Two-dimensional Version Research Articles

A global two-dimensional version of Smale’s cancellation theorem via spectral sequences

In this article, Conley’s connection matrix theory and a spectral sequence analysis of a filtered Morse chain complex $(C,{\rm\Delta})$ are used to study global continuation results for flows on surfaces. The briefly described unfoldings of Lyapunov graphs have been proved to be a well-suited combinatorial tool to keep track of continuations. The novelty herein is a global dynamical cancellation theorem inferred from the differentials of the spectral sequence $(E^{r},d^{r})$. The local version of this theorem relates differentials $d^{r}$ of the $r$th page $E^{r}$ to Smale’s theorem on cancellation of critical points.

The use of interpolating element-free Galerkin technique for solving 2D generalized Benjamin–Bona–Mahony–Burgers and regularized long-wave equations on non-rectangular domains with error estimate

In this paper a numerical technique is proposed for solving the nonlinear generalized Benjamin–Bona–Mahony–Burgers and regularized long-wave equations. Firstly, we obtain a time discrete scheme by approximating time derivative via a finite difference formula, then we use the interpolating element-free Galerkin approach to approximate the spatial derivatives. The element-free Galerkin method uses a weak form of the considered equation that is similar to the finite element method with the difference that in the element-free Galerkin method test and trial functions are moving least squares approximation shape functions. Also, in the element-free Galerkin method, we do not use any triangular, quadrangular or other type of meshes. It is a global method while finite element method is a local one. The element free Galerkin method is not a truly meshless method and for integration employs a background mesh. We prove that the time discrete scheme is unconditionally stable and convergent in time variable using the energy method. We show that convergence order of the time discrete scheme is O(τ). Since the shape functions of moving least squares approximation do not have Kronecker delta property, we cannot implement the essential boundary condition, directly. Thus, the improved moving least squares shape functions that have the mentioned property are employed. An error estimate for the method proposed in the current paper is obtained. Also, the two-dimensional version of both equations on different complex geometries is solved. The aim of this paper is to show that the meshless method based on the weak form is also suitable for the treatment of the nonlinear partial differential equations and to obtain an error bound for the new method. Numerical examples confirm the efficiency of the proposed scheme.

From Cultural Translation to Translations inside Photographs (1860–1930)

A significant proportion of photography in Spain, from Charles Clifford's earliest days in the 1850s to 1920s industrial images, aimed to offer its audience characteristic images of the country. This being so, much Spanish photography deals in cultural translation, seeking to transfer the realities of a society and culture into a fairly small, two-dimensional version, and then to transmit that version onward to an audience. The transfer from three-dimensional objects to two-dimensional forms has another consequence, however, regarding the effects created by those shapes. I explore here how the resulting forms and structures involve a whole series of other transformations and metamorphoses within the photographs, and of things crossing or being borne across the image (translatio). Cultural translation gives rise to translations within photographs. More still, that specific variety of translatio becomes a central subject and aesthetic characteristic of the images themselves.

Sensitivity of convection to observed variation in aerosol size distributions and composition at a rural site in the southeastern United States

We present a sensitivity analysis to determine the impact of variations in aerosol physical and chemical characteristics, as observed in the field in the southeastern United States, on convective cloud microphysics. Scenarios reflecting changes in aerosol properties, observed over the Piedmont of Virginia and associated with new particle formation and growth events, were evaluated using the two-dimensional version of the Goddard Cumulus Ensemble Model with detailed spectral-bin microphysics. Two aerosol size distributions represented early and late stages of particle growth, and two aerosol chemical compositions (norpinic acid and ammonium sulfate) represented extremes in aerosol hygroscopicity observed at the field site, for a total of four scenarios. The chosen compositions reflect inferred local changes in aerosol composition over short time scales. Variations in the aerosol size distribution and composition resulted in substantial variation in the total number of cloud condensation nuclei (CCN) produced in the four case studies. Cases with high CCN concentrations developed larger, more vigorous clouds with more precipitation generated by both warm and cold rain processes. Greater numbers of drops were propelled aloft and formed an extensive ice anvil that produced a large area of stratiform rain. Convection was enhanced by increasing aerosols despite decreases in precipitation efficiency. In contrast, lower CCN concentrations developed smaller clouds with suppressed cold rain processes and less total precipitation. The relatively small increase in CCN concentration associated with the increase in aerosol hygroscopicity resulted in an increase in accumulated modeled precipitation of 12 % after 180 min of simulation time in both high and low CCN cases. The increase in accumulated precipitation due to the substantial increase in CCN concentration associated with growth of the aerosol size distribution was 93 % for both aerosol compositions. The timing of the onset of precipitation was not affected by aerosol concentration or composition.

