The Clairaut's theorem on rotational surfaces in pseudo-Euclidean 4-space with index 2

Effects of magnetic field, binary particle loading and rotational conic surface on phase change process in a PCM filled cylinder

In this paper, we give a new approach to the rotational minimal surfaces in $4$-dimensional Euclidean space $\mathbb{R}^4$. One type of these surfaces is obtained by the composition of two families of rotations in orthogonal planes. For these surfaces, we give a new parameterization. Using this parametrization, we find new examples of rotational minimal surfaces and rotational surfaces with zero Gaussian curvature.

Weingarten surfaces are those whose principal curvatures satisfy a functional relation, whose set of solutions is called the curvature diagram or the W-diagram of the surface. Making use of the notion of geometric linear momentum of a plane curve, we propose a new approach to the study of rotational Weingarten surfaces in Euclidean 3-space. Our contribution consists of reducing any type of Weingarten condition on a rotational surface to a first-order differential equation on the momentum of the generatrix curve. In this line, we provide two new classification results involving a cubic and an hyperbola in the W-diagram of the surface characterizing, respectively, the non-degenerated quadric surfaces of revolution and the elasticoids, defined as the rotational surfaces generated by the rotation of the Euler elastic curves around their directrix line. As another application of our approach, we deal with the problem of prescribing mean or Gauss curvature on rotational surfaces in terms of arbitrary continuous functions depending on distance from the surface to the axis of revolution. As a consequence, we provide simple new proofs of some classical results concerning rotational surfaces, such as Euler’s theorem about minimal ones, Delaunay’s theorem on constant mean curvature ones, and Darboux’s theorem about constant Gauss curvature ones.

In the four-dimensional pseudo-Euclidean space with neutral metric there are three types of rotational surfaces with two-dimensional axis — rotational surfaces of elliptic, hyperbolic or parabolic type. A surface whose mean curvature vector field is lightlike is said to be quasi-minimal. In this paper we classify all rotational quasi-minimal surfaces of elliptic, hyperbolic and parabolic type, respectively.

We consider the Randers space (Vn, Fb) obtained by perturbing the Euclidean metric by a translation, Fb = α + β, where α is the Euclidean metric and β is a 1-form with norm b, 0 ≤ b < 1. We introduce the concept of a hypersurface with constant mean curvature in the direction of a unitary normal vector field. We obtain the ordinary differential equation that characterizes the rotational surfaces (V3, Fb) of constant mean curvature (cmc) in the direction of a unitary normal vector field. These equations reduce to the classical equation of the rotational cmc surfaces in Euclidean space, when b = 0. It also reduces to the equation that characterizes the minimal rotational surfaces in (V3, Fb) when H = 0, obtained by M. Souza and K. Tenenblat. Although the differential equation depends on the choice of the normal direction, we show that both equations determine the same rotational surface, up to a reflection. We also show that the round cylinders are cmc surfaces in the direction of the unitary normal field. They are generated by the constant solution of the differential equation. By considering the equation as a nonlinear dynamical system, we provide a qualitative analysis, for . Using the concept of stability and considering the linearization around the single equilibrium point (the constant solution), we verify that the solutions are locally asymptotically stable spirals. This is proved by constructing a Lyapunov function for the dynamical systemand by determining the basin of stability of the equilibrium point. The surfaces of rotation generated by such solutions tend asymptotically to one end of the cylinder.

To tackle the problems in adjusting and controlling shapes of rotation surfaces, a new efficient method for quickly constructing generalized Bezier rotation surfaces with multiple shape parameters is proposed. Firstly, following the important idea of transfinite vectored rational interpolating function, the shape-adjustable generalized Bezier rotation surfaces are constructed using a generalized Bezier curve with multiple shape parameters. Secondly, the explicit function expression of the shape-adjustable generalized Bezier rotation surfaces is presented. The new rotation surfaces inherit the outstanding properties of the Bezier rotation surfaces, with a good performance on adjusting their local shapes by changing the shape parameters. Finally, some properties of the new rotation surfaces are discussed, and the influence rules of the shape parameters on the new rotation surfaces are studied. The modeling examples illustrate that the shape-adjustable generalized Bezier rotation surfaces provide a valuable way for the design of rotation surfaces.

