Related papers: Exact coherent states underlying chaotic falling-f…
We report that many exact invariant solutions of the Navier-Stokes equations for both pipe and channel flows are well represented by just few modes of the model of McKeon & Sharma J. Fl. Mech. 658, 356 (2010). This model provides modes that…
The aim of this work is to establish the existence of invariant manifolds in complex systems. Considering trajectory curves integral of multiple time scales dynamical systems of dimension two and three (predator-prey models, neuronal…
A dynamical systems approach to turbulence envisions the flow as a trajectory through a high-dimensional state space transiently visiting the neighbourhoods of unstable simple invariant solutions (E. Hopf, Commun. Appl. Maths 1, 303, 1948).…
Motivated by recent experimental and numerical studies of coherent structures in wall-bounded shear flows, we initiate a systematic exploration of the hierarchy of unstable invariant solutions of the Navier-Stokes equations. We construct a…
Falling liquid films are complex physical processes modelled by infinite-dimensional dynamical systems. The interfacial behaviour is characterised by the Reynolds number, a nondimensional parameter, and above a critical value the uniform…
We consider the unsteady regimes of an acoustically-driven jet that forces a recirculating flow through successive reflections on the walls of a square cavity. The specific question being addressed is to know whether the system can sustain…
Dynamical systems with translational or rotational symmetry arise frequently in studies of spatially extended physical systems, such as Navier-Stokes flows on periodic domains. In these cases, it is natural to express the state of the fluid…
The stability of liquid films coating the walls of a parallel-plate channel and sheared by a pressure-driven gas flow along the channel centre is studied. The films are susceptible to a long-wavelength instability, whose dynamic behaviour…
We have found an infinite dimensional manifold of exact solutions of the Navier-Stokes loop equation for the Wilson loop in decaying Turbulence in arbitrary dimension $d >2$. This solution family is equivalent to a fractal curve in complex…
In this work we study the evolution of the interface between two different fluids in two concentric cylinders when the velocity is given by the Navier-Stokes equation and one of the fluids is thin. We present a formal asymptotic derivation…
We consider a two-dimensional motion of a thin film flowing down an inclined plane under the influence of the gravity and the surface tension. In order to investigate the stability of such flow, it is hard to treat the Navier--Stokes…
We investigate a model of high-dimensional dynamical variables with all-to-all interactions that are random and non-reciprocal. We characterize its phase diagram and show that the model can exhibit chaotic dynamics. We show that the…
This paper is concerned with a model for the dynamics of a single species in a one-dimensional heterogeneous environment. The environment consists of two kinds of patches, which are periodically alternately arranged along the spatial axis.…
Turbulence in wall-bounded shear flows is increasingly understood through exact coherent structures (ECS) -- invariant solutions of the Navier-Stokes equations that act as organising centres in the high-dimensional state space. Here we…
We consider free surface instabilities of films flowing on inverted substrates within the framework of lubrication approximation. We allow for the presence of fronts and related contact lines, and explore the role which they play in…
Exact coherent states of a linearly stable, plane parallel shear flow confined between stationary stress-free walls and driven by a sinusoidal body force (a flow first introduced by F. Waleffe, Phys. Fluids 9, 883 (1997)) are computed using…
We consider a class of singularly perturbed 2-component reaction-diffusion equations which admit bistable traveling front solutions, manifesting as sharp, slow-fast-slow, interfaces between stable homogeneous rest states. In many example…
We investigate whether a strongly turbulent flow with intermittent large-scale reorganizations admits a compact state-space description. As a representative high-dimensional chaotic system we consider two-dimensional Rayleigh--B\'enard…
Recent progress indicates that highly symmetric recurring solutions of the Navier-Stokes equations, such as equilibria and periodic orbits, provide a skeleton for turbulence dynamics in state-space. Many of these solutions have been found…
We present accurate and mathematically consistent formulations of a diffuse-interface model for two-phase flow problems involving rapid evaporation. The model addresses challenges including discontinuities in the density field by several…