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Production of building materials emits 11% of global carbon dioxide (CO2) emission. The greenhouse gas emission from the construction industry has been tried to mininmize from early 1980s; but after four decades of development, it is not fully sustainable. Cement is the second most consumed material in the world, after water and cement production contributes for 8% of global CO2 emission. We produced a greener cement from abundantly available waste: fly ash, blast furnace lag, and rice husk ash to significantly minimize the greenhouse gas emission. Discarded aluminium foil becomes one of most landfilling waste that has high potential for recycling. On other hand, cement carbonation is a curing method that stores significant amount of CO2 into cement with lesser cost and energy compared to commercial carbon sequestration. Therefore, we incorporate aluminium foil waste and CO2 waste from industry to improve the engineering and environmental performance of the cement. We compared changes in carbonation when using gaseous carbon dioxide (gCO2) and supercritical carbon dioxide (scCO2) and found that the scCO2 condition achieves higher compressive strength and yielded a stronger barrier against leaching. Hence, this carbon cured cement can be widely used in underground applications, where the heavy metal leaching is a critical issue. Projections show our greener cement reducing CO2 emission by 55% compared to Portland cement and reducing direct costs by 35%. Also, our cement ultimately reduces hydrogen gas demand by recycling aluminium, which releases pure hydrogen during the production process, and this effect reduces annual CO2 emission by 35 million tonnes from this hydrogen production alone. Adopted globally, the system would permanently store 72 million tonnes of CO2 in a stable composite annually. On whole, our cement production significantly reduces the energy requirement for cement manufacturing and releases future energy, hydrogen gas, as by product.

Funder: King's College, University of Cambridge; Id: http://dx.doi.org/10.13039/501100000648, We present a shallow, isothermal, Newtonian model for the transient interaction of lava flows with topography. Numerical integrations and simple mathematical approximations are deployed to quantify how topography controls lava thicknesses and flow speeds. Considering idealized topographic features, we show that modest depressions thicken and accelerate the flow—even far downstream—whilst mounds have the opposite effect. However, deep ponds of lava form in depressions of sufficient amplitude, which introduces a long timescale for lava to fill the depression and hence the accelerated downstream flow may never be attained. Relatively large mounds completely divert the lava, providing protected lava‐free regions for homes and infrastructure. There can, however, be hazardous, deep, fast flow around the edges of the mound, owing to diversion. Additionally, we show that our model accurately predicts the lava‐free region that has been observed in the eruption 35 kyr ago at Marcath Volcano, Nevada.

The behaviour of granular column collapses is associated with the dynamics of geohazards, such as debris flows, landslides and pyroclastic flows, yet their underlying physics is still not well understood. In this paper, we explore granular column collapses using the sphero-polyhedral discrete element method, where the system contains two types of particles with different frictional properties. We impose three different mixing ratios and multiple different particle frictional coefficients, which lead to different run-out distances and deposition heights. Based on our previous work and a simple mixture theory, we propose a new effective initial aspect ratio for the bi-frictional granular mixture, which helps unify the description of the relative run-out distances. We analyse the kinematics of bi-frictional granular column collapses and find that deviations from classical power-law scaling in both the dimensionless terminal time and the dimensionless time when the system reaches the maximum kinetic energy may result from differences in the initial solid fraction and initial structures. To clarify the influence of initial states, we further decrease the initial solid fraction of granular column collapses, and propose a trial function to quantitatively describe its influence. Due to the utilization of a simple mixture theory of contact occurrence probability, this study can be associated with the friction-dependent rheology of granular systems and friction-induced granular segregations, and further generalized to applications with multiple species of particles in various natural and engineering mixtures.

We evaluate theoretically and confirm experimentally the shape of the fluid envelope resulting from the input of relatively heavy fluid at a constant rate from a point source at the base of a homogeneous porous medium. In three dimensions an initially expanding hemisphere transitions into a gravity current flowing over the assumed rigid, horizontal and impermeable bottom of the porous medium. A range of increasing transition times occurs if defined by extrapolation of the relationships in the two extreme regimes (hemispherical shape and thin-layer gravity current) so that they intersect, for: the ratio of buoyancy to fluid resistance; the horizontal extent of the fluid; the ratio of height at the centre to the radius; and just the height at the centre. Corresponding results are derived for two-dimensional geometries. In this case, we conduct a series of laboratory experiments demonstrating the transition between the radial and gravity current regimes both in terms of form and propagation rate. The results are extrapolated briefly to two-layer systems, in order to begin to understand effects due to vertically heterogeneous pore structures. We sketch, and verify by experiment, that an expanding hemisphere in a lower layer can reach a much more permeable upper layer and flow through it as a gravity current, thereby uniting the two regimes.

Previous analyses of the flow of low-Reynolds-number, viscous gravity currents down inclined planes are investigated further and extended. Particular emphasis is on the motion of the fluid front and tail, which previous analyses treated somewhat cavalierly. We obtain reliable, approximate, analytic solutions in these regions, the accuracies of which are satisfactorily tested against our numerical evaluations. The solutions show that the flow has several time scales determined by the inclination angle, $\alpha$ . At short times, the influence of initial and boundary conditions is important and the flow is governed by both the pressure gradient and the direct action of gravity due to inclination. Later on, the areas where the boundary conditions are important shrink. This fact explains why previous solutions, being inaccurate near the front and the tail, described experimental data with high accuracy. At larger times, of the order of $\alpha ^{-5/2}$ , the influence of the pressure gradient may be neglected and the fluid profile converges to the square-root shape predicted in previous works. Important extensions of our approach are also outlined.

