scholarly journals Exact solution to a Liouville equation with Stuart vortex distribution on the surface of a torus

2019 ◽  
Vol 475 (2224) ◽  
pp. 20180666 ◽  
Author(s):  
Takashi Sakajo
Keyword(s):  
Exact Solution ◽  
Liouville Equation ◽  
Spherical Surface ◽  
Gauss Curvature ◽  
Beltrami Operator ◽  
Incompressible Euler Equation ◽  
Minor Radius ◽  
Link Type ◽  
Toroidal Surface ◽  
Major Radius

A steady solution of the incompressible Euler equation on a toroidal surface T R , r of major radius R and minor radius r is provided. Its streamfunction is represented by an exact solution to the modified Liouville equation, ∇ T R , r 2 ψ = c   e d ψ + ( 8 / d ) κ , where ∇ T R , r 2 and κ denote the Laplace–Beltrami operator and the Gauss curvature of the toroidal surface respectively, and c , d are real parameters with cd  < 0. This is a generalization of the flows with smooth vorticity distributions owing to Stuart (Stuart 1967 J. Fluid Mech. 29 , 417–440. ( doi:10.1017/S0022112067000941 )) in the plane and Crowdy (Crowdy 2004 J. Fluid Mech. 498 , 381–402. ( doi:10.1017/S0022112003007043 )) on the spherical surface. The flow consists of two point vortices at the innermost and the outermost points of the toroidal surface on the same line of a longitude, and a smooth vorticity distribution centred at their antipodal position. Since the surface of a torus has non-constant curvature and a handle structure that are different geometric features from the plane and the spherical surface, we focus on how these geometric properties of the torus affect the topological flow structures along with the change of the aspect ratio α  =  R / r . A comparison with the Stuart vortex on the flat torus is also made.

scholarly journals Approximating pointwise products of quasimodes

2020 ◽  
Vol 32 (3) ◽  
pp. 541-552
Author(s):  
Mei Ling Jin
Keyword(s):  
Vector Space ◽  
Harmonic Analysis ◽  
Riemannian Manifolds ◽  
Spectral Projection ◽  
Beltrami Operator ◽  
Singular Potentials ◽  
Link Type ◽  
Right Hand ◽  
Low Degree ◽  
The Right

AbstractWe obtain approximation bounds for products of quasimodes for the Laplace–Beltrami operator on compact Riemannian manifolds of all dimensions without boundary. We approximate the products of quasimodes uv by a low-degree vector space {B_{n}}, and we prove that the size of the space {\dim(B_{n})} is small. In this paper, we first study bilinear quasimode estimates of all dimensions {d=2,3}, {d=4,5} and {d\geq 6}, respectively, to make the highest frequency disappear from the right-hand side. Furthermore, the result of the case {\lambda=\mu} of bilinear quasimode estimates improves {L^{4}} quasimodes estimates of Sogge and Zelditch in [C. D. Sogge and S. Zelditch, A note on L^{p}-norms of quasi-modes, Some Topics in Harmonic Analysis and Applications, Adv. Lect. Math. (ALM) 34, International Press, Somerville 2016, 385–397] when {d\geq 8}. And on this basis, we give approximation bounds in {H^{-1}}-norm. We also prove approximation bounds for the products of quasimodes in {L^{2}}-norm using the results of {L^{p}}-estimates for quasimodes in [M. Blair, Y. Sire and C. D. Sogge, Quasimode, eigenfunction and spectral projection bounds for Schrodinger operators on manifolds with critically singular potentials, preprint 2019, https://arxiv.org/abs/1904.09665]. We extend the results of Lu and Steinerberger in [J. F. Lu and S. Steinerberger, On pointwise products of elliptic eigenfunctions, preprint 2018, https://arxiv.org/abs/1810.01024v2] to quasimodes.

scholarly journals Green's function for the Laplace–Beltrami operator on a toroidal surface

