Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences Research articles
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Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences RSS feed -- recent Research articles articles1471-2946Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences1364-5021<![CDATA[Buckling of a stiff thin film on an elastic graded compliant substrate]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2208/20170410?rss=1
The buckling of a stiff film on a compliant substrate has attracted much attention due to its wide applications such as thin-film metrology, surface patterning and stretchable electronics. An analytical model is established for the buckling of a stiff thin film on a semi-infinite elastic graded compliant substrate subjected to in-plane compression. The critical compressive strain and buckling wavelength for the sinusoidal mode are obtained analytically for the case with the substrate modulus decaying exponentially. The rigorous finite element analysis (FEA) is performed to validate the analytical model and investigate the postbuckling behaviour of the system. The critical buckling strain for the period-doubling mode is obtained numerically. The influences of various material parameters on the results are investigated. These results are helpful to provide physical insights on the buckling of elastic graded substrate-supported thin film.
]]>2017-12-13T00:05:16-08:00info:doi/10.1098/rspa.2017.0410hwp:master-id:royprsa;rspa.2017.04102017-12-13Research articles47322082017041020170410<![CDATA[On modelling the interaction between two rotating bodies with statistically distributed features: an application to dressing of grinding wheels]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2208/20170466?rss=1
The mechanisms of interaction between bodies with statistically arranged features present characteristics common to different abrasive processes, such as dressing of abrasive tools. In contrast with the current empirical approach used to estimate the results of operations based on attritive interactions, the method we present in this paper allows us to predict the output forces and the topography of a simulated grinding wheel for a set of specific operational parameters (speed ratio and radial feed-rate), providing a thorough understanding of the complex mechanisms regulating these processes. In modelling the dressing mechanisms, the abrasive characteristics of both bodies (grain size, geometry, inter-space and protrusion) are first simulated; thus, their interaction is simulated in terms of grain collisions. Exploiting a specifically designed contact/impact evaluation algorithm, the model simulates the collisional effects of the dresser abrasives on the grinding wheel topography (grain fracture/break-out). The method has been tested for the case of a diamond rotary dresser, predicting output forces within less than 10% error and obtaining experimentally validated grinding wheel topographies. The study provides a fundamental understanding of the dressing operation, enabling the improvement of its performance in an industrial scenario, while being of general interest in modelling collision-based processes involving statistically distributed elements.
]]>2017-12-06T00:05:14-08:00info:doi/10.1098/rspa.2017.0466hwp:master-id:royprsa;rspa.2017.04662017-12-06Research articles47322082017046620170466<![CDATA[The complex and quaternionic quantum bit from relativity of simultaneity on an interferometer]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2208/20170596?rss=1
The patterns of fringes produced by an interferometer have long been important testbeds for our best contemporary theories of physics. Historically, interference has been used to contrast quantum mechanics with classical physics, but recently experiments have been performed that test quantum theory against even more exotic alternatives. A physically motivated family of theories are those where the state space of a two-level system is given by a sphere of arbitrary dimension. This includes classical bits, and real, complex and quaternionic quantum theory. In this paper, we consider relativity of simultaneity (i.e. that observers may disagree about the order of events at different locations) as applied to a two-armed interferometer, and show that this forbids most interference phenomena more complicated than those of complex quantum theory. If interference must depend on some relational property of the setting (such as path difference), then relativity of simultaneity will limit state spaces to standard complex quantum theory, or a subspace thereof. If this relational assumption is relaxed, we find one additional theory compatible with relativity of simultaneity: quaternionic quantum theory. Our results have consequences for current laboratory interference experiments: they have to be designed carefully to avoid rendering beyond-quantum effects invisible by relativity of simultaneity.
]]>2017-12-06T00:05:18-08:00info:doi/10.1098/rspa.2017.0596hwp:master-id:royprsa;rspa.2017.05962017-12-06Research articles47322082017059620170596<![CDATA[On the characterization of vortex configurations in the steady rotating Bose-Einstein condensates]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2208/20170602?rss=1
Motivated by experiments in atomic Bose–Einstein condensates (BECs), we compare predictions of a system of ordinary differential equations (ODEs) for dynamics of one and two individual vortices in the rotating BECs with those of the Gross–Pitaevskii mean-field model written as a partial differential equation (PDE). In particular, we characterize orbitally stable vortex configurations in a symmetric harmonic trap due to a cubic repulsive interaction and a steady rotation. The ODE system is analysed in detail and the PDE model is approximated numerically. Good agreement between the two models is established in the semi-classical (Thomas–Fermi) limit that corresponds to the BECs at large values of the chemical potential.
]]>2017-12-06T00:05:13-08:00info:doi/10.1098/rspa.2017.0602hwp:master-id:royprsa;rspa.2017.06022017-12-06Research articles47322082017060220170602<![CDATA[On stochastic differential equations with arbitrarily slow convergence rates for strong approximation in two space dimensions]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2207/20170104?rss=1
In a recent article (Jentzen et al. 2016 Commun. Math. Sci.14, 1477–1500 (doi:10.4310/CMS.2016.v14.n6.a1)), it has been established that, for every arbitrarily slow convergence speed and every natural number d{4,5,...}, there exist d-dimensional stochastic differential equations with infinitely often differentiable and globally bounded coefficients such that no approximation method based on finitely many observations of the driving Brownian motion can converge in absolute mean to the solution faster than the given speed of convergence. In this paper, we strengthen the above result by proving that this slow convergence phenomenon also arises in two (d=2) and three (d=3) space dimensions.
]]>2017-11-29T00:05:34-08:00info:doi/10.1098/rspa.2017.0104hwp:master-id:royprsa;rspa.2017.01042017-11-29Research articles47322072017010420170104<![CDATA[Rolling resistance of shallow granular deformation]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2207/20170375?rss=1
Experiments are conducted to measure the resistance experienced by light cylinders rolling over flat beds of granular media. Sand and glass spheres are used for the beds. The trajectories of the rolling cylinders are determined through optical tracking, and velocity and acceleration data are inferred through fits to these trajectories. The rolling resistance is dominated by a velocity-independent component, but a velocity-dependent drag exceeding the expected strength of air drag is also observed. The results are compared to a theoretical model based on a cohesionless Mohr–Coulomb rheology for a granular medium in the presence of gravity. The model idealizes the flow pattern underneath the rolling cylinder as a plastically deforming zone in front of a rigidly rotating plug attached to the cylinder, as proposed previously for cylinders rolling on perfectly cohesive plastic media. The leading-order, rate-independent rolling resistance observed experimentally is well reproduced by the model predictions.
]]>2017-11-29T00:05:34-08:00info:doi/10.1098/rspa.2017.0375hwp:master-id:royprsa;rspa.2017.03752017-11-29Research articles47322072017037520170375<![CDATA[Local fields and effective conductivity tensor of ellipsoidal particle composite with anisotropic constituents]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2207/20170472?rss=1
An accurate semi-analytical solution of the conductivity problem for a composite with anisotropic matrix and arbitrarily oriented anisotropic ellipsoidal inhomogeneities has been obtained. The developed approach combines the superposition principle with the multipole expansion of perturbation fields of inhomogeneities in terms of ellipsoidal harmonics and reduces the boundary value problem to an infinite system of linear algebraic equations for the induced multipole moments of inhomogeneities. A complete full-field solution is obtained for the multi-particle models comprising inhomogeneities of diverse shape, size, orientation and properties which enables an adequate account for the microstructure parameters. The solution is valid for the general-type anisotropy of constituents and arbitrary orientation of the orthotropy axes. The effective conductivity tensor of the particulate composite with anisotropic constituents is evaluated in the framework of the generalized Maxwell homogenization scheme. Application of the developed method to composites with imperfect ellipsoidal interfaces is straightforward. Their incorporation yields probably the most general model of a composite that may be considered in the framework of analytical approach.
]]>2017-11-29T00:05:34-08:00info:doi/10.1098/rspa.2017.0472hwp:master-id:royprsa;rspa.2017.04722017-11-29Research articles47322072017047220170472<![CDATA[An instability of the standard model of cosmology creates the anomalous acceleration without dark energy]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2207/20160887?rss=1
We identify the condition for smoothness at the centre of spherically symmetric solutions of Einstein’s original equations without the cosmological constant or dark energy. We use this to derive a universal phase portrait which describes general, smooth, spherically symmetric solutions near the centre of symmetry when the pressure p=0. In this phase portrait, the critical k=0 Friedmann space–time appears as a saddle rest point which is unstable to spherical perturbations. This raises the question as to whether the Friedmann space–time is observable by redshift versus luminosity measurements looking outwards from any point. The unstable manifold of the saddle rest point corresponding to Friedmann describes the evolution of local uniformly expanding space–times whose accelerations closely mimic the effects of dark energy. A unique simple wave perturbation from the radiation epoch is shown to trigger the instability, match the accelerations of dark energy up to second order and distinguish the theory from dark energy at third order. In this sense, anomalous accelerations are not only consistent with Einstein’s original theory of general relativity, but are a prediction of it without the cosmological constant or dark energy.