The interaction of gravitationally driven, free-surface flows of viscous fluid with topographic features is investigated theoretically. The motion is studied in the regime where the depth of the flow is much smaller than the streamwise extent of the topography. A lubrication model of the motion is developed, integrated numerically and analysed asymptotically. For small mounds, it is shown that the flow surmounts the obstacles, but for larger mounds the flow is deflected around it and can form dry zones in its wake into which fluid does not flow, as well as forming deeper ponded regions upstream. Which of these phenomena prevails is shown to depend upon the amplitude of the mound height and the thickness of the oncoming flow relative to the streamwise length scale over which the topography varies. By using numerical and asymptotic results, we demonstrate that relatively wide mounds lead to the development of deep ponds of material upstream, which may lead to flow overtopping if the mound is not sufficiently high. These insights can be used to inform the design of barriers that defend built infrastructures from lava flows, and it is shown how this model can also provide an upper bound on the force exerted by the flow on them.

A wide range of initial-value problems in fluid mechanics in particular, and in the physical sciences in general, are described by nonlinear partial differential equations. Recourse must often be made to numerical solutions, but a powerful, well-established technique is to solve the problem in terms of similarity variables. A disadvantage of the similarity solution is that it is almost always independent of any specific initial conditions, with the solution to the full differential equation approaching the similarity solution for times $t\gg t_{\ast }$, for some $t_{\ast }$. But what is $t_{\ast }$? In this paper we consider the situation of viscous gravity currents and obtain useful formulae for the time of approach, $\unicode[STIX]{x1D70F}(p)$, for a number of different initial shapes, where $p$ is the percentage disagreement between the radius of the current as determined by the full numerical solution of the governing partial differential equation and the similarity solution normalised by the similarity solution. We show that for any initial shape of volume $V,\unicode[STIX]{x1D70F}\propto 1/(\unicode[STIX]{x1D6FD}V^{1/3}\unicode[STIX]{x1D6FE}_{0}^{8/3}p)$ (as $p\downarrow 0$), where $\unicode[STIX]{x1D6FD}=g\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}/(3\unicode[STIX]{x1D707})$, with $g$ representing the acceleration due to gravity, $\unicode[STIX]{x0394}\unicode[STIX]{x1D70C}$ the density difference between the gravity current and the ambient, $\unicode[STIX]{x1D707}$ the dynamic viscosity of the fluid that makes up the gravity current and $\unicode[STIX]{x1D6FE}_{0}$ the initial aspect ratio. This framework can used in many other situations, including where it is not an initial condition (in time) that is studied but one valid for specified values at a special spatial coordinate.

Co-auteur étranger; International audience; Granular flows that propagate down a mountainside, then reach the sea, a lake or a river and finally, travel underwater, is a common event on the Earth’s surface. To help the description of such events, laboratory experiments on gas-fluidized granular flows entering water are performed, analyzed, and compared to those propagating in air. The originality of this study lies in the fluidization process, which improves the laboratory modelling of geophysical flows by taking their high mobility into account. Qualitatively, the presence of the water body promotes the generation of a granular jet over the water surface, a leading and largest wave, and a particle-driven gravity current underwater. Hydrodynamic forces mainly play a dissipative role by slowing and reducing the spreading of the granular mass underwater, but a low amount of grains are still transported by the turbulent fluid as a gravity current far away. The temporal evolution of the granular jet and the particle-driven gravity current are well described by ballistic motion theory and scaling laws of homogeneous gravity currents, respectively. Most currents propagate with a constant flow-front velocity along the horizontal bottom, which is controlled by the flow height depending on the water depth. In constrast, the bulk volume concentration of particles in the current is estimated to be nearly constant, interpreted as a critical concentration above which the excess of particles cannot be maintained by the turbulent fluid. This experimental study highlights the complexity of the dynamics and deposits of granular masses when they encounter a water body.

We investigate granular column collapses with different column cross-sections and associate the cross-section shape influence with a finite-size analysis. Previous research, confined to initially circular configurations, reviewed the importance of granular column collapse studies and concluded that the run-out distance scales with the initial aspect ratio. In this work, granular columns with three different types of initial column cross-sections (square, equilateral triangular, and rectangular cross-sections) are simulated using the discrete element method (DEM). We explore how non-circular cross-sections lead to different run-out distances. Based on the previously obtained finite-size analysis, we further link the initial radius in different directions to the relative size of a column and perform the finite-size analysis to explain the cross-section influence. In the end, a universal relationship, which includes frictional properties, relative sizes, and the cross-section influence, is proposed. Our results are shown to have direct relevance to various natural and engineering systems.

The collapse of granular columns, which sheds light on the kinematics, dynamics, and deposition morphology of mass-driven flows, is crucial for understanding complex flows in both natural and engineering systems, such as debris flows and landslides. However, our research shows that a strong size effect and cross-section shape influence exist in this test. Thus, it is essential to better understand these effects. In this study, we explore the influence of both relative column sizes and cross-section shapes on the run-out behavior of collapsed granular columns and analyze their influence on the deposition morphology with the discrete element method (DEM) with Voronoi-based spheropolyhedron particles. We link the size effect that occurs in granular column collapse problems to the finite-size scaling functions and investigate the characteristic correlation length associated with the granular column collapses. The collapsing behavior of granular columns with different cross-section shapes is also studied, and we find that particles tend to accumulate in the direction normal to the edge of the cross-section instead of the vertex of it. The differences in the run-out behavior in different directions when the cross-section is no longer a circle can also be explained by the finite-size analysis we have performed in this study. We believe that such a study is crucial for us to better understand how granular material flows, how it deposits, and how to consider the size effect in the rheology of granular flows.

The leakage from a fracture network to a surrounding medium during drainage, or backflow, driven by elastic relaxation, is considered. A network model is extended to include the effects of permeable boundaries, with the permeation through the wall assumed to be proportional to the local pressure. The regimes in which leakage is dominant relative to the parallel flow along the channel are evaluated at different times. Results show that, when the aperture of the channel is large enough, the parallel flow is greater than the permeation through the wall, and the channel thickness decreases in time, $t$ , with a $t^{-1/3}$ behaviour, as reported previously. However, when the aperture is small, the channel thickness decreases exponentially in time. An asymptotic investigation of the solution for a single fracture is performed and extended to network systems. The study provides insight into the influence leakage may have on squeezing-induced flows, which is relevant to natural and engineering systems.

