Conical Surface Research Articles

Positive definite functions on a regular domain

Abstract We define positive and strictly positive definite functions on a domain and study these functions on a list of regular domains. The list includes the unit ball, conic surface, hyperbolic surface, solid hyperboloid and the simplex. Each of these domains is embedded in a quadrant or a union of quadrants of the unit sphere by a distance-preserving map, from which characterizations of positive definite and strictly positive definite functions are derived for these regular domains.

Development of a Cone Penetration Testing Apparatus with a Textured Shaft

The anisotropy of shear resistance depending on friction direction can be selectively utilized in geotechnical structures. For instance, deep foundations and soil nailing, which are subject to axial loads, benefit from increased load transfer due to greater shear resistance. In contrast, minimal shear resistance is desirable in applications such as pile driving and soil sampling. Previous studies explored the shear resistance by interface between soil and surface asperities of a plate inspired by the geometry of snake scales. In this study, the interface friction anisotropy based on the load direction of cones with surface asperities is evaluated. First, a laboratory model chamber and a small-scale cone system are developed to quantitatively assess shear resistance under two load directions (penetration ⟶ pull-out). A preliminary test is conducted to analyze the boundary effects for the size of the model chamber and the distance between cones by confirming similar penetration resistance values at four cone penetration points. The interface shear behavior between the cone surface and the surrounding sand is quantitatively analyzed using cones with various asperity geometries under constant vertical stress. The results show that penetration resistance and pull-out resistance are increased with a higher height, shorter length of asperity and shearing direction with a decreasing height of surface asperity.

Inversion formulas for the attenuated conical Radon transform: Plane and cylinder case

Since the invention of Compton camera imaging, the conical Radon transform, which maps a given function defined on 3-dimensional Euclidean space to its surface integrals over cones, has been studied intensively. The problem of recovering such a function from its unrestricted conical Radon transform is overdetermined, since the set of all cones depends on the three dimensions of the vertex, the two dimensions of the central axis, and the one-dimensional opening angle. Therefore, various types of restricted conical Radon transforms have also been studied. However, most of these studies have neglected the attenuation of the medium. This article presents the study of attenuated conical Radon transforms with vertices on a plane (referred to as the plane case) or the cylinder (referred to as the cylinder case) in a 3-dimensional space. In all cases, the function is integrated over conical surfaces with an additional weight that decreases with the distance to the vertex of the cones. The main results provide explicit inversion formulas for the attenuated conical Radon transform in the plane and in the cylinder case.

Bерифікація алгоритму методу пробних частинок на задачі осьового обтікання конуса розрідженим газовим струменем

This paper is concerned with an aerodynamic calculation of supersonic gas plume flows and the determination of the forces they exert on obstacles. The paper presents a development of the test particle statistical method (TPSM) to numerically simulate supersonic gas plumes over a wide range of flow conditions. The work is based on the idea of a combined approach, i.e., the use of the gas-dynamic parameter distribution at the nozzle exit or on the conventional boundary of the dense core of the plume as input data for a TPSM algorithm adapted from homogeneous flows to plume ones. Combining methods of continual aerodynamics (inside or near the nozzle, where a continuum flow takes place) and the TPSM (where the motion is described on a molecular-kinetic level) allows one to solve supersonic plume efflux problems for arbitrarily rarefied plumes. The TPSM plume algorithm was tested to verify its reliability on the problem of axial flow past a cone. At the initial stage of the use of the combined approach, consideration was given to a rather rarefied gas flow, for which the gas-dynamic parameters at the nozzle exit can be used as TPSM input data. The force distribution over the cone surface and the static pressure upstream of the cone were calculated. The TPSM results were found to be in satisfactory agreement with the available direct simulation Monte-Carlo and experimental data. It was concluded that using the plume velocity and density distributions at the continual zone exit found from the Navier–Stokes equations as TPSM input data would significantly improve the expected results. This use of the TPSM in an aerodynamic calculation of gas plumes is the first in Ukraine. The TPM offers saving in computational resources: the TPSM running time depends on a variety of factors, but it is many times shorter than that of the direct simulation Monte Carlo method.

Ferroelectric Smectic C Liquid Crystal Phase with Spontaneous Polarization in the Direction of the Director.