]]>2017-11-22T00:05:31-08:00info:doi/10.1098/rspa.2016.0887hwp:master-id:royprsa;rspa.2016.08872017-11-22Research articles47322072016088720160887<![CDATA[Fluid flow in porous media using image-based modelling to parametrize Richards' equation]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2207/20170178?rss=1
The parameters in Richards' equation are usually calculated from experimentally measured values of the soil–water characteristic curve and saturated hydraulic conductivity. The complex pore structures that often occur in porous media complicate such parametrization due to hysteresis between wetting and drying and the effects of tortuosity. Rather than estimate the parameters in Richards' equation from these indirect measurements, image-based modelling is used to investigate the relationship between the pore structure and the parameters. A three-dimensional, X-ray computed tomography image stack of a soil sample with voxel resolution of 6 μm has been used to create a computational mesh. The Cahn–Hilliard–Stokes equations for two-fluid flow, in this case water and air, were applied to this mesh and solved using the finite-element method in COMSOL Multiphysics. The upscaled parameters in Richards' equation are then obtained via homogenization. The effect on the soil–water retention curve due to three different contact angles, 0°, 20° and 60°, was also investigated. The results show that the pore structure affects the properties of the flow on the large scale, and different contact angles can change the parameters for Richards' equation.
]]>2017-11-22T00:05:35-08:00info:doi/10.1098/rspa.2017.0178hwp:master-id:royprsa;rspa.2017.01782017-11-22Research articles47322072017017820170178<![CDATA[Geometric description of a discrete power function associated with the sixth Painleve equation]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2207/20170312?rss=1
In this paper, we consider the discrete power function associated with the sixth Painlevé equation. This function is a special solution of the so-called cross-ratio equation with a similarity constraint. We show in this paper that this system is embedded in a cubic lattice with W~(3A1(1)) symmetry. By constructing the action of W~(3A1(1)) as a subgroup of W~(D4(1)), i.e. the symmetry group of P_{VI}, we show how to relate W~(D4(1)) to the symmetry group of the lattice. Moreover, by using translations in W~(3A1(1)), we explain the odd–even structure appearing in previously known explicit formulae in terms of the function.
]]>2017-11-22T00:05:35-08:00info:doi/10.1098/rspa.2017.0312hwp:master-id:royprsa;rspa.2017.03122017-11-22Research articles47322072017031220170312<![CDATA[The grasshopper problem]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2207/20170494?rss=1
We introduce and physically motivate the following problem in geometric combinatorics, originally inspired by analysing Bell inequalities. A grasshopper lands at a random point on a planar lawn of area 1. It then jumps once, a fixed distance d, in a random direction. What shape should the lawn be to maximize the chance that the grasshopper remains on the lawn after jumping? We show that, perhaps surprisingly, a disc-shaped lawn is not optimal for any d>0. We investigate further by introducing a spin model whose ground state corresponds to the solution of a discrete version of the grasshopper problem. Simulated annealing and parallel tempering searches are consistent with the hypothesis that, for d<^{–1/2}, the optimal lawn resembles a cogwheel with n cogs, where the integer n is close to (arcsin(d/2))–1. We find transitions to other shapes for d–1/2.
]]>2017-11-22T00:05:34-08:00info:doi/10.1098/rspa.2017.0494hwp:master-id:royprsa;rspa.2017.04942017-11-22Research articles47322072017049420170494<![CDATA[Theory of energy harvesting from heartbeat including the effects of pleural cavity and respiration]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2207/20170615?rss=1
Self-powered implantable devices with flexible energy harvesters are of significant interest due to their potential to solve the problem of limited battery life and surgical replacement. The flexible electronic devices made of piezoelectric materials have been employed to harvest energy from the motion of biological organs. Experimental measurements show that the output voltage of the device mounted on porcine left ventricle in chest closed environment decreases significantly compared to the case of chest open. A restricted-space deformation model is proposed to predict the impeding effect of pleural cavity, surrounding tissues, as well as respiration on the efficiency of energy harvesting from heartbeat using flexible piezoelectric devices. The analytical solution is verified by comparing theoretical predictions to experimental measurements. A simple scaling law is established to analyse the intrinsic correlations between the normalized output power and the combined system parameters, i.e. the normalized permitted space and normalized electrical load. The results may provide guidelines for optimization of in vivo energy harvesting from heartbeat or the motions of other biological organs using flexible piezoelectric energy harvesters.
]]>2017-11-22T00:05:34-08:00info:doi/10.1098/rspa.2017.0615hwp:master-id:royprsa;rspa.2017.06152017-11-22Research articles47322072017061520170615<![CDATA[On the buckling of an elastic holey column]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2207/20170477?rss=1
We report the results of a numerical and theoretical study of buckling in elastic columns containing a line of holes. Buckling is a common failure mode of elastic columns under compression, found over scales ranging from metres in buildings and aircraft to tens of nanometers in DNA. This failure usually occurs through lateral buckling, described for slender columns by Euler’s theory. When the column is perforated with a regular line of holes, a new buckling mode arises, in which adjacent holes collapse in orthogonal directions. In this paper, we firstly elucidate how this alternate hole buckling mode coexists and interacts with classical Euler buckling modes, using finite-element numerical calculations with bifurcation tracking. We show how the preferred buckling mode is selected by the geometry, and discuss the roles of localized (hole-scale) and global (column-scale) buckling. Secondly, we develop a novel predictive model for the buckling of columns perforated with large holes. This model is derived without arbitrary fitting parameters, and quantitatively predicts the critical strain for buckling. We extend the model to sheets perforated with a regular array of circular holes and use it to provide quantitative predictions of their buckling.
]]>2017-11-15T00:05:25-08:00info:doi/10.1098/rspa.2017.0477hwp:master-id:royprsa;rspa.2017.04772017-11-15Research articles47322072017047720170477<![CDATA[On the analysis of the double Hopf bifurcation in machining processes via centre manifold reduction]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2207/20170502?rss=1
The single-degree-of-freedom model of orthogonal cutting is investigated to study machine tool vibrations in the vicinity of a double Hopf bifurcation point. Centre manifold reduction and normal form calculations are performed to investigate the long-term dynamics of the cutting process. The normal form of the four-dimensional centre subsystem is derived analytically, and the possible topologies in the infinite-dimensional phase space of the system are revealed. It is shown that bistable parameter regions exist where unstable periodic and, in certain cases, unstable quasi-periodic motions coexist with the equilibrium. Taking into account the non-smoothness caused by loss of contact between the tool and the workpiece, the boundary of the bistable region is also derived analytically. The results are verified by numerical continuation. The possibility of (transient) chaotic motions in the global non-smooth dynamics is shown.
]]>2017-11-15T00:05:25-08:00info:doi/10.1098/rspa.2017.0502hwp:master-id:royprsa;rspa.2017.05022017-11-15Research articles47322072017050220170502<![CDATA[Advection and Taylor-Aris dispersion in rivulet flow]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2207/20170524?rss=1
Motivated by the need for a better understanding of the transport of solutes in microfluidic flows with free surfaces, the advection and dispersion of a passive solute in steady unidirectional flow of a thin uniform rivulet on an inclined planar substrate driven by gravity and/or a uniform longitudinal surface shear stress are analysed. Firstly, we describe the short-time advection of both an initially semi-infinite and an initially finite slug of solute of uniform concentration. Secondly, we describe the long-time Taylor–Aris dispersion of an initially finite slug of solute. In particular, we obtain the general expression for the effective diffusivity for Taylor–Aris dispersion in such a rivulet, and discuss in detail its different interpretations in the special case of a rivulet on a vertical substrate.
]]>2017-11-15T00:05:27-08:00info:doi/10.1098/rspa.2017.0524hwp:master-id:royprsa;rspa.2017.05242017-11-15Research articles47322072017052420170524<![CDATA[Lagrangian averaging with geodesic mean]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2207/20170558?rss=1
This paper revisits the derivation of the Lagrangian averaged Euler (LAE), or Euler-α equations in the light of an intrinsic definition of the averaged flow map as the geodesic mean on the volume-preserving diffeomorphism group. Under the additional assumption that first-order fluctuations are statistically isotropic and transported by the mean flow as a vector field, averaging of the kinetic energy Lagrangian of an ideal fluid yields the LAE Lagrangian. The derivation presented here assumes a Euclidean spatial domain without boundaries.
