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Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences RSS feed -- recent issues1471-2946Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences1364-5021<![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[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[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[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[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[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[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[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<![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[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[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[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[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[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[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[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[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[Correction to 'On a new class of electroelastic bodies. I]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2205/20170523?rss=1
2017-09-06T00:09:07-07:00info:doi/10.1098/rspa.2017.0523hwp:master-id:royprsa;rspa.2017.05232017-09-06Corrections47322052017052320170523<![CDATA[Whitham modulation theory for the Kadomtsev- Petviashvili equation]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2204/20160695?rss=1
The genus-1 Kadomtsev–Petviashvili (KP)-Whitham system is derived for both variants of the KP equation; namely the KPI and KPII equations. The basic properties of the KP-Whitham system, including symmetries, exact reductions and its possible complete integrability, together with the appropriate generalization of the one-dimensional Riemann problem for the Korteweg–de Vries equation are discussed. Finally, the KP-Whitham system is used to study the linear stability properties of the genus-1 solutions of the KPI and KPII equations; it is shown that all genus-1 solutions of KPI are linearly unstable, while all genus-1 solutions of KPII are linearly stable within the context of Whitham theory.
]]>2017-08-02T00:08:23-07:00info:doi/10.1098/rspa.2016.0695hwp:master-id:royprsa;rspa.2016.06952017-08-02Research articles47322042016069520160695<![CDATA[A variation on the Donsker-Varadhan inequality for the principal eigenvalue]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2204/20160877?rss=1
The purpose of this short paper is to give a variation on the classical Donsker–Varadhan inequality, which bounds the first eigenvalue of a second-order elliptic operator on a bounded domain by the largest mean first exit time of the associated drift–diffusion process via 1≥1supxExc.Instead of looking at the mean of the first exit time, we study quantiles: let dp,:->R≥0 be the smallest time t such that the likelihood of exiting within that time is p, then 1≥log(1/p)supxdp,(x).Moreover, as p->0, this lower bound converges to _{1}.
]]>2017-08-23T00:08:55-07:00info:doi/10.1098/rspa.2016.0877hwp:master-id:royprsa;rspa.2016.08772017-08-23Research articles47322042016087720160877<![CDATA[Recursive modular modelling methodology for lumped-parameter dynamic systems]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2204/20160891?rss=1
This paper proposes a novel approach to the modelling of lumped-parameter dynamic systems, based on representing them by hierarchies of mathematical models of increasing complexity instead of a single (complex) model. Exploring the multilevel modularity that these systems typically exhibit, a general recursive modelling methodology is proposed, in order to conciliate the use of the already existing modelling techniques. The general algorithm is based on a fundamental theorem that states the conditions for computing projection operators recursively. Three procedures for these computations are discussed: orthonormalization, use of orthogonal complements and use of generalized inverses. The novel methodology is also applied for the development of a recursive algorithm based on the Udwadia–Kalaba equation, which proves to be identical to the one of a Kalman filter for estimating the state of a static process, given a sequence of noiseless measurements representing the constraints that must be satisfied by the system.
]]>2017-08-30T00:09:15-07:00info:doi/10.1098/rspa.2016.0891hwp:master-id:royprsa;rspa.2016.08912017-08-30Research articles47322042016089120160891<![CDATA[Rebound mechanics of micrometre-scale, spherical particles in high-velocity impacts]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2204/20160936?rss=1
The impact mechanics of micrometre-scale metal particles with flat metal surfaces is investigated for high-velocity impacts ranging from 50 m s^{–1} to more than 1 km s^{–1}, where impact causes predominantly plastic deformation. A material model that includes high strain rate and temperature effects on the yield stress, heat generation due to plasticity, material damage due to excessive plastic strain and heat transfer is used in the numerical analysis. The coefficient of restitution e is predicted by the classical work using elastic–plastic deformation analysis with quasi-static impact mechanics to be proportional to Vi–1/4 and Vi–1/2 for the low and moderate impact velocities that span the ranges of 0–10 and 10–100 m s^{–1}, respectively. In the elastic–plastic and fully plastic deformation regimes the particle rebound is attributed to the elastic spring-back that initiates at the particle–substrate interface. At higher impact velocities (0.1–1 km s^{–1}) e is shown to be proportional to approximately Vi–1. In this deeply plastic deformation regime various deformation modes that depend on plastic flow of the material including the time lag between the rebound instances of the top and bottom points of particle and the lateral spreading of the particle are identified. In this deformation regime, the elastic spring-back initiates subsurface, in the substrate.
]]>2017-08-16T00:08:59-07:00info:doi/10.1098/rspa.2016.0936hwp:master-id:royprsa;rspa.2016.09362017-08-16Research articles47322042016093620160936<![CDATA[Model selection for dynamical systems via sparse regression and information criteria]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2204/20170009?rss=1
We develop an algorithm for model selection which allows for the consideration of a combinatorially large number of candidate models governing a dynamical system. The innovation circumvents a disadvantage of standard model selection which typically limits the number of candidate models considered due to the intractability of computing information criteria. Using a recently developed sparse identification of nonlinear dynamics algorithm, the sub-selection of candidate models near the Pareto frontier allows feasible computation of Akaike information criteria (AIC) or Bayes information criteria scores for the remaining candidate models. The information criteria hierarchically ranks the most informative models, enabling the automatic and principled selection of the model with the strongest support in relation to the time-series data. Specifically, we show that AIC scores place each candidate model in the strong support, weak support or no support category. The method correctly recovers several canonical dynamical systems, including a susceptible-exposed-infectious-recovered disease model, Burgers’ equation and the Lorenz equations, identifying the correct dynamical system as the only candidate model with strong support.
]]>2017-08-30T00:09:15-07:00info:doi/10.1098/rspa.2017.0009hwp:master-id:royprsa;rspa.2017.00092017-08-30Research articles47322042017000920170009<![CDATA[Organic molecule fluorescence as an experimental test-bed for quantum jumps in thermodynamics]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2204/20170099?rss=1
We demonstrate with an experiment how molecules are a natural test bed for probing fundamental quantum thermodynamics. Single-molecule spectroscopy has undergone transformative change in the past decade with the advent of techniques permitting individual molecules to be distinguished and probed. We demonstrate that the quantum Jarzynski equality for heat is satisfied in this set-up by considering the time-resolved emission spectrum of organic molecules as arising from quantum jumps between states. This relates the heat dissipated into the environment to the free energy difference between the initial and final state. We demonstrate also how utilizing the quantum Jarzynski equality allows for the detection of energy shifts within a molecule, beyond the relative shift.
]]>2017-08-30T00:09:15-07:00info:doi/10.1098/rspa.2017.0099hwp:master-id:royprsa;rspa.2017.00992017-08-30Research articles47322042017009920170099<![CDATA[Initial boundary-value problem for the spherically symmetric Einstein equations with fluids with tangential pressure]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2204/20170113?rss=1
We prove that, for a given spherically symmetric fluid distribution with tangential pressure on an initial space-like hypersurface with a time-like boundary, there exists a unique, local in time solution to the Einstein equations in a neighbourhood of the boundary. As an application, we consider a particular elastic fluid interior matched to a vacuum exterior.
]]>2017-08-09T00:08:43-07:00info:doi/10.1098/rspa.2017.0113hwp:master-id:royprsa;rspa.2017.01132017-08-09Research articles47322042017011320170113<![CDATA[Designing the optimal bit: balancing energetic cost, speed and reliability]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2204/20170117?rss=1
We consider the challenge of operating a reliable bit that can be rapidly erased. We find that both erasing and reliability times are non-monotonic in the underlying friction, leading to a trade-off between erasing speed and bit reliability. Fast erasure is possible at the expense of low reliability at moderate friction, and high reliability comes at the expense of slow erasure in the underdamped and overdamped limits. Within a given class of bit parameters and control strategies, we define ‘optimal’ designs of bits that meet the desired reliability and erasing time requirements with the lowest operational work cost. We find that optimal designs always saturate the bound on the erasing time requirement, but can exceed the required reliability time if critically damped. The non-trivial geometry of the reliability and erasing time scales allows us to exclude large regions of parameter space as suboptimal. We find that optimal designs are either critically damped or close to critical damping under the erasing procedure.
