Three-dimensional Vector Fields Research Articles

A low-temperature high resolution scanning tunneling microscope with a three-dimensional magnetic vector field operating in ultrahigh vacuum

We present a low-temperature ultrahigh vacuum (UHV) scanning tunneling microscope setup with a combination of a superconducting solenoid coil and two split-pair magnets, providing a rotatable magnetic field up to 500 mT applicable in all spatial directions. An absolute field maximum of B=7 T(3 T) can be applied perpendicular (parallel) to the sample surface. The instrument is operated at a temperature of 4.8 K. Topographic and spectroscopic measurements on tungsten carbide and indium antimonide revealed a z-noise of 300 fm(pp), which barely changes in magnetic field. The microscope is equipped with a tip exchange mechanism and a lateral sample positioning stage, which allows exact positioning of the tip with an accuracy of 5 microm prior to the measurement. Additional contacts to the sample holder allow, e.g., the application of an additional gate voltage. The UHV part of the system contains versatile possibilities of in situ sample and tip preparation as well as low-energy electron diffraction and Auger analysis.

Conical soliton escape into a third dimension of a surface vortex

We present an exact three-dimensional solitonic solution to a sine-Gordon-type Euler-Lagrange equation that describes a configuration of a three-dimensional vector field n constrained to a surface p-vortex, with a prescribed polar tilt angle on a planar substrate and escaping into the third dimension in the bulk. The solution is relevant to characterization of a schlieren texture in nematic liquid-crystal films with tangential (in-plane) substrate alignment. The solution is identical to a section of a point defect discovered many years ago by Saupe [Mol. Cryst. Liq. Cryst. 21, 211 (1973)], when latter is restricted to a surface.

Lagrangian coherent structures in the human carotid artery bifurcation

The carotid artery bifurcation is known as a site of atheromatous plaque formation which is closely related to hemodynamics. To investigate the fluid mechanics inside the bifurcation, a transparent model of the carotid geometry was built to estimate the feasibility of using stereoscopic particle image velocimetry (PIV) in a complex three-dimensional geometry. As a first approach, steady inflow conditions are considered. Velocity data are acquired in cross-sectional planes and combined to yield the full three-dimensional velocity vector field in the region of the bifurcation. The finite-time Lyapunov exponent (FTLE) is used as a criterion to reveal the complex flow structure and is found to be particularly efficient in discriminating between reverse flow and recirculation regions. The Lagrangian criterion is also computed with time-resolved, two-component PIV measurements obtained by increasing the Reynolds number up to the onset of unsteadiness. The FTLE field produces in this case a detailed visualization of the instability development.

Multisensor Hot Wire Vorticity Probe Measurements of the Formation Field of Two Corotating Vortices

Experimental evidence is reported, regarding the formation of a pair of co-rotating tip vortices by a split wing configuration, consisting of two half wings at equal and opposite angles of attack. Simultaneous measurements of the three-dimensional vector fields of velocity and vorticity were conducted on a cross plane at a downstream distance corresponding to 0.3 cord lengths (near wake), using an in-house constructed 12-sensor hot wire anemometry vorticity probe. The probe consists of three closely separated orthogonal 4-wire velocity sensor arrays, measuring simultaneously the three-dimensional velocity vector at three closely spaced locations on a cross plane of the flow filed. This configuration makes possible the estimation of spatial velocity derivatives by means of a forward difference scheme of first order accuracy. Velocity measurements obtained with an X-wire are also presented for comparison. In this near wake location, the flow field is dictated by the pressure distribution established by the flow around the wings, mobilizing large masses of air and leading to the roll up of fluid sheets. Fluid streams penetrating between the wings collide, creating on the cross plane flow a stagnation point and an “impermeable” line joining the two vortex centres. Along this line fluid is directed towards the two vortices, expanding their cores and increasing their separation distance. This feeding process generates a dipole of opposite sign streamwise mean vorticity within each vortex. The rotational flow within the vortices obligates an adverse streamwise pressure gradient leading to a significant streamwise velocity deficit characterizing the vortices. The turbulent flow field is the result of temporal changes in the intensity of the vortex formation and changes in the position of the cores (wandering).

Quantitative three-dimensional elasticity imaging from quasi-static deformation: a phantom study

We present a methodology to image and quantify the shear elastic modulus of three-dimensional (3D) breast tissue volumes held in compression under conditions similar to those of a clinical mammography system. Tissue phantoms are made to mimic the ultrasonic and mechanical properties of breast tissue. Stiff lesions are created in these phantoms with size and modulus contrast values, relative to the background, that are within the range of values of clinical interest. A two-dimensional ultrasound system, scanned elevationally, is used to acquire 3D images of these phantoms as they are held in compression. From two 3D ultrasound images, acquired at different compressed states, a three-dimensional displacement vector field is measured. The measured displacement field is then used to solve an inverse problem, assuming the phantom material to be an incompressible, linear elastic solid, to recover the shear modulus distribution within the imaged volume. The reconstructed values are then compared to values measured independently by direct mechanical testing.