Three-dimensional particle-resolved direct numerical simulation study of behaviors of a particle settling near a vertical wall

Three-dimensional particle-resolved direct numerical simulation based on the multi-direct forcing and Immersed Boundary method is applied to investigate the indirect particle–wall interactions on the behaviors of a particle settling near a vertical wall in a viscous Newtonian fluid. The particle settling in unbounded domain is also simulated for the references of comparison. It is found that the sideways drift and the anomalous rotation are affected by the initial lateral position of the particle relative to the wall. The oscillation of the sedimentation velocity, rotation shifting and the three-dimensional zigzag migration are closely associated with the vortex shedding, depending on the particle terminal Reynolds number. A particle with small solid/fluid density ratio moves in a narrower range compared to a dense particle with the same Reynolds number when it is being drifted away from the wall, but if it is free of wall effect the light particle is to be more influenced by the eventual unsteadiness of its wake. It's interesting to find that the particle–fluid interaction force is also linked closely with the anomalous rotation rate. It can be proved that the particle–fluid interactions are stronger in three-dimensional cases than two-dimensional versions. First, the particle motion in the direction free of limitations is confined in a limited range due to the wall effect. Second, the rotation behavior is stronger in three-dimensional simulations. Third, the particle settling near a wall with large enough Reynolds number reaches an equilibrium stand-away distance away from the wall in three-dimensional simulations.

Simultaneous integer values of pairs of quadratic forms

Abstract We prove that a pair of integral quadratic forms in five or more variables will simultaneously represent “almost all” pairs of integers that satisfy the necessary local conditions, provided that the forms satisfy a suitable nonsingularity condition. In particular such forms simultaneously attain prime values if the obvious local conditions hold. The proof uses the circle method, and in particular pioneers a two-dimensional version of a Kloosterman refinement.

Design of Bi-Tortuous, Anisotropic Graphite Anodes for Fast Ion-Transport in Li-Ion Batteries

Thick Li-ion battery electrodes with high ion transport rates could enable batteries that cost less and that have higher gravimetric and volumetric energy density, because they require fewer inactive cell-components. Finding ways to increase ion transport rates in thick electrodes would be especially valuable for electrodes made with graphite platelets, which have been shown to have tortuosities in the thru-plane direction about 3 times higher than in the in-plane direction. Here, we predict that bi-tortuous electrode structures (containing electrolyte-filled macro-pores embedded in micro-porous graphite) can enhance ion transport and can achieve double the discharge capacity compared to an unstructured electrode at the same average porosity. We introduce a new two-dimensional version of porous-electrode theory with anisotropic ion transport to investigate these effects and to interpret the mechanisms by which performance enhancements arise. From this analysis we determine criteria for the design of bi-tortuous graphite anodes, including the particular volume fraction of macro-pores that maximizes discharge capacity (approximately 20 vol.%) and a threshold spacing interval (half the electrode's thickness) below which only marginal enhancement in discharge capacity is obtained. We also report the sensitivity of performance with respect to cycling rate, electrode thickness, and average porosity/electroactive-material loading.

Two species competitive model with the Allee effect

We consider the following system of difference equations: , , , where a, b, c, d are positive constants and are initial conditions. This system has interesting dynamics and can have up to nine equilibrium points. The most complex and perhaps most interesting case is the case of nine equilibrium points, four of which are local attractors, four of which are saddle points, and one of which is a repeller. Using recent results of Kulenović and Merino we are able to characterize the basins of attractions of all local attractors and thus to describe the global dynamics of this system. This case can be considered as a two-dimensional version of the Allee effect for competitive systems. MSC:39A10, 39A30, 37G35.