]]>2017-08-23T00:08:55-07:00info:doi/10.1098/rspa.2017.0117hwp:master-id:royprsa;rspa.2017.01172017-08-23Research articles47322042017011720170117<![CDATA[Characterization of columnar inertial modes in rapidly rotating spheres and spheroids]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2204/20170181?rss=1
We consider fluid-filled spheres and spheroidal containers of eccentricity in rapid rotation, as a proxy for the interior dynamics of stars and planets. The fluid motion is assumed to be quasi-geostrophic (QG): horizontal motions are invariant parallel to the rotation axis z, a characteristic which is handled by use of a stream function formulation which additionally enforces mass conservation and non-penetration at the boundary. By linearizing about a quiescent background state, we investigate a variety of methods to study the QG inviscid inertial wave modes which are compared with fully three-dimensional (3D) calculations. We consider the recently proposed weak formulation of the inviscid system valid in spheroids of arbitrary eccentricity, to which we present novel closed-form polynomial solutions. Our modal solutions accurately represent, in both spatial structure and frequency, the most z-invariant of the inertial wave modes in a spheroid, and constitute a simple basis set for the analysis of rotationally dominated fluids. We further show that these new solutions are more accurate than those of the classical axial-vorticity equation, which is independent of and thus fails to properly encode the container geometry. We also consider the effects of viscosity for the cases of both no-slip and stress-free boundary conditions for a spherical container. Calculations performed under the columnar approximation are compared with 3D solutions and excellent agreement has been found despite fundamental differences in the two formulations.
]]>2017-08-09T00:08:43-07:00info:doi/10.1098/rspa.2017.0181hwp:master-id:royprsa;rspa.2017.01812017-08-09Research articles47322042017018120170181<![CDATA[Effective potentials in nonlinear polycrystals and quadrature formulae]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2204/20170213?rss=1
This study presents a family of estimates for effective potentials in nonlinear polycrystals. Noting that these potentials are given as averages, several quadrature formulae are investigated to express these integrals of nonlinear functions of local fields in terms of the moments of these fields. Two of these quadrature formulae reduce to known schemes, including a recent proposition (Ponte Castañeda 2015 Proc. R. Soc. A471, 20150665 (doi:10.1098/rspa.2015.0665)) obtained by completely different means. Other formulae are also reviewed that make use of statistical information on the fields beyond their first and second moments. These quadrature formulae are applied to the estimation of effective potentials in polycrystals governed by two potentials, by means of a reduced-order model proposed by the authors (non-uniform transformation field analysis). It is shown how the quadrature formulae improve on the tangent second-order approximation in porous crystals at high stress triaxiality. It is found that, in order to retrieve a satisfactory accuracy for highly nonlinear porous crystals under high stress triaxiality, a quadrature formula of higher order is required.
]]>2017-08-30T00:09:15-07:00info:doi/10.1098/rspa.2017.0213hwp:master-id:royprsa;rspa.2017.02132017-08-30Research articles47322042017021320170213<![CDATA[Numerical simulations of drumlin formation]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2204/20170220?rss=1
We summarize the present form of the instability theory for drumlin formation, which describes the coupled subglacial flow of ice, water and sediment. This model has evolved over the last 20 years, and is now at the point where it can predict instabilities corresponding to ribbed moraine, drumlins and mega-scale glacial lineations, but efforts to provide numerical solutions of the model have been limited. The present summary adds some slight nuances to previously published versions of the theory, notably concerning the constitutive description of the subglacial water film and its flow. A new numerical method is devised to solve the model, and we show that it can be solved for realistic values of most of the parameters, with the exception of that corresponding to the water film thickness. We show that evolved bedforms can be three-dimensional and of the correct sizes, and we explore to some extent the variation of the solutions with the model’s parameters.
]]>2017-08-30T00:09:15-07:00info:doi/10.1098/rspa.2017.0220hwp:master-id:royprsa;rspa.2017.02202017-08-30Research articles47322042017022020170220<![CDATA[Temperature profile in a liquid-vapour interface near the critical point]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2204/20170229?rss=1
Thanks to an expansion with respect to densities of energy, mass and entropy, we discuss the concept of thermocapillary fluid for inhomogeneous fluids. The non-convex state law valid for homogeneous fluids is modified by adding terms taking account of the gradients of these densities. This seems more realistic than Cahn and Hilliard’s model which uses a density expansion in mass-density gradient only. Indeed, through liquid–vapour interfaces, realistic potentials in molecular theories show that entropy density and temperature do not vary with the mass density as it would do in bulk phases. In this paper, we prove using a rescaling process near the critical point, that liquid–vapour interfaces behave essentially in the same way as in Cahn and Hilliard’s model.
]]>2017-08-09T00:08:43-07:00info:doi/10.1098/rspa.2017.0229hwp:master-id:royprsa;rspa.2017.02292017-08-09Research articles47322042017022920170229<![CDATA[Flexural edge waves generated by steady-state propagation of a loaded rectilinear crack in an elastically supported thin plate]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2204/20170265?rss=1
The problem of a rectilinear crack propagating at constant speed in an elastically supported thin plate and acted upon by an equally moving load is considered. The full-field solution is obtained and the spotlight is set on flexural edge wave generation. Below the critical speed for the appearance of travelling waves, a threshold speed is met which marks the transformation of decaying edge waves into edge waves propagating along the crack and dying away from it. Yet, besides these, and for any propagation speed, a pair of localized edge waves, which rapidly decay behind the crack tip, is also shown to exist. These waves are characterized by a novel dispersion relation and fade off from the crack line in an oscillatory manner, whence they play an important role in the far field behaviour. Dynamic stress intensity factors are obtained and, for speed close to the critical speed, they show a resonant behaviour which expresses the most efficient way to channel external work into the crack. Indeed, this behaviour is justified through energy considerations regarding the work of the applied load and the energy release rate. Results might be useful in a wide array of applications, ranging from fracturing and machining to acoustic emission and defect detection.
]]>2017-08-30T00:09:14-07:00info:doi/10.1098/rspa.2017.0265hwp:master-id:royprsa;rspa.2017.02652017-08-30Research articles47322042017026520170265<![CDATA[The structure of turbulence in a rapid tidal flow]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2204/20170295?rss=1
The structure of turbulence in a rapid tidal flow is characterized through new observations of fundamental statistical properties at a site in the UK which has a simple geometry and sedate surface wave action. The mean flow at the Sound of Islay exceeded 2.5 m s^{–1} and the turbulent boundary layer occupied the majority of the water column, with an approximately logarithmic mean velocity profile identifiable close to the seabed. The anisotropic ratios, spectral scales and higher-order statistics of the turbulence generally agree well with values reported for two-dimensional open channels in the laboratory and other tidal channels, therefore providing further support for the application of universal models. The results of the study can assist in developing numerical models of turbulence in rapid tidal flows such as those proposed for tidal energy generation.
]]>2017-08-23T00:08:55-07:00info:doi/10.1098/rspa.2017.0295hwp:master-id:royprsa;rspa.2017.02952017-08-23Research articles47322042017029520170295<![CDATA[Nonlinear electroelasticity: material properties, continuum theory and applications]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2204/20170311?rss=1
In the last few years, it has been recognized that the large deformation capacity of elastomeric materials that are sensitive to electric fields can be harnessed for use in transducer devices such as actuators and sensors. This has led to the reassessment of the mathematical theory that is needed for the description of the electromechanical (in particular, electroelastic) interactions for purposes of material characterization and prediction. After a review of the key experiments concerned with determining the nature of the electromechanical interactions and a discussion of the range of applications to devices, we provide a short account of the history of developments in the nonlinear theory. This is followed by a succinct modern treatment of electroelastic theory, including the governing equations and constitutive laws needed for both material characterization and the analysis of general electroelastic coupling problems. For illustration, the theory is then applied to two simple representative boundary-value problems that are relevant to the geometries of activation devices; in particular, (a) a rectangular plate and (b) a circular cylindrical tube, in each case with compliant electrodes on the major surfaces and a potential difference between them. In (a), an electric field is generated normal to the major surfaces and in (b), a radial electric field is present. This is followed by a short section in which other problems addressed on the basis of the general theory are described briefly.