]]>2017-11-15T00:43:18-08:00info:doi/10.1098/rspa.2017.0558hwp:master-id:royprsa;rspa.2017.05582017-11-15Research articles47322072017055820170558<![CDATA[Continuum modelling of pantographic sheets for out-of-plane bifurcation and vibrational analysis]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2207/20170636?rss=1
A nonlinear two-dimensional (2D) continuum with a latent internal structure is introduced as a coarse model of a plane network of beams which, in turn, is assumed as a model of a pantographic structure made up by two families of equispaced beams, superimposed and connected by pivots. The deformation measures of the beams of the network and that of the 2D body are introduced and the former are expressed in terms of the latter by making some kinematical assumptions. The expressions for the strain and kinetic energy densities of the network are then introduced and given in terms of the kinematic quantities of the 2D continuum. To account for the modelling abilities of the 2D continuum in the linear range, the eigenmode and eigenfrequencies of a given specimen are determined. The buckling and post-buckling behaviour of the same specimen, subjected to two different loading conditions are analysed as tests in the nonlinear range. The problems have been solved numerically by means of the COMSOL Multiphysics finite element software.
]]>2017-11-15T03:22:36-08:00info:doi/10.1098/rspa.2017.0636hwp:master-id:royprsa;rspa.2017.06362017-11-15Research articles47322072017063620170636<![CDATA[Ducks in space: from nonlinear absolute instability to noise-sustained structures in a pattern-forming system]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2207/20170018?rss=1
A subcritical pattern-forming system with nonlinear advection in a bounded domain is recast as a slow–fast system in space and studied using a combination of geometric singular perturbation theory and numerical continuation. Two types of solutions describing the possible location of stationary fronts are identified, whose origin is traced to the onset of convective and absolute instability when the system is unbounded. The former are present only for non-zero upstream boundary conditions and provide a quantitative understanding of noise-sustained structures in systems of this type. The latter correspond to the onset of a global mode and are present even with zero upstream boundary conditions. The role of canard trajectories in the nonlinear transition between these states is clarified and the stability properties of the resulting spatial structures are determined. Front location in the convective regime is highly sensitive to the upstream boundary condition, and its dependence on this boundary condition is studied using a combination of numerical continuation and Monte Carlo simulations of the partial differential equation. Statistical properties of the system subjected to random or stochastic boundary conditions at the inlet are interpreted using the deterministic slow–fast spatial dynamical system.
]]>2017-11-08T00:05:22-08:00info:doi/10.1098/rspa.2017.0018hwp:master-id:royprsa;rspa.2017.00182017-11-08Research articles47322072017001820170018<![CDATA[A post Gurney quantum mechanical perspective on the electrolysis of water: ion neutralization in solution]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2207/20170371?rss=1
Electron fluxes crossing the interface between a metallic conductor and an aqueous environment are important in many fields; hydrogen production, environmental scanning tunnelling microscopy, scanning electrochemical microscopy being some of them. Gurney (Gurney 1931 Proc. R. Soc. Lond.134, 137 (doi:10.1098/rspa.1931.0187)) provided in 1931 a scheme for tunnelling during electrolysis and outlined conditions for it to occur. We measure the low-voltage current flows between gold electrodes in pure water and use the time-dependent behaviour at voltage switch-on and switch-off to evaluate the relative contribution to the steady current arising from tunnelling of electrons between the electrodes and ions in solution and from the neutralization of ions adsorbed onto the electrode surface. We ascribe the larger current contribution to quantum tunnelling of electrons to and from ions in solution near the electrodes. We refine Gurney's barrier scheme to include solvated electron states and quantify energy differences using updated information. We show that Gurney's conditions would prevent the current flow at low voltages we observe but outline how the ideas of Marcus (Marcus 1956 J. Chem. Phys.24, 966–978 (doi:10.1063/1.1742723)) concerning solvation fluctuations enable the condition to be relaxed. We derive an average barrier tunnelling model and a multiple pathways tunnelling model and compare predictions with measurements of the steady-state current–voltage relation. The tunnelling barrier was found to be wide and low in agreement with other experimental studies. Applications as a biosensing mechanism are discussed that exploit the fast tunnelling pathways along molecules in solution.
]]>2017-11-08T00:05:22-08:00info:doi/10.1098/rspa.2017.0371hwp:master-id:royprsa;rspa.2017.03712017-11-08Research articles47322072017037120170371<![CDATA[Force direction patterns promote whole body stability even in hip-flexed walking, but not upper body stability in human upright walking]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2207/20170404?rss=1
Directing the ground reaction forces to a focal point above the centre of mass of the whole body promotes whole body stability in human and animal gaits similar to a physical pendulum. Here we show that this is the case in human hip-flexed walking as well. For all upper body orientations (upright, 25°, 50°, maximum), the focal point was well above the centre of mass of the whole body, suggesting its general relevance for walking. Deviations of the forces' lines of action from the focal point increased with upper body inclination from 25 to 43 mm root mean square deviation (RMSD). With respect to the upper body in upright gait, the resulting force also passed near a focal point (17 mm RMSD between the net forces' lines of action and focal point), but this point was 18 cm below its centre of mass. While this behaviour mimics an unstable inverted pendulum, it leads to resulting torques of alternating sign in accordance with periodic upper body motion and probably provides for low metabolic cost of upright gait by keeping hip torques small. Stabilization of the upper body is a consequence of other mechanisms, e.g. hip reflexes or muscle preflexes.
]]>2017-11-08T00:05:22-08:00info:doi/10.1098/rspa.2017.0404hwp:master-id:royprsa;rspa.2017.04042017-11-08Research articles47322072017040420170404<![CDATA[Symmetric rearrangement of groundwater-fed streams]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2207/20170539?rss=1
Streams shape landscapes through headward growth and lateral migration. When these streams are primarily fed by groundwater, recent work suggests that their tips advance to maximize the symmetry of the local Laplacian field associated with groundwater flow. We explore the extent to which such forcing is responsible for the lateral migration of streams by studying two features of groundwater-fed streams in Bristol, Florida: their confluence angle near junctions and their curvature. First, we find that, while streams asymptotically form a 72° angle near their tips, they simultaneously exhibit a wide 120° confluence angle within approximately 10 m of their junctions. We show that this wide angle maximizes the symmetry of the groundwater field near the junction. Second, we argue that streams migrate laterally within valleys and present a new spectral analysis method to relate planform curvature to the surrounding groundwater field. Our results suggest that streams migrate laterally in response to fluxes from the surrounding groundwater table, providing evidence of a new mechanism that complements Laplacian growth at their tips.
]]>2017-11-08T00:59:42-08:00info:doi/10.1098/rspa.2017.0539hwp:master-id:royprsa;rspa.2017.05392017-11-08Research articles47322072017053920170539<![CDATA[Three-dimensional quasi-periodic shifted Green function throughout the spectrum, including Wood anomalies]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2207/20170242?rss=1
This work, part II in a series, presents an efficient method for evaluation of wave scattering by doubly periodic diffraction gratings at or near what are commonly called ‘Wood anomaly frequencies’. At these frequencies, there is a grazing Rayleigh wave, and the quasi-periodic Green function ceases to exist. We present a modification of the Green function by adding two types of terms to its lattice sum. The first type are transversely shifted Green functions with coefficients that annihilate the growth in the original lattice sum and yield algebraic convergence. The second type are quasi-periodic plane wave solutions of the Helmholtz equation which reinstate certain necessary grazing modes without leading to blow-up at Wood anomalies. Using the new quasi-periodic Green function, we establish, for the first time, that the Dirichlet problem of scattering by a smooth doubly periodic scattering surface at a Wood frequency is uniquely solvable. We also present an efficient high-order numerical method based on this new Green function for scattering by doubly periodic surfaces at and around Wood frequencies. We believe this is the first solver able to handle Wood frequencies for doubly periodic scattering problems in three dimensions. We demonstrate the method by applying it to acoustic scattering.
]]>2017-11-01T02:09:49-07:00info:doi/10.1098/rspa.2017.0242hwp:master-id:royprsa;rspa.2017.02422017-11-01Research articles47322072017024220170242<![CDATA[Age distribution dynamics with stochastic jumps in mortality]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2207/20170451?rss=1
While deterministic age distribution models have been extensively studied and applied in various disciplines, little work has been devoted to understanding the role of stochasticity in birth and mortality terms. In this paper, we analyse a stochastic M’Kendrick–von Foerster equation in which jumps in mortality represent intense losses of population due to external events. We present explicit solutions for the probability density functions of the age distribution and the total population and for the temporal dynamics of their moments. We also derive the dynamics of the mean age of the population and its harmonic mean. The framework is then used to calculate the age distribution of salt in the soil root zone, where the accumulation of salt by atmospheric deposition is counteracted by plant uptake and by jump losses due to percolation events.
]]>2017-11-01T02:09:49-07:00info:doi/10.1098/rspa.2017.0451hwp:master-id:royprsa;rspa.2017.04512017-11-01Research articles47322072017045120170451<![CDATA[Bidispersive vertical convection]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2207/20170481?rss=1
A bidispersive porous material is one which has usual pores but additionally contains a system of micro pores. We consider a fluid-saturated bidispersive porous medium in the vertical layer x(–1/2,1/2) with gravity in the –z (downward) direction. The walls of the layer are maintained at different constant temperatures. A suitable Rayleigh number is defined and we derive a global stability threshold below which no instability may arise. We additionally show that the porous layer is stable for all Rayleigh numbers provided the initial temperature gradient is bounded in a precise sense.
