Transition to disordered front propagation

Subcritical transitions whereby a dynamical system abruptly transitions from an ordered, invariant state to unsteady, disordered dynamics have intrigued scientists for over a century. Although the ordered basic state of the system is stable to infinitesimal perturbations, above a threshold value of the driving parameter, finite amplitude perturbations can trigger a sudden change to complex, continuously evolving
dynamics. In the seminal interpretation of the transition to turbulence in shear flows, excursions from the stable state correspond to the transient exploration of the stable manifolds of weakly unstable states of the system or so-called edge states, which determine the basin boundary separating initial conditions decaying to laminar flow from those growing to turbulence.

The proposed research will explore whether this dynamical scenario applies to a broader class of systems exhibiting subcritical transitions to disorder. A prime candidate is the two-phase displacement flow in a rectangular (Hele-Shaw) channel which has the key advantage that the only nonlinearity arises due to the presence of the interface between the two fluids so that different states can be distinguished using only the interfacial morphology. Moreover, the behaviour of the system can be described by a depth-averaged set of equations that is more amenable to analysis than the full Navier-Stokes equations required to describe shear flow. In order to pursue the idea that when the driving flux is sufficiently large the complex time-dependent behaviour of the propagating front can be interpreted as an exploration of the stable manifolds of edge states, we need to establish whether these perturbation-driven excursions are always transient or become self-sustaining above a threshold flux. In other words, is a localised perturbation sufficient to trigger disorder or are spatially-distributed perturbations required to generate long-term disordered pattern formation?

We propose to answer this question with a systematic experimental investigation in our large-scale experimental facility funded by EPSRC project grant EP/P026044/1 by testing the response of the steadily propagating Saffman-Taylor finger for a wide range of driving fluxes to both localised and spatially-distributed perturbations. The findings from the experiments will inform the model, which will in turn accelerate the identification of the transition scenario in this system.