]]>2017-08-02T00:08:23-07:00info:doi/10.1098/rspa.2017.0311hwp:master-id:royprsa;rspa.2017.03112017-08-02Review articles47322042017031120170311<![CDATA[Mittag-Leffler synchronization of delayed fractional-order bidirectional associative memory neural networks with discontinuous activations: state feedback control and impulsive control schemes]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2204/20170322?rss=1
This paper is concerned with the drive–response synchronization for a class of fractional-order bidirectional associative memory neural networks with time delays, as well as in the presence of discontinuous activation functions. The global existence of solution under the framework of Filippov for such networks is firstly obtained based on the fixed-point theorem for condensing map. Then the state feedback and impulsive controllers are, respectively, designed to ensure the Mittag-Leffler synchronization of these neural networks and two new synchronization criteria are obtained, which are expressed in terms of a fractional comparison principle and Razumikhin techniques. Numerical simulations are presented to validate the proposed methodologies.
]]>2017-08-02T00:08:23-07:00info:doi/10.1098/rspa.2017.0322hwp:master-id:royprsa;rspa.2017.03222017-08-02Research articles47322042017032220170322<![CDATA[Elastic deformation of twinned microstructures]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2204/20170330?rss=1
Many crystalline materials exhibit twinned microstructures, where well-defined orientation relationships define the special symmetry between different, elastically anisotropic twin variants. When such twins are subjected to external loading, additional internal stresses necessarily occur at the twin boundaries in order to maintain compatibility. These compatibility stresses are constant inside each variant in repeating stacks of twins and considerably affect the local stress state. In this paper, we use anisotropic linear elasticity to derive general analytical solutions for compatibility stresses in a stack of twin variants in arbitrary materials, for arbitrary variant volume fractions and twin types, subjected to arbitrary applied stresses. By considering two examples, growth twins in electrodeposited Cu and B19' martensite twins in the shape memory alloy NiTi, we further demonstrate that compatibility stresses can considerably alter the preferred slip systems for dislocation plasticity as well as the effective macroscopic behaviour of twinned microstructures.
]]>2017-08-16T00:08:59-07:00info:doi/10.1098/rspa.2017.0330hwp:master-id:royprsa;rspa.2017.03302017-08-16Research articles47322042017033020170330<![CDATA[Adaptive compliant structures for flow regulation]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2204/20170334?rss=1
This paper introduces conceptual design principles for a novel class of adaptive structures that provide both flow regulation and control. While of general applicability, these design principles, which revolve around the idea of using the instabilities and elastically nonlinear behaviour of post-buckled panels, are exemplified through a case study: the design of a shape-adaptive air inlet. The inlet comprises a deformable post-buckled member that changes shape depending on the pressure field applied by the surrounding fluid, thereby regulating the inlet aperture. By tailoring the stress field in the post-buckled state and the geometry of the initial, stress-free configuration, the deformable section can snap through to close or open the inlet completely. Owing to its inherent ability to change shape in response to external stimuli—i.e. the aerodynamic loads imposed by different operating conditions—the inlet does not have to rely on linkages and mechanisms for actuation, unlike conventional flow-controlling devices.
]]>2017-08-16T00:08:59-07:00info:doi/10.1098/rspa.2017.0334hwp:master-id:royprsa;rspa.2017.03342017-08-16Research articles47322042017033420170334<![CDATA[A tale of two nested elastic rings]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2204/20170340?rss=1
Elastic rods in contact provide a rich paradigm for understanding shape and deformation in interacting elastic bodies. Here, we consider the problem of determining the static solutions of two nested elastic rings in the plane. If the inner ring is longer than the outer ring, it will buckle creating a space between the two rings. This deformation can be further influenced by either adhesion between the rings or if pressure is applied externally or internally. We obtain an exact solution of this problem when both rings are assumed inextensible and unshearable. Through a variational formulation of the problem, we identify the boundary conditions at the contact point and use the Kirchhoff analogy to give exact solutions of the problems in terms of elliptic functions. The role of both adhesion and pressure is explored.
]]>2017-08-09T00:08:43-07:00info:doi/10.1098/rspa.2017.0340hwp:master-id:royprsa;rspa.2017.03402017-08-09Research articles47322042017034020170340<![CDATA[Spectral derivation of the classic laws of wall-bounded turbulent flows]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2204/20170354?rss=1
We show that the classic laws of the mean-velocity profiles (MVPs) of wall-bounded turbulent flows—the ‘law of the wall,’ the ‘defect law’ and the ‘log law’—can be predicated on a sufficient condition with no manifest ties to the MVPs, namely that viscosity and finite turbulent domains have a depressive effect on the spectrum of turbulent energy. We also show that this sufficient condition is consistent with empirical data on the spectrum and may be deemed a general property of the energetics of wall turbulence. Our findings shed new light on the physical origin of the classic laws and their immediate offshoot, Prandtl’s theory of turbulent friction.
]]>2017-08-09T00:08:43-07:00info:doi/10.1098/rspa.2017.0354hwp:master-id:royprsa;rspa.2017.03542017-08-09Research articles47322042017035420170354<![CDATA[Multi-parameter actuation of a neutrally stable shell: a flexible gear-less motor]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2204/20170364?rss=1
We have designed and tested experimentally a morphing structure consisting of a neutrally stable thin cylindrical shell driven by a multi-parameter piezoelectric actuation. The shell is obtained by plastically deforming an initially flat copper disc, so as to induce large isotropic and almost uniform inelastic curvatures. Following the plastic deformation, in a perfectly isotropic system, the shell is theoretically neutrally stable, having a continuous set of stable cylindrical shapes corresponding to the rotation of the axis of maximal curvature. Small imperfections render the actual structure bistable, giving preferred orientations. A three-parameter piezoelectric actuation, exerted through micro-fibre-composite actuators, allows us to add a small perturbation to the plastic inelastic curvature and to control the direction of maximal curvature. This actuation law is designed through a geometrical analogy based on a fully nonlinear inextensible uniform-curvature shell model. We report on the fabrication, identification and experimental testing of a prototype and demonstrate the effectiveness of the piezoelectric actuators in controlling its shape. The resulting motion is an apparent rotation of the shell, controlled by the voltages as in a ‘gear-less motor’, which is, in reality, a precession of the axis of principal curvature.
]]>2017-08-16T00:08:59-07:00info:doi/10.1098/rspa.2017.0364hwp:master-id:royprsa;rspa.2017.03642017-08-16Research articles47322042017036420170364<![CDATA[Origin of the onset of meandering of a straight river]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2204/20170376?rss=1
In this paper, to explore the origin of the onset of meandering of a straight river, we, first, analyse the linear stability of a straight river. We discover that the natural perturbation modes of a straight river maintain an equilibrium state by confining themselves to an onset wavenumber band that is dependent on the flow regimes, aspect ratio, relative roughness number and Shields number. Then, we put forward a phenomenological description of the onset of meandering of a straight river. Its mechanism is governed by turbulent flow, with counter-rotation of neighbouring large-scale or macro-turbulent eddies in succession to generate the processes of alternating erosion and deposition of sediment grains of the riverbed. This concept is explained by a theorem (universal scaling law) stemming from the phenomenology of a turbulent energy cascade to provide a quantitative insight into the criterion for the onset of meandering of a straight river. It is revealed from this universal scaling law that, at the onset of meandering of a river, the longitudinal riverbed slope is a unique function of the river width, flow discharge and sediment grain size. This unique functional relationship is corroborated by the data obtained from the measurements in natural and model rivers.
]]>2017-08-16T00:08:59-07:00info:doi/10.1098/rspa.2017.0376hwp:master-id:royprsa;rspa.2017.03762017-08-16Research articles47322042017037620170376<![CDATA[Some applications of mathematics in golf]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2204/20170392?rss=1
At its core, like many other sports, golf is a game of integers. The minimization of the number of strokes played is generally what determines the winner, whether each of these are associated with the shortest of putts or the longest of drives. The outcomes of these shots are influenced by very slight changes, but hopefully in a deterministic sense. Understanding the mechanics of golf necessitates the development of models and this is coupled more often than not to the use of statistics. In essence, the individual aspects of the sport can be modelled adequately via fairly simplistic models, but the presence of a human at one end of the kinematic chain has a significant impact on the variability of the entire process. In this paper, we will review some of the ways that mathematics has been used to develop the understanding of the physical processes involved in the sport, including some of the analysis which is exploited within the Equipment Rules. We will also discuss some of the future challenges.