In the previous study, the existence of an unidentified ferroelectric smectic phase is demonstrated in the low-temperature region of the ferroelectric smectic A phase, where the layer spacing decreases with decreasing temperature. In the present study, the phase is identified by taking 2D X-ray diffraction images of a magnetically oriented sample while allowing it to rotate and constructed a 3D reciprocal space with the sample rotation angle as the third axis for the whole picture of the reciprocal lattice vectors originating from the smectic structure. Consequently, circular diffraction images are obtained when the reciprocal lattice vectors are evenly distributed on the conical surface at a certain inclination angle in the reciprocal space. This result provides clear evidence that the phase in question is smectic C. The polarization properties also showed that the observed smectic C phase has spontaneous polarization in the direction parallel to the director and is identified as ferroelectric smectic C. These results provide a new type of classification for liquid crystalline phases that has been established over many years and is a significant contribution to the basic science of soft matter research.

Steady secondary flow in a turbulent boundary layer past a slender axisymmetric body

The turbulent boundary layer in a viscous incompressible fluid developing longitudinally past the surface of a thin cone or cylinder at a finite distance from the laminar–turbulent transition zone is studied. The characteristic Reynolds number, determined from the external flow velocity and the length of the body, is assumed to be large, and the thickness of the boundary layer is small and comparable to the radius of the body. The asymptotic method of multiple scales is used to find solutions to the Navier–Stokes equations. Instead of the traditional decomposition of the solution into time-averaged values and their fluctuations, the velocities and pressure are expressed as an asymptotic series consisting of steady and perturbed terms. As a result, the viscous steady flow (‘secondary’) that arises in the boundary layer as a mandatory component of fast turbulent fluctuations was described. Analytical and numerical solutions for the radial steady velocity are presented, describing the self-induced suction of fluid from the external flow into the boundary layer. Further analytical solutions are obtained for the longitudinal and circumferential velocities, which differ markedly from the laminar regime. The solutions found are somewhat similar to the degenerate (one-dimensional) case of self-sustaining longitudinal thin structures in turbulent shear flows. A qualitative comparison with direct numerical simulations is presented.

Harmonic extension through cylindrical and conical surfaces

This article describes recent progress concerning the extension of harmonic functions which vanish on a cylindrical or conical surface. The results, which are based on detailed analyses of various expansions of Green functions, can be viewed as partial generalizations of the well-known Schwarz reflection principle to surfaces other than hyperplanes and spheres. Some unexpected features emerge which suggest avenues for further exploration.

Numerical and theoretical modeling of water droplet impact on hydrophilic and superhydrophobic cones

The phenomenon of droplet impingement on solid surfaces is prevalent in various natural and industrial contexts. Research on impact dynamics on conical surfaces keeps emerging, with superhydrophobic cones receiving more attention than hydrophilic ones. This study systematically investigates water droplet impact dynamics on both hydrophilic and superhydrophobic cones using a two-phase numerical solver under different Weber numbers (We) and cone angles (φ). Three distinct phases are identified in the We–φ map to describe the different outcomes on each type of cones. Generally, deposition occurs ultimately on hydrophilic cones, whereas rebounding is observed on superhydrophobic ones. The maximum spreading area βAmax on hydrophilic cones depends only slightly on φ but consistently increases with We, following a scaling law of We0.5 at higher We. In contrast, on superhydrophobic cones, βAmax increases significantly with both We and φ, and the exponent in the scaling laws of βAmax with respect to We increases notably as φ increases. Three characteristic times are defined to describe important motion states on both types of cones. Corresponding scaling laws for each time with We are established. Two theoretical models are developed to predict the maximum spreading position for droplets on hydrophilic cones and the rebound position on superhydrophobic cones, respectively. Gravitational potential energy is included in the energy budget for both models, and an auxiliary viscous dissipation due to spontaneous spreading is accounted for the hydrophilic case. Satisfactory agreement between the theoretical and numerical results is achieved.

Circular motion of a charged particle on the inner surface of a frictionless cone under the influence of an electric field due to an electric dipole

A common problem provided in elementary physics textbooks is the analysis of the circular motion of a point-like particle affected by a gravitational field on the frictionless inner surface of an inverted cone. In this research, we study the motion of a charged particle on the same surface subject to an electric field generated by an electric dipole placed at the bottom of the cone. We study negatively and positively charged particles and analyse the dynamics of both. We calculate the physical quantities such as speed, angular velocity, angular momentum, and mechanical energy. Furthermore, we determine all possible distances travelled by the particles measured from the dipole. We also discuss the stability of the circular orbit using the effective potential. Finally, we consider the particle’s motion on a generic surface formed by the revolution of the curve ρz around the vertical axis. We determine the differential equation of ρz , in which the kinematic variables including speed v, angular velocity ω, and angular momentum L are constant.