2013 ◽  
Vol 469 (2149) ◽  
pp. 20120479 ◽  
Author(s):  
Christopher C. Green ◽  
Jonathan S. Marshall
Keyword(s):  
Green’S Function ◽  
Green's Function ◽  
Vortex Dynamics ◽  
Stereographic Projection ◽  
Three Dimensional ◽  
Beltrami Operator ◽  
Toroidal Surface ◽  
Torus Surface ◽  
Dilogarithm Function ◽  
Prime Function

Green's function for the Laplace–Beltrami operator on the surface of a three-dimensional ring torus is constructed. An integral ingredient of our approach is the stereographic projection of the torus surface onto a planar annulus. Our representation for Green's function is written in terms of the Schottky–Klein prime function associated with the annulus and the dilogarithm function. We also consider an application of our results to vortex dynamics on the surface of a torus.

coils is also presented in Section 2. The simulation produced by the Helmholtz coils, which damages the results about the optimum parameter for Helmholtz uniform magnetic field. Therefore, theoretical coils are presented in Section 3. Finally, conclusion parameters of Helmholtz coils cannot be used and further work are outlined in Section 4. directly in edge detection. By extracting temperature information using COMSOL via the AC/DC module, we can confirm which optimum parameters 2 METHODOLOGY AND EXPERIMENT SETUP of Helmholtz coils can produce most effective Helmholtz coils are a special arrangement of air-excitation for edge detection. cored coils, and they are all used as a means of This simulation is conducted using COMSOL generating magnetic fields that are uniform over a multiphysics FEM simulation software via the volume (Cakir ). According to Biot-Savart AC/DC module. Fig. 1 shows the constitution of law, magnetic flux density at any point on the axis of Helmholtz coils testing, where r is the minor radius Helmholtz coils can be calculated from Equation (1) of Helmholtz coils, r is the major radius of (Bronaugh )： Helmholtz coils, h is the sample height, d is the N Ir N Ir distance between Helmholtz coils edge and sample H  H H   (1) edge, and z is the distance between Helmholtz coils. 2  r a  2  r a  The physical characteristics of the model to be simulated and studied are given in Table 1. The According to the definition of Helmholtz coils, geometry of the sample is 40502 mm ; the r  r  r , N  N 1 and 2a  2a  r . major and minor radii of Helmholtz coils are equal Using Taylor series expansion and calculating the to 10 mm and 2 mm, respectively, and the turns differential of H (0) (when z  0 ), after some equal 1. The excitation module is a small period (0.3 s) of high-frequency current (256 kHz). manipulation, Equation (1) becomes Table 1. Electrical and thermal parameters for steel  144  z   used in the simulation H ( z )  H (0)   1   125  r   (2)     

2015 ◽  
pp. 200-202
Keyword(s):  
Edge Detection ◽  
Fem Simulation ◽  
Taylor Series Expansion ◽  
Simulation Software ◽  
Magnetic Flux Density ◽  
Minor Radius ◽  
Helmholtz Coils ◽  
Sample Height ◽  
Definition Of ◽  
Major Radius

scholarly journals Purely entropic self-assembly of the bicontinuous Ia 3 d gyroid phase in equilibrium hard-pear systems

2017 ◽  
Vol 7 (4) ◽  
pp. 20160161 ◽  
Author(s):  
Philipp W. A. Schönhöfer ◽  
Laurence J. Ellison ◽  
Matthieu Marechal ◽  
Douglas J. Cleaver ◽  
Gerd E. Schröder-Turk
Keyword(s):  
Phase Diagram ◽  
Minimal Surface ◽  
Lipid Bilayers ◽  
Self Assembly ◽  
Particle Shape ◽  
Voronoi Tessellation ◽  
Gauss Curvature ◽  
Constant Density ◽  
Continuous Change ◽  
Link Type