We study the dynamics of proppants carried by fluid driven into an evolving penny-shaped fracture. The behaviour of the slurry flow is investigated in two phases: pressurised injection and elastic closure. During injection the slurry is modelled using a frictional rheology that takes into account the shear-induced migration and jamming of the proppants. Making pragmatic assumptions of negligible toughness and cross-fracture fluid slip, we find self-similar solutions supporting a range of proppant concentration profiles. In particular, we define an effective viscosity, which equates the fracture evolution of a slurry flow with a given proppant volume fraction, to a Newtonian flow with a particular viscosity. Using this framework, we are able to make predictions about the geometry of the growing fracture and the significance of tip screen-out. In the closure phase, proppants are modelled as incompressible and radially immobile within the narrowing fracture. The effects of proppant concentration on the geometry of the residual propped fracture are explored in full. The results have important applications to industrial fracking and geological dike formation by hot, intruding magma., 20 pages, 9 figures

Motivated by shallow ocean waves propagating over coral reefs, we investigate the drift velocities due to surface wave motion in an effectively inviscid fluid that overlies a saturated porous bed of finite depth. Previous work in this area either neglects the large-scale flow between layers (Phillips in Flow and reactions in permeable rocks, Cambridge University Press, Cambridge, 1991) or only considers the drift above the porous layer (Monismith in Ann Rev Fluid Mech 39:37–55, 2007). Overcoming these limitations, we propose a model where flow is described by a velocity potential above the porous layer and by Darcy’s law in the porous bed, with derived matching conditions at the interface between the two layers. Both a horizontal and a novel vertical drift effect arise from the damping of the porous bed, which requires the use of a complex wavenumber k. This is in contrast to the purely horizontal second-order drift first derived by Stokes (Trans Camb Philos Soc 8:441–455, 1847) when working with solely a pure fluid layer. Our work provides a physical model for coral reefs in shallow seas, where fluid drift both above and within the reef is vitally important for maintaining a healthy reef ecosystem (Koehl et al. In: Proceedings of the 8th International Coral Reef Symposium, vol 2, pp 1087–1092, 1997; Monismith in Ann Rev Fluid Mech 39:37–55, 2007). We compare our model with field measurements by Koehl and Hadfield (J Mar Syst 49:75–88, 2004) and also explain the vertical drift effects as documented by Koehl et al. (Mar Ecol Prog Ser 335:1–18, 2007), who measured the exchange between a coral reef layer and the (relatively shallow) sea above.

The behaviour of a viscous drop squeezed between two horizontal planes is treated by both theory and experiment. When the squeezing force F is constant and surface tension is neglected, the theory predicts ultimate growth of the radius a~ t^{1/8}, in excellent agreement with our experiment. Surface tension at the drop boundary is included in the analysis, although negligibly small in the squeezing experiments. The circular evolution is found to be stable under small perturbations. If the force is reversed (F < 0), so that the plates are pulled or levered apart, the boundary of the drop is subject to a fingering instability. The effect of a trapped air bubble at the centre of the drop is then considered. The annular evolution of the drop under constant squeezing is still found to follow a `one-eighth' power law, but this is unstable, the instability originating at the boundary of the air bubble. If the plates are drawn apart, the evolution is still subject to the fingering instability driven from the outer boundary of the annulus. This instability is realised experimentally by levering the plates apart at one corner: fingering develops at the outer boundary and spreads rapidly to the interior as the levering is slowly increased. At a later stage, small cavitation bubbles appear in the very low pressure region far from the point of leverage. This exotic behaviour is discussed in the light of the foregoing theoretical analysis., 23 pages, 12 figures, submitted to J. Fluid Mech

The flow of viscous fluid from a line source on an inclined plane containing a topographic depression is studied using a lubrication model. The steady flow is perturbed by the depression. We focus on the regime in which the flow thickness is small relative to the lengthscale of the depression. For shallow depressions, the flow is only slightly perturbed. However, for deeper depressions a large pond of fluid with a horizontal free-surface is observed. This ponding of fluid is associated with the topography having uphill regions, which in the absence of inertia can only be surmounted by developing a deep fluid layer. The accumulation of fluid in the depression leads to a focusing of the flow, which is observed even far downstream of the depression.

Free-surface flows of viscous liquid down an inclined plane and past cylinders of various cross-sections are investigated theoretically and experimentally. The cylinders are oriented with their axis perpendicular to the plane and are sufficiently tall that they are not overtopped. A lubrication model is applied to derive the steady governing equation for the flow depth, which is integrated numerically and analyzed asymptotically to calculate how the depth of a steady uniform flow is perturbed as it flows past the cylinder. Flows past cylinders that are narrow relative to the depth of the oncoming flow are only slightly perturbed, but for relatively wide cylinders, there forms a pond of nearly stationary fluid upstream of the cylinder and a dry region in which there is no fluid downstream of the cylinder. The structure of the flow in the regime of a relatively wide cylinder depends in detail upon the curvature of the cylinder at the upstream stagnation point. For flows past cylinders with circular and square cross-sections, the maximum flow depth occurs at the upstream stagnation point. Its numerical value may be predicted analytically on the basis of the asymptotic expressions and exhibits different dependencies upon the variables that characterize the motion. In addition, wedge-shaped obstructions are analyzed for which the flow depth increases along the wedge wall and the maximum flow depth occurs away from the upstream stagnation point. The results from new laboratory experiments of flows past circular cylinders are reported and these corroborate the theory, confirming the occurrence of both pond and dry regions. The investigation has direct relevance to the deflection of lava flows by barriers and buildings and the theory is employed to deduce simplified asymptotic expressions of the force exerted on the cylinders.

The steady lateral spreading of a free-surface viscous flow down an inclined plane around a vertex from which the channel width increases linearly with downstream distance is investigated analytically, numerically and experimentally. From the vertex the channel wall opens by an angle α to the downslope direction and the viscous fluid spreads laterally along it before detaching. The motion is modelled using lubrication theory and the distance at which the flow detaches is computed as a function of α using analytical and numerical methods. Far downslope after detachment, it is shown that the motion is accurately modelled in terms of a similarity solution. Moreover, the detachment point is well approximated by a simple expression for a broad range of opening angles. The results are corroborated through a series of laboratory experiments and the implication for the design of barriers to divert lava flows are discussed. This article is part of the theme issue ‘Stokes at 200 (Part 1)’.