]]>2017-11-01T02:09:49-07:00info:doi/10.1098/rspa.2017.0481hwp:master-id:royprsa;rspa.2017.04812017-11-01Research articles47322072017048120170481<![CDATA[Swelling-induced wrinkling in layered gel beams]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2207/20170454?rss=1
Gels are widely employed in smart mechanical devices and biomedical applications. Swelling-induced bending actuation can be obtained by means of a simple bilayer gel beam. We show that this system can also exhibit wrinkling patterns of potential interest for structural morphing and sensing. We study swelling-induced wrinkling at the extrados of a bilayer gel beam with the softer layer on top. The bent configuration at finite strain is recovered first and, starting from it, a linear perturbation analysis is performed. We delimit the zone corresponding to wrinkling modes in a parameter plane encompassing a mechanical stiffness ratio and a geometric top layer to total height ratio. Interestingly, we observe that surface instability precedes and envelopes wrinkling modes of finite wavelength. Finally, we discuss the effect of changes in stiffness and of the Flory–Huggins parameters on the size of the wrinkling domain.
]]>2017-11-01T02:09:49-07:00info:doi/10.1098/rspa.2017.0454hwp:master-id:royprsa;rspa.2017.04542017-11-01Research articles47322072017045420170454<![CDATA[Multi-scale mechanics of granular solids from grain-resolved X-ray measurements]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2207/20170491?rss=1
This work discusses an experimental technique for studying the mechanics of three-dimensional (3D) granular solids. The approach combines 3D X-ray diffraction and X-ray computed tomography to measure grain-resolved strains, kinematics and contact fabric in the bulk of a granular solid, from which continuum strains, grain stresses, interparticle forces and coarse-grained elasto-plastic moduli can be determined. We demonstrate the experimental approach and analysis of selected results on a sample of 1099 stiff, frictional grains undergoing multiple uniaxial compression cycles. We investigate the inter-particle force network, elasto-plastic moduli and associated length scales, reversibility of mechanical responses during cyclic loading, the statistics of microscopic responses and microstructure–property relationships. This work serves to highlight both the fundamental insight into granular mechanics that is furnished by combined X-ray measurements and describes future directions in the field of granular materials that can be pursued with such approaches.
]]>2017-11-01T02:09:49-07:00info:doi/10.1098/rspa.2017.0491hwp:master-id:royprsa;rspa.2017.04912017-11-01Research articles47322072017049120170491<![CDATA[Nonlinear mode interaction in equal-leg angle struts susceptible to cellular buckling]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2207/20170583?rss=1
A variational model that describes the interactive buckling of a thin-walled equal-leg angle strut under pure axial compression is presented. A formulation combining the Rayleigh–Ritz method and continuous displacement functions is used to derive a system of differential and integral equilibrium equations for the structural component. Solving the equations using numerical continuation reveals progressive cellular buckling (or snaking) arising from the nonlinear interaction between the weak-axis flexural buckling mode and the strong-axis flexural–torsional buckling mode for the first time—the resulting behaviour being highly unstable. Physical experiments conducted on 10 cold-formed steel specimens are presented and the results show good agreement with the variational model.
]]>2017-11-01T07:43:36-07:00info:doi/10.1098/rspa.2017.0583hwp:master-id:royprsa;rspa.2017.05832017-11-01Research articles47322072017058320170583<![CDATA[Critical ignition conditions in exothermically reacting systems: first-order reactions]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2206/20170145?rss=1
In this paper, the comparative analysis of the thermal explosion (TE) critical conditions on the planes temperature–conversion degree and temperature–time was conducted. It was established that the ignition criteria are almost identical only at relatively small values of Todes parameter. Otherwise, the results of critical conditions analysis on the plane temperature–conversion degree may be wrong. The asymptotic method of critical conditions calculation for the first-order reactions was proposed (taking into account the reactant consumption). The degeneration conditions of TE were determined. The calculation of critical conditions for specific first-order reaction was made. The comparison of the analytical results obtained with the results of numerical calculations and experimental data showed that they are in good agreement.
]]>2017-10-25T00:05:19-07:00info:doi/10.1098/rspa.2017.0145hwp:master-id:royprsa;rspa.2017.01452017-10-25Research articles47322062017014520170145<![CDATA[On three-dimensional misorientation spaces]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2206/20170274?rss=1
Determining the local orientation of crystals in engineering and geological materials has become routine with the advent of modern crystallographic mapping techniques. These techniques enable many thousands of orientation measurements to be made, directing attention towards how such orientation data are best studied. Here, we provide a guide to the visualization of misorientation data in three-dimensional vector spaces, reduced by crystal symmetry, to reveal crystallographic orientation relationships. Domains for all point group symmetries are presented and an analysis methodology is developed and applied to identify crystallographic relationships, indicated by clusters in the misorientation space, in examples from materials science and geology. This analysis aids the determination of active deformation mechanisms and evaluation of cluster centres and spread enables more accurate description of transformation processes supporting arguments regarding provenance.
]]>2017-10-25T00:05:19-07:00info:doi/10.1098/rspa.2017.0274hwp:master-id:royprsa;rspa.2017.02742017-10-25Research articles47322062017027420170274<![CDATA[Comparing the constructions of Goldberg, Fuller, Caspar, Klug and Coxeter, and a general approach to local symmetry-preserving operations]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2206/20170267?rss=1
The use of operations on polyhedra possibly dates back to the ancient Greeks, who were the first to describe the Archimedean solids, which can be constructed from the Platonic solids by local symmetry-preserving operations (e.g. truncation) on the solid. By contrast, the results of decorations of polyhedra, e.g. by Islamic artists and by Escher, have been interpreted as decorated polyhedra—and not as new and different polyhedra. Only by interpreting decorations as combinatorial operations does it become clear how closely these two approaches are connected. In this article, we first sketch and compare the operations of Goldberg, Fuller, Caspar & Klug and Coxeter to construct polyhedra with icosahedral symmetry, where all faces are pentagons or hexagons and all vertices have three neighbours. We point out and correct an error in Goldberg’s construction. In addition, we transform the term symmetry-preserving into an exact requirement. This goal, symmetry-preserving, could also be obtained by taking global properties into account, e.g. the symmetry group itself, so we make precise the terms local and operation. As a result, we can generalize Goldberg’s approach to a systematic one that uses chamber operations to encompass all local symmetry-preserving operations on polyhedra.
]]>2017-10-25T00:05:19-07:00info:doi/10.1098/rspa.2017.0267hwp:master-id:royprsa;rspa.2017.02672017-10-25Research articles47322062017026720170267<![CDATA[The motion of a vortex on a closed surface of constant negative curvature]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2206/20170447?rss=1
The purpose of this work is to present an algorithm to determine the motion of a single hydrodynamic vortex on a closed surface of constant curvature and of genus greater than one. The algorithm is based on a relation between the Laplace–Beltrami Green function and the heat kernel. The algorithm is used to compute the motion of a vortex on the Bolza surface. This is the first determination of the orbits of a vortex on a closed surface of genus greater than one. The numerical results show that all the 46 vortex equilibria can be explicitly computed using the symmetries of the Bolza surface. Some of these equilibria allow for the construction of the first two examples of infinite vortex crystals on the hyperbolic disc. The following theorem is proved: ‘a Weierstrass point of a hyperellitic surface of constant curvature is always a vortex equilibrium’.
]]>2017-10-25T00:05:19-07:00info:doi/10.1098/rspa.2017.0447hwp:master-id:royprsa;rspa.2017.04472017-10-25Research articles47322062017044720170447<![CDATA[A Fokker-Planck reaction model for the epitaxial growth and shape transition of quantum dots]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2206/20170308?rss=1
We construct a Fokker–Planck reaction (FPR) model to investigate the dynamics of the coupled epitaxial growth and shape transition process of an array of quantum dots (QDs). The FPR model is based on a coupled system of Fokker–Planck equations wherein the distribution of each island type is governed by its own Fokker–Planck equation for growth, with reaction terms describing the shape transitions between islands of different types including asymmetric shapes. The reaction terms for the shape transitions depend on the island size and are determined from explicit calculations of the lowest barrier pathway for each shape transition. This mean-field model enables us to consider the kinetics of asymmetric shape transitions and study the evolution of island shape distributions during the coupled growth and transition process. Asymmetric metastable shapes play a crucial role in the dynamics, with asymmetric QDs comprising up to 10% of the population, and with up to 100% of the shape transitions passing through asymmetric shapes. Moreover, we find that the characteristic multimodal distribution of pyramid/dome QD coarsening can be eliminated at sufficiently high temperature and deposition rate.
