A droplet of oil may walk horizontally while bouncing vertically when placed on a vertically vibrating bath of the same liquid. Each bounce of the droplet creates a localized decaying standing wave, which in turn guides the horizontal motion of the droplet, resulting in a self-propelled wave–particle entity. We show that for certain spatial forms of the waves, Lorenz-like dynamical systems emerge from the trajectory equation of the wave–particle entity. Understanding the dynamics of the wave–particle entity in terms of Lorenz-like systems may prove to be useful in rationalizing emergent statistical behavior from underlying chaotic dynamics in hydrodynamic quantum analogs of walking droplets.

A millimetre-sized wave-particle entity in the form of a walking droplet can emerge on the surface of a vertically vibrating liquid bath. Using a simple theoretical model of this wave-particle entity, we have shown that applying a small constant bias force to the wave-particle entity can result in a net drift of the wave-particle entity in a direction opposite to the applied force. Such paradoxical behaviors are typically observed in non-equilibrium systems driven by noise or periodic driving but here we have shown that this behaviour can also arise in a memory-driven system of a self-propelled wave-particle entity.


  • R. N. Valani, B. Harding and Y. M. Stokes, Bifurcations and dynamics in inertial focusing of particles in curved rectangular ducts (under review) arXiv
  • R. N. Valani, B. Harding and Y. M. Stokes, Bifurcations in inertial focusing of a particle suspended in flow through curved rectangular ducts (under review) arXiv
  • R. N. Valani and D. M. Paganin, Deterministic Active Matter Generated Using Strange Attractors (under review) arXiv
  • R. N. Valani, A. C. Slim, D. M. Paganin, T. P. Simula, and T. Vo, Unsteady dynamics of a classical particle-wave entity, Physical Review E, 104, 015106 (2021) arXiv

A droplet bouncing on the surface of a vertically vibrating liquid bath can walk horizontally, guided by the waves it generates on each impact. This results in a self-propelled classical particle-wave entity. By using a one-dimensional theoretical pilot-wave model with a generalized wave form, we investigate the dynamics of this particle-wave entity. We explore the dynamical and statistical aspects of unsteady walking and show an equivalence between the droplet dynamics and the Lorenz system, as well as making connections with the Langevin equation and deterministic diffusion.

Vertically vibrating a liquid bath at two frequencies having a constant relative phase difference can give rise to self-propelled superwalking droplets on the liquid surface. We have numerically investigated such superwalking droplets in the regime where the phase difference varies slowly with time. Our simulations uncover three different types of intermittent droplet motion: back-and-forth, forth-and-forth, and irregular stop-and-go motion, which we explore in detail. Our findings lay a foundation for further studies of dynamically driven droplets, whereby the droplet’s motion may be guided by engineering arbitrary time-dependent phase difference functions.

A new class of self-propelled droplets, coined superwalkers, has been shown to emerge when a bath of silicone oil is vibrated simultaneously at a given frequency and its subharmonic tone with a relative phase difference between them. To understand the emergence of superwalking droplets, we explore their vertical and horizontal dynamics by extending previously established theoretical models for walkers driven by a single frequency to superwalkers driven by two frequencies. We provide an explanation for the emergence of superwalking at two frequencies and compare our simulated superwalkers with experiments.


Millimetre-sized walking droplets that emerge on the surface of a vertically vibrating liquid bath can exhibit certain features that were previously thought to be exclusive to the microscopic quantum realm. This thesis investigates a new class of walking droplets, coined superwalkers, and also studies the rich dynamical behaviour of droplets in a generalised pilot-wave framework. Insights from this research may help further our understanding in the areas of hydrodynamic quantum analogues and active matter.


walker is a droplet of liquid that self-propels on the free surface of an oscillating bath of the same liquid through feedback between the droplet and its wave field. We have studied walking droplets in the presence of two driving frequencies and have observed a new class of walking droplets, which we coin superwalkers. Superwalkers may be more than double the size of the largest walkers, may travel at more than triple the speed of the fastest ones, and enable a plethora of novel multidroplet behaviors.


A droplet bouncing on the surface of a vibrating liquid bath can move horizontally guided by the wave it produces on impacting the bath. The wave itself is modified by the environment, and thus, the interactions of the moving droplet with the surroundings are mediated through the wave. Using a description for walking droplets as a theoretical pilot-wave model, we investigate the dynamics of two interacting identical, in-phase bouncing droplets theoretically and numerically. A remarkably rich range of behaviors is encountered as a function of the two system parameters. We explore these regimes and others and the bifurcations between them through analytic and numerical linear stability analyses and through fully nonlinear numerical simulation.

We present a numerical study of two-droplet pair correlations for in-phase droplets walking on a vibrating bath. Two such walkers are launched toward a common point of intersection. As they approach, their carrier waves may overlap and the droplets have a non-zero probability of forming a two-droplet bound state. Three generic types of two-droplet correlations are observed: promenading, orbiting, and chasing pair of walkers. For certain parameters, the droplets may become correlated for certain initial path differences and remain uncorrelated for others, while in other cases, the droplets may never produce droplet pairs. These observations pave the way for further studies of strongly correlated many-droplet behaviors in the hydrodynamical quantum analogs of bouncing and walking droplets.

We have studied statistical mechanics of a gas of vortices in two dimensions. We introduce a new observable—a condensate fraction of Onsager vortices—to quantify the emergence of the vortex condensate. The condensation of Onsager vortices is most transparently observed in a single vortex species system and occurs due to a competition between solid body rotation (see vortex lattice) and potential flow (see multiple quantum vortex state). We propose an experiment to observe the condensation transition of the vortices in such a single vortex species system.


  • R. N. Valani and Kerry Hourigan, A numerical study of flow past a forced oscillating cylinder, AIAA student conference (2016)
  • R. N. Valani, A numerical study of flow past a forced oscillating circular cylinder at low Reynolds number, Final Year Project Thesis, Monash University (2016)

A numerical study of flow past a cylinder oscillating transverse, inline and at an angle to the incoming flow was performed at low Reynolds number. The forcing frequency and the forcing amplitude ratio were varied independently and rich dynamical responses including periodic, quasiperiodic and chaotic responses were recorded and compared to the wake topology.


  • R. Valani, A. Slim and J. Miller, Wave particle duality in multiple bouncing fluid droplets, AMSI research report (2015)

Using a simple theoretical model of a self-propelled wave-particle duality inspired from walking droplets, we theoretically and numerically explore the dynamics of two and three droplets. Using our analysis we were able to find stationary states, parallel walking states and orbiting states for two droplets as well as stationary states for three droplets which were confirmed numerically and also found experimentally. By investigating the dynamics of three droplets, various trajectories and exotic orbits were identified.