The methodology of geometrical modeling of surface deformation and surface rotation [surface Euler–Lagrange deformation of the first kind (Lagrange surface strain), linearized surface rotation tensor, and surface Euler–Lagrange deformation tensor of the second kind (Lagrangian tensor of change of curvature)] is applied for the analysis of space geodetic data (IERS-JGS/ITRF97: 390 stations of type VLBI, SLR, GPS, DORIS) in Europe and in the Mediterranean area. The present-day surface deformation and rotation patterns are compared with seismotectonic features for the test area, which extends from the Atlantic Ocean in the west to the Black Sea in the east, and from Fennoscandia in the north as far as the northern border of the African Plate in the south.

The ground motions recorded in the near‐field regions of earthquakes reveal that, along with translational motions the rotational motions can severely damage structures. To capture these ground rotations more realistically, the present study makes a novel attempt to model the Earth medium as a horizontally layered reduced micropolar half‐space. Layers in this medium are characterized by five material constants: Lame's constant (), Eringen's shear modulus (), density (), rotational moment of inertia (), and an additional material constant (). Cylindrical coordinate system is used to analytically derive the Green's functions and obtain the response of the layered Earth medium subjected to a point earthquake force. The analytically obtained surface translations and rotations involve integrations over the wavenumber and frequency parameters and hence, are evaluated numerically. These solutions converge to corresponding classical elastic medium when the micropolar medium parameters obey the relations and , where is the maximum frequency considered in the simulation. Thereafter, ground motions are simulated for several combinations of earthquake magnitudes and epicenter distances by modeling the Earth medium as an eight layered reduced micropolar half‐space. Further, ground motions for the 2012 Wutai earthquake were simulated using the proposed model and compared with the response of classical elastic medium. The comparisons indicate that the surface rotations are significantly higher in the case of reduced micropolar medium. The proposed formulation can be extended to simulate ground motions for finite‐fault seismic sources.

The accumulation of ice on the aero-engine inlet compromises engine safety. Traditional hot air anti-icing systems, which utilize bleed air, require substantial energy, decreasing engine performance and increasing emissions. Superhydrophobic materials have shown potential in reducing energy consumption when combined with these systems. Research indicates that superhydrophobic surfaces on stationary components significantly reduce anti-icing energy consumption by altering runback water flow behavior. However, for rotating aero-engine components, the effectiveness of superhydrophobic surfaces and the influence of surface wettability on runback water flow remain unclear due to centrifugal and Coriolis forces. This study investigates the runback water flow behavior on aero-engine rotating spinner surfaces with varying wettabilities in a straight-flow spray wind tunnel. The results demonstrated that centrifugal force reduces the amount of runback water on the rotating spinner compared to the stationary surface, forming rivulet flows deflected opposite to the direction of rotation. Furthermore, wettability significantly affects the flow characteristics of runback water on rotating surfaces. As the contact angle increases, the liquid water on the rotating spinner transitions from continuous film flow to rivulet and bead-like flows. Notably, the superhydrophobic surface prevents water adhesion, indicating its potential for anti-icing on rotating components. In addition, the interaction between rotational speed and surface wettability enhances the effects, with both increased rotational speed and larger contact angles contributing to higher liquid water flow velocities, promoting the rapid formation and detachment of rivulet and bead-like flows.

In this work, we define the rotational surface with a light-like axis in conformally flat pseudo-spaces $\left(\mathbb{E}_3^1\right)_\lambda$, where $\lambda$ is a radial-type conformal factor. We relate the principal curvatures of a non-degenerate surface that belongs to conformally equivalent spaces $\left(\mathbb{E}_3^1\right)_\lambda$ and $\mathbb{R}_1^3$, based on the radial conformal factor. Thus, we establish a relationship between the radial conformal factor and the profile curve of the rotational flat surface in $\left(\mathbb{E}_3^1\right)_\lambda$, but also for that of the rotational surface with zero extrinsic curvature.