We investigate a model of hard pear-shaped particles which forms the bicontinuous Ia d structure by entropic self-assembly, extending the previous observations of Barmes et al. (2003 Phys. Rev. E 68 , 021708. ( doi:10.1103/PhysRevE.68.021708 )) and Ellison et al. (2006 Phys. Rev. Lett. 97 , 237801. ( doi:10.1103/PhysRevLett.97.237801 )). We specifically provide the complete phase diagram of this system, with global density and particle shape as the two variable parameters, incorporating the gyroid phase as well as disordered isotropic, smectic and nematic phases. The phase diagram is obtained by two methods, one being a compression–decompression study and the other being a continuous change of the particle shape parameter at constant density. Additionally, we probe the mechanism by which interdigitating sheets of pears in these systems create surfaces with negative Gauss curvature, which is needed to form the gyroid minimal surface. This is achieved by the use of Voronoi tessellation, whereby both the shape and volume of Voronoi cells can be assessed in regard to the local Gauss curvature of the gyroid minimal surface. Through this, we show that the mechanisms prevalent in this entropy-driven system differ from those found in systems which form gyroid structures in nature (lipid bilayers) and from synthesized materials (di-block copolymers) and where the formation of the gyroid is enthalpically driven. We further argue that the gyroid phase formed in these systems is a realization of a modulated splay-bend phase in which the conventional nematic has been predicted to be destabilized at the mesoscale due to molecular-scale coupling of polar and orientational degrees of freedom.

scholarly journals The stability and nonlinear evolution of quasi-geostrophic toroidal vortices

2019 ◽  
Vol 863 ◽  
pp. 60-78 ◽  
Author(s):  
Jean N. Reinaud ◽  
David G. Dritschel
Keyword(s):  
Nonlinear Evolution ◽  
Three Dimensional ◽  
Azimuthal Mode ◽  
Stability Properties ◽  
Central Vortex ◽  
Minor Radius ◽  
Toroidal Vortices ◽  
The Stability ◽  
Major Radius ◽  
Vortex Array

We investigate the linear stability and nonlinear evolution of a three-dimensional toroidal vortex of uniform potential vorticity under the quasi-geostrophic approximation. The torus can undergo a primary instability leading to the formation of a circular array of vortices, whose radius is approximately the same as the major radius of the torus. This occurs for azimuthal instability mode numbers $m\geqslant 3$, on sufficiently thin tori. The number of vortices corresponds to the azimuthal mode number of the most unstable mode growing on the torus. This value of $m$ depends on the ratio of the torus’ major radius to its minor radius, with thin tori favouring high mode $m$ values. The resulting array is stable when $m=4$ and $m=5$ and unstable when $m=3$ and $m\geqslant 6$. When $m=3$ the array has barely formed before it collapses towards its centre with the ejection of filamentary debris. When $m=6$ the vortices exhibit oscillatory staggering, and when $m\geqslant 7$ they exhibit irregular staggering followed by substantial vortex migration, e.g. of one vortex to the centre when $m=7$. We also investigate the effect of an additional vortex located at the centre of the torus. This vortex alters the stability properties of the torus as well as the stability properties of the circular vortex array formed from the primary toroidal instability. We show that a like-signed central vortex may stabilise a circular $m$-vortex array with $m\geqslant 6$.

Micromagnetic simulation study of magnetization reversal in torus-shaped permalloy nanorings

2017 ◽  
Vol 31 (22) ◽  
pp. 1750162 ◽  
Author(s):  
Amaresh Chandra Mishra ◽  
R. Giri
Keyword(s):  
Hysteresis Loop ◽  
Simulation Study ◽  
Magnetization Reversal ◽  
Hysteresis Loops ◽  
Vortex State ◽  
Micromagnetic Simulation ◽  
Hysteresis Curve ◽  
Minor Radius ◽  
Maximum Value ◽  
Major Radius

Using micromagnetic simulation, the magnetization reversal of soft permalloy rings of torus shape with major radius R varying within 20–100 nm has been investigated. The minor radius r of the torus rings was increased from 5 nm up to a maximum value r[Formula: see text] such that R- r[Formula: see text] = 10 nm. Micromagnetic simulation of in-plane hysteresis curve of these nanorings revealed that in the case of very thin rings (r [Formula: see text] 10 nm), the remanent state is found to be an onion state, whereas for all other rings, the remanent state is a vortex state. The area of the hysteresis loop was found to be decreasing gradually with the increment of r. The normalized area under the hysteresis loops (A[Formula: see text]) increases initially with increment of r. It attains a maximum for a certain value of r = r0 and again decreases thereafter. This value r0 increases as we decrease R and as a result, this peak feature is hardly visible in the case of smaller rings (rings having small R).