Although we humans have known since the first smokey campfires of prehistory that our activities might alter our local surroundings, the nineteenth century saw the first indications that humankind might alter the global environment; what we currently know as anthropogenic climate change. We are now celebrating the bicentenaries of three figures with a hand in the birth of climate science. George Stokes, John Tyndall and John Ruskin were born in August 1819, August 1820 and February 1819, respectively. We look back from the perspective of two centuries following their births. We outline their contributions to climate science: understanding the equations of fluid motion and the recognition of the need to collect global weather data together with comprehending the role in regulating terrestrial temperature played by gases in the atmosphere. This knowledge was accompanied by fears of the Earth’s regression to another ice age, together with others that industrialization was ruining humankind’s health, morals and creativity. The former fears of global cooling were justified but seem strange now that the balance has tipped so far the other way towards global warming; the latter, on the other hand, today seem very prescient. This article is part of the theme issue ‘Stokes at 200 (Part 1)’.

We develop a model for predicting the flow resulting from the relaxation of pre-strained, fluid-filled, elastic network structures. This model may be useful for understanding relaxation processes in various systems, e.g. deformable microfluidic systems or by-products from hydraulic fracturing operations. The analysis is aimed at elucidating features that may provide insight on the rate of fluid drainage from fracturing operations. The model structure is a bifurcating network made of fractures with uniform length and elastic modulus, which allows for general self-similar branching and variation in fracture length and rigidity between fractures along the flow path. A late-time $t^{-1/3}$ power law is attained and the physical behaviour can be classified into four distinct regimes that describe the late-time dynamics based on the location of the bulk of the fluid volume (which shifts away from the outlet as branching is increased) and pressure drop (which shifts away from the outlet as rigidity is increased upstream) along the network. We develop asymptotic solutions for each of the regimes, predicting the late-time flux and evolution of the pressure distribution. The effects of the various parameters on the outlet flux and the network’s drainage efficiency are investigated and show that added branching and a decrease in rigidity upstream tend to increase drainage time.

We study experimentally the flow of light granular material along the free surface of a liquid of greater density. Despite a rich set of related geophysical and environmental phenomena, such as the spreading of calved ice, volcanic ash, debris and industrial wastes, there are few previous studies on this topic. We conduct a series of lock-release experiments of buoyant spherical beads into a rectangular tank initially filled with either fresh or salt water, and record the time evolution of the interface shape and the front location of the current of beads. We find that following the release of the lock the front location obeys a power-law behaviour during an intermediate time period before the nose of beads reaches a maximum runout distance within a finite time. We investigate the dependence of the scaling exponent and runout distance on the total amount of beads, the initial lock length, and the properties of the liquid that fills the tank in the experiments. Scaling arguments are provided to collapse the experimental data into universal curves, which can be used to describe the front dynamics of buoyant granular flows with different size and buoyancy effects and initial lock aspect ratios.

Hydraulic fracturing for production of oil and gas from shale formations releases fluid waste, by-products that must be managed carefully to avoid significant harm to human health and the environment. These fluids are presumed to result from a variety of fracture relaxation processes, and are commonly referred to as ‘flowback’ and ‘produced water’, depending primarily on the time scale of their appearance. Here, a model is presented for investigating the dynamics of backflows caused by the elastic relaxation of a pre-strained medium, namely a single fracture and two model fracture network systems: a single bifurcated channel and its generalization for $n$ bifurcated fracture generations. Early- and late-time asymptotic solutions are obtained for the model problems and agree well with numerical solutions. In the late-time period, the fracture apertures and backflow rates exhibit a time dependence of $t^{-1/3}$ and $t^{-4/3}$, respectively. In addition, the pressure distributions collapse to universal curves when scaled by the maximum pressure in the system, which we calculate as a function of $n$. The pressure gradient along the network is steepest near the outlet while the bulk of the network serves as a ‘reservoir’. Fracture networks with larger $n$ are less efficient at evicting fluids, manifested through a longer time required for a given fractional reduction of the initial volume. The developed framework may be useful for informing engineering design and environmental regulations.

We consider the instantaneous release of a finite volume of fluid in a porous medium saturated with a second, immiscible fluid of different density. The resulting two-phase gravity current exhibits a rich array of behaviours due to both the residual trapping of fluid as the current recedes and the differing effects of surface tension between advancing and receding regions of the current. We develop a framework for the evolution of such a current with particular focus on the large-scale implications of the form of the constitutive relation between residual trapping and initial saturation. Pore-scale hysteresis within the current is represented by families of scanning curves relating capillary pressure and relative permeability to saturation. In the resulting vertically integrated model, all capillary effects are incorporated within specially defined saturation and flux functions specific to each region. In the long-time limit, when the height of the current and the saturations within it are low, the saturation and flux functions can be approximated by mathematically convenient power laws. If the trapping model is approximately linear at low saturations, the equations admit a similarity solution for the propagation rate and height profile of the late-time gravity current. We also solve the governing partial differential equation numerically for the nonlinear Land’s trapping model, which is commonly used in studies of two-phase flows. Our investigation suggests that for trapping relations for which the proportion of trapped to initial fluid saturation increases and tends to unity as the initial saturation decreases, both of which are properties of Land’s model, a gravity current slows and eventually stops. This trapping behaviour has important applications, for example to the ultimate distance contaminants or stored carbon dioxide may travel in the subsurface.

