Research Case Study - Multiscale Analysis of Spatially Extended Systems
Spatially extended systems (SES), i.e. infinite-dimensional dynamical systems described through partial differential equations (PDEs) deterministic or stochastic in large or unbounded domains, are typically characterized by the presence of a wide range of characteristic length- and timescales which often leads to complex spatiotemporal behaviour. They arise frequently as mathematical models of a large variety of natural phenomena and technological applications.
The systematic derivation of reduced low-dimensional models for the effective dynamics of SES poses a great scientific challenge with important technological applications. The primary aim of this project is the development of state-of-the-art efficient methods for mode reduction and coarse-graining of SES, both deterministic and stochastic.
Other highlights include:
(a) Statistical mechanics of inhomogeneous classical fluids and dynamic density-functional theory (DDFT). We have unified previous DDFTs and by using multiplescale methods–singular perturbation theory we have rigorously formulated a general DDFT that takes into account inertia and hydrodynamic interactions [1,2,3], the combined effect of which was neglected in previous theories, even though they strongly influence non-equilibrium properties of the system as shown in Fig. 1.
(b) Upscaling of phase field models for interfacial dynamics in strongly heterogeneous domains. We have examined the problem of upscaling the CahnHilliard (CH) equation for perfo- rated/strongly heterogeneous domains. An effective macroscopic CH equation for such domains was derived rigorously using singular perturbation techniques, averaging and homogenisation theory. Our results were applied to wetting dynamics in porous media and to a single channel with strongly heterogeneous walls [4] and provide rigorous justification of phase-field models often postulated on an ad-hoc basis. Classical results such as Taylor-Aris dispersion are simple byproducts of our analysis.
Find out more at the Complex Multiphase Systems Group website
1. B.D. Goddard, G.A. Pavliotis & S. Kalliadasis, “The overdamped limit of dynamic density functional theory: Rigorous results,” SIAM Multiscale Model. Simul. 10 633–663 (2012).
2. B.D. Goddard, A. Nold, N. Savva, G.A. Pavliotis & S. Kalliadasis, “General Dynamical Density Functional Theory for Classical Fluids,” Phys. Rev. Lett. 109 Art. No. 120603 (2012).
3. B.D. Goddard, A. Nold, N. Savva, P. Yatsyshin & S. Kalliadasis, “Unification of dynamic density functional theory for colloidal fluids to include inertia and hydrodynamic interactions: derivation and numerical experiments,” J. Phys.: Condens. Matter 25 Art. No. 035101 (2013).
4. M. Schmuck, M. Pradas, G.A. Pavliotis & S. Kalliadasis, “Upscaled phase-field models for interfacial dynamics in strongly heterogeneous domains,” Proc. R. Soc. A 468 3705-3724 (2012).