]]>2017-08-02T00:08:23-07:00info:doi/10.1098/rspa.2017.0392hwp:master-id:royprsa;rspa.2017.03922017-08-02Special feature47322042017039220170392<![CDATA[Unsteady solute dispersion in small blood vessels using a two-phase Casson model]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2204/20170427?rss=1
This study explores the transport of a solute in an unsteady blood flow in small arteries with and without absorption at the wall. The Casson fluid model is suitable for blood flow in small vessels. Owing to the aggregation of red cells in the central region of the small vessels, a two-phase model is considered in this investigation. Using the generalized dispersion model (Sankarasubramanian & Gill 1973 Proc. R. Soc. Lond. A333, 115–132. (doi:10.1098/rspa.1973.0051)), the convection, dispersion and mean concentration of the solute are analysed at all times in small arteries of different radii. The effects of the yield stress, wall absorption, the amplitude of the fluctuating pressure gradient component, the peripheral layer thickness, the Womersley frequency parameter, the Schmidt number and the Peclet number on the dispersion process are discussed. A comparative study of solute dispersion among single- and two-phase fluid models is presented. For small vessels, a significant difference between these models is observed during the solute dispersion; however, this difference becomes insignificant for large vessels. The mean concentration of solute reduces with increasing radius of the vessels. The present investigation is more realistic for understanding the transportation process of drugs in blood flow in small arteries using the non-Newtonian fluid model.
]]>2017-08-16T00:08:59-07:00info:doi/10.1098/rspa.2017.0427hwp:master-id:royprsa;rspa.2017.04272017-08-16Research articles47322042017042720170427<![CDATA[A phase-plane analysis of localized frictional waves]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2203/20160606?rss=1
Sliding frictional interfaces at a range of length scales are observed to generate travelling waves; these are considered relevant, for example, to both earthquake ground surface movements and the performance of mechanical brakes and dampers. We propose an explanation of the origins of these waves through the study of an idealized mechanical model: a thin elastic plate subject to uniform shear stress held in frictional contact with a rigid flat surface. We construct a nonlinear wave equation for the deformation of the plate, and couple it to a spinodal rate-and-state friction law which leads to a mathematically well-posed problem that is capable of capturing many effects not accessible in a Coulomb friction model. Our model sustains a rich variety of solutions, including periodic stick–slip wave trains, isolated slip and stick pulses, and detachment and attachment fronts. Analytical and numerical bifurcation analysis is used to show how these states are organized in a two-parameter state diagram. We discuss briefly the possible physical interpretation of each of these states, and remark also that our spinodal friction law, though more complicated than other classical rate-and-state laws, is required in order to capture the full richness of wave types.
]]>2017-07-05T00:08:16-07:00info:doi/10.1098/rspa.2016.0606hwp:master-id:royprsa;rspa.2016.06062017-07-05Research articles47322032016060620160606<![CDATA[Direct linearizing transform for three-dimensional discrete integrable systems: the lattice AKP, BKP and CKP equations]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2203/20160915?rss=1
A unified framework is presented for the solution structure of three-dimensional discrete integrable systems, including the lattice AKP, BKP and CKP equations. This is done through the so-called direct linearizing transform, which establishes a general class of integral transforms between solutions. As a particular application, novel soliton-type solutions for the lattice CKP equation are obtained.
]]>2017-07-12T00:08:17-07:00info:doi/10.1098/rspa.2016.0915hwp:master-id:royprsa;rspa.2016.09152017-07-12Research articles47322032016091520160915<![CDATA[Gyro-elastic beams for the vibration reduction of long flexural systems]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2203/20170136?rss=1
The paper presents a model of a chiral multi-structure incorporating gyro-elastic beams. Floquet–Bloch waves in periodic chiral systems are investigated in detail, with the emphasis on localization and the formation of standing waves. It is found that gyricity leads to low-frequency standing modes and generation of stop-bands. A design of an earthquake protection system is offered here, as an interesting application of vibration isolation. Theoretical results are accompanied by numerical simulations in the time-harmonic regime.
]]>2017-07-19T00:08:13-07:00info:doi/10.1098/rspa.2017.0136hwp:master-id:royprsa;rspa.2017.01362017-07-19Research articles47322032017013620170136<![CDATA[Modulational instability in the full-dispersion Camassa-Holm equation]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2203/20170153?rss=1
We determine the stability and instability of a sufficiently small and periodic travelling wave to long-wavelength perturbations, for a nonlinear dispersive equation which extends a Camassa–Holm equation to include all the dispersion of water waves and the Whitham equation to include nonlinearities of medium-amplitude waves. In the absence of the effects of surface tension, the result qualitatively agrees with the Benjamin–Feir instability of a Stokes wave. In the presence of the effects of surface tension, it qualitatively agrees with those from formal asymptotic expansions of the physical problem and improves upon that for the Whitham equation, predicting the critical wave number at the strong surface tension limit. We discuss the modulational stability and instability in the Camassa–Holm equation and other related models.
]]>2017-07-19T00:08:13-07:00info:doi/10.1098/rspa.2017.0153hwp:master-id:royprsa;rspa.2017.01532017-07-19Research articles47322032017015320170153<![CDATA[Multi-fractal characterization of bacterial swimming dynamics: a case study on real and simulated Serratia marcescens]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2203/20170154?rss=1
To add to the current state of knowledge about bacterial swimming dynamics, in this paper, we study the fractal swimming dynamics of populations of Serratia marcescens bacteria both in vitro and in silico, while accounting for realistic conditions like volume exclusion, chemical interactions, obstacles and distribution of chemoattractant in the environment. While previous research has shown that bacterial motion is non-ergodic, we demonstrate that, besides the non-ergodicity, the bacterial swimming dynamics is multi-fractal in nature. Finally, we demonstrate that the multi-fractal characteristic of bacterial dynamics is strongly affected by bacterial density and chemoattractant concentration.
]]>2017-07-12T00:08:17-07:00info:doi/10.1098/rspa.2017.0154hwp:master-id:royprsa;rspa.2017.01542017-07-12Research articles47322032017015420170154<![CDATA[Modelling the fear of crime]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2203/20170156?rss=1
How secure people feel in a particular region is obviously linked to the actual crime suffered in that region but the exact relationship between crime and its fear is quite subtle. Two regions may have the same crime rate but their local perception of security may differ. Equally, two places may have the same perception of security even though one may have a significantly lower crime rate. Furthermore, a negative perception might persist for many years, even when crime rates drop. Here, we develop a model for the dynamics of the perception of security of a region based on the distribution of crime suffered by the population using concepts similar to those used for opinion dynamics. Simulations under a variety of conditions illustrate different scenarios and help us determine the impact of suffering more, or less, crime. The inhomogeneous concentration of crime together with a memory loss process is incorporated into the model for the perception of security, and results explain why people are often more fearful than actually victimized; why a region is perceived as being insecure despite a low crime rate; and why a decrease in the crime rate might not significantly improve the perception of security.
]]>2017-07-12T00:08:17-07:00info:doi/10.1098/rspa.2017.0156hwp:master-id:royprsa;rspa.2017.01562017-07-12Research articles47322032017015620170156<![CDATA[Interferometric visibility and coherence]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2203/20170170?rss=1
Recently, the basic concept of quantum coherence (or superposition) has gained a lot of renewed attention, after Baumgratz et al. (Phys. Rev. Lett.113, 140401. (doi:10.1103/PhysRevLett.113.140401)), following Åberg (http://arxiv.org/abs/quant-ph/0612146), have proposed a resource theoretic approach to quantify it. This has resulted in a large number of papers and preprints exploring various coherence monotones, and debating possible forms for the resource theory. Here, we take the view that the operational foundation of coherence in a state, be it quantum or otherwise wave mechanical, lies in the observation of interference effects. Our approach here is to consider an idealized multi-path interferometer, with a suitable detector, in such a way that the visibility of the interference pattern provides a quantitative expression of the amount of coherence in a given probe state. We present a general framework of deriving coherence measures from visibility, and demonstrate it by analysing several concrete visibility parameters, recovering some known coherence measures and obtaining some new ones.
