Nonlinear dynamics of microscale interfacial flows and model nonlinear partial differential equations

This Overseas Travel Grant (OTG) proposal seeks funding to enable visits by the PI to Princeton U. and Tokyo U. of Science to undertake research in the following two subprojects:

1. Droplet motion with dynamic droplet variation (Princeton U.).

Droplet motion is ubiquitous in a wide spectrum of natural phenomena and technological applications. From the way various surfaces such as plant leaves and windows of our houses interact with rain droplets, to the rapidly growing field of micro- and nanofluidics. A problem of technological significance is that of polymer electrolyte membrane (PEM) fuel cells currently being investigated primarily experimentally at Princeton U. This system involves multiphase flows in complex geometries and in particular droplets that emerge from pores and grow into gas flow channels.

The full problem is quite involved and a simpler prototype, that of a droplet on a solid substrate with a small pore from which liquid can be pumped in or out (thus, emulating the growth process of droplets in the PEM cells), will be considered. Of particular interest are the effects of substrate disorder, either chemical or topographical and influence of noise. Indeed, in the PEM fuel cells the droplets are constantly in contact with disordered substrates and they are also subjected to fluctuations which are naturally present in the system. Despite its simplicity, this problem has a rather complex dynamics as we discuss in the Case of Support.

2. Nonlinear forecasting analysis of complex spatiotemporal behavior in spatially extended systems (SES) (Tokyo U. of Science).

SES are infinite-dimensional dynamical systems described through partial differential equations (PDEs) deterministic or stochastic in large or unbounded domains, and are typically characterized by the presence of a wide range of characteristic length and time scales which often leads to complex spatiotemporal behavior. An example of such systems is the generalized Kuramoto-Sivashinsky (gKS) equation, a prototype that retains the fundamental elements of any nonlinear process that involves wave evolution in one dimension. The equation has been reported in a wide variety of physical and technological contexts, from plasma and geophysical phenomena to falling liquid films.

The deterministic gKS equation has received considerable attention over the years. One of the main findings is that sufficiently strong dispersion tends to regularise the spatiotemporal chaos of the KS equation in favor of spatially periodic cellular structures. The noisy gKS equation also appears in a wide variety of physical and technological contexts, e.g. evolution of solid films by sputtering. The proposed OTG seeks to explore the effects on noise on the gKS equation and to establish conditions under which it is possible to distinguish between the chaotic behavior of the gKS equation (for small dispersion) and the stochastic effects induced by noise.

A related problem is that of synchronization in noisy SES. Synchronization is central to many applications and natural phenomena, from electric circuits to biological systems, e.g. the cooperative behavior of living beings. Here we shall examine synchronization in noisy SES using a system of coupled noisy gKS equations as a prototype.

It is useful to recall first the two Themes of the proposed research. Theme 1: Droplet motion with dynamic volume variation; Theme 2: Nonlinear forecasting analysis of complex spatiotemporal behavior of spatially extended systems (SES).

The economic and societal impact of the proposed research in Theme 1 will be realised through the development of modelling and predictive tools for small-scale complex fluid processes, crucial in a wide spectrum of technological applications, e.g. small-scale separations, lab-on-chip systems and microengineered devices in general. Such systems are key to several rapidly developing technologies in the UK's smart materials, specialty manufacturing sectors and biomedical industries. Future developments in these sectors will require sophisticated mathematical and computational techniques, as proposed here, which can contribute to many aspects, from improvements in product and design all the way to production.

The research will also lead to the development of state-of-the-art numerical methodologies for the accurate and reliable multiscale simulations of fluid flow in confined microgeometry. These codes will be of benefit to the control and optimisation of industrial processes and devices that exploit microscale flows.

A central aim of Theme 2 is to understand the effects of noise and synchronization in SES. This could be potentially crucial in several technological applications that can be approximated as noisy SES, e.g. evolution of thin solid films with sputtering and nanostructuring of solid surfaces by ion beam errosion. Such systems are key to several rapidly developing technologies in the UK, including smart materials and printing industries to name but a few. Understanding the effects of noise could offer the possibility to control their behavior, e.g. to stably confine particles in very small spaces, or to subject them to controlled forces. In this respect, noise could be used as an "engineering tool" to extract fundamental knowledge but also offer design solutions. Moreover, Theme 2 goes beyond technological applications, since noisy SES appear as model systems in a wide spectrum problems from biology to social dynamics.

Finally, the PI has a strong track record of knowledge transfer through close collaborations with industry, short courses, workshops and training of high-calibre researchers. The impact plan we have devised is based on a range of routes to maximise the likelihood of success and to reach as wide a community as possible: links with other projects and interactions with industrial partners through other projects, publication in leading journals, conference presentations, provision of a web-based dissemination and research colloquium.