The separation of volcanic ash and sulfur dioxide (SO2) gas is sometimes observed during volcanic eruptions. The exact conditions under which separation occurs are not fully understood but the phenomenon is of importance because of the effects volcanic emissions have on aviation, on the environment, and on the earth's radiation balance. The eruption of Grímsvötn, a subglacial volcano under the Vatnajökull glacier in Iceland during 21–28 May 2011 produced one of the most spectacular examples of ash and SO2 separation, which led to errors in the forecasting of ash in the atmosphere over northern Europe. Satellite data from several sources coupled with meteorological wind data and photographic evidence suggest that the eruption column was unable to sustain itself, resulting in a large deposition of ash, which left a low-level ash-rich atmospheric plume moving southwards and then eastwards towards the southern Scandinavian coast and a high-level predominantly SO2 plume travelling northwards and then spreading eastwards and westwards. Here we provide observational and modelling perspectives on the separation of ash and SO2 and present quantitative estimates of the masses of ash and SO2 that erupted, the directions of transport, and the likely impacts. We hypothesise that a partial column collapse or sloughing fed with ash from pyroclastic density currents (PDCs) occurred during the early stage of the eruption, leading to an ash-laden gravity intrusion that was swept southwards, separated from the main column. Our model suggests that water-mediated aggregation caused enhanced ash removal because of the plentiful supply of source water from melted glacial ice and from entrained atmospheric water. The analysis also suggests that ash and SO2 should be treated with separate source terms, leading to improvements in forecasting the movement of both types of emissions.

Transport of dense fluid by an inclined gravity current can control the vertical density structure of the receiving basin in many natural and industrial settings. A case familiar to many is a lake fed by river water that is dense relative to the lake water. In laboratory experiments, we pulsed dye into the basin inflow to visualise the transport pathway of the inflow fluid through the basin. We also measured the evolving density profile as the basin filled. The experiments confirmed previous observations that when the turbulent gravity current travelled through ambient fluid of uniform density, only entrainment into the dense current occurred. When the gravity current travelled through the stratified part of the ambient fluid, however, the outer layers of the gravity current outflowed from the current by a peeling detrainment mechanism and moved directly into the ambient fluid over a large range of depths. The prevailing model of a filling box flow assumes that a persistently entraining gravity current entrains fluid from the basin as the current descends to the deepest point in the basin. This model, however, is inconsistent with the transport pathway observed in visualisations and poorly matches the stratifications measured in basin experiments. The main contribution of the present work is to extend the prevailing filling box model by incorporating the observed peeling detrainment. The analytical expressions given by the peeling detrainment model match the experimental observations of the density profiles more closely than the persistently entraining model. Incorporating peeling detrainment into multiprocess models of geophysical systems, such as lakes, will lead to models that better describe inflow behaviour.

In this letter, we focus on the size effect of granular column collapses, which are potentially connected to the dynamics of complex geophysical flows, even if the link between microscopic structures of granular assemblies and their macroscopic behaviors is still not well understood. Using the sphero-polyhedral discrete element method (DEM), we show that the column radius/grain size ratio has a strong influence on the collapse behavior. A finite-size analysis, which is inspired by a phase transition around an inflection point, is performed to obtain a general scaling equation with critical exponents for run-out distances. We further link the size effect with the strong force network and formalize a correlation length scale that exponentially scales with the effective aspect ratio. Such a scaling solution shows similarities with the percolation problem of two-dimensional random networks and can be extended to other similar natural and engineering systems.

Granular column collapses result in an array of flow phenomena and deposition morphologies, the understanding of which brings insights into studying granular flows in both natural and engineering systems. Guided by experiments, we carried out computational studies with the discrete element method (DEM) to identify fundamental links between the macro-scale behavior and micro-scale properties of granular columns. A dimensionless number combining particle and bulk properties of the column, $$\alpha _{\text {eff}}$$ , was found from dimensional analysis to determine three collapse regimes (quasi-static, inertial, and liquid-like), revealing universal trends of flow regimes and deposition morphologies under different conditions. This dimensionless number may represent physically the competing inertial and frictional effects that govern the behavior of the granular column collapse, including energy conversion and dissipation. The finding is important for understanding quantitatively the flow of granular materials and their deposits.

We present the second half of the papers from the Stokes 200 symposium celebrating the bicentenary of George Gabriel Stokes. This article is part of the theme issue ‘Stokes at 200 (part 2)’.

Sir George Gabriel Stokes PRS was for 30 years an inimitable Secretary of the Royal Society and its President from 1885 to 1890. Two hundred years after his birth, Stokes is a towering figure in physics and applied mathematics; fluids, asymptotics, optics, acoustics among many other fields. At the Stokes 200 meeting, held at Pembroke College, Cambridge from 15–18th September 2019, an invited audience of about 100 discussed the state of the art in all the modern research fields that have sprung from his work in physics and mathematics, along with the history of how we have got from Stokes’ contributions to where we are now. This theme issue is based on work presented at the Stokes 200 meeting. In bringing together people whose work today is based upon Stokes’ own, we aim to emphasize his influence and legacy at 200 to the community as a whole. This article is part of the theme issue ‘Stokes at 200 (Part 1)’.

Summary In a recent article, Ball and Huppert (J. Fluid Mech., 874, 2019) introduced a novel method for ascertaining the characteristic timescale over which the similarity solution to a given time-dependent nonlinear differential equation converges to the actual solution, obtained by numerical integration, starting from given initial conditions. In this article, we apply this method to a range of different partial differential equations describing propagating gravity currents of fixed volume as well as modifying the techniques to apply to situations for which convergence to the numerical solution is oscillatory, as appropriate for gravity currents propagating at large Reynolds numbers. We investigate properties of convergence in all of these cases, including how different initial geometries affect the rate at which the two solutions agree. It is noted that geometries where the flow is no longer unidirectional take longer to converge. A method of time-shifting the similarity solution is introduced to improve the accuracy of the approximation given by the similarity solution, and also provide an upper bound on the percentage disagreement over all time.