]]>2017-07-26T00:08:18-07:00info:doi/10.1098/rspa.2017.0170hwp:master-id:royprsa;rspa.2017.01702017-07-26Research articles47322032017017020170170<![CDATA[The role of molybdenum in suppressing cold dwell fatigue in titanium alloys]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2203/20170189?rss=1
We test a hypothesis to explain why Ti-6242 is susceptible to cold dwell fatigue (CDF), whereas Ti-6246 is not. The hypothesis is that, in Ti-6246, substitutional Mo-atoms in α-Ti grains trap vacancies, thereby limiting creep relaxation. In Ti-6242, this creep relaxation enhances the loading of grains unfavourably oriented for slip and they subsequently fracture. Using density functional theory to calculate formation and binding energies between Mo-atoms and vacancies, we find no support for the hypothesis. In the light of this result, and experimental observations of the microstructures in these alloys, we agree with the recent suggestion (Qiu et al. 2014 Metall. Mater. Trans. A45, 6075–6087. (doi:10.1007/s11661-014-2541-5)) that Ti-6246 has a much smaller susceptibility to CDF because it has a smaller grain size and a more homogeneous distribution of grain orientations. We propose that the reduction of the susceptibility to CDF of Ti-6242 at temperatures above about 200°C is due to the activation of <c+a> slip in ‘hard’ grains, which reduces the loading of grain boundaries.
]]>2017-07-05T00:08:08-07:00info:doi/10.1098/rspa.2017.0189hwp:master-id:royprsa;rspa.2017.01892017-07-05Research articles47322032017018920170189<![CDATA[A note on flow reversal in a wavy channel filled with anisotropic porous material]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2203/20170193?rss=1
Viscous flow through a symmetric wavy channel filled with anisotropic porous material is investigated analytically. Flow inside the porous bed is assumed to be governed by the anisotropic Brinkman equation. It is assumed that the ratio of the channel width to the wavelength is small (i.e. ^{2}<<1). The problem is solved up to O(^{2}) assuming that ^{2}^{2}<<1, where is the anisotropic ratio. The key purpose of this paper is to study the effect of anisotropic permeability on flow near the crests of the wavy channel which causes flow reversal. We present a detailed analysis of the flow reversal at the crests. The ratio of the permeabilities (anisotropic ratio) is responsible for the flow separation near the crests of the wall where viscous forces are effective. For a flow configuration (say, low amplitude parameter) in which there is no separation if the porous media is isotropic, introducing anisotropy causes flow separation. On the other hand, interestingly, flow separation occurs even in the case of isotropic porous medium if the amplitude parameter a is large.
]]>2017-07-12T00:08:17-07:00info:doi/10.1098/rspa.2017.0193hwp:master-id:royprsa;rspa.2017.01932017-07-12Research articles47322032017019320170193<![CDATA[Effects of boundary conditions on bistable behaviour in axisymmetrical shallow shells]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2203/20170230?rss=1
Multistable shells are thin-walled structures that have more than one stable state of self-stress. We consider isotropic axisymmetrical shallow shells of arbitrary polynomial shapes using a Föppl–von Kármán analytical model. By employing a Rayleigh–Ritz approach, we identify stable shapes from local minima in the strain energy formulation, and we formally characterize the level of influence of the boundary conditions on the critical geometry for achieving bistable inversion—an effect not directly answered in the literature. Systematic insight is afforded by connecting the boundary to ground through sets of extensional and rotational linear springs. For typical cap-like shells, it is shown that bistability is generally enhanced when the extensional spring stiffness increases and when the rotational spring stiffness decreases, i.e. when boundary movements in-plane are resisted but when their rotations are not; however, for certain other shapes and large in-plane stiffness values, bistability can be enhanced by resisting but not entirely preventing edge rotations. Our predictions are furnished as detailed regime maps of the critical geometry, which are accurately correlated against finite-element analysis. Furthermore, the suitabilities of single degree-of-freedom models, for which solutions are achieved in closed form, are evaluated and compared to our more accurate predictions.
]]>2017-07-19T00:08:13-07:00info:doi/10.1098/rspa.2017.0230hwp:master-id:royprsa;rspa.2017.02302017-07-19Research articles47322032017023020170230<![CDATA[Trigonal curves and algebro-geometric solutions to soliton hierarchies I]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2203/20170232?rss=1
This is the first part of a study, consisting of two parts, on Riemann theta function representations of algebro-geometric solutions to soliton hierarchies. In this part, using linear combinations of Lax matrices of soliton hierarchies, we introduce trigonal curves by their characteristic equations, explore general properties of meromorphic functions defined as ratios of the Baker–Akhiezer functions, and determine zeros and poles of the Baker–Akhiezer functions and their Dubrovin-type equations. We analyse the four-component AKNS soliton hierarchy in such a way that it leads to a general theory of trigonal curves applicable to construction of algebro-geometric solutions of an arbitrary soliton hierarchy.
]]>2017-07-05T00:08:08-07:00info:doi/10.1098/rspa.2017.0232hwp:master-id:royprsa;rspa.2017.02322017-07-05Research articles47322032017023220170232<![CDATA[Trigonal curves and algebro-geometric solutions to soliton hierarchies II]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2203/20170233?rss=1
This is a continuation of a study on Riemann theta function representations of algebro-geometric solutions to soliton hierarchies. In this part, we straighten out all flows in soliton hierarchies under the Abel–Jacobi coordinates associated with Lax pairs, upon determining the Riemann theta function representations of the Baker–Akhiezer functions, and generate algebro-geometric solutions to soliton hierarchies in terms of the Riemann theta functions, through observing asymptotic behaviours of the Baker–Akhiezer functions. We emphasize that we analyse the four-component AKNS soliton hierarchy in such a way that it leads to a general theory of trigonal curves applicable to construction of algebro-geometric solutions of an arbitrary soliton hierarchy.
]]>2017-07-05T00:08:08-07:00info:doi/10.1098/rspa.2017.0233hwp:master-id:royprsa;rspa.2017.02332017-07-05Research articles47322032017023320170233<![CDATA[Piecewise affine stress-free martensitic inclusions in planar nonlinear elasticity]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2203/20170235?rss=1
We consider a partial differential inclusion problem which models stress-free martensitic inclusions in an austenitic matrix, based on the standard geometrically nonlinear elasticity theory. We show that for specific parameter choices there exist piecewise affine continuous solutions for the square-to-oblique and the hexagonal-to-oblique phase transitions. This suggests that for specific crystallographic parameters the hysteresis of the phase transformation will be particularly small.
]]>2017-07-05T00:08:16-07:00info:doi/10.1098/rspa.2017.0235hwp:master-id:royprsa;rspa.2017.02352017-07-05Research articles47322032017023520170235<![CDATA[Modulational instability, beak-shaped rogue waves, multi-dark-dark solitons and dynamics in pair-transition-coupled nonlinear Schrödinger equations]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2203/20170243?rss=1
The integrable coupled nonlinear Schrödinger equations with four-wave mixing are investigated. We first explore the conditions for modulational instability of continuous waves of this system. Secondly, based on the generalized N-fold Darboux transformation (DT), beak-shaped higher-order rogue waves (RWs) and beak-shaped higher-order rogue wave pairs are derived for the coupled model with attractive interaction in terms of simple determinants. Moreover, we derive the simple multi-dark-dark and kink-shaped multi-dark-dark solitons for the coupled model with repulsive interaction through the generalizing DT. We explore their dynamics and classifications by different kinds of spatial–temporal distribution structures including triangular, pentagonal, ‘claw-like’ and heptagonal patterns. Finally, we perform the numerical simulations to predict that some dark solitons and RWs are stable enough to develop within a short time. The results would enrich our understanding on nonlinear excitations in many coupled nonlinear wave systems with transition coupling effects.
]]>2017-07-26T00:08:18-07:00info:doi/10.1098/rspa.2017.0243hwp:master-id:royprsa;rspa.2017.02432017-07-26Research articles47322032017024320170243<![CDATA[A non-local asymptotic theory for thin elastic plates]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2203/20170249?rss=1
The three-dimensional dynamic non-local elasticity equations for a thin plate are subject to asymptotic analysis assuming the plate thickness to be much greater than a typical microscale size. The integral constitutive relations, incorporating the variation of an exponential non-local kernel across the thickness, are adopted. Long-wave low-frequency approximations are derived for both bending and extensional motions. Boundary layers specific for non-local behaviour are revealed near the plate faces. It is established that the effect of the boundary layers leads to the first-order corrections to the bending and extensional stiffness in the classical two-dimensional plate equations.