Explicit solution of one boundary value problem in the full coupled theory of elasticity for solids with double porosity

Our goal was to consider the two-dimensional version of the full coupled linear equilibrium theory of elasticity for materials with double porosity and to construct explicitly the solutions of BVPs, in the form of absolutely and uniformly convergent series that is useful in engineering practice. In this paper, the Neumann-type BVPs of statics for an elastic circle and for a plane with circular hole are considered. The uniqueness theorems of the considered boundary value problems are proved.

An implicit RBF meshless approach for solving the time fractional nonlinear sine-Gordon and Klein–Gordon equations

In this paper, we propose a numerical method for the solution of time fractional nonlinear sine-Gordon equation that appears extensively in classical lattice dynamics in the continuum media limit and Klein–Gordon equation which arises in physics. In this method we first approximate the time fractional derivative of the mentioned equations by a scheme of order O(τ3−α),1<α<2 then we will use the Kansa approach to approximate the spatial derivatives. We solve the two-dimensional version of these equations using the method presented in this paper on different domains such as rectangular and non-rectangular domains. Also, we prove the unconditional stability and convergence of the time discrete scheme. We show that convergence order of the time discrete scheme is O(τ). We solve these fractional PDEs on different non-rectangular domains. The aim of this paper is to show that the meshless method based on the radial basis functions and collocation approach is also suitable for the treatment of the nonlinear time fractional PDEs. The results of numerical experiments are compared with analytical solutions to confirm the accuracy and efficiency of the presented scheme.

Effects of Tides and Currents on Tsunami Propagation in Large Rivers: Columbia River, United States

AbstractTo better understand the effects of river flow and tidal stage on the propagation of realistic tsunami waves up large rivers, simulations of the Columbia River (United States) were carried out. The two-dimensional depth-averaged version of the advanced circulation model was used. The model was forced with open boundary tides, tidal potential, and river flow. Simulations without tsunami waves demonstrated excellent agreement with numerous tidal gauging stations in the domain. A Cascadia Subduction Zone earthquake and associated tsunami were then considered by prescribing a rapid temporal change in the bathymetric/topographic grid. The simulations were repeated over a wide range of river flow conditions and a wide range of tidal phases. In each simulation, the tsunami was characterized in terms of the variation of its amplitude with upriver distance, how much land was inundated, and its propagation speed up the estuary. Although river flow is a dominant mechanism in the Columbia River in terms of st...

Evaluation of a Cascade-Based Acoustic Model for Fan Tonal Noise Prediction

This study aims at evaluating an analytical model for the prediction of tonal fan noise created by the rotor–stator interaction. The methodology consists of comparing unsteady flow simulations with the cascade-based acoustic model to quantify the influence of some technological effects not included in the model. The simulations are performed with a dedicated turbomachinery flow solver Turb’Flow on a simplified stator vane cascade. They allow discussing the effects of both model assumptions of no vane thickness and inviscid flow on the predictions of the acoustic sources as well as on the modal acoustic powers radiated within the duct. The Kutta condition is found to be efficient to locally represent the viscosity effects, and the vane thickness tends to moderately modify the distributions of the sources and the amplitudes of the duct modes. Moreover, a three-dimensional decomposition of the aerodynamic excitation is proposed and coupled with the three-dimensional analytical cascade response. This fully three-dimensional acoustic model is evaluated by comparisons with an unsteady simulation of a realistic axial compressor stage. An improvement in the prediction of the acoustic sources and powers is clearly shown with respect to the previously used two-dimensional version of the model.

Gravity wave emission in an atmosphere-like configuration of the differentially heated rotating annulus experiment

AbstractA finite-volume model of the classic differentially heated rotating annulus experiment is used to study the spontaneous emission of gravity waves (GWs) from jet stream imbalances, which may be an important source of these waves in the atmosphere and for which no satisfactory parameterisation exists. Experiments were performed using a classic laboratory configuration as well as using a much wider and shallower annulus with a much larger temperature difference between the inner and outer cylinder walls. The latter configuration is more atmosphere-like, in particular since the Brunt–Väisälä frequency is larger than the inertial frequency, resulting in more realistic GW dispersion properties. In both experiments, the model is initialised with a baroclinically unstable axisymmetric state established using a two-dimensional version of the code, and a low-azimuthal-mode baroclinic wave featuring a meandering jet is allowed to develop. Possible regions of GW activity are identified by the horizontal velocity divergence and a modal decomposition of the small-scale structures of the flow. Results indicate GW activity in both annulus configurations close to the inner cylinder wall and within the baroclinic wave. The former is attributable to boundary layer instabilities, while the latter possibly originates in part from spontaneous GW emission from the baroclinic wave.