We consider the steady flow of a viscous compressible gas through an axisymmetric or two-dimensional porous medium whose properties in the direction of the flow are sufficiently slowly varying. The study is partly motivated by a number of different applications in the Earth sciences, including the release of magmatic volatiles from a magma chamber beneath an active volcano and the discharge of geothermal fluids. The results are also relevant to evaluating the consequences of an accidental release of carbon dioxide from a storage reservoir within the Earth, as might happen at a carbon capture and storage (CCS) site. We consider both slow, thermally equilibrated, flows and fast, adiabatic flows. Because the flow is compressible, it is the mass (and not the volume) flux which is conserved along the flow. We determine this constant mass flux and the velocity and pressure fields, both of which vary with position along the flow, as a function of all the physical parameters. We find that the resultant pressure gradient in the medium is largest at the far, low-pressure end of the conduit because the velocity is largest at that end due to the smallest density being associated with the smallest pressures. This means that the pressure in the permeable conduit is always larger than the linear pressure distribution which joins the given pressures at depth and at the surface, as would be the situation if the flow were incompressible. The detailed pressure distribution is shown to depend on the variation with depth of the quantity $\unicode[STIX]{x1D707}T/(ka^{2})$, where $\unicode[STIX]{x1D707}$ is the dynamic viscosity of the vapour, $T$ is the external temperature, $k$ the permeability and $\unicode[STIX]{x03C0}a^{2}$ the cross-sectional area of the conduit. The resultant mass flux is determined to be proportional to the mean along the flow of $\unicode[STIX]{x1D707}T/(ka^{2})$. We present two numerical illustrations of the results.

In his famous paper of 1847 (Stokes GG. 1847 On the theory of oscillatory waves. Trans. Camb. Phil. Soc. 8 , 441–455), Stokes introduced the drift effect of particles in a fluid that is undergoing wave motion. This effect, now known as Stokes drift, is the result of differences between the Lagrangian and Eulerian velocities of the fluid element and has been well-studied, both in the laboratory and as a mechanism of mass transport in the oceans. On a smaller scale, it is of vital importance to the hydrodynamics of coral reefs to understand drift effects arising from waves on the ocean surface, transporting nutrients and oxygen to the complex ecosystems within. A new model is proposed for a class of coral reefs in shallow seas, which have a permeable layer of depth-varying permeability. We then note that the behaviour of the waves above the reef is only affected by the permeability at the top of the porous layer, and not its properties within, which only affect flow inside the porous layer. This model is then used to describe two situations found in coral reefs; namely, algal layers overlying the reef itself and reef layers whose permeability decreases with depth. This article is part of the theme issue ‘Stokes at 200 (part 2)’.

The low-Reynolds-number gravitational settling of a sphere through a fluid interface is investigated experimentally. By varying the viscosity ratio between the two fluids and the Bond number, two different modes of interfacial deformation are observed: a tailing mode and a film drainage mode. In the tailing mode, the interface deforms significantly as the sphere approaches, and the sphere becomes enveloped by a layer of the upper fluid. A tail forms, connecting the sphere to the bulk of the upper phase. In the film drainage mode, the interface deforms much less and the sphere impacts onto the interface, which either ruptures to form a contact line on the sphere or leaves a very thin wetting film. Additionally, two types of sinking profiles are observed: steady sinking, where the sphere velocity changes monotonically as it sinks, and stalled sinking, where the sphere's progress is inhibited by the interface, before it accelerates into the lower fluid. We present a regime diagram showing the different behaviors. Finally, the dependence of the sinking time on the Bond number and viscosity ratio is investigated. For the film drainage regime a simple scaling law is deduced; the tailing regime exhibits more complicated dynamics, possibly explained by a multistage sinking process.

We study the flow and leakage of gravity currents injected into an unsaturated (dry), vertically confined porous layer containing a localized outlet or leakage point in its lower boundary. The leakage is driven by the combination of the gravitational hydrostatic pressure head of the current above the outlet and the pressure build-up from driving fluid downstream of the leakage point. Model solutions illustrate transitions towards one of three long-term regimes of flow, depending on the value of a dimensionless parameter $D$, which, when positive, represents the ratio of the hydrostatic head above the outlet for which gravity-driven leakage balances the input flux, to the depth of the medium. If $D\leqslant 0$, the input flux is insufficient to accumulate any fluid above the outlet and fluid migrates directly through the leakage pathway. If $0, some fluid propagates downstream of the outlet but retains a free surface above it. The leakage rate subsequently approaches the input flux asymptotically but much more gradually than if $D\leqslant 0$. If $D>1$, the current fills the entire depth of the medium above the outlet. Confinement then fixes gravity-driven leakage at a constant rate but introduces a new force driving leakage in the form of the pressure build-up associated with mobilizing fluid downstream of the outlet. This causes the leakage rate to approach the injection rate faster than would occur in the absence of the confining boundary. This conclusion is in complete contrast to fluid-saturated media, where confinement can potentially reduce long-term leakage by orders of magnitude. Data from a new series of laboratory experiments confirm these predictions.

Hydraulic fracturing is a procedure by which a fracture is initiated and propagates due to pressure (hydraulic loading) applied by a fluid introduced inside the fracture. In this study, we focus on a crack driven by an incompressible Newtonian fluid, injected at a constant rate into an elastic matrix. The injected fluid creates a radial fracture that propagates along a plane. We investigate this type of fracture both theoretically and experimentally. Our experimental apparatus uses a brittle and transparent polyacrylamide hydrogel matrix. Using this medium, we examine the rate of radial crack growth, fracture aperture, shape of the crack tip and internal fluid flow field. Our range of experimental parameters allows us to exhibit two distinct fracturing regimes, and the transition between these, in which the rate of radial crack propagation is dominated by either viscous flow within the fracture or the material toughness. Measurements of the profiles near the crack tip provide additional evidence of the viscosity-dominated and toughness-dominated regimes, and allow us to observe the transition from the viscous to the toughness regime as the crack propagates. Particle image velocimetry measurements show that the flow in the crack is radial, as expected in the viscous regime and in the early stages of the toughness regime. However, at later times in the toughness regime, circulation cells are observed in the flow within the crack that destroy the radial symmetry of the flow field.

