Analysis and applications of dynamic density functional theory
Classical ﬂuid mechanics and, in particular, the general compressible Navier-Stokes-Fourier equations, have long been of great use in the prediction and understanding of the ﬂow of ﬂuids in various scenarios. While the classical theory is well established in increasingly rigorous mathematical frameworks, the atomistic properties and microscopic processes of ﬂuids must be considered by other means. A central problem in ﬂuid mechanics concerns capturing microscopic eﬀects in meso/macroscopic continuum models. With more attention given to the non-Newtonian properties of many naturally occurring ﬂuid ﬂows, resolving the gaps between the atomistic viewpoint and the continuum approach of Navier-Stokes-Fourier is a rich and open ﬁeld. This thesis centres on the modelling, analysis and computation of one continuum method designed to resolve the highly multiscale nature of non-equilibrium ﬂuid ﬂow on the particle scale: Dynamic Density Functional Theory (DDFT). A generalised version of DDFT is derived from ﬁrst principles to include: driven ﬂow, inertia and hydrodynamic interactions (HI) and it is observed that the equations reproduce known dynamics in heuristic overdamped and inviscid limits. Also included are rigorous, analytical derivations of the short-range lubrication forces on particles at low Reynolds number, with accompanying asymptotic theory, uniformly valid in the entire regime of particle distances and size ranges, which were previously unknown. As well as demonstrating an improvement on known classical results, these calculations were determined necessary to comply with the continuous nature of the integro-diﬀerential equations for DDFT. The numerical implementation of the driven, inertial equations with short range HI for a range of colloidal systems in conﬁning geometries is also included by developing the pseudo-spectral collocation scheme 2DChebClass . A further area of interest for non-equilibrium ﬂuids is mathematical well–posedness. This thesis provides, for the ﬁrst time, the existence and uniqueness of weak solutions to an overdamped DDFT with HI, as well as a rigorous investigation of its equilibrium behaviour.