]]>2017-07-12T00:08:17-07:00info:doi/10.1098/rspa.2017.0249hwp:master-id:royprsa;rspa.2017.02492017-07-12Research articles47322032017024920170249<![CDATA[Mathematical and computational modelling of skin biophysics: a review]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2203/20170257?rss=1
The objective of this paper is to provide a review on some aspects of the mathematical and computational modelling of skin biophysics, with special focus on constitutive theories based on nonlinear continuum mechanics from elasticity, through anelasticity, including growth, to thermoelasticity. Microstructural and phenomenological approaches combining imaging techniques are also discussed. Finally, recent research applications on skin wrinkles will be presented to highlight the potential of physics-based modelling of skin in tackling global challenges such as ageing of the population and the associated skin degradation, diseases and traumas.
]]>2017-07-26T00:08:18-07:00info:doi/10.1098/rspa.2017.0257hwp:master-id:royprsa;rspa.2017.02572017-07-26Review article47322032017025720170257<![CDATA[Error bounds for the asymptotic expansion of the Hurwitz zeta function]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2203/20170363?rss=1
In this paper, we reconsider the large-a asymptotic expansion of the Hurwitz zeta function (s,a). New representations for the remainder term of the asymptotic expansion are found and used to obtain sharp and realistic error bounds. Applications to the asymptotic expansions of the polygamma functions, the gamma function, the Barnes G-function and the s-derivative of the Hurwitz zeta function (s,a) are provided. A detailed discussion on the sharpness of our error bounds is also given.
]]>2017-07-05T00:08:16-07:00info:doi/10.1098/rspa.2017.0363hwp:master-id:royprsa;rspa.2017.03632017-07-05Research articles47322032017036320170363<![CDATA[Is a time symmetric interpretation of quantum theory possible without retrocausality?]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20160607?rss=1
Huw Price has proposed an argument that suggests a time symmetric ontology for quantum theory must necessarily be retrocausal, i.e. it must involve influences that travel backwards in time. One of Price's assumptions is that the quantum state is a state of reality. However, one of the reasons for exploring retrocausality is that it offers the potential for evading the consequences of no-go theorems, including recent proofs of the reality of the quantum state. Here, we show that this assumption can be replaced by a different assumption, called -mediation, that plausibly holds independently of the status of the quantum state. We also reformulate the other assumptions behind the argument to place them in a more general framework and pin down the notion of time symmetry involved more precisely. We show that our assumptions imply a timelike analogue of Bell's local causality criterion and, in doing so, give a new interpretation of timelike violations of Bell inequalities. Namely, they show the impossibility of a (non-retrocausal) time symmetric ontology.
]]>2017-06-21T00:08:19-07:00info:doi/10.1098/rspa.2016.0607hwp:master-id:royprsa;rspa.2016.06072017-06-21Research articles47322022016060720160607<![CDATA[Liquid toroidal drop under uniform electric field]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20160633?rss=1
The problem of a stationary liquid toroidal drop freely suspended in another fluid and subjected to an electric field uniform at infinity is addressed analytically. Taylor’s discriminating function implies that, when the phases have equal viscosities and are assumed to be slightly conducting (leaky dielectrics), a spherical drop is stationary when Q=(2R^{2}+3R+2)/(7R^{2}), where R and Q are ratios of the phases’ electric conductivities and dielectric constants, respectively. This condition holds for any electric capillary number, Ca_{E}, that defines the ratio of electric stress to surface tension. Pairam and Fernández-Nieves showed experimentally that, in the absence of external forces (Ca_{E}=0), a toroidal drop shrinks towards its centre, and, consequently, the drop can be stationary only for some Ca_{E}>0. This work finds Q and Ca_{E} such that, under the presence of an electric field and with equal viscosities of the phases, a toroidal drop having major radius and volume 4/3 is qualitatively stationary—the normal velocity of the drop’s interface is minute and the interface coincides visually with a streamline. The found Q and Ca_{E} depend on R and , and for large , e.g. ≥3, they have simple approximations: Q~(R^{2}+R+1)/(3R^{2}) and CaE~33/2 (6 ln +2 ln[96]–9)/(12 ln +4 ln[96]–17) (R+1)2/(R–1)2.
]]>2017-06-21T03:05:36-07:00info:doi/10.1098/rspa.2016.0633hwp:master-id:royprsa;rspa.2016.06332017-06-21Research articles47322022016063320160633<![CDATA[Nonlinear model identification and spectral submanifolds for multi-degree-of-freedom mechanical vibrations]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20160759?rss=1
In a nonlinear oscillatory system, spectral submanifolds (SSMs) are the smoothest invariant manifolds tangent to linear modal subspaces of an equilibrium. Amplitude–frequency plots of the dynamics on SSMs provide the classic backbone curves sought in experimental nonlinear model identification. We develop here, a methodology to compute analytically both the shape of SSMs and their corresponding backbone curves from a data-assimilating model fitted to experimental vibration signals. This model identification utilizes Taken’s delay-embedding theorem, as well as a least square fit to the Taylor expansion of the sampling map associated with that embedding. The SSMs are then constructed for the sampling map using the parametrization method for invariant manifolds, which assumes that the manifold is an embedding of, rather than a graph over, a spectral subspace. Using examples of both synthetic and real experimental data, we demonstrate that this approach reproduces backbone curves with high accuracy.
]]>2017-06-14T00:07:50-07:00info:doi/10.1098/rspa.2016.0759hwp:master-id:royprsa;rspa.2016.07592017-06-14Research articles47322022016075920160759<![CDATA[On the regularization of impact without collision: the Painleve paradox and compliance]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20160773?rss=1
We consider the problem of a rigid body, subject to a unilateral constraint, in the presence of Coulomb friction. We regularize the problem by assuming compliance (with both stiffness and damping) at the point of contact, for a general class of normal reaction forces. Using a rigorous mathematical approach, we recover impact without collision (IWC) in both the inconsistent and the indeterminate Painlevé paradoxes, in the latter case giving an exact formula for conditions that separate IWC and lift-off. We solve the problem for arbitrary values of the compliance damping and give explicit asymptotic expressions in the limiting cases of small and large damping, all for a large class of rigid bodies.
]]>2017-06-14T00:07:50-07:00info:doi/10.1098/rspa.2016.0773hwp:master-id:royprsa;rspa.2016.07732017-06-14Research articles47322022016077320160773<![CDATA[Energy-based analysis of biomolecular pathways]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20160825?rss=1
Decomposition of biomolecular reaction networks into pathways is a powerful approach to the analysis of metabolic and signalling networks. Current approaches based on analysis of the stoichiometric matrix reveal information about steady-state mass flows (reaction rates) through the network. In this work, we show how pathway analysis of biomolecular networks can be extended using an energy-based approach to provide information about energy flows through the network. This energy-based approach is developed using the engineering-inspired bond graph methodology to represent biomolecular reaction networks. The approach is introduced using glycolysis as an exemplar; and is then applied to analyse the efficiency of free energy transduction in a biomolecular cycle model of a transporter protein [sodium-glucose transport protein 1 (SGLT1)]. The overall aim of our work is to present a framework for modelling and analysis of biomolecular reactions and processes which considers energy flows and losses as well as mass transport.
]]>2017-06-21T00:08:19-07:00info:doi/10.1098/rspa.2016.0825hwp:master-id:royprsa;rspa.2016.08252017-06-21Research articles47322022016082520160825<![CDATA[Knotted fields and explicit fibrations for lemniscate knots]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20160829?rss=1
We give an explicit construction of complex maps whose nodal lines have the form of lemniscate knots. We review the properties of lemniscate knots, defined as closures of braids where all strands follow the same transverse (1, ) Lissajous figure, and are therefore a subfamily of spiral knots generalizing the torus knots. We then prove that such maps exist and are in fact fibrations with appropriate choices of parameters. We describe how this may be useful in physics for creating knotted fields, in quantum mechanics, optics and generalizing to rational maps with application to the Skyrme–Faddeev model. We also prove how this construction extends to maps with weakly isolated singularities.