]]>2017-10-18T00:05:20-07:00info:doi/10.1098/rspa.2017.0308hwp:master-id:royprsa;rspa.2017.03082017-10-18Research articles47322062017030820170308<![CDATA[A bioinspired study on the compressive resistance of helicoidal fibre structures]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2206/20170538?rss=1
Helicoidal fibre structures are widely observed in natural materials. In this paper, an integrated experimental and analytical approach was used to investigate the compressive resistance of helicoidal fibre structures. First, helicoidal fibre-reinforced composites were created using three-dimensionally printed helicoids and polymeric matrices, including plain, ring-reinforced and helix-reinforced helicoids. Then, load–displacement curves under monotonic compression tests were collected to measure the compressive strengths of helicoidal fibre composites. Fractographic characterization was performed using an X-ray microtomographer and scanning electron microscope, through which crack propagations in helicoidal structures were illustrated. Finally, mathematical modelling was performed to reveal the essential fibre architectures in the compressive resistance of helicoidal fibre structures. This work reveals that fibre–matrix ratios, helix pitch angles and interlayer rotary angles are critical to the compressive resistance of helicoidal structures.
]]>2017-10-12T02:18:05-07:00info:doi/10.1098/rspa.2017.0538hwp:master-id:royprsa;rspa.2017.05382017-10-12Research articles47322062017053820170538<![CDATA[Indentation of a floating elastic sheet: geometry versus applied tension]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2206/20170335?rss=1
The localized loading of an elastic sheet floating on a liquid bath occurs at scales from a frog sitting on a lily pad to a volcano supported by the Earth’s tectonic plates. The load is supported by a combination of the stresses within the sheet (which may include applied tensions from, for example, surface tension) and the hydrostatic pressure in the liquid. At the same time, the sheet deforms, and may wrinkle, because of the load. We study this problem in terms of the (relatively weak) applied tension and the indentation depth. For small indentation depths, we find that the force–indentation curve is linear with a stiffness that we characterize in terms of the applied tension and bending stiffness of the sheet. At larger indentations, the force–indentation curve becomes nonlinear and the sheet is subject to a wrinkling instability. We study this wrinkling instability close to the buckling threshold and calculate both the number of wrinkles at onset and the indentation depth at onset, comparing our theoretical results with experiments. Finally, we contrast our results with those previously reported for very thin, highly bendable membranes.
]]>2017-10-11T00:28:31-07:00info:doi/10.1098/rspa.2017.0335hwp:master-id:royprsa;rspa.2017.03352017-10-11Research articles47322062017033520170335<![CDATA[Nonlinear mechanics of non-rigid origami: an efficient computational approach]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2206/20170348?rss=1
Origami-inspired designs possess attractive applications to science and engineering (e.g. deployable, self-assembling, adaptable systems). The special geometric arrangement of panels and creases gives rise to unique mechanical properties of origami, such as reconfigurability, making origami designs well suited for tunable structures. Although often being ignored, origami structures exhibit additional soft modes beyond rigid folding due to the flexibility of thin sheets that further influence their behaviour. Actual behaviour of origami structures usually involves significant geometric nonlinearity, which amplifies the influence of additional soft modes. To investigate the nonlinear mechanics of origami structures with deformable panels, we present a structural engineering approach for simulating the nonlinear response of non-rigid origami structures. In this paper, we propose a fully nonlinear, displacement-based implicit formulation for performing static/quasi-static analyses of non-rigid origami structures based on ‘bar-and-hinge’ models. The formulation itself leads to an efficient and robust numerical implementation. Agreement between real models and numerical simulations demonstrates the ability of the proposed approach to capture key features of origami behaviour.
]]>2017-10-11T00:28:31-07:00info:doi/10.1098/rspa.2017.0348hwp:master-id:royprsa;rspa.2017.03482017-10-11Research articles47322062017034820170348<![CDATA[Extended lubrication theory: improved estimates of flow in channels with variable geometry]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2206/20170234?rss=1
Lubrication theory is broadly applicable to the flow characterization of thin fluid films and the motion of particles near surfaces. We offer an extension to lubrication theory by starting with Stokes equations and considering higher-order terms in a systematic perturbation expansion to describe the fluid flow in a channel with features of a modest aspect ratio. Experimental results qualitatively confirm the higher-order analytical solutions, while numerical results are in very good agreement with the higher-order analytical results. We show that the extended lubrication theory is a robust tool for an accurate estimate of pressure drop in channels with shape changes on the order of the channel height, accounting for both smooth and sharp changes in geometry.
]]>2017-10-04T00:05:17-07:00info:doi/10.1098/rspa.2017.0234hwp:master-id:royprsa;rspa.2017.02342017-10-04Research articles47322062017023420170234<![CDATA[Modelling wave-induced sea ice break-up in the marginal ice zone]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2206/20170258?rss=1
A model of ice floe break-up under ocean wave forcing in the marginal ice zone (MIZ) is proposed to investigate how floe size distribution (FSD) evolves under repeated wave break-up events. A three-dimensional linear model of ocean wave scattering by a finite array of compliant circular ice floes is coupled to a flexural failure model, which breaks a floe into two floes provided the two-dimensional stress field satisfies a break-up criterion. A closed-feedback loop algorithm is devised, which (i) solves the wave-scattering problem for a given FSD under time-harmonic plane wave forcing, (ii) computes the stress field in all the floes, (iii) fractures the floes satisfying the break-up criterion, and (iv) generates an updated FSD, initializing the geometry for the next iteration of the loop. The FSD after 50 break-up events is unimodal and near normal, or bimodal, suggesting waves alone do not govern the power law observed in some field studies. Multiple scattering is found to enhance break-up for long waves and thin ice, but to reduce break-up for short waves and thick ice. A break-up front marches forward in the latter regime, as wave-induced fracture weakens the ice cover, allowing waves to travel deeper into the MIZ.
]]>2017-10-04T00:05:17-07:00info:doi/10.1098/rspa.2017.0258hwp:master-id:royprsa;rspa.2017.02582017-10-04Research articles47322062017025820170258<![CDATA[Kinematic dynamos in spheroidal geometries]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2206/20170432?rss=1
The kinematic dynamo problem is solved numerically for a spheroidal conducting fluid of possibly large aspect ratio with an insulating exterior. The solution method uses solenoidal representations of the magnetic field and the velocity by spheroidal toroidal and poloidal fields in a non-orthogonal coordinate system. Scaling of coordinates and fields to a spherical geometry leads to a modified form of the kinematic dynamo problem with a geometric anisotropic diffusion and an anisotropic current-free condition in the exterior, which is solved explicitly. The scaling allows the use of well-developed spherical harmonic techniques in angle. Dynamo solutions are found for three axisymmetric flows in oblate spheroids with semi-axis ratios 1≤a/c≤25. For larger aspect ratios strong magnetic fields may occur in any region of the spheroid, depending on the flow, but the external fields for all three flows are weak and concentrated near the axis or periphery of the spheroid.
]]>2017-10-04T00:05:17-07:00info:doi/10.1098/rspa.2017.0432hwp:master-id:royprsa;rspa.2017.04322017-10-04Research articles47322062017043220170432<![CDATA[The problem of camouflaging via mirror reflections]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2205/20170147?rss=1
This work is related to billiards and their applications in geometric optics. It is known that perfectly invisible bodies with mirror surface do not exist. It is, therefore, natural to search for bodies that are, in a sense, close to invisible. We introduce a visibility index of a body measuring the mean angle of deviation of incident light rays, and derive a lower estimate for this index. This estimate is a function of the body’s volume and of the minimal radius of a ball containing the body. This result is far from being final and opens a possibility for further research.
]]>2017-09-26T07:50:32-07:00info:doi/10.1098/rspa.2017.0147hwp:master-id:royprsa;rspa.2017.01472017-09-26Research articles47322052017014720170147<![CDATA[The steady aerodynamics of aerofoils with porosity gradients]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2205/20170266?rss=1
This theoretical study determines the aerodynamic loads on an aerofoil with a prescribed porosity distribution in a steady incompressible flow. A Darcy porosity condition on the aerofoil surface furnishes a Fredholm integral equation for the pressure distribution, which is solved exactly and generally as a Riemann–Hilbert problem provided that the porosity distribution is Hölder-continuous. The Hölder condition includes as a subset any continuously differentiable porosity distributions that may be of practical interest. This formal restriction on the analysis is examined by a class of differentiable porosity distributions that approach a piecewise, discontinuous function in a certain parametric limit. The Hölder-continuous solution is verified in this limit against analytical results for partially porous aerofoils in the literature. Finally, a comparison made between the new theoretical predictions and experimental measurements of SD7003 aerofoils presented in the literature. Results from this analysis may be integrated into a theoretical framework to optimize turbulence noise suppression with minimal impact to aerodynamic performance.