A general mathematical model for manufacturing Roots pump rotors using conical milling cutters

The rotors are the key components in the Roots pump, and the rotor manufacture time will decide the pump price. In special cases, such as when one rotor shaft has two stages, respectively, with claw-type and Roots profiles, the traditional forming method is unsuitable for manufacturing the Roots rotor stage. Instead, more suitable manufacturing methods, such as planning or end-mill milling, can be applied. However, these two methods take time and effort. Therefore, this study proposes a general mathematical approach for Roots rotor manufacturing using the conical milling cutter as the key tool, which can reduce manufacturing time. The contact points and meshing angles between the rotor and the conical milling cutter can be calculated based on contact conditions. Then, a 3D conical milling cutter surface is obtained based on the cutter profile as a straight line, and the conical cutter is designed and considered to avoid undercutting in the rotor cutting process. The generated Roots rotor can be obtained by applying cutting simulation from the proposed mathematical model. The normal deviations of the rotor can be found by comparing the generated rotor profile to the datum rotor profile. In addition, the influence of axis errors of the conical milling cutter on the tooth profiles of the Roots pump rotor is investigated and discussed. The experimental results based on the proposed mathematical model are obtained and expressed to confirm the practicability of the proposed method.

Droplet Friction on Superhydrophobic Surfaces Scales With Liquid-Solid Contact Fraction.

It is generally assumed that contact angle hysteresis of superhydrophobic surfaces scales with liquid-solid contact fraction, however, its experimental verification has been problematic due to the limited accuracy of contact angle and sliding angle goniometry. Advances in cantilever-based friction probes enable accurate droplet friction measurements down to the nanonewton regime, thus suiting much better for characterizing the wetting of superhydrophobic surfaces than contact angle hysteresis measurements. This work quantifies the relationship between droplet friction and liquid-solid contact fraction, through theory and experimental validation. Well-defined micropillar and microcone structures are used as model surfaces to provide a wide range of different liquid-solid contact fractions. Micropillars are known to be able to hold the water on top of them, and a theoretical analysis together with confocal laser scanning microscopy shows that despite the spiky nature of the microcones droplets do not sink into the conical structure either, rendering a diminishingly small liquid-solid contact fraction. Droplet friction characterization with a micropipette force sensor technique reveals a strong dependence of the droplet friction on the contact fraction, and the dependency is described with a simple physical equation, despite the nearly three-orders-of-magnitude difference in liquid-solid contact fraction between the sparsest cone surface and the densest pillar surface.

Shannon entropy of a particle on a conical surface

In this work we solve the time-independent Schrödinger equation of a particle restricted to move on the surface of a circular cone of finite height. The energy eigenvalues, as well as the corresponding wave functions, are obtained analytically as a function of, r and ϕ, the radial distance to the apex, 0≤r≤r0, and the angular variable around the axis of the cone. We compute the Shannon entropy of this system in both configuration and momentum space as a function of r0 and θ0, the angular semi-aperture of the cone. In configuration space, the Shannon entropy decreases, signalling a more pronounced localization, as either r0 or θ0 diminish; in momentum space, an opposite behaviour happens, i.e., the Shannon entropy increases when either, r0 or θ0, decrease. We also compute the radial standard deviation; we find that the Shannon entropy better describes the localization-delocalization phenomena. The present results agree with those previously published for a particle confined to a circle of radius r0, which corresponds to θ0=π/2 in the present case.

Constant sectional curvature surfaces with a semi-symmetric non-metric connection

Consider the Euclidean space R3 endowed with a canonical semi-symmetric non-metric connection determined by a vector field C∈X(R3). We study surfaces when the sectional curvature with respect to this connection is constant. In case that the surface is cylindrical, we obtain full classification when the rulings are orthogonal or parallel to C. If the surface is rotational, we prove that the rotation axis is parallel to C and we classify all conical rotational surfaces with constant sectional curvature. Finally, for the particular case 12 of the sectional curvature, the existence of rotational surfaces orthogonally intersecting the rotation axis is also obtained.

