Spiral Solutions Research Articles

Variety of unsymmetric multibranched logarithmic vortex spirals

Building on work of Prandtl and Alexander, we study logarithmic vortex spiral solutions of the two-dimensional incompressible Euler equations. We consider multi-branched spirals that are not symmetric, including mixtures of sheets and continuum vorticity. We find that non-trivial solutions allow only sheets, that there is a large variety of such solutions, but that only the Alexander spirals with three or more symmetric branches appear to yield convergent Biot–Savart integral.

Spiral Lambert’s Problem

Lambert’s problem is solved for the case of a spacecraft accelerated by a continuous thrust. The solution is based on the family of generalized logarithmic spirals, which provides a fully analytic description of the dynamics including the time of flight and involves two conservation laws. The structure of the solution yields a collection of properties that are closely related to those of the Keplerian case. A minimum-energy spiral is found, with pairs of conjugate spirals bifurcating from it. Thanks to the integral of motion related to the energy, the solutions are classified as elliptic, parabolic, and hyperbolic. The maximum acceleration reached along the transfer can be solved in closed form. The problem of designing a low-thrust trajectory between two bodies reduces to solving two equations with two unknowns. Double-time opportunity transfers appear naturally thanks to the symmetry properties of the generalized logarithmic spirals. Comparing the Keplerian and spiral pork-chop plots in an Earth–Mars example shows that the spiral solution might increase the mass fraction delivered to the final orbit, thanks to reducing of the magnitude of the impulsive maneuvers at departure and arrival.

Algebraic spiral solutions of the 2d incompressible Euler equations

We construct a class of self-similar 2d incompressible Euler solutions that have initial vorticity of mixed sign. The regions of positive and negative vorticity form algebraic spirals.

Quantum spirals

Quantum systems often exhibit fundamental incapability to entertain vortex. The Meissner effect, a complete expulsion of the magnetic field (the electromagnetic vorticity), for instance, is taken to be the defining attribute of the superconducting state. Superfluidity is another, close-parallel example; fluid vorticity can reside only on topological defects with a limited (quantized) amount. Recent developments in the Bose–Einstein condensates produced by particle traps further emphasize this characteristic. We show that the challenge of imparting vorticity to a quantum fluid can be met through a nonlinear mechanism operating in a hot fluid corresponding to a thermally modified Pauli–Schrödinger spinor field. The thermal baroclinic effect is represented by a nonlinear, non-Hermitian Hamiltonian, which, in conjunction with spin vorticity, leads to new interesting quantum states; a spiral solution is explicitly worked out in a simple field-free model.

Role of forcing in large-time behavior of vorticity equation solutions on a sphere

The nonlinear barotropic vorticity equation (BVE) describing the vortex dynamics of viscous incompressible and forced fluid on a rotating sphere is considered. The asymptotic behavior of solutions of nonstationary BVE as t → ∞ is studied. Particular forms of the external vorticity source are given that guarantee the existence of a bounded attractive set in the phase space of solutions. The asymptotic behavior of the BVE solutions is shown to depend on both the structure and the smoothness of external forcing. Three types of sufficient conditions for global asymptotic stability of smooth and weak BVE solutions are also given. Simple attractive sets of a viscous incompressible fluid on a sphere under quasi-periodic polynomial forcing are considered. Each attractive set represents a BVE quasi-periodic solution of the complex (2n + 1)-dimensional subspace Hn of homogeneous spherical polynomials of degree n. The Hausdorff dimension of its trajectory, being an open spiral densely wound around a 2n-dimensional torus in Hn, equals to 2n. As the generalized Grashof number G becomes small enough then the domain of attraction of such spiral solution is expanded from Hn to the entire BVE phase space. It is shown that for a given G, there exists an integer nG such that each spiral solution generated by a forcing of Hn with n ≥ nG is globally asymptotically stable. Thus we demonstrate the difference in the asymptotic behavior of solutions in the cases, then Grashof number G is fixed and bounded, but the forcing is stationary or non-stationary. Whereas the dimension of the fluid attractor under a stationary forcing is limited above by G, the dimension of the spiral attractive solution (equal to 2n) may become arbitrarily large as the degree n of the quasi-periodic polynomial forcing grows. Since the small-scale quasi-periodic functions, unlike the stationary ones, more adequately depict the forcing in the barotropic atmosphere, this result is of meteorological interest and shows that the dimension of attractive sets depends not only on the forcing amplitude, but also on its spatial and temporal structure. This example also shows that the search of a finite-dimensional global attractor in the barotropic atmosphere is not well justified.