]]>2017-09-26T07:50:32-07:00info:doi/10.1098/rspa.2017.0266hwp:master-id:royprsa;rspa.2017.02662017-09-26Research articles47322052017026620170266<![CDATA[A priori estimation of memory effects in reduced-order models of nonlinear systems using the Mori-Zwanzig formalism]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2205/20170385?rss=1
Reduced models of nonlinear dynamical systems require closure, or the modelling of the unresolved modes. The Mori–Zwanzig procedure can be used to derive formally closed evolution equations for the resolved physics. In these equations, the unclosed terms are recast as a memory integral involving the time history of the resolved variables. While this procedure does not reduce the complexity of the original system, these equations can serve as a mathematically consistent basis to develop closures based on memory approximations. In this scenario, knowledge of the memory kernel is paramount in assessing the validity of a memory approximation. Unravelling the memory kernel requires solving the orthogonal dynamics, which is a high-dimensional partial differential equation that is intractable, in general. A method to estimate the memory kernel a priori, using full-order solution snapshots, is proposed. The key idea is to solve a pseudo orthogonal dynamics equation, which has a convenient Liouville form, instead. This ersatz arises from the assumption that the semi-group of the orthogonal dynamics is a composition operator for one observable. The method is exact for linear systems. Numerical results on the Burgers and Kuramoto–Sivashinsky equations demonstrate that the proposed technique can provide valuable information about the memory kernel.
]]>2017-09-26T07:50:32-07:00info:doi/10.1098/rspa.2017.0385hwp:master-id:royprsa;rspa.2017.03852017-09-26Research articles47322052017038520170385<![CDATA[Density of convex intersections and applications]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2205/20160919?rss=1
In this paper, we address density properties of intersections of convex sets in several function spaces. Using the concept of -convergence, it is shown in a general framework, how these density issues naturally arise from the regularization, discretization or dualization of constrained optimization problems and from perturbed variational inequalities. A variety of density results (and counterexamples) for pointwise constraints in Sobolev spaces are presented and the corresponding regularity requirements on the upper bound are identified. The results are further discussed in the context of finite-element discretizations of sets associated with convex constraints. Finally, two applications are provided, which include elasto-plasticity and image restoration problems.
]]>2017-09-20T01:05:12-07:00info:doi/10.1098/rspa.2016.0919hwp:master-id:royprsa;rspa.2016.09192017-09-20Research articles47322052016091920160919<![CDATA[On the Keller-Rubinow model for Liesegang ring formation]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2205/20170128?rss=1
We study the model of Keller & Rubinow (Keller & Rubinow 1981 J. Chem. Phys74, 5000–5007. (doi:10.1063/1.441752)) describing the formation of Liesegang rings due to Ostwald's supersaturation mechanism. Keller and Rubinow provided an approximate solution both for the growth and equilibration of the first band, and also for the formation of secondary bands, based on a presumed asymptotic limit. However, they did not provide a parametric basis for the assumptions in their solution, nor did they provide any numerical corroboration, particularly of the secondary band formation. Here, we provide a different asymptotic solution, based on a specific parametric limit, and we show that the growth and subsequent cessation of the first band can be explained. We also show that the model is unable to explain the formation of finite width secondary bands, and we confirm this result by numerical computation. We conclude that the model is not fully posed, lacking a transition variable which can describe the hysteretic switch across the nucleation threshold.
]]>2017-09-20T01:05:12-07:00info:doi/10.1098/rspa.2017.0128hwp:master-id:royprsa;rspa.2017.01282017-09-20Research articles47322052017012820170128<![CDATA[On the topology of the Lorenz system]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2205/20170374?rss=1
We present a new paradigm for three-dimensional chaos, and specifically for the Lorenz equations. The main difficulty in these equations and for a generic flow in dimension 3 is the existence of singularities. We show how to use knot theory as a way to remove the singularities. Specifically, we claim: (i) for certain parameters, the Lorenz system has an invariant one-dimensional curve, which is a trefoil knot. The knot is a union of invariant manifolds of the singular points. (ii) The flow is topologically equivalent to an Anosov flow on the complement of this curve, and moreover to a geodesic flow. (iii) When varying the parameters, the system exhibits topological phase transitions, i.e. for special parameter values, it will be topologically equivalent to an Anosov flow on a knot complement. Different knots appear for different parameter values and each knot controls the dynamics at nearby parameters.
]]>2017-09-20T00:09:30-07:00info:doi/10.1098/rspa.2017.0374hwp:master-id:royprsa;rspa.2017.03742017-09-20Research articles47322052017037420170374<![CDATA[Can a quantum state over time resemble a quantum state at a single time?]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2205/20170395?rss=1
The standard formalism of quantum theory treats space and time in fundamentally different ways. In particular, a composite system at a given time is represented by a joint state, but the formalism does not prescribe a joint state for a composite of systems at different times. If there were a way of defining such a joint state, this would potentially permit a more even-handed treatment of space and time, and would strengthen the existing analogy between quantum states and classical probability distributions. Under the assumption that the joint state over time is an operator on the tensor product of single-time Hilbert spaces, we analyse various proposals for such a joint state, including one due to Leifer and Spekkens, one due to Fitzsimons, Jones and Vedral, and another based on discrete Wigner functions. Finding various problems with each, we identify five criteria for a quantum joint state over time to satisfy if it is to play a role similar to the standard joint state for a composite system: that it is a Hermitian operator on the tensor product of the single-time Hilbert spaces; that it represents probabilistic mixing appropriately; that it has the appropriate classical limit; that it has the appropriate single-time marginals; that composing over multiple time steps is associative. We show that no construction satisfies all these requirements. If Hermiticity is dropped, then there is an essentially unique construction that satisfies the remaining four criteria.
]]>2017-09-20T00:09:30-07:00info:doi/10.1098/rspa.2017.0395hwp:master-id:royprsa;rspa.2017.03952017-09-20Research articles47322052017039520170395<![CDATA[Irreversible particle motion in surfactant-laden interfaces due to pressure-dependent surface viscosity]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2205/20170346?rss=1
The surface shear viscosity of an insoluble surfactant monolayer often depends strongly on its surface pressure. Here, we show that a particle moving within a bounded monolayer breaks the kinematic reversibility of low-Reynolds-number flows. The Lorentz reciprocal theorem allows such irreversibilities to be computed without solving the full nonlinear equations, giving the leading-order contribution of surface pressure-dependent surface viscosity. In particular, we show that a disc translating or rotating near an interfacial boundary experiences a force in the direction perpendicular to that boundary. In unbounded monolayers, coupled modes of motion can also lead to non-intuitive trajectories, which we illustrate using an interfacial analogue of the Magnus effect. This perturbative approach can be extended to more complex geometries, and to two-dimensional suspensions more generally.
]]>2017-09-20T01:05:12-07:00info:doi/10.1098/rspa.2017.0346hwp:master-id:royprsa;rspa.2017.03462017-09-20Research articles47322052017034620170346<![CDATA[Uncertainties in the 2004 Sumatra-Andaman source through nonlinear stochastic inversion of tsunami waves]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2205/20170353?rss=1
Numerical inversions for earthquake source parameters from tsunami wave data usually incorporate subjective elements to stabilize the search. In addition, noisy and possibly insufficient data result in instability and non-uniqueness in most deterministic inversions, which are barely acknowledged. Here, we employ the satellite altimetry data for the 2004 Sumatra–Andaman tsunami event to invert the source parameters. We also include kinematic parameters that improve the description of tsunami generation and propagation, especially near the source. Using a finite fault model that represents the extent of rupture and the geometry of the trench, we perform a new type of nonlinear joint inversion of the slips, rupture velocities and rise times with minimal a priori constraints. Despite persistently good waveform fits, large uncertainties in the joint parameter distribution constitute a remarkable feature of the inversion. These uncertainties suggest that objective inversion strategies should incorporate more sophisticated physical models of seabed deformation in order to significantly improve the performance of early warning systems.
]]>2017-09-20T01:05:12-07:00info:doi/10.1098/rspa.2017.0353hwp:master-id:royprsa;rspa.2017.03532017-09-20Research articles47322052017035320170353<![CDATA[Stochastic partial differential fluid equations as a diffusive limit of deterministic Lagrangian multi-time dynamics]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2205/20170388?rss=1
In Holm (Holm 2015 Proc. R. Soc. A471, 20140963. (doi:10.1098/rspa.2014.0963)), stochastic fluid equations were derived by employing a variational principle with an assumed stochastic Lagrangian particle dynamics. Here we show that the same stochastic Lagrangian dynamics naturally arises in a multi-scale decomposition of the deterministic Lagrangian flow map into a slow large-scale mean and a rapidly fluctuating small-scale map. We employ homogenization theory to derive effective slow stochastic particle dynamics for the resolved mean part, thereby obtaining stochastic fluid partial equations in the Eulerian formulation. To justify the application of rigorous homogenization theory, we assume mildly chaotic fast small-scale dynamics, as well as a centring condition. The latter requires that the mean of the fluctuating deviations is small, when pulled back to the mean flow.
