The primary tool for predicting infectious disease spread and intervention effectiveness is the mass action susceptible -infected -recovered model of Kermack & McKendrick. Its usefulness derives largely from its conceptual and mathematical simplicity; however, it incorrectly assumes that all individuals have the same contact rate and partnerships are fleeting. In this study, we introduce edge-based compartmental modelling, a technique eliminating these assumptions. We derive simple ordinary differential equation models capturing social heterogeneity (heterogeneous contact rates) while explicitly considering the impact of partnership duration. We introduce a graphical interpretation allowing for easy derivation and communication of the model and focus on applying the technique under different assumptions about how contact rates are distributed and how long partnerships last.

We numerically characterize the temporal regimes for solutal convection from almost first contact to high dissolved solute concentration in a two-dimensional ideal porous layer for Rayleigh numbers $\mathcal{R}$ between $100$ and $5\times 10^4$. The lower boundary is impenetrable. The upper boundary is saturated with dissolved solute and either impermeable or partially permeable to fluid flow. In the impermeable case, initially there is pure diffusion of solute away from the upper boundary, followed by the birth and growth of convective fingers. Eventually fingers interact and merge, generating complex downwelling plumes. Once the inter-plume spacing is sufficient, small protoplumes reinitiate on the boundary layer and are swept into the primary plumes. The flow is now in a universal regime characterized by a constant (dimensionless) dissolution flux $F=0.017$ (the rate at which solute dissolves from the upper boundary). The horizontally averaged concentration profile stretches as a simple self-similar wedge beneath a diffusive horizontal boundary layer. Throughout, the plume width broadens proportionally to $\sqrt{t}$, where $t$ is (dimensionless) time. The above behaviour is parameter independent; the Rayleigh number only controls when transition occurs to a final $\mathcal{R}$-dependent shut-down regime. For the constant-flux and shut-down regimes, we rigourously derive upscaled equations connecting the horizontally averaged concentration, vertical advective flux and plume widths. These are partially complete; a universal expression for the plume width remains elusive. We complement these governing equations with phenomenological boundary conditions based on a marginally stable diffusive boundary layer at the top and zero advective flux at the bottom. Making appropriate approximations in each regime, we find good agreement between predictions from this model and simulated results for both solutal and thermal convection. In the partially permeable upper boundary case, fluid from the convecting layer can penetrate an overlying separate-phase-solute bearing layer where it immediately saturates. The regime diagram remains almost the same as for the impermeable case, but the dissolution flux is significantly augmented. Our work is motivated by dissolution of carbon dioxide relevant to geological storage, and we conclude with a simple flux parameterization for inclusion in gravity current models and suggest that the upscaled equations could lay the foundation for accurate inclusion of dissolution in reservoir simulators.

Motivated by convection in the context of geological carbon-dioxide (CO2) storage, we present an experimental study of dissolution-driven convection in a Hele–Shaw cell for Rayleigh numbers \documentclass[12pt]{minimal}\begin{document}$\mathcal {R}$\end{document}R in the range \documentclass[12pt]{minimal}\begin{document}$100 < \mathcal {R}< 1700$\end{document}100<R<1700. We use potassium permanganate (KMnO4) in water as an analog for CO2 in brine and infer concentration profiles at high spatial and temporal resolution and accuracy from transmitted light intensity. We describe behavior from first contact up to 65% average saturation and measure several global quantities including dissolution flux, average concentration, amplitude of perturbations away from pure one-dimensional diffusion, and horizontally averaged concentration profiles. We show that the flow evolves successively through distinct regimes starting with a simple one-dimensional diffusional profile. This is followed by linear growth in which fingers are initiated and grow quasi-exponentially, independently of one-another. Once the fingers are well-established, a flux-growth regime begins as fresh fluid is brought to the interface and contaminated fluid removed, with the flux growing to a local maximum. During this regime, fingers still propagate independently. However, beyond the flux maximum, fingers begin to interact and zip together from the root down in a merging regime. Several generations of merging occur before only persistent primary fingers remain. Beyond this, the reinitiation regime begins with new fingers created between primary existing ones before merging into them. Through appropriate scaling, we show that the regimes are universal and independent of layer thickness (equivalently \documentclass[12pt]{minimal}\begin{document}$\mathcal {R}$\end{document}R) until the fingers hit the bottom. At this time, progression through these regimes is interrupted and the flow transitions to a saturating regime. In this final regime, the flux gradually decays in a manner well described by a Howard-style phenomenological model.

Motivated by convection in the context of geological carbon dioxide sequestration, we present the conditions for free, dissolution-driven convection in a horizontal, ideal porous layer from a time-dependent, pure-diffusion base state. We assume that solute as a separate phase is instantaneously placed in the pores above a given horizontal level at time zero, and gradually diffuses into the underlying liquid. As the concentration of dissolved solute in the liquid increases, its density increases and the system may eventually become gravitationally unstable and convection may begin. We define the amplitude of a perturbation as the mean square of the difference of the concentration profile and the pure-diffusion profile. To identify instability, we calculate the maximum possible instantaneous growth rate of the amplitude over all possible infinitesimal and finite perturbations. Instability exists where this growth rate is positive. We consider two scenarios. In the first scenario, the underlying liquid cannot penetrate into the upper region occupied by the separate, solute-rich phase. In this case, no instability is possible for thin porous layers corresponding to Rayleigh–Darcy numbers less than 32.50. For thicker layers, convection can occur at finite, nonzero times and wavenumbers. The earliest possible onset becomes independent of layer thickness for Rayleigh–Darcy numbers above about 75, and occurs at a time of approximately 47.9D(μϕ/[KΔρ(1−ci∗/cm∗)g])2 after solute placement, where D is the effective diffusivity, μ the liquid viscosity, ϕ the porosity and K the permeability of the medium, g the acceleration due to gravity, Δρ=ρm−ρ0 with ρm the density of saturated liquid and ρ0 the density of pure liquid, cm∗ the maximum solute concentration, and ci∗ the initial average concentration. In the second scenario, the liquid phase in the upper layer remains connected; the liquid beneath can penetrate the horizontal boundary between the two layers and immediately becomes saturated. In this case, the onset occurs sooner and increasingly so for thicker layers. We also present the mode profiles and discuss the implications for sequestration.

A new class of travelling droplets, coined superwalkers, have recently been shown to emerge when a bath of silicone oil is vibrated simultaneously at a given frequency and its subharmonic tone with a relative phase difference (Valani et al. 2019). To understand the emergence of superwalking droplets, here we explore their vertical and horizontal dynamics using an extension of the theoretical model for walking droplets of Moláček & Bush (2013a,b). We show that driving the bath at two frequencies with an appropriate phase difference lowers every second peak and raises the intermediate peaks in the bath motion, allowing large droplets that could otherwise not walk to clear every second peak and bounce and walk in a similar manner to normal walkers. We find that the droplet's vertical and horizontal dynamics are strongly influenced by the relative height difference between successive peaks of the bath motion, a parameter that is controlled by the phase difference. Comparison of speed-size characteristics between simulations and experiments shows good agreement for small-to moderate-sized superwalkers. A novel behaviour of superwalking droplets, stop-and-go motion, is also captured in our simulations. Key words:Recently, a new class of walking droplets, coined superwalkers, have been observed (Valani et al. 2019). These emerge when the bath is driven simultaneously at two frequencies, f and f /2, with a relative phase difference ∆φ. For a commonly studied †