On spiral solutions to generalized crystalline motion with a rotating tip motion

In our previous paper we proposed a crystalline motion of spiral-shaped polygonal curves with a tip motion as a simple model of a step motion on a crystal surface under screw dislocation and discussed global existence of spiral solutions to the proposed model. In this paper we extend the previous results for generalized crystalline curvature flow with a suitable tip motion. We show that solution curves never intersect a trajectory of a tip and has no self-intersections. We also show that any facet never disappear during time evolution. Finally we show a time-global existence of the spiral-shaped solutions.

Pythagorean hodograph spline spirals that match [formula omitted] Hermite data from circles

A construction is given for a G3 piecewise rational Pythagorean hodograph convex spiral which interpolates two G3 Hermite data associated with two non-concentric circles, one being inside the other. The spiral solution is of degree 7 and is the involute of a G2 convex curve, referred to as the evolute solution, with prescribed length, and composed of two PH quartic curves. Conditions for G3 continuous contact with circles are then studied and it turns out that an ordinary cusp at each end of the evolute solution is required. Thus, geometric properties of a family of PH polynomial quartics, allowing to generate such an ordinary cusp at one end, are studied. Finally, a constructive algorithm is described with illustrative examples.

Core solutions of rigidly rotating spiral waves in highly excitable media.

Analytical spiral wave solutions for reaction-diffusion equations play an important role in studying spiral wave dynamics. In this paper, we focus on such analytical solutions in the case of highly excitable media. We present numerical evidence that, for rigidly rotating spiral waves in highly excitable media, the species values in the spiral core region do harmonic oscillations but not relaxation ones, and their amplitudes grow linearly with the distance from the rotation center. An analytical solution is proposed to describe such spiral wave dynamics, and the quantitative comparisons between the numerical results and the analytical solutions show that the proposed spiral core solution works well in highly excitable media.

Algebraic spiral solutions of 2d incompressible Euler

We consider self-similar solutions of the 2d incompressible Euler equations. We construct a class of solutions with vorticity forming algebraic spirals near the origin, in analogy to vortex sheets rolling up into algebraic spirals.

Fully nonlinear mode competitions of nearly bicritical spiral or Taylor vortices in Taylor-Couette flow

Interactions between nearly bicritical modes in Taylor-Couette flow, which have been concerned with the framework of weakly nonlinear theory, are extended to fully nonlinear Navier-Stokes computation. For this purpose, a standard Newton solver for axially periodic flows is generalized to compute any mixed solutions having up to two phases, which typically arise from interactions of two spiral or Taylor vortex modes. Also, a simple theory is developed in order to classify the mixed solutions. With these methods, we elucidate pattern formation phenomena, which have been observed in a Taylor-Couette flow experiment. Focusing on the counter-rotating parameter range, all possible classes of interaction of various solutions with different azimuthal and axial wave numbers are considered within our computational restriction, and we observe numerous connection branches, e.g., footbridge solutions. Some of the mixed solutions result in a three-dimensional wavy spiral solution with axial relative periodicity or an axially doubly periodic toroidally closed vortex solution. The possible connection of the former solution family to spiral turbulence, which has been observed in highly counter-rotating Taylor-Couette flow, is discussed.

Chiral magnetization configurations in magnetic nanostructures in the presence of Dzyaloshinskii-Moriya interactions

Many low-dimensional systems, such as nanoscale islands, thin films, and multilayers, as well as bulk systems, such as multiferroics, are characterized by the lack of inversion symmetry, a fact that may give rise to a Dzyaloshinskii-Moriya (DM) interaction. For sufficient strength, the DM interaction will favor spiral spin configurations of definite chirality. In order to harness such systems for applications, it is important to understand the conditions under which these spiral spin configurations form and how they can be controlled via an external field. Here, we present exact solutions of the 1D magnetization profiles in such systems for arbitrary material parameters in closed form. Determining the energy per unit length exactly, we are able to present the critical strength of the DM interaction, at which spiral solutions are energetically favorable. These magnetization profiles, in general, take the form of a domain wall or soliton lattice, with all solitons having the same chirality, whose sign is dictated by DM interaction. Conversely, given an energetically favorable spiral solution, we determine quantitatively how the magnetization profile changes as a function of the applied field.

