Stokes at 200 (part 2).

We present the second half of the papers from the Stokes200 symposium celebrating the bicentenary of George Gabriel Stokes. This article is part of the theme issue 'Stokes at 200 (part 2)'.

Stokes drift in coral reefs with depth-varying permeability.

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)'.

Shallow free-surface Stokes flow around a corner.

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)'.

Stokes at 200: a celebration of the remarkable achievements of Sir George Gabriel Stokes two hundred years after his birth.

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 Stokes200 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 Stokes200 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)'.

Stokes, Tyndall, Ruskin and the nineteenth-century beginnings of climate science.

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)'.

Symmetric coalescence of two hydraulic fractures.

The formation of a fracture network is a key process for many geophysical and industrial practices from energy resource recovery to induced seismic management. We focus on the initial stage of a fracture network formation using experiments on the symmetric coalescence of two equal coplanar, fluid-driven, penny-shaped fractures in a brittle elastic medium. Initially, the fractures propagate independently of each other. The fractures then begin to interact and coalesce, forming a bridge between them. Within an intermediate period after the initial contact, most of the fracture growth is localized along this bridge, perpendicular to the line connecting the injection sources. Using light attenuation and particle image velocimetry to measure both the fracture aperture and velocity field, we characterize the growth of this bridge. We model this behavior using a geometric volume conservation argument dependent on the symmetry of the interaction, with a 2D approximation for the bridge. We also verify experimentally the scaling for the bridge growth and the shape of the thickness profile along the bridge. The influence of elasticity and toughness of the solid, injection rate of the fluid, and initial location of the fractures are captured by our scaling.

Maximal liquid bridges between horizontal cylinders.

We investigate two-dimensional liquid bridges trapped between pairs of identical horizontal cylinders. The cylinders support forces owing to surface tension and hydrostatic pressure that 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 with 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 with 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, using the linearized Laplace-Young equation. These solutions analytically verify the above-mentioned relationships obtained for the maximization of the trapping capacity.

Elastic Relaxation of Fluid-Driven Cracks and the Resulting Backflow.

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.

Interface pinning of immiscible gravity-exchange flows in porous media.

We study the gravity-exchange flow of two immiscible fluids in a porous medium and show that, in contrast with the miscible case, a portion of the initial interface remains pinned at all times. We elucidate, by means of micromodel experiments, the pore-level mechanism responsible for capillary pinning at the macroscale. We propose a sharp-interface gravity-current model that incorporates capillarity and quantitatively explains the experimental observations, including the x~t(1/2) spreading behavior at intermediate times and the fact that capillarity stops a finite-release current. Our theory and experiments suggest that capillary pinning is potentially an important, yet unexplored, trapping mechanism during CO(2) sequestration in deep saline aquifers.

Pouring viscous fluid out of a tipped container in minimal time.

Emptying a container partially filled with viscous fluid can be a frustratingly slow process. It takes time for the fluid to even begin discharging after tipping the container to develop a draining film on the interior surface. To study the effects of the shape and the tipping angle of the container, we predict the time required for the fluid to begin discharging in two simple geometries. In addition, the volume of the fluid yet to be discharged at subsequent times is predicted to decrease as t(-1/2) or t(-1) for flow driven along a plane or corner, respectively. We compare these theoretical predictions with laboratory experiments and discuss how viscous fluids could be poured out most effectively.

Shallow granular flows.

Many processes in geophysical and industrial settings involve the flow of granular materials down a slope. In order to investigate the granular dynamics, we report a series of laboratory experiments conducted by releasing grains at a steady rate from a localized source on a rough inclined plane. Different types of dense granular flow are observed by varying the flow rate at the source and the slope of the inclined plane. The two cases of steady flow confined by levees and the flow of avalanches down the plane are examined. The width of the steady flow increases linearly with the prescribed flow rate, which does not appreciably affect the characteristic depth or surface velocity of the bulk flow. When the flow rate is just below that required for sustaining the steady flow, avalanches are triggered at regular intervals. The avalanches maintain their shape, size, and speed down the inclined plane. We propose a simple model of steady flow that is consistent with our observations and discuss the challenges associated with the theoretical treatment of avalanche dynamics.

Extreme natural hazards: population growth, globalization and environmental change.

Mankind is becoming ever more susceptible to natural disasters, largely as a consequence of population growth and globalization. It is likely that in the future, we will experience several disasters per year that kill more than 10,000 people. A calamity with a million casualties is just a matter of time. This situation is mainly a consequence of increased vulnerability. Climate change may also be affecting the frequency of extreme weather events as well as the vulnerability of coastal areas due to sea-level rise. Disastrous outcomes can only increase unless better ways are found to mitigate the effects through improved forecasting and warning, together with more community preparedness and resilience. There are particular difficulties with extreme events, which can affect several countries, while the largest events can have global consequences. The hazards of supervolcanic eruptions and asteroid impacts could cause global disaster with threats to civilization and deaths of billions of people. Although these are very rare events, they will happen and require consideration. More frequent and smaller events in the wrong place at the wrong time could have very large human, environmental and economic effects. A sustained effort is needed to identify places at risk and take steps to apply science before the events occur.

Collapses of two-dimensional granular columns.

The first detailed quantitative observations of the two-dimensional collapse of a granular column along a horizontal channel are presented for a variety of materials. Together with the complementary study for the axisymmetric situation, we conclude that for granular collapses the generally accepted approaches, that are highly dependent on frictional parameters, do not describe the main flow phenomena. The motion divides in two main flow regimes at a approximately 1.8, where the aspect ratio a = hi/di and hi and di are the initial height and width of the column. We describe the details of collapse by emphasizing the sequential occurrence of a main spreading followed by a final avalanching phase. For the low a regime, a < 1.8, we derive descriptions of the final geometry by direct physical arguments. For the large a regime, a > 1.8, we determine that nearly all details of the collapse, including the position of the flow front as a function of time, the emplacement time, the self-similar final profiles, and especially their maximum vertical and horizontal extension, are established during the spreading phase and can be expressed in terms of the initial geometrical parameters but are independent of basal and internal friction parameters.

The role of volatiles in magma chamber dynamics.

Many andesitic volcanoes exhibit effusive eruption activity, with magma volumes as large as 10(7)-10(9) m(3) erupted at rates of 1-10 m(3) x s(-1) over periods of years or decades. During such eruptions, many complex cycles in eruption rates have been observed, with periods ranging from hours to years. Longer-term trends have also been observed, and are thought to be associated with the continuing recharge of magma from deep in the crust and with waning of overpressure in the magma reservoir. Here we present a model which incorporates effects due to compressibility of gas in magma. We show that the eruption duration and volume of erupted magma may increase by up to two orders of magnitude if the stored internal energy associated with dissolved volatiles can be released into the magma chamber. This mechanism would be favoured in shallow chambers or volatile-rich magmas and the cooling of magma by country rock may enhance this release of energy, leading to substantial increases in eruption rate and duration.