The equilibration time $\unicode[STIX]{x1D70F}$ in response to a change in flux from $Q$ to $\unicode[STIX]{x1D6EC}Q$ after an injection period $T$ applied to either a low-Reynolds-number gravity current or one propagating through a porous medium, in both axisymmetric and one-dimensional geometries, is shown to be of the form $\unicode[STIX]{x1D70F}=Tf(\unicode[STIX]{x1D6EC})$, independent of all the remaining physical parameters. Numerical solutions are used to investigate $f(\unicode[STIX]{x1D6EC})$ for each of these situations and compare very well with experimental results in the case of an axisymmetric current propagating over a rigid horizontal boundary. Analysis of the relaxation towards self-similarity provides an illuminating connection between the excess (deficit) volume from early times and an asymptotically equivalent shift in time origin, and hence a good quantitative estimate of $\unicode[STIX]{x1D70F}$. The case $\unicode[STIX]{x1D6EC}=0$ of equilibration after ceasing injection at time $T$ is a singular limit. Extensions to high-Reynolds-number currents and to the case of a constant-volume release followed by constant-flux injection are discussed briefly.

We study the two-dimensional flow and leakage of buoyant fluid injected at a constant volumetric rate into a fluid-saturated porous medium confined vertically by horizontal boundaries. The upper boundary contains a localized vertical fracture that allows fluid to leak into an open or partially confined porous layer above. The rate of leakage is modelled as proportional to the combined action of the gravitational hydrostatic head of the current below the fracture and the background pressure introduced by the injection. After the injected current reaches the fracture, leakage is initially controlled kinematically by the rate at which injected fluid flows towards the fracture. Once the rate at which buoyant fluid flows towards the fracture exceeds a critical value, the current overshoots the fracture and leakage switches to being controlled dynamically by the pressure drop across the fracture. Two long-term regimes of flow can emerge. In one, the current approaches a steady height above the lower boundary and essentially all fluid injected into the medium leaks at long times. In the other, the current accumulates to fill the entire depth of the medium below the fracture. Only a fraction of the injected fluid then leaks at long times, implying significantly greater long-term storage than has been proposed from studies of leakage from unconfined media. An understanding of the flow regimes is obtained using numerical solutions and analysis of long-term similarity solutions. The implications of our results to the geological storage of carbon dioxide is discussed.

Gravity-driven flows in the subsurface have attracted recent interest in the context of geological carbon dioxide (CO2) storage, where supercritical CO2 is captured from the flue gas of power plants and injected underground into deep saline aquifers. After injection, the CO2 will spread and migrate as a buoyant gravity current relative to the denser, ambient brine. Although the CO2 and the brine are immiscible, the impact of capillarity on CO2 spreading and migration is poorly understood. We previously studied the early time evolution of an immiscible gravity current, showing that capillary pressure hysteresis pins a portion of the macroscopic fluid-fluid interface and that this can eventually stop the flow. Here we study the full lifetime of such a gravity current. Using tabletop experiments in packings of glass beads, we show that the horizontal extent of the pinned region grows with time and that this is ultimately responsible for limiting the migration of the current to a finite distance. We also find that capillarity blunts the leading edge of the current, which contributes to further limiting the migration distance. Using experiments in etched micromodels, we show that the thickness of the blunted nose is controlled by the distribution of pore-throat sizes and the strength of capillarity relative to buoyancy. We develop a theoretical model that captures the evolution of immiscible gravity currents and predicts the maximum migration distance. By applying this model to representative aquifers, we show that capillary pinning and blunting can exert an important control on gravity currents in the context of geological CO2 storage.

We present a theoretical and experimental study of viscous flows injected into a porous medium that is confined vertically by horizontal impermeable boundaries and filled with an ambient fluid of different density and viscosity. General three-dimensional equations describing such flows are developed, showing that the dynamics can be affected by two separate contributions: spreading due to gradients in hydrostatic pressure, and that due to the pressure drop introduced by the injection. In the illustrative case of a two-dimensional injection of fluid at a constant volumetric rate, the injected fluid initially forms a viscous gravity current insensitive both to the depth of the medium and to the viscosity of the ambient fluid. Beyond a characteristic time scale, the dynamics transition to being dominated by the injection pressure, and the injected fluid eventually intersects the second boundary to form a second moving contact line. Three different late-time asymptotic regimes can emerge, depending on whether the viscosity of the injected fluid is less than, equal to or greater than that of the ambient fluid. With a less viscous injection, the flow undergoes a slow decay towards a similarity solution in which the two contact lines extend linearly in time with differing prefactors. Perturbations from this long-term state are shown to decay algebraically with time. Equal viscosities result in both contact lines approaching the same leading-order asymptotic position but with a first-order correction to the distance between them that expands as $t^{1/2}$ due to gravitational spreading. For a more viscous injection, the distance between the contact lines approaches a constant value, with perturbations decaying exponentially. Data from a new series of laboratory experiments confirm these theoretical predictions.

We present a theoretical and experimental study of the propagation of gravity currents in porous media with variations in the topography over which they flow, motivated in part by the sequestration of carbon dioxide in saline aquifers. We consider cases where the height of the topography slopes upwards in the direction of the flow and is proportional to the$n\text{th} $power of the horizontal distance from a line or point source of a constant volumetric flux. In two-dimensional cases with$n\gt 1/ 2$, the current evolves from a self-similar form at early times, when the effects of variations in topography are negligible, towards a late-time regime that has an approximately horizontal upper surface and whose evolution is dictated entirely by the geometry of the topography. For$n\lt 1/ 2$, the transition between these flow regimes is reversed. We compare our theoretical results in the case$n= 1$with data from a series of laboratory experiments in which viscous glycerine is injected into an inclined Hele-Shaw cell, obtaining good agreement between the theoretical results and the experimental data. In the case of axisymmetric topography, all topographic exponents$n\gt 0$result in a transition from an early-time similarity solution towards a topographically controlled regime that has an approximately horizontal free surface. We also analyse the evolution over topography that can vary with different curvatures and topographic exponents between the two horizontal dimensions, finding that the flow transitions towards a horizontally topped regime at a rate which depends strongly on the ratio of the curvatures along the principle axes. Finally, we apply our mathematical solutions to the geophysical setting at the Sleipner field, concluding that topographic influence is unlikely to explain the observed non-axisymmetric flow.