Local multifractal detrended fluctuation analysis for tea breeds identification

In recent years, the popular multifractal detrended fluctuation analysis (MF-DFA) is extended to two-dimensional (2D) version, which has been applied in some field of image processing. In this paper, based on the 2D MF-DFA, a novel multifractal estimation method for images, which we called the local multifractal detrended fluctuation analysis (LMF-DFA), is proposed to recognize and distinguish 20 types of tea breeds. A set of new multifractal descriptors, namely the local multifractal fluctuation exponents is defined to portray the local scaling properties of a surface. After collecting 10 tea leaves for each breed and photographing them to standard images, the LMF-DFA method is used to extract characteristic parameters for the images. Our analysis finds that there are significant differences among the different tea breeds' characteristic parameters by analysis of variance. Both the proposed LMF-DFA exponents and another classic parameter, namely the exponent based on capacity measure method have been used as features to distinguish the 20 tea breeds. The comparison results illustrate that the LMF-DFA estimation can differentiate the tea breeds more effectively and provide more satisfactory accuracy.

Mathematical model of stem cell differentiation and tissue regeneration with stochastic noise.

Differentiation and self-renewal of stem cells is an essential process for the maintenance of tissue composition. The promise of novel medical therapies combined with the complexity of this process encourage us to employ numerical and mathematical methods. This will allow us to understand better the mechanisms which regulate stem cell behaviour. Perturbations to the cellular environment may have an influence on the death rate, proliferation rate and on the fraction of self-renewal at every stage of differentiation. In this paper, we present mathematical study of the effect of stochastic noise on the process of tissue regeneration. Here, a system of Itô stochastic differential equations with linear diffusion coefficients that is based on a deterministic model of multistage cell lineages is investigated. Numerical simulations of the stochastic model are shown for a different number of stages of differentiation. Interactions between the noise, added to the different stages, are characterised using numerical simulations. The long-time behaviour of the two-dimensional version of the model is fully characterised; asymptotic stability of the related Markov semigroup is proved using the theory of the Markov semigroups and the method of the Khasminskií function.

Heat transfer and film cooling effectiveness on the squealer tip of a turbine blade

Detailed heat/mass transfer coefficients and film-cooling effectiveness were measured on the tip and inner rim surfaces of a blade with a squealer rim. The test blade was a two-dimensional version of a modern first-stage gas turbine rotor blade with a squealer rim. The experimental apparatus was equipped with a linear cascade of three blades, and the axial chord length (Cx) was 237 mm with a turning angle of 126°, the mainstream Reynolds number based on the axial chord and inlet velocity was 1.5 × 105. In addition, three different types of blade tip surfaces were equipped with a single row of film-cooling holes along the camber line, near the pressure and suction-side rim. The blowing ratio was fixed at 1.5. High heat transfer rates were observed near the leading edge on the tip surface due to reattached flow. Furthermore, heat transfer on both inner side surfaces was higher than that on the tip surface. High film cooling effectiveness was observed in the middle region (0.1 < X/Cx < 0.6) due to stagnation of the film cooling. Ultimately, a proper cooling system is suggested to reduce the thermal load and enhance the film cooling effectiveness in the squealer tip.

The numerical solution of nonlinear high dimensional generalized Benjamin–Bona–Mahony–Burgers equation via the meshless method of radial basis functions

In this paper a numerical technique is proposed for solving the nonlinear generalized Benjamin–Bona–Mahony–Burgers equation. Firstly, we obtain a time discrete scheme by approximating the first-order time derivative via forward finite difference formula, then we use Kansa’s approach to approximate the spatial derivatives. We prove that the time discrete scheme is unconditionally stable and convergent in time variable using the energy method. Also, we show that convergence order of the time discrete scheme is O(τ). We solve the two-dimensional version of this equation using the method presented in this paper on different geometries such as the rectangular, triangular and circular domains and also the three-dimensional case is solved on the cubical and spherical domains. The aim of this paper is to show that the meshless method based on the radial basis functions and collocation approach is also suitable for the treatment of the nonlinear partial differential equations. Also, several test problems including the three-dimensional case are given. Numerical examples confirm the efficiency of the proposed scheme.

