fluctuations can significantly affect the mixing behavior; the ensemble-averaged solution shows a departure from the deterministic solution. We conclude that for such systems, it is important to account for thermal fluctuations in order to correctly capture their physical behavior.