The separation of volcanic ash and sulphur dioxide (SO2) gas is sometimes observed during volcanic eruptions. The exact conditions under which separation occurs are not fully understood but the phenomenon is of importance because of the effects volcanic emissions have on aviation, on the environment and to the earth's radiation balance. The eruption of Grimsvotn, a subglacial volcano under the Vatnajokull glacier in Iceland during 21–28 May 2011 produced one of the most spectacular examples of ash and SO2 separation that led to errors in the forecasting of ash in the atmosphere over northern Europe. Satellite data from several sources coupled with meteorological wind data and photographic evidence suggest that the eruption column was unable to sustain itself, resulting in a large deposition of ash which left a low level ash-rich atmospheric plume moving southwards and then eastwards towards the southern Scandanavian coast, and a high level predominantly SO2 plume travelling northwards and then spreading eastwards and westwards. Here we provide observational and modelling perspectives on the separation of ash and SO2 and present quantitative estimates of the masses of ash and SO2 erupted, the directions of transport, and the likely impacts. We hypothesise that a partial column collapse or perhaps several occurred during the early stage of the eruption leading to an ash-laden gravity intrusion that was swept southwards, separated from the main column. Our model suggests that water-mediated aggregation caused enhanced ash removal because of the plentiful supply of source water from exsolved magmatic water, from melted glacial ice and from entrained atmospheric water. The analysis also suggests that ash and SO2 should be treated with separate source terms, leading to improvements in forecasting the movement of both types of emissions.

River water flowing as an interflow was investigated using field data, collected in Lake Iseo (Italy), and theory. A theory for the lateral falling mechanism of plunging was developed for inflows when the initial densimetric Froude number (Fr0) is slightly larger than unity. The ratio of the river width to the offshore extent of the plunge region was equal to Fr0. The mixing ratio in the plunge region was 0.06. Theoretical results were quantitatively consistent with the length scale and mixing ratio of the observed plunge region. The progression of the inflow was interpreted as: initially a laterally falling plunge region with little mixing; followed by a steep underflow region with substantial mixing; and finally an intrusion. The intrusion was at first controlled dynamically by an inertia-buoyancy force balance. Further from the liftoff point, turbulent mixing effects dominated over those due to inertia. Ultimately, the intrusion diffused into the adjacent layers in such a way that the interflow fluid was effectively indistinguishable from the lake water.

Cracks filled with fluid propagation when the pressurized fluid is injected into the crack. Subsequently, when the fluid inlet is exposed to a lower pressure, the fluid flows backwards (backflow) and the crack closes due to the elastic relaxation of the solid. Here we study the dynamics of the crack closure during the backflow. We find that the crack radius remains constant and the fluid volume in the crack decreases with time in a power-law manner at late times. The balance between the viscous stresses in the fluid and elastic stresses in the fluid and the elastic stresses in the solid yields a scaling law that agrees with the experimental results for different fluid viscosities, Young's moduli of the solid, and initial radii of the cracks. Furthermore, we visualize the time-dependent crack shapes, and the convergence to a universal dimensionless shape demonstrates the self-similarity of the crack shapes during the backflow process.

We investigate two-dimensional liquid bridges trapped between pairs of identical horizontal cylinders. The cylinders support forces due to surface tension and hydrostatic pressure which balance the weight of the liquid. The shape of the liquid bridge is determined by analytically solving the nonlinear Laplace-Young equation. Parameters that maximize the trapping capacity (defined as the cross-sectional area of the liquid bridge) are then determined. The results show that these parameters can be approximated with simple relationships when the radius of the cylinders is small compared to the capillary length. For such small cylinders, liquid bridges with the largest cross sectional area occur when the centre-to-centre distance between the cylinders is approximately twice the capillary length. The maximum trapping capacity for a pair of cylinders at a given separation is linearly related to the separation when it is small compared to the capillary length. The meniscus slope angle of the largest liquid bridge produced in this regime is also a linear function of the separation. We additionally derive approximate solutions for the profile of a liquid bridge making use of the linearized Laplace-Young equation. These solutions analytically verify the above relationships obtained for the maximization of the trapping capacity., Comment: 14 pages, 7 figures, submitted to Proceedings of the Royal Society A: Mathematical, Physical & Engineering Sciences

We consider the two-dimensional buoyancy driven flow of a fluid injected into a saturated semi-infinite porous medium bounded by a horizontal barrier in which a single line sink, representing a fissure some distance from the point of injection, allows leakage of buoyant fluid. Our studies are motivated by the geological sequestration of carbon dioxide (CO2) and the possibility that fissures in the cap rock may compromise the safe long-term storage of CO2. A theoretical model is presented that accounts for leakage through the fissure using two parameters, which characterize leakage driven both by the hydrostatic pressure within the overriding fluid and by the buoyancy of the fluid within the fissure. We determine numerical solutions for the evolution of both the gravity current within the porous medium and the volume of fluid that has escaped through the fissure as a function of time. A quantity of considerable practical interest is the efficiency of storage, which we define as the amount of fluid remaining in the porous medium relative to the amount injected. This efficiency scales like t−1/2 at late times, indicating that the efficiency of storage ultimately tends to zero. We confirm the results of our model by comparison with an analogue laboratory experiment and discuss the implications of our two-dimensional model of leakage from a fissure for the geological sequestration of CO2.

Laboratory experiments have shown that the steady flow of granular material down a rough inclined plane has a surface that is not parallel to the plane, but has a curvature across the slope with the height increasing toward the middle of the flow. We study this observation by postulating a new granular rheology, similar to that of a second order fluid. This model is applied to the experiments using a shallow water approximation, given that the depth of the flow is much smaller than the width. The model predicts that a second normal stress difference allows cross-slope height variations to develop in regions with considerable cross-slope velocity shear, consistent with the experiments. The model also predicts the development of lateral eddies, which are yet to be observed.