Diffractive optical elements on non-planar surfaces for the formation of color 3D images

This paper is dedicated to the development of synthesis methods for diffractive optical elements (DOEs) on non-planar surfaces. To form 3D images, a set of color 2D frames is used, visible to an observer from different observation points. The developed DOEs can be called a stereogram with a 360-deg view when the DOE is illuminated with white light. Methods are developed for computing the phase function of the DOE and are applicable to a wide range of convex surfaces. However, the manufacturing technology used limits this range to conical and cylindrical surfaces. The microrelief of the flat DOE is produced using electron-beam lithography. A video recorded using optical elements produced demonstrates the capabilities of the technologies developed. These DOEs can be used to protect brands against counterfeit.

Compact broadband high-resolution real-time four-dimensional imaging spectrometer

A broadband high-resolution real-time four-dimensional imaging spectrometer (HRRFDIS) is presented, which can acquire both broadband fine spectra and high-resolution three-dimensional (3D) spatial images of a 3D object in real time. The HRRFDIS consists of a first microlens array arranged in a plane to achieve orthographic view spatial imaging, a second microlens array arranged on a conical surface to measure the depth and to achieve 360-degree side-view spatial imaging, multiple optical fibers, a collimating microlens array arranged in a straight line, a parallel planar transmission grating pair to obtain high spectral resolution over a broadband spectral range, and an area-array detector. Compared with the scanning four-dimensional imaging spectrometer (FDIS), the HRRFDIS can obtain a broadband high-resolution four-dimensional dataset using only one frame of data, and it is more stable, compact, small-sized, and lightweight. Compared to the staring FDIS using a liquid crystal filter and requiring at least one modulation period of liquid crystal to acquire a complete hyperspectral image, the HRRFDIS can acquire a complete broadband hyperspectral image in real time. Compared to existing snapshot FDIS, the HRRFDIS can achieve much higher spectral resolution, especially over a broadband spectral range. The HRRFDIS is a unique concept that is the first to obtain both high-resolution broadband spectral information and high-resolution 3D spatial information in real time, to the best of our knowledge. The HRRFDIS will be suitable for real-time measurement of 3D objects in the ultraviolet to infrared spectral range.

An overdetermined problem related to the Finsler p$p$‐Laplacian

AbstractIn this paper, we consider the Finsler ‐Laplacian torsion equation. The domain of the problem is bounded by a conical surface supporting a Neumann‐type condition, and an unknown surface supporting both a Dirichlet and a Neumann condition. The case when the cone coincides with the punctured space is included. We show that the existence of a weak solution implies that the unknown surface lies on the boundary of a Finsler‐ball. Incidentally, some properties of the Finsler–Minkowski norms are proved here under mild smoothness assumptions.

Numerical investigation on the performance of combinational spike and transpiration cooling concept for nose cone under hypersonic conditions

Abstract Aiming at solving the thermal protection problems of the nose cone for hypersonic vehicles, a novel combinational spike and transpiration cooling concept is proposed in this paper. Although the temperature of the nose cone structure can be effectively reduced by traditional transpiration cooling, the cooling effect is not ideal due to the extremely high heat flux and pressure near the stagnation point. In the combinational configuration, the bow shock wave at the leading edge of the nose cone is pushed away by the spike, so that the pressure and heat flux along the nose cone surface are reduced obviously, and the cooling effect is also improved significantly. The flow field generated by this innovative concept has been thoroughly examined through numerical simulations employing the compressible Reynolds-Averaged Navier-Stokes (RANS) equations in conjunction with the shear stress transport (SST) k-ω turbulence model. The fluid-thermal interaction is considered by using the conjugate heat transfer approach. The performance of three configurations based on the nose cone are numerically investigated, including transpiration cooling, spike, combinational spike and transpiration cooling. The numerical results show that, compared with the traditional transpiration cooling, the temperature at the stagnation point of the nose cone for the combinational configuration is reduced by about 64%, which proves the feasibility and high efficiency of the novel combinational concept. With the increase of coolant mass flow, the temperature along the nose cone surface for combinational configuration decreases, while the magnitude of the temperature reduction decreases. Therefore, under certain incoming flow conditions, the appropriate coolant mass flow should be considered to achieve better thermal protection effect.