In this paper, we study on the characterizations of loxodromes on the rotational surfaces satisfying some special geometric properties such as having constant Gaussian curvature and a constant ratio of principal curvatures (CRPC rotational surfaces). First, we give the parametrizations of loxodromes parametrized by arc-length parameter on any rotational surfaces in $\mathbb{E}^{3}$ and then, we calculate the curvature and the torsion of such loxodromes. Then, we give the parametrizations of loxodromes on rotational surfaces with constant Gaussian curvature. Also, we investigate the loxodromes on the CRPC rotational surfaces. Moreover, we give the parametrizations of loxodromes on the minimal rotational surface which is a special case of CRPC rotational surfaces. Finally, we give some visual examples to strengthen our main results via Wolfram Mathematica.

The work presents research on the tasks of computer modeling of surfaces by the inscribed or described cone and additional structural conditions. Among the structural conditions, in particular, can be distinguished the contact line of the described cone with the surface that is modeled. In the case when the surface that model is the surface of the second order (quadriса), the contact line should also be a second order curve (conica). It is this additional condition that is considered in these studies. In this work is showed an algorithm for modeling the rotation surface. We want to pay attention to such a moment that the plane in which the contact line is located must be perpendicular to at least one of the two planes of the inscribed (described) cone. It is clear that in any case, the task is to find the meridian of the surface of rotation and the search for the axis of this surface. The work shows that the normal to the cone, drawn anywhere in the contact line, intersects with a perpendicular, which is made from out from the center of the circular section, if the circular section is drawn through the same contact section. The considered geometric task will be the basis of the applied task, when it is necessary to modeling the surface by conjugating the surface of the second order of the general type and surface of the rotation. Connecting of the surfaces will take place along a flat second-order curves. This task has already been considered in one of the works of the authors, but it was solved by means of 3D modeling and was implemented in the Solid Works system. The presented work deals with tasks that can occur with 2D computer modeling or manual modeling by graphic methods. A graphical toolkit is proposed, which allows solving spatial problems of constructing a surface of rotation for a given inscribed or described cone in 2D space. Namely, the task of finding the meridian and axis of the surface of rotation, including the length of this axis, is presented. The problem has an important application in the conjugation of second-order surfaces. In particular, it is shown that for a given contact section and a given cone, only one surface of rotation can be constructed. However, it can be conjugated with a one-parameter set of surfaces of general appearance.

Lithium abundance A(Li) and surface rotation are good diagnostic tools to probe the internal mixing and angular momentum transfer in stars. We explore the relation between surface rotation, A(Li) and age in a sample of seismic solar-analogue (SA) stars and study their possible binary nature. We select a sample of 18 SA observed by the NASA Kepler satellite for an in-depth analysis. Their seismic properties and surface rotation are well constrained from previous studies. About 53 hours of high-resolution spectroscopy were obtained to derive fundamental parameters and A(Li). These values were combined and confronted with seismic masses, radii and ages, as well as surface rotation periods. We identify a total of 6 binary systems. A well-defined relation between A(Li) and rotation was obtained. With models constrained by the characterisation of the individual mode frequencies for single stars, we identify a sequence of three SA with similar mass (~1.1Mo) and stellar ages ranging between 1 to 9 Gyr. Within the realistic estimate of ~7% for the mass uncertainty, we find a good agreement between the measured A(Li) and the predicted A(Li) evolution from a grid of models calculated with the Toulouse-Geneva stellar evolution code, which includes rotational internal mixing, calibrated to reproduce solar chemical properties. We present A(Li) for a consistent spectroscopic survey of SA with a mass of 1.00+/-0.15Mo, and characterised through asteroseismology and surface rotation rates based on Kepler observations. The correlation between A(Li) and P_rot supports the gyrochronological concept for stars younger than the Sun. The consensus between measured A(Li) for solar analogues with model grids, calibrated onto the Sun's chemical properties suggests that these targets share the same internal physics. In this light, the solar Li and rotation rate appear to be normal for a star like the Sun.

Mean curvatures and Gauss maps of a pair of isometric helicoidal and rotation surfaces in Minkowski 3-space

The construction of λμ-B-spline curves and its application to rotational surfaces