Holographic and time-resolving ability of pulse-pair two-dimensional velocity interferometry.

Previous velocity interferometers used at research laboratories for shock physics experiments measured target motion at a point or many points on a line on the target. Recently, a two-dimensional (2d) version (2d-velocity interferometer system for any reflector) has been demonstrated using a pair of ultrashort (3 ps) pulses for illumination, separated by 268 ps. We have discovered new abilities for this instrument, by treating the complex output image as a hologram. For data taken in an out of focus configuration, we can Fourier process to bring narrow features such as cracks into sharp focus, which are otherwise completely blurred. This solves a practical problem when using high numerical aperture optics having narrow depth of field to observe moving surface features such as cracks. Furthermore, theory predicts that the target appearance (position and reflectivity) at two separate moments in time are recorded by the main and conjugate images of the same hologram, and are partially separable during analysis for narrow features. Hence, for the cracks we bring into refocus, we can make a two-frame movie with a subnanosecond frame period. Longer and shorter frame periods are possible with different interferometer delays. Since the megapixel optical detectors we use have superior spatial resolution over electronic beam based framing cameras, this technology could be of great use in studying microscopic three-dimensional-behavior of targets at ultrafast times scales. Demonstrations on shocked silicon are shown.

Bulldozing of granular material

AbstractWe investigate the bulldozing motion of a granular sandpile driven forwards by a vertical plate. The problem is set up in the laboratory by emplacing the pile on a table rotating underneath a stationary plate; the continual circulation of the bulldozed material allows the dynamics to be explored over relatively long times, and the variation of the velocity with radius permits one to explore the dependence on bulldozing speed within a single experiment. We measure the time-dependent surface shape of the dune for a range of rotation rates, initial volumes and radial positions, for four granular materials, ranging from glass spheres to irregularly shaped sand. The evolution of the dune can be separated into two phases: a rapid initial adjustment to a state of quasi-steady avalanching perpendicular to the blade, followed by a much slower phase of lateral spreading and radial migration. The quasi-steady avalanching sets up a well-defined perpendicular profile with a nearly constant slope. This profile can be scaled by the depth against the bulldozer to collapse data from different times, radial positions and experiments onto common ‘master curves’ that are characteristic of the granular material and depend on the local Froude number. The lateral profile of the dune along the face of the bulldozer varies more gradually with radial position, and evolves by slow lateral spreading. The spreading is asymmetrical, with the inward progress of the dune eventually arrested and its bulk migrating to larger radii. A one-dimensional depth-averaged model recovers the nearly linear perpendicular profile of the dune, but does not capture the finer nonlinear details of the master curves. A two-dimensional version of the model leads to an advection–diffusion equation that reproduces the lateral spreading and radial migration. Simulations using the discrete element method reproduce in more quantitative detail many of the experimental findings and furnish further insight into the flow dynamics.

A constitutive model for cytoskeletal contractility of smooth muscle cells

The constitutive model presented in this article aims to describe the main bio-chemo-mechanical features involved in the contractile response of smooth muscle cells, in which the biochemical response is modelled by extending the four-state Hai–Murphy model to isotonic contraction of the cells and the mechanical response is mainly modelled based on the phosphorylation-dependent hyperbolic relation between isotonic shortening strain rate and tension. The one-dimensional version of the model is used to simulate shortening-induced deactivation with good agreement with selected experimental measurements. The results suggest that the Hai–Murphy biochemical model neglects the strain rate effect on the kinetics of cross-bridge interactions with actin filaments in the isotonic contractions. The two-dimensional version and three-dimensional versions of the model are developed using the homogenization method under finite strain continuum mechanics framework. The two-dimensional constitutive model is used to simulate swine carotid media strips under electrical field stimulation, experimentally investigated by Singer and Murphy, and contraction of a hollow airway and a hollow arteriole buried in a soft matrix subjected to multiple calcium ion stimulations. It is found that the transverse deformation may have significant influence on the response of the swine carotid medium. In both cases, the orientation of the maximal value of attached myosin is aligned with the orientation of maximum principal stress.