]]>2017-06-07T01:27:08-07:00info:doi/10.1098/rspa.2016.0829hwp:master-id:royprsa;rspa.2016.08292017-06-07Research articles47322022016082920160829<![CDATA[The cutting of metals via plastic buckling]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20160863?rss=1
The cutting of metals has long been described as occurring by laminar plastic flow. Here we show that for metals with large strain-hardening capacity, laminar flow mode is unstable and cutting instead occurs by plastic buckling of a thin surface layer. High speed in situ imaging confirms that the buckling results in a small bump on the surface which then evolves into a fold of large amplitude by rotation and stretching. The repeated occurrence of buckling and folding manifests itself at the mesoscopic scale as a new flow mode with significant vortex-like components—sinuous flow. The buckling model is validated by phenomenological observations of flow at the continuum level and microstructural characteristics of grain deformation and measurements of the folding. In addition to predicting the conditions for surface buckling, the model suggests various geometric flow control strategies that can be effectively implemented to promote laminar flow, and suppress sinuous flow in cutting, with implications for industrial manufacturing processes. The observations impinge on the foundations of metal cutting by pointing to the key role of stability of laminar flow in determining the mechanism of material removal, and the need to re-examine long-held notions of large strain deformation at surfaces.
]]>2017-06-07T00:07:57-07:00info:doi/10.1098/rspa.2016.0863hwp:master-id:royprsa;rspa.2016.08632017-06-07Research articles47322022016086320160863<![CDATA[Regularly configured structures with polygonal prisms for three-dimensional auxetic behaviour]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20160926?rss=1
We report here structures, constructed with regular polygonal prisms, that exhibit negative Poisson’s ratios. In particular, we show how we can construct such a structure with regular n-gonal prism-shaped unit cells that are again built with regular n-gonal component prisms. First, we show that the only three possible values for n are 3, 4 and 6 and then discuss how we construct the unit cell again with regular n-gonal component prisms. Then, we derive Poisson’s ratio formula for each of the three structures and show, by analysis and numerical verification, that the structures possess negative Poisson’s ratio under certain geometric conditions.
]]>2017-06-14T00:07:51-07:00info:doi/10.1098/rspa.2016.0926hwp:master-id:royprsa;rspa.2016.09262017-06-14Research articles47322022016092620160926<![CDATA[Nonlinear evolution of a thin anodic film]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20160930?rss=1
The formation of pores in anodic aluminium oxide films is treated with a model equation. The model treats the oxide layer as a thin viscous liquid in two dimensions. Surface tension on the top boundary, electrostriction due to the external electric field and mass flow through the bottom boundary due to oxide formation are all included. Viscous flow is treated with the creeping flow assumption. The model equation is solved numerically using a Fourier spectral method in space and Adams–Bashforth/Adams–Moulton methods in time. Initial conditions include sinusoidal shapes as well as random shapes. The results show that pores form at the trough of the initial sinusoidal shape. Random shapes get smoothed before forming pore structures with spacing different than predicted by linear theory.
]]>2017-06-07T00:07:57-07:00info:doi/10.1098/rspa.2016.0930hwp:master-id:royprsa;rspa.2016.09302017-06-07Research articles47322022016093020160930<![CDATA[Wave reflection and transmission in multiply stented blood vessels]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20170015?rss=1
Closed circulatory systems display an exquisite balance between vascular elasticity and viscous fluid effects, to induce pulse-smoothing and avoid resonance during the cardiac cycle. Stents in the arterial tree alter this balance through stiffening and because a periodic structure is introduced, capable of interacting with the fluid in a complex way. While the former feature has been investigated, the latter received no attention so far. But periodic structures are the building blocks of metamaterials, known for their ‘non-natural’ behaviour. Thus, the investigation of a stent's periodic microstructure dynamical interactions is crucial to assess possible pathological responses. A one-dimensional fluid–structure interaction model, simple enough to allow an analytical solution for situations of interest involving one or two interacting stents, is introduced. It is determined: (i) whether or not frequency bands exist in which reflected blood pulses are highly increased and (ii) if these bands are close to the characteristic frequencies of arteries and finally, (iii) if the internal structure of the stent can sensibly affect arterial blood dynamics. It is shown that, while the periodic structure of an isolated stent can induce anomalous reflection only in pathological conditions, the presence of two interacting stents is more critical, and high reflection can occur at frequencies not far from the physiological values.
]]>2017-06-07T01:27:08-07:00info:doi/10.1098/rspa.2017.0015hwp:master-id:royprsa;rspa.2017.00152017-06-07Research articles47322022017001520170015<![CDATA[Turbulent boundary layer under the control of different schemes]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20170038?rss=1
This work explores experimentally the control of a turbulent boundary layer over a flat plate based on wall perturbation generated by piezo-ceramic actuators. Different schemes are investigated, including the feed-forward, the feedback, and the combined feed-forward and feedback strategies, with a view to suppressing the near-wall high-speed events and hence reducing skin friction drag. While the strategies may achieve a local maximum drag reduction slightly less than their counterpart of the open-loop control, the corresponding duty cycles are substantially reduced when compared with that of the open-loop control. The results suggest a good potential to cut down the input energy under these control strategies. The fluctuating velocity, spectra, Taylor microscale and mean energy dissipation are measured across the boundary layer with and without control and, based on the measurements, the flow mechanism behind the control is proposed.
]]>2017-06-28T00:08:07-07:00info:doi/10.1098/rspa.2017.0038hwp:master-id:royprsa;rspa.2017.00382017-06-28Research articles47322022017003820170038<![CDATA[Ultrasonic defect characterization using parametric-manifold mapping]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20170056?rss=1
The aim of ultrasonic non-destructive evaluation includes the detection and characterization of defects, and an understanding of the nature of defects is essential for the assessment of structural integrity in safety critical systems. In general, the defect characterization challenge involves an estimation of defect parameters from measured data. In this paper, we explore the extent to which defects can be characterized by their ultrasonic scattering behaviour. Given a number of ultrasonic measurements, we show that characterization information can be extracted by projecting the measurement onto a parametric manifold in principal component space. We show that this manifold represents the entirety of the characterization information available from far-field harmonic ultrasound. We seek to understand the nature of this information and hence provide definitive statements on the defect characterization performance that is, in principle, extractable from typical measurement scenarios. In experiments, the characterization problem of surface-breaking cracks and the more general problem of elliptical voids are studied, and a good agreement is achieved between the actual parameter values and the characterization results. The nature of the parametric manifold enables us to explain and quantify why some defects are relatively easy to characterize, whereas others are inherently challenging.
]]>2017-06-07T01:27:08-07:00info:doi/10.1098/rspa.2017.0056hwp:master-id:royprsa;rspa.2017.00562017-06-07Research articles47322022017005620170056<![CDATA[The rolling suitcase instability: a coupling between translation and rotation]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20170076?rss=1
A two-wheel suitcase or trolley can exhibit undamped rocking oscillations from one wheel to the other when pulled fast enough. We study this instability both experimentally—with a toy model of a suitcase rolling on a treadmill—and theoretically. The suitcase oscillates only if a finite perturbation is applied. This is because intrinsic dissipation occurs when the supporting wheel switches. When unstable, the suitcase either increasingly rocks until overturning or reaches a stable limit cycle. The friction force at the rolling wheels constrains wheels to roll without slipping. This constraint imposes a coupling between the translational motion and the three-dimensional rotational motion of the suitcase that drives the rocking instability. The same behaviours are observed in the experiments and in the simulations. The asymptotic scaling laws we observe in the simulations are explained by means of a simplified model where the coupling force is explicit.
]]>2017-06-21T00:08:19-07:00info:doi/10.1098/rspa.2017.0076hwp:master-id:royprsa;rspa.2017.00762017-06-21Research articles47322022017007620170076<![CDATA[Addressing the discrepancy of finding the equilibrium melting point of silicon using molecular dynamics simulations]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20170084?rss=1
We performed molecular dynamics simulations to study the equilibrium melting point of silicon using (i) the solid–liquid coexistence method and (ii) the Gibbs free energy technique, and compared our novel results with the previously published results obtained from the Monte Carlo (MC) void-nucleated melting method based on the Tersoff-ARK interatomic potential (Agrawal et al. Phys. Rev. B72, 125206. (doi:10.1103/PhysRevB.72.125206)). Considerable discrepancy was observed (approx. 20%) between the former two methods and the MC void-nucleated melting result, leading us to question the applicability of the empirical MC void-nucleated melting method to study a wide range of atomic and molecular systems. A wider impact of the study is that it highlights the bottleneck of the Tersoff-ARK potential in correctly estimating the melting point of silicon.
