When a shallow layer of inviscid fluid flows over a substrate, the fluid particle trajectories are, to leading order in the layer thickness, geodesics on the two-dimensional curved space of the substrate. Since the two-dimensional geodesic equation i
s a two degree-of-freedom autonomous Hamiltonian system, it can exhibit chaos, depending on the shape of the substrate. We find chaotic behaviour for a range of substrates.
We investigate experimentally the mixing dynamics in a channel flow with a finite stirring region undergoing chaotic advection. We study the homogenization of dye in two variants of an eggbeater stirring protocol that differ in the extent of their mi
xing region. In the first case, the mixing region is separated from the side walls of the channel, while in the second it extends to the walls. For the first case, we observe the onset of a permanent concentration pattern that repeats over time with decaying intensity. A quantitative analysis of the concentration field of dye confirms the convergence to a self-similar pattern, akin to the strange eigenmodes previously observed in closed flows. We model this phenomenon using an idealized map, where an analysis of the mixing dynamics explains the convergence to an eigenmode. In contrast, for the second case the presence of no-slip walls and separation points on the frontier of the mixing region leads to non-self-similar mixing dynamics.
A stirring device consisting of a periodic motion of rods induces a mapping of the fluid domain to itself, which can be regarded as a homeomorphism of a punctured surface. Having the rods undergo a topologically-complex motion guarantees at least a m
inimum amount of stretching of material lines, which is important for chaotic mixing. We use topological considerations to describe the nature of the injection of unmixed material into a central mixing region, which takes place at injection cusps. A topological index formula allow us to predict the possible types of unstable foliations that can arise for a fixed number of rods.
The mixing effectiveness, i.e., the enhancement of molecular diffusion, of a flow can be quantified in terms of the suppression of concentration variance of a passive scalar sustained by steady sources and sinks. The mixing enhancement defined this w
ay is the ratio of the RMS fluctuations of the scalar mixed by molecular diffusion alone to the (statistically steady-state) RMS fluctuations of the scalar density in the presence of stirring. This measure of the effectiveness of the stirring is naturally related to the enhancement factor of the equivalent eddy diffusivity over molecular diffusion, and depends on the Peclet number. It was recently noted that the maximum possible mixing enhancement at a given Peclet number depends as well on the structure of the sources and sinks. That is, the mixing efficiency, the effective diffusivity, or the eddy diffusion of a flow generally depends on the sources and sinks of whatever is being stirred. Here we present the results of particle-based simulations quantitatively confirming the source-sink dependence of the mixing enhancement as a function of Peclet number for a model flow.
The transfer of heat between the air and surrounding soil in underground tunnels ins investigated, as part of the analysis of environmental conditions in underground rail systems. Using standard turbulent modelling assumptions, flow profiles are obta
ined in both open tunnels and in the annulus between a tunnel wall and a moving train, from which the heat transfer coefficient between the air and tunnel wall is computed. The radial conduction of heat through the surrounding soil resulting from changes in the temperature of air in the tunnel are determined. An impulse change and an oscillating tunnel air temperature are considered separately. The correlations between fluctuations in heat transfer coefficient and air temperature are found to increase the mean soil temperature. Finally, a model for the coupled evolution of the air and surrounding soil temperature along a tunnel of finite length is given.
The Cahn-Hilliard equation describes phase separation in binary liquids. Here we study this equation with spatially-varying sources and stirring, or advection. We specialize to symmetric mixtures and time-independent sources and discuss stirring stra
tegies that homogenize the binary fluid. By measuring fluctuations of the composition away from its mean value, we quantify the amount of homogenization achievable. We find upper and lower bounds on our measure of homogenization using only the Cahn-Hilliard equation and the incompressibility of the advecting flow. We compare these theoretical bounds with numerical simulations for two model flows: the constant flow, and the random-phase sine flow. Using the sine flow as an example, we show how our bounds on composition fluctuations provide a measure of the effectiveness of a given stirring protocol in homogenizing a phase-separating binary fluid.
