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

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

Thin-sheet flow between coalescing bubbles

When two spherical bubbles touch, a hole is formed in the fluid sheet between them, and capillary pressure acting on its tightly curved edge drives an outward radial flow which widens the hole joining the bubbles. Recent images of the early stages of this process (Paulsen et al., Nat. Commun., vol. 5, 2014) show that the radius of the hole $r_{\!E}$ at time $t$ grows proportional to $t^{1/2}$, and that the rate is dependent on the fluid viscosity. Here, we explain this behaviour in terms of similarity solutions to a third-order system of radial extensional-flow equations for the thickness and velocity of the sheet of fluid between the bubbles, and determine the growth rate as a function of the Ohnesorge number $\mathit{Oh}$. The initially quadratic sheet profile allows the ratio of viscous and inertial effects to be independent of time. We show that the sheet is slender for $r_{\!E}\ll a$ if $\mathit{Oh}\gg 1$, where $a$ is the bubble radius, but only slender for $r_{\!E}\ll \mathit{Oh}^{2}a$ if $\mathit{Oh}\ll 1$ due to a compressional boundary layer of length $L\propto \mathit{Oh}\,r_{\!E}$, after which there is a change in the structure but not the speed of the retracting sheet. For $\mathit{Oh}\ll 1$, the detailed analysis justifies a simple momentum-balance argument, which gives the analytic prediction $r_{\!E}\sim (32a{\it\gamma}/3{\it\rho})^{1/4}t^{1/2}$, where ${\it\gamma}$ is the surface tension and ${\it\rho}$ is the density.J.P.M. acknowledges an Engineering and Physical Sciences Research Council studentship. C.R.A. and O.A.B. acknowledge the Donors of the American Chemical Society Petroleum Research Fund for partial support of this research. All data accompanying this publication are directly available within the publication.This is the accepted manuscript of a paper published in the Journal of Fluid Mechanics (Munro JP, Anthony CR, Basaran OA, Lister JR, Journal of Fluid Mechanics, 2015, 773, doi:10.1017/jfm.2015.253). The final version is available at http://dx.doi.org/10.1017/jfm.2015.25

Nondecaying Hydrodynamic Interactions along Narrow Channels.

Particle-particle interactions are of paramount importance in every multibody system as they determine the collective behavior and coupling strength. Many well-known interactions such as electrostatic, van der Waals, or screened Coulomb interactions, decay exponentially or with negative powers of the particle spacing r. Similarly, hydrodynamic interactions between particles undergoing Brownian motion decay as 1/r in bulk, and are assumed to decay in small channels. Such interactions are ubiquitous in biological and technological systems. Here we confine two particles undergoing Brownian motion in narrow, microfluidic channels and study their coupling through hydrodynamic interactions. Our experiments show that the hydrodynamic particle-particle interactions are distance independent in these channels. This finding is of fundamental importance for the interpretation of experiments where dense mixtures of particles or molecules diffuse through finite length, water-filled channels or pore networks.U.F.K. was supported by an ERC starting grant (PassMembrane 261101). S.P. acknowledges funding from a Leverhulme Early Career Fellowship. K.M. was supported by a grant from the EPSRC. E.L. was supported by Marie Curie CIG grant from EU.This is the author accepted manuscript. The final version is available from APS via http://dx.doi.org/10.1103/PhysRevLett.115.03830

Liquid ropes: a geometrical model for thin viscous jet instabilities.

Thin, viscous fluid threads falling onto a moving belt behave in a way reminiscent of a sewing machine, generating a rich variety of periodic stitchlike patterns including meanders, W patterns, alternating loops, and translated coiling. These patterns form to accommodate the difference between the belt speed and the terminal velocity at which the falling thread strikes the belt. Using direct numerical simulations, we show that inertia is not required to produce the aforementioned patterns. We introduce a quasistatic geometrical model which captures the patterns, consisting of three coupled ordinary differential equations for the radial deflection, the orientation, and the curvature of the path of the thread's contact point with the belt. The geometrical model reproduces well the observed patterns and the order in which they appear as a function of the belt speed.P.-T. B. was partially funded by the ERC Grant No. SIMCOMICS 280117.This is the author accepted manuscript. The final version is available from APS via http://dx.doi.org/10.1103/PhysRevLett.114.17450