Analytical Solution to the Steady Navier-Stokes Equation for a Supersonic Cone in the Area of Boundary Layer Behind the Shock Wave by Means Tunnel Mathematics

The means of tunnel mathematics (the theory of functions of a spatial complex variable) allow for an analytical solution to the problem of supersonic flow around a cone in the area of the boundary layer and beyond. The peculiar feature of the Navier-Stokes equations is that they allow for the determination of analytical velocity fields of fluids only for a small number of simple problems. Of course, the problem of the supersonic motion of fluid around a cone is not included in this number. Tunnel mathematics is a method for finding analytical vector velocity fields for steady flows of fluids with axial symmetry. The Navier-Stokes equations are then used to determine pressure and temperature distributions. The main theorem of tunnel mathematics allows for the determination of these distributions for planes z = const (similar to constructing slices of a brain in an MRI procedure). By collecting these “slices,” we can obtain full space distributions of pressure and temperature around a supersonic cone. At this stage of investigation, the conclusions obtained through tunnel mathematics make it possible to qualitatively assess the thickness of the boundary layer on the cone's surface, the shape of the shock wave, and whether the shock wave intersects the boundary layer. First of all, we focused on ensuring that the resulting solutions corresponded to the physical pattern of phenomena. No doubt, solutions obtained through tunnel mathematics must be confirmed experimentally.

Anisotropic conformal change of conic pseudo-Finsler surfaces, I*

Abstract The present work is devoted to investigate anisotropic conformal transformation of conic pseudo-Finsler surfaces $(M,F)$, that is, $ F(x,y)\longmapsto \overline{F}(x,y)=e^{\phi(x,y)}F(x,y)$, where the function $\phi(x,y)$ depends on both position $x$ and direction $y$, contrary to the ordinary (isotropic) conformal transformation which depends on position only. If $F$ is a pseudo-Finsler metric, the above transformation does not yield necessarily a pseudo-Finsler metric. Consequently, we find out necessary and sufficient condition for a (conic) pseudo-Finsler surface $(M,F)$ to be transformed to a (conic) pseudo-Finsler surface $(M,\overline{F})$ under the transformation $\overline{F}=e^{\phi(x,y)}F$. In general dimension, it is extremely difficult to find the anisotropic conformal change of the inverse metric tensor in a tensorial form. However, by using the modified Berwald frame on a Finsler surface, we obtain the change of the components of the inverse metric tensor in a tensorial form. This progress enables us to study the transformation of the Finslerian geometric objects and the geometric properties associated with the transformed Finsler function $\overline{F}$. In contrast to isotropic conformal transformation, we have a non-homothetic conformal factor $\phi(x,y)$ that preserves the geodesic spray. Also, we find out some invariant geometric objects under the anisotropic conformal change. Furthermore, we investigate a sufficient condition for $\overline{F}$ to be dually flat or/and projectively flat. Finally, we study some special cases of the conformal factor $\phi(x,y)$. Various examples are provided whenever the situation needs.

Transition delay in a Mach 6 boundary layer using steady blowing and suction strips

Direct numerical simulations (DNS) were carried out to investigate flow control for transition delay using steady blowing/suction strips at the wall of a flared cone at Mach 6 and zero angle of attack. For the numerical investigations of the transition control strategy, the flared cone geometry and the flow conditions of the experiments in the Boeing/Air Force Office of Scientific Research (AFOSR) Mach 6 Quiet Tunnel (BAM6QT) at Purdue University were chosen. For the DNS, transition was initiated by introducing random disturbances at the inflow of the computational domain, emulating ‘natural’ transition in wind-tunnel experiments caused by free-stream noise. In both wind-tunnel experiments and numerical simulations, streamwise ‘hot’ streaks were found on the surface of the flared cone, which are caused by a nonlinear interaction of an axisymmetric second-mode wave and a pair of oblique waves of the same frequency (‘fundamental resonance’). The objective of the flow control strategy proposed here is to delay the transition onset, and thus mitigate the negative consequences associated with the nonlinear transition stages, i.e. the development of hot streaks and large wall-pressure amplitudes that were observed in experiments and DNS. Our previous so-called ‘controlled’ transition simulations have shown that flow control using steady blowing and suction strips can lead to a significant delay of the hot streak development on the surface of the flared cone. The simulation results presented in this paper show that this flow control strategy remains effective, even in a natural transition scenario characterized by broadband disturbances.