]]>2017-06-07T00:07:57-07:00info:doi/10.1098/rspa.2017.0084hwp:master-id:royprsa;rspa.2017.00842017-06-07Research articles47322022017008420170084<![CDATA[Analytical solutions for two-dimensional Stokes flow singularities in a no-slip wedge of arbitrary angle]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20170134?rss=1
An analytical method to find the flow generated by the basic singularities of Stokes flow in a wedge of arbitrary angle is presented. Specifically, we solve a biharmonic equation for the stream function of the flow generated by a point stresslet singularity and satisfying no-slip boundary conditions on the two walls of the wedge. The method, which is readily adapted to any other singularity type, takes full account of any transcendental singularities arising at the corner of the wedge. The approach is also applicable to problems of plane strain/stress of an elastic solid where the biharmonic equation also governs the Airy stress function.
]]>2017-06-07T00:07:57-07:00info:doi/10.1098/rspa.2017.0134hwp:master-id:royprsa;rspa.2017.01342017-06-07Research articles47322022017013420170134<![CDATA[Extensional channel flow revisited: a dynamical systems perspective]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20170151?rss=1
Extensional self-similar flows in a channel are explored numerically for arbitrary stretching–shrinking rates of the confining parallel walls. The present analysis embraces time integrations, and continuations of steady and periodic solutions unfolded in the parameter space. Previous studies focused on the analysis of branches of steady solutions for particular stretching–shrinking rates, although recent studies focused also on the dynamical aspects of the problems. We have adopted a dynamical systems perspective, analysing the instabilities and bifurcations the base state undergoes when increasing the Reynolds number. It has been found that the base state becomes unstable for small Reynolds numbers, and a transitional region including complex dynamics takes place at intermediate Reynolds numbers, depending on the wall acceleration values. The base flow instabilities are constitutive parts of different codimension-two bifurcations that control the dynamics in parameter space. For large Reynolds numbers, the restriction to self-similarity results in simple flows with no realistic behaviour, but the flows obtained in the transition region can be a valuable tool for the understanding of the dynamics of realistic Navier–Stokes solutions.
]]>2017-06-07T01:27:08-07:00info:doi/10.1098/rspa.2017.0151hwp:master-id:royprsa;rspa.2017.01512017-06-07Research articles47322022017015120170151<![CDATA[A free-boundary model of diffusive valley growth: theory and observation]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20170159?rss=1
Valleys that form around a stream head often develop characteristic finger-like elevation contours. We study the processes involved in the formation of these valleys and introduce a theoretical model that indicates how shape may inform the underlying processes. We consider valley growth as the advance of a moving boundary travelling forward purely through linearly diffusive erosion, and we obtain a solution for the valley shape in three dimensions. Our solution compares well to the shape of slowly growing groundwater-fed valleys found in Bristol, Florida. Our results identify a new feature in the formation of groundwater-fed valleys: a spatially variable diffusivity that can be modelled by a fixed-height moving boundary.
]]>2017-06-21T00:08:19-07:00info:doi/10.1098/rspa.2017.0159hwp:master-id:royprsa;rspa.2017.01592017-06-21Research articles47322022017015920170159<![CDATA[Windowed Green function method for the Helmholtz equation in the presence of multiply layered media]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20170161?rss=1
This paper presents a new methodology for the solution of problems of two- and three-dimensional acoustic scattering (and, in particular, two-dimensional electromagnetic scattering) by obstacles and defects in the presence of an arbitrary number of penetrable layers. Relying on the use of certain slow-rise windowing functions, the proposed windowed Green function approach efficiently evaluates oscillatory integrals over unbounded domains, with high accuracy, without recourse to the highly expensive Sommerfeld integrals that have typically been used to account for the effect of underlying planar multilayer structures. The proposed methodology, whose theoretical basis was presented in the recent contribution (Bruno et al. 2016 SIAM J. Appl. Math.76, 1871–1898. (doi:10.1137/15M1033782)), is fast, accurate, flexible and easy to implement. Our numerical experiments demonstrate that the numerical errors resulting from the proposed approach decrease faster than any negative power of the window size. In a number of examples considered in this paper, the proposed method is up to thousands of times faster, for a given accuracy, than corresponding methods based on the use of Sommerfeld integrals.
]]>2017-06-14T00:07:51-07:00info:doi/10.1098/rspa.2017.0161hwp:master-id:royprsa;rspa.2017.01612017-06-14Research articles47322022017016120170161<![CDATA[Non-reciprocal wave propagation in modulated elastic metamaterials]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20170188?rss=1
Time-reversal symmetry for elastic wave propagation breaks down in a resonant mass-in-mass lattice whose inner-stiffness is weakly modulated in space and in time in a wave-like fashion. Specifically, one-way wave transmission, conversion and amplification as well as unidirectional wave blocking are demonstrated analytically through an asymptotic analysis based on coupled mode theory and numerically thanks to a series of simulations in harmonic and transient regimes. High-amplitude modulations are then explored in the homogenization limit where a non-standard effective mass operator is recovered and shown to take negative values over unusually large frequency bands. These modulated metamaterials, which exhibit either non-reciprocal behaviours or non-standard effective mass operators, offer promise for applications in the field of elastic wave control in general and in one-way conversion/amplification in particular.
]]>2017-06-28T00:08:07-07:00info:doi/10.1098/rspa.2017.0188hwp:master-id:royprsa;rspa.2017.01882017-06-28Research articles47322022017018820170188<![CDATA[The effect of coherent stirring on the advection-condensation of water vapour]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20170196?rss=1
Atmospheric water vapour is an essential ingredient of weather and climate. The key features of its distribution can be represented by kinematic models which treat it as a passive scalar advected by a prescribed flow and reacting through condensation. Condensation acts as a sink that maintains specific humidity below a prescribed, space-dependent saturation value. To investigate how the interplay between large-scale advection, small-scale turbulence and condensation controls moisture distribution, we develop simple kinematic models which combine a single circulating flow with a Brownian-motion representation of turbulence. We first study the drying mechanism of a water-vapour anomaly released inside a vortex at an initial time. Next, we consider a cellular flow with a moisture source at a boundary. The statistically steady state attained shows features reminiscent of the Hadley cell such as boundary layers, a region of intense precipitation and a relative humidity minimum. Explicit results provide a detailed characterization of these features in the limit of strong flow.
]]>2017-06-28T00:08:07-07:00info:doi/10.1098/rspa.2017.0196hwp:master-id:royprsa;rspa.2017.01962017-06-28Research articles47322022017019620170196<![CDATA[New conformal mapping for adaptive resolving of the complex singularities of Stokes wave]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20170198?rss=1
A new highly efficient method is developed for computation of travelling periodic waves (Stokes waves) on the free surface of deep water. A convergence of numerical approximation is determined by the complex singularities above the free surface for the analytical continuation of the travelling wave into the complex plane. An auxiliary conformal mapping is introduced which moves singularities away from the free surface thus dramatically speeding up numerical convergence by adapting the numerical grid for resolving singularities while being consistent with the fluid dynamics. The efficiency of that conformal mapping is demonstrated for the Stokes wave approaching the limiting Stokes wave (the wave of the greatest height) which significantly expands the family of numerically accessible solutions. It allows us to provide a detailed study of the oscillatory approach of these solutions to the limiting wave. Generalizations of the conformal mapping to resolve multiple singularities are also introduced.
]]>2017-06-21T00:08:19-07:00info:doi/10.1098/rspa.2017.0198hwp:master-id:royprsa;rspa.2017.01982017-06-21Research articles47322022017019820170198<![CDATA[Correction to 'Homogenized boundary conditions and resonance effects in Faraday cages]]>
http://rspa.royalsocietypublishing.org/cgi/content/short/473/2202/20170331?rss=1
2017-06-14T00:07:51-07:00info:doi/10.1098/rspa.2017.0331hwp:master-id:royprsa;rspa.2017.03312017-06-14Corrections47322022017033120170331