Vortex-induced vibrations of an elastically mounted sphere with three degrees of freedom at Re = 300: hysteresis and vortex shedding modes

AbstractFluid–structure interaction (FSI) simulations are carried out to investigate vortex-induced vibrations of a sphere, mounted on elastic supports in all three spatial directions. The reduced velocity (${U}^{\ast } $) is systematically varied in the range ${U}^{\ast } = 4\text{{\ndash}} 9$, while the Reynolds number and reduced mass are held fixed at $\mathit{Re}= 300$ and ${m}^{\ast } = 2$, respectively. In the lock-in regime, two distinct branches are observed in the response curve, each corresponding to a distinct type of vortex shedding, namely, hairpin and spiral vortices. While shedding of hairpin vortices has been observed in several previous investigations of stationary and vibrating spheres, the shedding of intertwined, longitudinal spiral vortices in the wake of a vibrating sphere is reported herein for the first time. When the wake is in the hairpin shedding mode, the sphere moves along a linear path in the transverse plane, while when spiral vortices are shed, the sphere vibrates along a circular orbit. In the spiral mode branch, the simulations reveal hysteresis in the response amplitude at the beginning of the lock-in regime. Lower-amplitude vibrations are found as the sphere sheds hairpin vortices for increasing ${U}^{\ast } $ up until the beginning of the synchronization regime. On the other hand, higher-amplitude oscillations persist for the spiral mode as ${U}^{\ast } $ is decreased from the point of the start of the synchronization. The hairpin mode is found to be unstable for the value of reduced velocity where the spiral and hairpin solution branches merge together. When this point is approached along the hairpin solution branch, the sphere naturally transitions from shedding hairpin vortices and moving along a linear path to shedding spiral vortices and moving along a circular path in the transverse plane. The spiral mode was not observed in the work of Horowitz & Williamson (J. Fluid Mech., vol. 651, 2010, pp. 251–294), who studied experimentally the vibration modes of a freely rising or falling sphere and only reported zigzag vibrations. Our results suggest that this apparent discrepancy between experiments and simulations should be attributed to the fact that, for the range of governing parameters considered in the simulations, the elastic supports act to suppress streamwise vibrations, thus subjecting the sphere to a nearly axisymmetric elasticity constraint and enabling it to vibrate transversely along a circular path.

On the interpolation of concentric curvature elements

A convex G 2 Hermite interpolation problem of concentric curvature elements is considered in this paper. It is first proved that there is no spiral arc solution with turning angle less than or equal to π and then, that any convex solution admits at least two vertices. The curvature and the evolute profiles of such an interpolant are analyzed. In particular, conditions for the existence of a G 2 convex interpolant with prescribed extremal curvatures are given.

Symmetric Spiral Patterns on Spheres

Spiral patterns on the surface of a sphere have been seen in laboratory experiments and in numerical simulations of reaction-diffusion equations and convection. We classify the possible symmetries of spirals on spheres, which are quite different from the planar case since spirals typically have tips at opposite points on the sphere. We concentrate on the case where the system has an additional sign-change symmetry, in which case the resulting spiral patterns do not rotate. Spiral patterns arise through a mode interaction between spherical harmonics of degrees $\ell$ and $\ell+1$. Using the methods of equivariant bifurcation theory, possible symmetry types are determined for each $\ell$. For small values of $\ell$, the center manifold equations are constructed, and spiral solutions are found explicitly. Bifurcation diagrams are obtained showing how spiral states can appear at secondary bifurcations from primary solutions or at tertiary bifurcations. The results are consistent with numerical simulations of a model pattern-forming system.