]]>2017-09-20T01:05:12-07:00info:doi/10.1098/rspa.2017.0388hwp:master-id:royprsa;rspa.2017.03882017-09-20Research articles47322052017038820170388<![CDATA[Toroidal drop under electric field: arbitrary drop-to-ambient fluid viscosity ratio]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2205/20170379?rss=1
In the absence of external forces, a liquid toroidal drop freely suspended in another fluid shrinks towards its centre. It is shown that if the two phases are slightly conducting viscous incompressible fluids with the drop-to-ambient fluid ratios of electric conductivities, dielectric constants and viscosities to be 1/R, Q and , respectively, then the toroidal drop with volume 4/3 and having major radius can become almost stationary when subjected to a uniform electric field aligned with the drop’s axis of symmetry. In this case, Q and electric capillary number Ca_{E} that defines the ratio of electric stress to surface tension, are functions of R, and and are found analytically. Those functions are relatively insensitive to , and for ≥3, they admit simple approximations, which coincide with those obtained recently for =1. Streamlines inside and outside the toroidal drop for the same R and but different are qualitatively similar.
]]>2017-09-20T01:05:12-07:00info:doi/10.1098/rspa.2017.0379hwp:master-id:royprsa;rspa.2017.03792017-09-20Research articles47322052017037920170379<![CDATA[Cautionary tales on air-quality improvement in Beijing]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2205/20170457?rss=1
The official air-quality statistic reported that Beijing had a 9.9% decline in the annual concentration of PM_{2.5} in 2016. While this statistic offered some relief for the inhabitants of the capital, we present several analyses based on Beijing's PM_{2.5} data of the past 4 years at 36 monitoring sites along with meteorological data of the past 7 years. The analyses reveal the air pollution situation in 2016 was not as rosy as the 9.9% decline would convey, and improvement if any was rather uncertain. The paper also provides an assessment on the city's PM_{2.5} situation in the past 4 years.
]]>2017-09-20T00:09:30-07:00info:doi/10.1098/rspa.2017.0457hwp:master-id:royprsa;rspa.2017.04572017-09-20Research articles47322052017045720170457<![CDATA[Elastodynamic image forces on screw dislocations in the presence of phase boundaries]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2205/20170484?rss=1
The elastodynamic image forces acting on straight screw dislocations in the presence of planar phase boundaries are derived. Two separate dislocations are studied: (i) the injected, non-moving screw dislocation and (ii) the injected (or pre-existing), generally non-uniformly moving screw dislocation. The image forces are derived for both the case of a rigid surface and of a planar interface between two homogeneous, isotropic phases. The case of a rigid interface is shown to be solvable employing Head's image dislocation construction. The case of the elastodynamic image force due to an interface is solved by deriving the reflected wave's contribution to the global solution across the interface. This entails obtaining the fundamental solution (Green's function) for a point unit force via Cagniard's method, and then applying the convolution theorem for a screw dislocation modelled as a force distribution. Complete, explicit formulae are provided when available. It is shown that the elastodynamic image forces are generally affected by retardation effects, and that those acting on the moving dislocations display a dynamic magnification that exceed the attraction (or repulsion) predicted in classical elastostatic calculations.
]]>2017-09-20T00:09:30-07:00info:doi/10.1098/rspa.2017.0484hwp:master-id:royprsa;rspa.2017.04842017-09-20Research articles47322052017048420170484<![CDATA[Fully localized post-buckling states of cylindrical shells under axial compression]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2205/20170177?rss=1
We compute nonlinear force equilibrium solutions for a clamped thin cylindrical shell under axial compression. The equilibrium solutions are dynamically unstable and located on the stability boundary of the unbuckled state. A fully localized single dimple deformation is identified as the edge state—the attractor for the dynamics restricted to the stability boundary. Under variation of the axial load, the single dimple undergoes homoclinic snaking in the azimuthal direction, creating states with multiple dimples arranged around the central circumference. Once the circumference is completely filled with a ring of dimples, snaking in the axial direction leads to further growth of the dimple pattern. These fully nonlinear solutions embedded in the stability boundary of the unbuckled state constitute critical shape deformations. The solutions may thus be a step towards explaining when the buckling and subsequent collapse of an axially loaded cylinder shell is triggered.
]]>2017-09-13T00:09:15-07:00info:doi/10.1098/rspa.2017.0177hwp:master-id:royprsa;rspa.2017.01772017-09-13Research articles47322052017017720170177<![CDATA[Meso-scale defect evaluation of selective laser melting using spatially resolved acoustic spectroscopy]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2205/20170194?rss=1
Developments in additive manufacturing technology are serving to expand the potential applications. Critical developments are required in the supporting areas of measurement and in process inspection to achieve this. CM247LC is a nickel superalloy that is of interest for use in aerospace and civil power plants. However, it is difficult to process via selective laser melting (SLM) as it suffers from cracking during rapid cooling and solidification. This limits the viability of CM247LC parts created using SLM. To quantify part integrity, spatially resolved acoustic spectroscopy (SRAS) has been identified as a viable non-destructive evaluation technique. In this study, a combination of optical microscopy and SRAS was used to identify and classify the surface defects present in SLM-produced parts. By analysing the datasets and scan trajectories, it is possible to correlate morphological information with process parameters. Image processing was used to quantify porosity and cracking for bulk density measurement. Analysis of surface acoustic wave data showed that an error in manufacture in the form of an overscan occurred. Comparing areas affected by overscan with a bulk material, a change in defect density from 1.17% in the bulk material to 5.32% in the overscan regions was observed, highlighting the need to reduce overscan areas in manufacture.
]]>2017-09-13T00:09:15-07:00info:doi/10.1098/rspa.2017.0194hwp:master-id:royprsa;rspa.2017.01942017-09-13Research articles47322052017019420170194<![CDATA[Rate-based structural health monitoring using permanently installed sensors]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2205/20170270?rss=1
Permanently installed sensors are becoming increasingly ubiquitous, facilitating very frequent in situ measurements and consequently improved monitoring of ‘trends’ in the observed system behaviour. It is proposed that this newly available data may be used to provide prior warning and forecasting of critical events, particularly system failure. Numerous damage mechanisms are examples of positive feedback; they are ‘self-accelerating’ with an increasing rate of damage towards failure. The positive feedback leads to a common time-response behaviour which may be described by an empirical relation allowing prediction of the time to criticality. This study focuses on Structural Health Monitoring of engineering components; failure times are projected well in advance of failure for fatigue, creep crack growth and volumetric creep damage experiments. The proposed methodology provides a widely applicable framework for using newly available near-continuous data from permanently installed sensors to predict time until failure in a range of application areas including engineering, geophysics and medicine.
]]>2017-09-13T00:09:15-07:00info:doi/10.1098/rspa.2017.0270hwp:master-id:royprsa;rspa.2017.02702017-09-13Research articles47322052017027020170270<![CDATA[Numerical studies of cavitation erosion on an elastic-plastic material caused by shock-induced bubble collapse]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2205/20170315?rss=1
We present a study of shock-induced collapse of single bubbles near/attached to an elastic–plastic solid using the free-Lagrange method, which forms the latest part of our shock-induced collapse studies. We simulated the collapse of 40 μm radius single bubbles near/attached to rigid and aluminium walls by a 60 MPa lithotripter shock for various scenarios based on bubble–wall separations, and the collapse of a 255 μm radius bubble attached to aluminium foil with a 65 MPa lithotripter shock. The coupling of the multi-phases, compressibility, axisymmetric geometry and elastic–plastic material model within a single solver has enabled us to examine the impingement of high-speed liquid jets from the shock-induced collapsing bubbles, which imposes an extreme compression in the aluminium that leads to pitting and plastic deformation. For certain scenarios, instead of the high-speed jet, a radially inwards flow along the aluminium surface contracts the bubble to produce a ‘mushroom shape’. This work provides methods for quantifying which parameters (e.g. bubble sizes and separations from the solid) might promote or inhibit erosion on solid surfaces.
]]>2017-09-13T00:09:15-07:00info:doi/10.1098/rspa.2017.0315hwp:master-id:royprsa;rspa.2017.03152017-09-13Research articles47322052017031520170315<![CDATA[On the differing growth mechanisms of black-smoker and Lost City-type hydrothermal vents]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2205/20170387?rss=1
Black smokers and Lost City-type springs are varieties of hydrothermal vents on the ocean floors that emit hot, acidic water and cool, alkaline water, respectively. While both produce precipitation structures as the issuing fluid encounters oceanic water, Lost City-type hydrothermal vents in particular have been implicated in the origin of life on the Earth. We present a parallel-velocity flow model for the radius and flow rate of a cylindrical jet of fluid that forms the template for the growth of a tube precipitated about itself and we compare the solution with previous laboratory experimental results from growth of silicate chemical gardens. We show that when the growth of the solid structure is determined by thermal diffusion, fluid flow is slow at the solid–liquid contact. However, in the case of chemical diffusive transport, the fluid jet effectively drags the liquid in the pores of the solid precipitate. These findings suggest a continuum in the diffusive growth rate of hydrothermal vent structures, where Lost City-type hydrothermal vents favour contact between the vent fluid and the external seawater. We explore the implications for the road to life.