scholarly journals Vortex crystals on the surface of a torus

2019 ◽  
Vol 377 (2158) ◽  
pp. 20180344 ◽  
Author(s):  
Takashi Sakajo
Keyword(s):  
Null Space ◽  
Spherical Surface ◽  
Particle Interaction ◽  
Singular Values ◽  
Point Vortices ◽  
Relative Configuration ◽  
Point Configurations ◽  
Toroidal Surface ◽  
The Matrix ◽  
Value Decomposition

Vortex crystals are equilibrium states of point vortices whose relative configuration is unchanged throughout the evolution. They are examples of stationary point configurations subject to a logarithmic particle interaction energy, which give rise to phenomenological models of pattern formations in incompressible fluids, superconductors, superfluids and Bose–Einstein condensates. In this paper, we consider vortex crystals rotating at a constant speed in the latitudinal direction on the surface of a torus. The problem of finding vortex crystals is formulated as a linear null equation A Γ  = 0 for a non-normal matrix A whose entities are derived from the locations of point vortices, and a vector Γ consisting of the strengths of point vortices and the latitudinal speed of rotation. Point configurations of vortex crystals are obtained numerically through the singular value decomposition by prescribing their locations and/or by moving them randomly so that the matrix A becomes rank deficient. Their strengths are taken from the null space corresponding to the zero singular values. The toroidal surface has a non-constant curvature and a handle structure, which are geometrically different from the plane and the spherical surface where vortex crystals have been constructed in the preceding studies. We find new vortex crystals that are associated with these toroidal geometry: (i) a polygonal arrangement of point vortices around the line of longitude; (ii) multiple latitudinal polygonal ring configurations of point vortices that are evenly arranged around the handle; and (iii) point configurations along helical curves corresponding to the fundamental group of the toroidal surface. We observe the strengths of point vortices and the behaviour of their distribution as the number of point vortices gets larger. Their linear stability is also examined. This article is part of the theme issue ‘Topological and geometrical aspects of mass and vortex dynamics’.

Determination of Size Distribution of Elliptical Microvessels from Size Distribution Measurement of Their Section Profiles

2003 ◽  
Vol 228 (1) ◽  
pp. 84-92 ◽  
Author(s):  
R.A. Krasnoperov ◽  
A.N. Gerasimov
Keyword(s):  
Size Distribution ◽  
Three Dimensional ◽  
Axial Ratio ◽  
Minor Radius ◽  
Transmission Electron ◽  
Section Profile ◽  
Ultrathin Sections ◽  
The One ◽  
Major Radius

In transmission electron microscopy, microvessels (MVs) are studied as profiles on ultrathin sections. To determine MV sizes from measurements made on MV profiles, an assumption must be made about MV shape, a circular cylinder being used to approximate the latter on limited lengths. However, this model is irrelevant in case MVs have some flatness. The elliptical cylinder model is preferable, although relationships between the cylinder profile (two-dimensional; 2D) and its true (three-dimensional; 3D) sizes are not yet known. We have obtained the 2D/3D functions that express the relationships between such profile sizes as the minor radius (Y), major radius (X), axial ratio (X/Y), area (S), and perimeter (P) on the one hand, and the corresponding MV sizes (Y0, X0, X0/Y0, S0, and P0) on the other. The 2D/3D functions make it possible to derive elliptical MV sizes from section profile size distributions, probability density functions (PDFs) for the latter being determined. We have applied the 2D/3D functions in studying axial ratios of thyroid hemocapillaries. A factual X/Y frequency histogram has been constructed and fitted by theoretical X/Y PDFs plotted for different sets of capillary sizes. The thyroid capillaries have been revealed to be clustered, 72.7% of them having X0/Y0 ≈ 1.6, 17.6%, X0/Y0 ≈ 1.0. and 9.7%, X0/Y0 ≈ 3.2. The proposed technique is instrumental in precise modeling of microclrculatory network geometry.

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