The asymptotic structure of a slender dragged viscous thread

The behaviour of a viscous thread as it falls onto a moving belt is analysed in the asymptotic limit of a slender thread. While the bending resistance of a slender thread is small, its effects are dynamically important near the contact point with the belt, where it changes the curvature and orientation of the thread. Steady flows are shown to fall into one of three distinct regimes, depending on whether the belt is moving faster than, slower than or close to the same speed as the free-fall velocity of the thread. The key dynamical balances in each regime are explained and the role of bending stresses is found to be qualitatively different. The asymptotic solutions exhibit the ‘backward-facing heel’ observed experimentally for low belt speeds, and provide the leading-order corrections to the stretching catenary in theory previously developed for high belt speeds. The asymptotic stability of the thread to the onset of meandering is also analysed. It is shown that the entire thread, rather than the bending boundary layer alone, governs the stability. A balance between the destabilising reaction forces near the belt and the restoring force of gravity on the remainder of the thread determines the onset of meandering, and an analytic estimate for the meandering frequency is thereby obtained. At leading order, neutral stability occurs with the belt moving a little more slowly than the free-fall velocity of the thread, not when the lower part of the thread begins to be under compression, but when the horizontal reaction force at the belt begins to be slightly against the direction of belt motion. The onset of meandering is the heel ‘losing its balance’

Climate Change and Biosphere Response: Unlocking the Collections Vault

Natural history collections (NHCs) are an important source of the long-term data needed to understand how biota respond to ongoing anthropogenic climate change. These include taxon occurrence data for ecological modeling, as well as information that can be used to reconstruct mechanisms through which biota respond to changing climates. The full potential of NHCs for climate change research cannot be fully realized until high-quality data sets are conveniently accessible for research, but this requires that higher priority be placed on digitizing the holdings most useful for climate change research (e.g., whole-biota studies, time series, records of intensively sampled common taxa). Natural history collections must not neglect the proliferation of new information from efforts to understand how present-day ecosystems are responding to environmental change. These new directions require a strategic realignment for many NHC holders to complement their existing focus on taxonomy and systematics. To set these new priorities, we need strong partnerships between NHC holders and global change biologists

Plethora of transitions during breakup of liquid filaments.

Thinning and breakup of liquid filaments are central to dripping of leaky faucets, inkjet drop formation, and raindrop fragmentation. As the filament radius decreases, curvature and capillary pressure, both inversely proportional to radius, increase and fluid is expelled with increasing velocity from the neck. As the neck radius vanishes, the governing equations become singular and the filament breaks. In slightly viscous liquids, thinning initially occurs in an inertial regime where inertial and capillary forces balance. By contrast, in highly viscous liquids, initial thinning occurs in a viscous regime where viscous and capillary forces balance. As the filament thins, viscous forces in the former case and inertial forces in the latter become important, and theory shows that the filament approaches breakup in the final inertial-viscous regime where all three forces balance. However, previous simulations and experiments reveal that transition from an initial to the final regime either occurs at a value of filament radius well below that predicted by theory or is not observed. Here, we perform new simulations and experiments, and show that a thinning filament unexpectedly passes through a number of intermediate transient regimes, thereby delaying onset of the inertial-viscous regime. The new findings have practical implications regarding formation of undesirable satellite droplets and also raise the question as to whether similar dynamical transitions arise in other free-surface flows such as coalescence that also exhibit singularities.The authors thank Dr. Pankaj Doshi for several insightful discussions. This work was supported by the Basic Energy Sciences program of the US Department of Energy (DE-FG02-96ER14641), Procter & Gamble USA, the Chevron Corporation, the UK Engineering and Physical Sciences Research Council (Grant EP/H018913/1), the John Fell Oxford University Press Research Fund, and the Royal Society.This is the final published version. It first appeared via PNAS at http://dx.doi.org/10.1073/pnas.141854111