An attracting singular-hyperbolic set containing a non trivial hyperbolic repeller

The chain recurrent set of an Anosov three-dimensional vector field (e.g. [3]) is an example of a non-transitive singular-hyperbolic attracting set containing hyperbolic repellers. Observe that this example has no singularities. In this paper we construct a C ∞ three-dimensional vector field that exhibits a singular-hyperbolic attracting set containing a singularity which is not Lorenz-like and a non trivial repeller hyperbolic set.

The bifurcation theory of magnetic monopoles in a charged two-condensate Bose–Einstein system

Magnetic monopoles, which are particle-like field configurations with which one can associate a topological charge, widely exist in various three-dimensional condensate systems. In this paper, by making use of Duan's topological current theory, we obtain the charge density of magnetic monopoles and their bifurcation theory in a charged two-condensate Bose–Einstein system. The evolution of magnetic monopoles is studied from the topological properties of a three-dimensional vector field. The magnetic monopoles are found generating or annihilating at the limit points and encountering, splitting or merging at the bifurcation points.

Asymmetry and transition to turbulence in a smooth axisymmetric constriction

The flow through a smooth axisymmetric constriction (a stenosis in medical applications) of 75% restriction in area is measured using stereoscopic and time-resolved particle image velocimetry (PIV) in the Reynolds number range Re ~ 100–1100. At low Reynolds numbers, steady flow results reveal an asymmetry of the flow downstream of the constriction. The jet emanating from the throat of the nozzle is deflected towards the wall causing the formation of a one-sided recirculation region. The asymmetry results from a Coanda-type wall attachment already observed in symmetric planar sudden expansion flows. When the Reynolds number is increased above the critical value of 400, the separation surface cannot remain attached and an unsteady flow regime begins. Low-frequency axial oscillations of the reattachment point are observed along with a slow swirling motion of the jet. The phenomenon is linked to a periodic discharge of the unstable recirculation region inducing alternating laminar and turbulent flow phases. The resulting flow is highly non-stationary and intermittent. Discrete wavelet transforms are used to discriminate between the large-scale motions of the mean flow and the vortical and turbulent fluctuations. Continuous wavelet transforms reveal the spectral structure of flow disturbances. Temporal measurements of the three velocity components in cross-sections are used with the Taylor hypothesis to qualitatively reconstruct the three-dimensional velocity vector fields, which are validated by comparing with two-dimensional PIV measurements in meridional planes. Visualizations of isosurfaces of the swirling strength criterion allow the identification of the topology of the vortices and highlight the formation and evolution of hairpin-like vortex structures in the flow. Finally, with further increase of the Reynolds number, the flow exhibits less intermittency and becomes stationary for Re ~ 900. Linear stochastic estimation identifies the predominance of vortex rings downstream of the stenosis before breakdown to turbulence.

Light scattering on random dielectric layers

Scattering of light by a random stack of dielectric layers represents a one-dimensional scattering problem, where the scattered field is a three-dimensional vector field. We investigate the dependence of the scattering properties (band gaps and Anderson localization) on the wavelength, strength of randomness and relative angle of the incident wave. There is a characteristic angular dependence of Anderson localization for wavelengths close to the thickness of the layers. In particular, the localization length varies non-monotonously with the angle. In contrast to Anderson localization, absorptive layers do not have this characteristic angular dependence.

Measurements of Three-Dimensional Flow in Microchannel With Complex Shape by Micro-Digital-Holographic Particle-Tracking Velocimetry

High time-resolution flow field measurement in two microchannels with a complex shape is performed by a micro-digital-holographic particle-tracking velocimetry (micro-DHPTV). The first microchannel has a Y junction that combines the flow of fluid from two inlets into one outlet. In this case, two laminar velocity profiles from the inlet regions merge into one laminar velocity profile. The second microchannel has a convergence region from where a fluid flows into a divergence region. At this region, two recirculation regions appear. Consequently, approximately 250 velocity vectors in both cases can be obtained instantaneously. For a microchannel with the convergence region, the two recirculation regions that appear at the divergence point are captured from a three-dimensional vector field, with which the axes of recircular vortices have some alignment. The reason why we can observe this phenomenon is that a three-dimensional velocity, including the depth direction, can be obtained by micro-DHPTV.

Bifurcation analysis of a new Lorenz-like chaotic system

In this paper we study the local codimension one and two bifurcations which occur in a family of three-dimensional vector fields depending on three parameters. An equivalent family, depending on five parameters, was recently proposed as a new chaotic system with a Lorenz-like butterfly shaped attractor and was studied mainly from a numerical point of view, for particular values of the parameters, for which computational evidences of the chaotic attractor was shown. In order to contribute to the understand of this new system we present an analytical study and the bifurcation diagrams of an equivalent three parameter system, showing the qualitative changes in the dynamics of its solutions, for different values of the parameters.