Immunosuppressive Proteins Isolated from Spiral and Coccoid Cytoplasmic Solutions of Helicobacter pylori

The aim of this study was to investigate the antiproliferative proteins that probably have a role in Helicobacter pylori evade of immune response and cause chronic infection disease and also to see if coccoid form had a role in its chronicity. H. pylori strain VacA s2/m2 positive and CagA negative, from a gastric biopsy of a patient with peptic ulcer disease, was isolated and cultured in brucella agar. Both spiral and coccoid forms were harvested and ruptured by sonication. The cytoplasmic solutions of both forms were collected and their fractions obtained by gel chromatography and preparative polyacrylamide gel electrophoresis. The fractions were analyzed by MTT assay for their antiproliferative activity. We isolated two proteins with a significant dose dependent antiproliferative activity that analyzed with sodium dodecyl sulfate-polyacrylamide gel electrophoresis, one of them that was urease positive showed two bands with 61 and 27 kDa, which is resumed to urease of H. pylori, another consist of 57 and 63 kDa. Helicobacterpylori secret some proteins like urease that inhibit immune cells proliferation response against its antigens.

Spiral solution to the two-dimensional transport equations

The existence of spiral solution for the two-dimensional transport equations is considered in the present paper. Based on the notion of generalized solutions in the sense of Lebesgue-stieltjes integral, the global weak solution of transport equations which includes δ-shocks and vacuum is constructed for some special initial data.

Secondary bifurcation of mixed-cross-spirals connecting travelling wave solutions

We present numerical results of secondarily forward bifurcating, stationary flow states that mediate transitions between travelling helical waves (spirals). These so-called mixed-cross-spirals (MCSs) can be seen as nonlinear superpositions of spiral solutions with different helicity and pitch. Thereby, the contribution of the respective spiral to the entire MCS varies continuously with the control parameters. Furthermore, MCSs connect the bifurcation branches of the involved spirals, even when both spiral pitches differ. This makes them interesting for pattern-forming systems in general. The bifurcation scenarios of MCSs differ from the well-studied cross-spiral-mediated transitions between mirror-symmetric left- and right-winding spirals, even in the case of MCS solutions that start and end in the same spiral branch.The structure, spatiotemporal dynamics, bifurcation behaviour and stability of MCSs are elucidated in detail for the axially periodic Taylor–Couette system as a prototypical reference for pattern formation. The results are obtained by solving the full Navier–Stokes equations with a combination code of a finite differences and a Galerkin method.

A real-time 3D motion planning and simulation scheme for nonholonomic systems

We propose a new motion planning and simulation scheme for nonholonomic systems in this paper to provide a practical solution for these application problems taking into account of real-time obstacle avoidance and the continuous curvature path generation simultaneously in 3D unknown environment. The proposed motion planning and simulation scheme generates the motion path using a new universal Euler spiral generation algorithm, which is locally optimal based on perceived points of view. The generated Euler spiral solution can be non-symmetrical and easily implemented while maintaining a C 2 continuous. It is therefore more flexible and powerful in dealing with dynamic situations in real-time, compared with current symmetrical Euler spirals solutions. Real-time solutions are particularly important in navigation in unknown environments. The universal Euler spiral algorithm proposed displays a smaller maximum curvature value and smaller mean square curvature value than the conventional symmetrical algorithm in tested cases. Another significant contribution of our work is the new motion planning scheme which extend current 2D based motion planning into three-dimensional (3D) space. In this paper, we have conducted experiments and describe simulation results including 3D motion trajectory modeling for a flight simulation.

End wall effects on the transitions between Taylor vortices and spiral vortices

We present numerical simulations as well as experimental results concerning transitions between Taylor vortices and spiral vortices in the Taylor-Couette system with rigid, nonrotating lids at the cylinder ends. These transitions are performed by wavy structures appearing via a secondary bifurcation out of Taylor vortices and spirals, respectively. In the presence of these axial end walls, pure spiral solutions do not occur as for axially periodic boundary conditions but are substituted by primary bifurcating, stable wavy spiral structures. Similarly to the periodic system, we found a transition from Taylor vortices to wavy spirals mediated by so-called wavy Taylor vortices and, on the other hand, a transition from wavy spirals to Taylor vortices triggered by a propagating defect. We furthermore observed and investigated the primary bifurcation of wavy spirals out of the basic circular Couette flow with Ekman vortices at the cylinder ends.

Acoustic black holes for relativistic fluids

We derive a new acoustic black hole metric from the Abelian Higgs model. In the non-relativistic limit, while the Abelian Higgs model becomes the Ginzburg-Landau model, the metric reduces to an ordinary Unruh type. We investigate the possibility of using (type I and II) superconductors as the acoustic black holes. We propose to realize experimental acoustic black holes by using spiral vortices solutions from the Navier-stokes equation in the non-relativistic classical fluids.