Coalescence of Liquid Drops

When two drops of radius $R$ touch, surface tension drives an initially singular motion which joins them into a bigger drop with smaller surface area. This motion is always viscously dominated at early times. We focus on the early-time behavior of the radius \rmn of the small bridge between the two drops. The flow is driven by a highly curved meniscus of length 2\pi \rmn and width \Delta\ll\rmn around the bridge, from which we conclude that the leading-order problem is asymptotically equivalent to its two-dimensional counterpart. An exact two-dimensional solution for the case of inviscid surroundings [Hopper, J. Fluid Mech. ${\bf 213}$, 349 (1990)] shows that \Delta \propto \rmn^3 and \rmn \sim (t\gamma/\pi\eta)\ln [t\gamma/(\eta R)]; and thus the same is true in three dimensions. The case of coalescence with an external viscous fluid is also studied in detail both analytically and numerically. A significantly different structure is found in which the outer fluid forms a toroidal bubble of radius \Delta \propto \rmn^{3/2} at the meniscus and \rmn \sim (t\gamma/4\pi\eta) \ln [t\gamma/(\eta R)]. This basic difference is due to the presence of the outer fluid viscosity, however small. With lengths scaled by $R$ a full description of the asymptotic flow for \rmn(t)\ll1 involves matching of lengthscales of order \rmn^2, \rmn^{3/2}, \rmn$, 1 and probably$\rmn^{7/4}$.Comment: 36 pages, including 9 figure

Convective shutdown in a porous medium at high Rayleigh number

Convection in a closed domain driven by a dense buoyancy source along the upper boundary soon starts to wane owing to the increase of the average interior density. In this paper, theoretical and numerical models are developed of the subsequent long period of shutdown of convection in a two-dimensional porous medium at high Rayleigh number Ra\mathit{Ra}. The aims of this paper are twofold. Firstly, the relationship between this slowly evolving ‘one-sided’ shutdown system and the statistically steady ‘two-sided’ Rayleigh–Bénard (RB) cell is investigated. Numerical measurements of the Nusselt number Nu\mathit{Nu} from an RB cell (Hewitt et al., Phys. Rev. Lett., vol. 108, 2012, 224503) are very well described by the simple parametrization Nu=2.75+0.0069Ra\mathit{Nu}= 2. 75+ 0. 0069\mathit{Ra}. This parametrization is used in theoretical box models of the one-sided shutdown system and found to give excellent agreement with high-resolution numerical simulations of this system. The dynamical structure of shutdown can also be accurately predicted by measurements from an RB cell. Results are presented for a general power-law equation of state. Secondly, these ideas are extended to model more complex physical systems, which comprise two fluid layers with an equation of state such that the solution that forms at the (moving) interface is more dense than either layer. The two fluids are either immiscible or miscible. Theoretical box models compare well with numerical simulations in the case of a flat interface between the fluids. Experimental results from a Hele-Shaw cell and numerical simulations both show that interfacial deformation can dramatically enhance the convective flux. The applicability of these results to the convective dissolution of geologically sequestered CO2{\mathrm{CO} }_{2} in a saline aquifer is discussed

How well do high resolution models reproduce tropical convection?

Cascade is a multi-institution project studying the temporal and spatial organization of tropical convective systems. While cloud resolving numerical models can reproduce the observed diurnal cycle of such systems they are sensitive to the chosen resolution. As part of this effort, we are comparing results from the Met. Office Unified Model to data from the Global Earth Radiation Budget satellite instrument over the African Monsoon Interdisciplinary Analyses region of North Africa. We use a variety of mathematical techniques to study the outgoing radiation and the evolution of properties such as the cloud size distribution. The effectiveness of various model resolutions is tested with a view to determining the optimum balance between resolution and the need to reproduce the observations

Privatised and Unprepared: The NHS Supply Chain

Months after the arrival of the Covid-19 pandemic, huge numbers of UK health and care workers still lack adequate personal protective equipment (PPE). The aim of this report is to expose the role that the privatisation of health and social care has played in this preventable catastrophe. Privatisation has created a system which is both chaotic and bureaucratic – both fragmented and sclerotic. There has been an outcry over PPE shortages in media coverage of the pandemic, but little has been said about privatisation. This is a serious oversight, which this report will address

Convective shutdown in a porous medium at high Rayleigh number

Convection in a closed domain driven by a dense buoyancy source along the upper boundary soon starts to wane owing to the increase of the average interior density. In this paper, theoretical and numerical models are developed of the subsequent long period of shutdown of convection in a two-dimensional porous medium at high Rayleigh number Ra\mathit{Ra}. The aims of this paper are twofold. Firstly, the relationship between this slowly evolving ‘one-sided’ shutdown system and the statistically steady ‘two-sided’ Rayleigh–Bénard (RB) cell is investigated. Numerical measurements of the Nusselt number Nu\mathit{Nu} from an RB cell (Hewitt et al., Phys. Rev. Lett., vol. 108, 2012, 224503) are very well described by the simple parametrization Nu=2.75+0.0069Ra\mathit{Nu}= 2. 75+ 0. 0069\mathit{Ra}. This parametrization is used in theoretical box models of the one-sided shutdown system and found to give excellent agreement with high-resolution numerical simulations of this system. The dynamical structure of shutdown can also be accurately predicted by measurements from an RB cell. Results are presented for a general power-law equation of state. Secondly, these ideas are extended to model more complex physical systems, which comprise two fluid layers with an equation of state such that the solution that forms at the (moving) interface is more dense than either layer. The two fluids are either immiscible or miscible. Theoretical box models compare well with numerical simulations in the case of a flat interface between the fluids. Experimental results from a Hele-Shaw cell and numerical simulations both show that interfacial deformation can dramatically enhance the convective flux. The applicability of these results to the convective dissolution of geologically sequestered CO2{\mathrm{CO} }_{2} in a saline aquifer is discussed

Evaporation effects in elastocapillary aggregation

We consider the effect of evaporation on the aggregation of a number of elastic objects due to a liquid’s surface tension. In particular, we consider an array of spring–block elements in which the gaps between blocks are filled by thin liquid films that evaporate during the course of an experiment. Using lubrication theory to account for the fluid flow within the gaps, we study the dynamics of aggregation. We find that a non-zero evaporation rate causes the elements to aggregate more quickly and, indeed, to contact within finite time. However, we also show that the final number of elements within each cluster decreases as the evaporation rate increases. We explain these results quantitatively by comparison with the corresponding two-body problem and discuss their relevance for controlling pattern formation in elastocapillary systems.KS and DV wish to acknowledge the support of the King Abdullah University of Science and Technology (KAUST; Award No. KUK-C1-013-04), and the John Fell Oxford University Press (OUP) Research Fund.This is the author accepted manuscript. It is currently under an indefinite embargo pending publication by Cambridge University Press