Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences current issue
http://rspa.royalsocietypublishing.org
Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences RSS feed -- current issue1471-2946August 2, 2017Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences1364-5021<![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[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[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[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[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[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