Piecewise-constant optimal control strategies for controlling the outbreak of COVID-19 in the Irish population.

We introduce a deterministic SEIR model and fit it to epidemiological data for the COVID-19 outbreak in Ireland. We couple the model to economic considerations - we formulate an optimal control problem in which the cost to the economy of the various non-pharmaceutical interventions is minimized, subject to hospital admissions never exceeding a threshold value corresponding to health-service capacity. Within the framework of the model, the optimal strategy of disease control is revealed to be one of disease suppression, rather than disease mitigation.

A geometric diffuse-interface method for droplet spreading.

This paper exploits the theory of geometric gradient flows to introduce an alternative regularization of the thin-film equation valid in the case of large-scale droplet spreading-the geometric diffuse-interface method. The method possesses some advantages when compared with the existing models of droplet spreading, namely the slip model, the precursor-film method and the diffuse-interface model. These advantages are discussed and a case is made for using the geometric diffuse-interface method for the purpose of numerical simulations. The mathematical solutions of the geometric diffuse interface method are explored via such numerical simulations for the simple and well-studied case of large-scale droplet spreading for a perfectly wetting fluid-we demonstrate that the new method reproduces Tanner's Law of droplet spreading via a simple and robust computational method, at a low computational cost. We discuss potential avenues for extending the method beyond the simple case of perfectly wetting fluids.

Flow-parametric regulation of shear-driven phase separation in two and three dimensions.

The Cahn-Hilliard equation with an externally prescribed chaotic shear flow is studied in two and three dimensions. The main goal is to compare and contrast the phase separation in two and three dimensions, using high-resolution numerical simulation as the basis for the study. The model flow is parametrized by its amplitudes (thereby admitting the possibility of anisotropy), length scales, and multiple time scales, and the outcome of the phase separation is investigated as a function of these parameters as well as the dimensionality. In this way, a parameter regime is identified wherein the phase separation and the associated coarsening phenomenon are not only arrested but in fact the concentration variance decays, thereby opening up the possibility of describing the dynamics of the concentration field using the theories of advection diffusion. This parameter regime corresponds to long flow correlation times, large flow amplitudes and small diffusivities. The onset of this hyperdiffusive regime is interpreted by introducing Batchelor length scales. A key result is that in the hyperdiffusive regime, the distribution of concentration (in particular, the frequency of extreme values of concentration) depends strongly on the dimensionality. Anisotropic scenarios are also investigated: for scenarios wherein the variance saturates (corresponding to coarsening arrest), the direction in which the domains align depends on the flow correlation time. Thus, for correlation times comparable to the inverse of the mean shear rate, the domains align in the direction of maximum flow amplitude, while for short correlation times, the domains initially align in the opposite direction. However, at very late times (after the passage of thousands of correlation times), the fate of the domains is the same regardless of correlation time, namely alignment in the direction of maximum flow amplitude. A theoretical model to explain these features is proposed. These features and the theoretical model carry over to the three-dimensional case, albeit that an extra degree of freedom pertains, such that the dynamics of the domain alignment in three dimensions warrant a more detailed consideration, also presented herein.

Dynamical effects and phase separation in cooled binary fluid films.

We study phase separation in thin films using a model based on the Navier-Stokes Cahn-Hilliard equations in the lubrication approximation, with a van der Waals potential to account for substrate-film interactions. We solve the resulting thin-film equations numerically and compare to experimental data. The model captures the qualitative features of real phase-separating fluids, in particular, how concentration gradients produce film thinning and surface roughening. The ultimate outcome of the phase separation depends strongly on the dynamical back reaction of concentration gradients on the flow, an effect we demonstrate by applying a shear stress at the film's surface. When the back reaction is small, the phase domain boundaries align with the direction of the imposed stress, while for larger back-reaction strengths, the domains align in the perpendicular direction.

Bubbles and filaments: stirring a Cahn-Hilliard fluid.

The advective Cahn-Hilliard equation describes the competing processes of stirring and separation in a two-phase fluid. Intuition suggests that bubbles will form on a certain scale, and previous studies of Cahn-Hilliard dynamics seem to suggest the presence of one dominant length scale. However, the Cahn-Hilliard phase-separation mechanism contains a hyperdiffusion term and we show that, by stirring the mixture at a sufficiently large amplitude, we excite the diffusion and overwhelm the segregation to create a homogeneous liquid. At intermediate amplitudes we see regions of bubbles coexisting with regions of hyperdiffusive filaments. Thus, the problem possesses two dominant length scales, associated with the bubbles and filaments. For simplicity, we use a chaotic flow that mimics turbulent stirring at large Prandtl number. We compare our results with the case of variable mobility, in which growth of bubble size is dominated by interfacial rather than bulk effects, and find qualitatively similar results.