A reconstruction approach for imaging in 3D cone beam vector field tomography.

3D cone beam vector field tomography (VFT) aims for reconstructing and visualizing the velocity field of a moving fluid by measuring line integrals of projections of the vector field. The data are obtained by ultrasound measurements along a scanning curve which surrounds the object. From a mathematical point of view, we have to deal with the inversion of the vectorial cone beam transform. Since the vectorial cone beam transform of any gradient vector field with compact support is identically equal to zero, we can only hope to reconstruct the solenoidal part of an arbitrary vector field. In this paper we will at first summarize important properties of the cone beam transform for three-dimensional solenoidal vector fields and then propose a solution approach based on the method of approximate inverse. In this context, we intensively make use of results from scalar 3D computerized tomography. The findings presented in the paper will continuously be illustrated by pictures from first numerical experiments done with exact, simulated data.

Topological invariants for singularities of real vector fields in dimension three

We give topological invariants for a wide class of abso-lutely isolated singularities of three-dimensional real vector fields. Our invariants are complete once the desingularization morphism $\pi$ is fixed. They are obtained in terms of a finite set of configurations depending only on the eigenvalues of the singularities.

Irrotational vector fields, partially invariant with respect to group of rotations

A geometrical description of three-dimensional irrotational vector fields which are partially invariant with respect to the SO(3) group of rotations is given. It is shown that the motion of a continuum, that satisfies the equation of continuity and is partially invariant with respect to SO(3), is necessarily rotational, that is, the curl of the velocity vector is non zero.

Generalized hyperboloid structures of polarization singularities in Laguerre-Gaussian vector fields

We present the propagation-dependent polarization vector fields by use of an isotropic microchip laser with the longitudinal-transverse coupling and the entanglement of the polarization states. With the coherent superposition of orthogonal circularly polarized vortex modes which are made up of two Laguerre-Gaussian modes with different order, the experimental three-dimensional vector fields can be reconstructed analytically. From the theoretical analyses, the generalized structures of singularities such as $V$ points, $C$ lines, and $L$ surfaces can be clearly demonstrated. Importantly, the projections of $C$ lines on the transverse plane are found to form the intriguing petal structures.

The Lyapunov exponents of generic zero divergence three-dimensional vector fields

AbstractWe prove that for a C1-generic (dense Gδ) subset of all the conservative vector fields on three-dimensional compact manifolds without singularities, we have for Lebesgue almost every (a.e.) point p∈M that either the Lyapunov exponents at p are zero or X is an Anosov vector field. Then we prove that for a C1-dense subset of all the conservative vector fields on three-dimensional compact manifolds, we have for Lebesgue a.e. p∈M that either the Lyapunov exponents at p are zero or p belongs to a compact invariant set with dominated splitting for the linear Poincaré flow.

Nonoscillating Projections for Trajectories of Vector Fields

The properties of oscillation for trajectories of three-dimensional vector fields are "visible" in the sense that they can be detected by means of plane projections. We also recover a three-dimensional version of the classical dichotomy "nonoscillation" versus "spiralling" for plane vector fields.

Degenerate bifurcations of nontwisted heterodimensional cycles with codimension 3

Bifurcations of heterodimensional cycles with highly degenerate conditions are studied by establishing a suitable local coordinate system in three-dimensional vector fields. The existence, coexistence and noncoexistence of the periodic orbit, homoclinic loop, heteroclinic loop and double periodic orbit are obtained under some generic hypotheses. The bifurcation surfaces and the existence regions are located; the number of the bifurcation surfaces exhibits variety and complexity of the bifurcation of degenerate heterodimensional cycles. The corresponding bifurcation graph is also drawn.

Orthogonal, solenoidal, three-dimensional vector fields for no-slip boundary conditions

Numerical calculations of turbulence, as well as theoretical turbulence closure techniques, often depend upon a spectral decomposition of the flow fields. Viscous fluid dynamical calculations generally assume no-slip boundary conditions. However, for incompressible flows, such spectral decompositions have been limited to two-dimensional situations. Here we present a method that yields orthogonal decompositions of incompressible, three-dimensional flow fields and apply it to periodic cylindrical and spherical no-slip boundaries.

Hyperboloid Structures Formed by Polarization Singularities in Coherent Vector Fields with Longitudinal-Transverse Coupling

The propagation-dependent polarization vector fields are experimentally created from an isotropic microchip laser with a longitudinal-transverse coupling and entanglement of the polarization states. The experimental three-dimensional coherent vector fields are analytically reconstructed with a coherent superposition of orthogonal circularly polarized vortex modes. Each polarized component is found to comprise two Laguerre-Gaussian modes with different topological charges. With the analytical representation, the polarization singularities, on which the electric polarization ellipse is purely circular (C lines) or purely linear (L surfaces), are explored. The C line singularities are found to form an intriguing hyperboloidal structure.