]]>2017-09-13T00:09:15-07:00info:doi/10.1098/rspa.2017.0387hwp:master-id:royprsa;rspa.2017.03872017-09-13Research articles47322052017038720170387<![CDATA[Singular diffusionless limits of double-diffusive instabilities in magnetohydrodynamics]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2205/20170344?rss=1
We study local instabilities of a differentially rotating viscous flow of electrically conducting incompressible fluid subject to an external azimuthal magnetic field. In the presence of the magnetic field, the hydrodynamically stable flow can demonstrate non-axisymmetric azimuthal magnetorotational instability (AMRI) both in the diffusionless case and in the double-diffusive case with viscous and ohmic dissipation. Performing stability analysis of amplitude transport equations of short-wavelength approximation, we find that the threshold of the diffusionless AMRI via the Hamilton–Hopf bifurcation is a singular limit of the thresholds of the viscous and resistive AMRI corresponding to the dissipative Hopf bifurcation and manifests itself as the Whitney umbrella singular point. A smooth transition between the two types of instabilities is possible only if the magnetic Prandtl number is equal to unity, Pm=1. At a fixed Pm=1, the threshold of the double-diffusive AMRI is displaced by finite distance in the parameter space with respect to the diffusionless case even in the zero dissipation limit. The complete neutral stability surface contains three Whitney umbrella singular points and two mutually orthogonal intervals of self-intersection. At these singularities, the double-diffusive system reduces to a marginally stable system which is either Hamiltonian or parity–time-symmetric.
]]>2017-09-13T00:57:51-07:00info:doi/10.1098/rspa.2017.0344hwp:master-id:royprsa;rspa.2017.03442017-09-13Research articles47322052017034420170344<![CDATA[Accuracy of the hypothetical sky-polarimetric Viking navigation versus sky conditions: revealing solar elevations and cloudinesses favourable for this navigation method]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2205/20170358?rss=1
According to Thorkild Ramskou's theory proposed in 1967, under overcast and foggy skies, Viking seafarers might have used skylight polarization analysed with special crystals called sunstones to determine the position of the invisible Sun. After finding the occluded Sun with sunstones, its elevation angle had to be measured and its shadow had to be projected onto the horizontal surface of a sun compass. According to Ramskou's theory, these sunstones might have been birefringent calcite or dichroic cordierite or tourmaline crystals working as polarizers. It has frequently been claimed that this method might have been suitable for navigation even in cloudy weather. This hypothesis has been accepted and frequently cited for decades without any experimental support. In this work, we determined the accuracy of this hypothetical sky-polarimetric Viking navigation for 1080 different sky situations characterized by solar elevation and cloudiness , the sky polarization patterns of which were measured by full-sky imaging polarimetry. We used the earlier measured uncertainty functions of the navigation steps 1, 2 and 3 for calcite, cordierite and tourmaline sunstone crystals, respectively, and the newly measured uncertainty function of step 4 presented here. As a result, we revealed the meteorological conditions under which Vikings could have used this hypothetical navigation method. We determined the solar elevations at which the navigation uncertainties are minimal at summer solstice and spring equinox for all three sunstone types. On average, calcite sunstone ensures a more accurate sky-polarimetric navigation than tourmaline and cordierite. However, in some special cases (generally at 35° ≤ ≤ 40°, 1 okta ≤ ≤ 6 oktas for summer solstice, and at 20° ≤ ≤ 25°, 0 okta ≤ ≤ 4 oktas for spring equinox), the use of tourmaline and cordierite results in smaller navigation uncertainties than that of calcite. Generally, under clear or less cloudy skies, the sky-polarimetric navigation is more accurate, but at low solar elevations its accuracy remains relatively large even at high cloudiness. For a given , the absolute value of averaged peak North uncertainties dramatically decreases with increasing until the sign (±) change of these uncertainties. For a given , this absolute value can either decrease or increase with increasing . The most advantageous sky situations for this navigation method are at summer solstice when the solar elevation and cloudiness are 35° ≤ ≤ 40° and 2 oktas ≤ ≤ 3 oktas.
]]>2017-09-13T00:57:51-07:00info:doi/10.1098/rspa.2017.0358hwp:master-id:royprsa;rspa.2017.03582017-09-13Research articles47322052017035820170358<![CDATA[Coordinate representation for non-Hermitian position and momentum operators]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2205/20170434?rss=1
In this paper, we undertake an analysis of the eigenstates of two non-self-adjoint operators q^ and p^ similar, in a suitable sense, to the self-adjoint position and momentum operators q^0 and p^0 usually adopted in ordinary quantum mechanics. In particular, we discuss conditions for these eigenstates to be biorthogonal distributions, and we discuss a few of their properties. We illustrate our results with two examples, one in which the similarity map between the self-adjoint and the non-self-adjoint is bounded, with bounded inverse, and the other in which this is not true. We also briefly propose an alternative strategy to deal with q^ and p^, based on the so-called quasi *-algebras.
]]>2017-09-13T00:09:15-07:00info:doi/10.1098/rspa.2017.0434hwp:master-id:royprsa;rspa.2017.04342017-09-13Research articles47322052017043420170434<![CDATA[Bitwise efficiency in chaotic models]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2205/20170144?rss=1
Motivated by the increasing energy consumption of supercomputing for weather and climate simulations, we introduce a framework for investigating the bit-level information efficiency of chaotic models. In comparison with previous explorations of inexactness in climate modelling, the proposed and tested information metric has three specific advantages: (i) it requires only a single high-precision time series; (ii) information does not grow indefinitely for decreasing time step; and (iii) information is more sensitive to the dynamics and uncertainties of the model rather than to the implementation details. We demonstrate the notion of bit-level information efficiency in two of Edward Lorenz’s prototypical chaotic models: Lorenz 1963 (L63) and Lorenz 1996 (L96). Although L63 is typically integrated in 64-bit ‘double’ floating point precision, we show that only 16 bits have significant information content, given an initial condition uncertainty of approximately 1% of the size of the attractor. This result is sensitive to the size of the uncertainty but not to the time step of the model. We then apply the metric to the L96 model and find that a 16-bit scaled integer model would suffice given the uncertainty of the unresolved sub-grid-scale dynamics. We then show that, by dedicating computational resources to spatial resolution rather than numeric precision in a field programmable gate array (FPGA), we see up to 28.6% improvement in forecast accuracy, an approximately fivefold reduction in the number of logical computing elements required and an approximately 10-fold reduction in energy consumed by the FPGA, for the L96 model.
]]>2017-09-06T00:09:07-07:00info:doi/10.1098/rspa.2017.0144hwp:master-id:royprsa;rspa.2017.01442017-09-06Research articles47322052017014420170144<![CDATA[Elastic field of a spherical inclusion with non-uniform eigenfields in second strain gradient elasticity]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2205/20170254?rss=1
In this paper, within the complete form of Mindlin’s second strain gradient theory, the elastic field of an isolated spherical inclusion embedded in an infinitely extended homogeneous isotropic medium due to a non-uniform distribution of eigenfields is determined. These eigenfields, in addition to eigenstrain, comprise eigen double and eigen triple strains. After the derivation of a closed-form expression for Green’s function associated with the problem, two different cases of non-uniform distribution of the eigenfields are considered as follows: (i) radial distribution, i.e. the distributions of the eigenfields are functions of only the radial distance of points from the centre of inclusion, and (ii) polynomial distribution, i.e. the distributions of the eigenfields are polynomial functions in the Cartesian coordinates of points. While the obtained solution for the elastic field of the latter case takes the form of an infinite series, the solution to the former case is represented in a closed form. Moreover, Eshelby’s tensors associated with the two mentioned cases are obtained.
]]>2017-09-06T00:09:07-07:00info:doi/10.1098/rspa.2017.0254hwp:master-id:royprsa;rspa.2017.02542017-09-06Research articles47322052017025420170254<![CDATA[Stability of the shear-thinning boundary-layer flow over a flat inclined plate]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2205/20170350?rss=1
In this study, we consider the boundary-layer flow of an inelastic non-Newtonian fluid over an inclined flat plate. Using two popular generalized Newtonian models, we determine base flow profiles and associated linear stability results for a range shear-thinning fluids. In addition to neutral stability curves, we also present results concerning the linear growth of the Tollmien–Schlichting waves as they propagate downstream. Furthermore, to gain an insight into the underlying physical mechanisms affecting the destabilization of the disturbances, an integral energy equation is derived and energy calculations are presented. Results from all three analyses suggest that the effect of shear-thinning will act to stabilize the boundary-layer flow. Consequently, it can be argued that the addition of shear-thinning agents could act as a passive control mechanism for flows of this nature.
]]>2017-09-06T00:09:07-07:00info:doi/10.1098/rspa.2017.0350hwp:master-id:royprsa;rspa.2017.03502017-09-06Research articles47322052017035020170350