On Minimal Hypersurfaces of a Unit Sphere

Minimal compact hypersurface in the unit sphere Sn+1 having squared length of shape operator A2<n are totally geodesic and with A2=n are Clifford hypersurfaces. Therefore, classifying totally geodesic hypersurfaces and Clifford hypersurfaces has importance in geometry of compact minimal hypersurfaces in Sn+1. One finds a naturally induced vector field w called the associated vector field and a smooth function ρ called support function on the hypersurface M of Sn+1. It is shown that a necessary and sufficient condition for a minimal compact hypersurface M in S5 to be totally geodesic is that the support function ρ is a non-trivial solution of static perfect fluid equation. Additionally, this result holds for minimal compact hypersurfaces in Sn+1, (n>2), provided the scalar curvature τ is a constant on integral curves of w. Yet other classification of totally geodesic hypersurfaces among minimal compact hypersurfaces in Sn+1 is obtained using the associated vector field w an eigenvector of rough Laplace operator. Finally, a characterization of Clifford hypersurfaces is found using an upper bound on the integral of Ricci curvature in the direction of the vector field Aw.

Linear and Nonlinear Electrostatic Excitations and Their Stability in a Nonextensive Anisotropic Magnetoplasma

In the present work, the propagation of (non)linear electrostatic waves is reported in a normal (electron–ion) magnetoplasma. The inertialess electrons follow a non-extensive q-distribution, while the positive inertial ions are assumed to be warm mobile. In the linear regime, the dispersion relation for both the fast and slow modes is derived, whose properties are analyzed parametrically, focusing on the effect of nonextensive parameter, component of parallel anisotropic ion pressure, component of perpendicular anisotropic ion pressure, and magnetic field strength. The reductive perturbation technique is employed for reducing the fluid equation of the present plasma model to a Zakharov–Kuznetsov (ZK) equation. The parametric role of physical parameters on the characteristics of the symmetry planar structures such solitary waves is investigated. Furthermore, the stability of the pulse soliton solution of the ZK equation against the oblique perturbations is investigated. Furthermore, the dependence of the instability growth rate on the related physical parameters is examined. The present investigation could be useful in space and astrophysical plasma systems.

Fluid Equation-Based and Data-Driven Simulation of Special Effects Animation

This paper analyzes the simulation of special effects animation through fluid equations and data-driven methods. This paper also considers the needs of computer fluid animation simulation in terms of computational accuracy and simulation efficiency, takes high real-time, high interactivity, and high physical accuracy of simulation algorithm as the research focus and target, and proposes a solution algorithm and acceleration scheme based on deep neural network framework for the key problems of simulation of natural phenomena including smoke and liquid. With the deep development of artificial intelligence technology, deep neural network models are widely used in research fields such as computer image classification, speech recognition, and fluid detail synthesis with their powerful data learning capability. Its stable and efficient computational model provides a new problem-solving approach for computerized fluid animation simulation. In terms of time series reconstruction, this paper adopts a tracking-based reconstruction method, including target tracking, 2D trajectory fitting and repair, and 3D trajectory reconstruction. For continuous image sequences, a linear dynamic model algorithm based on pyramidal optical flow is used to track the feature centers of the objects, and the spatial coordinates and motion parameters of the feature points are obtained by reconstructing the motion trajectories. The experimental results show that in terms of spatial reconstruction, the matching method proposed in this paper is more accurate compared with the traditional stereo matching algorithm; in terms of time series reconstruction, the error of target tracking reduced. Finally, the 3D motion trajectory of the point feature object and the motion pattern at a certain moment are shown, and the method in this paper obtains more ideal results, which proves the effectiveness of the method.

One-dimensional thermal equation modeled on a two-dimensional heat exchanger with variable solid-fluid interface

In this study, we are to present that a one-dimensional equation for vertically averaged temperature, modeled on a vertically thin, two-dimensional heat exchanger with variable top solid-fluid interface, recovers the two-dimensional thermal information, i.e. steady temperature and flux distribution on the top and temperature-fixed bottom faces. The relative error of these quantities is less than 5% with the maximum gradient of the height kept approximately below 0.5, while the computational time is reduced to 0.1–5%, when compared with direct two-dimensional computations, depending on the shape of the top face. The model equation, derived by the vertical averaging of the two-dimensional thermal conduction equation, is closed by an approximation that the heat exchanger is sufficiently thin in the sense that the second derivative of temperature with respect to the horizontal coordinate depends only on the coordinate. In this model equation, the fluid equation above the exchanger is decoupled by a conventional equation for the normal heat flux on the top surface. In principle, however, the coupling of the model and the fluid equation is possible through the temperature and heat flux on the top interface, recovered by the model equation. The type of mathematical modeling can be applicable to a wide variety of bodies with extremely small dimensions in some (coordinate-transformed) directions.

A Hybrid Real/Ideal Gas Mixture Computational Framework to Capture Wave Propagation in Liquid Rocket Combustion Chamber Conditions

The present work focuses on the development of new mathematical and numerical tools to deal with wave propagation problems in a realistic liquid rocket chamber environment. A simplified real fluid equation of state is here derived, starting from the literature. An approximate Riemann solver is then specifically derived for the selected conservation laws and primitive variables. Both the new equation of state and the new Riemann solver are embedded into an in-house one-dimensional CFD solver. The verification and validation of the new code against wave propagation problems are then performed, showing good behavior. Although such problems might be of interest for different applications, the present study is specifically oriented to the low order modeling of high-frequency combustion instability in liquid-propellant rocket engines.

Stability and Unbalance Analysis of Rigid Rotors Supported by Spiral Groove Bearings: Comparison of Different Approaches

The dynamic behavior of spiral-grooved gas bearing supported 4DoF rotors is investigated by means of linearized bearing force coefficients and full time-integrated transient analysis. The transient method consists of a state-space representation, which couples the equations of motion with the compressible thin film fluid equation. The linearized method is based on the perturbation analysis around a given eccentric shaft position ε, allowing to compute the static and linear dynamic bearing force coefficients at different excitation frequencies. The two methods are compared for a variation of test rotors and bearing geometries in a given compressibility number interval of Λ = [0,40]. The limitations and weaknesses of the linearized model are presented. It is shown that shafts with two symmetric herringbone-groove journal bearings have their maximum stability and load capacity if the center of gravity lays in the middle of the two bearings. For symmetric rotors (la/lb = 1) the two rigid modes, cylindrical and conical, are present and are influenced by the mass and transverse moment of inertia independently. For asymmetric rotors (la/lb &lt; 1) the stability region decreases and the modes have a mixed shape. It is no longer possible to clearly distinguish between pure cylindrical and pure conical mode shapes. The two methods predict the critical mass and critical transverse moment of inertias within a difference of &lt; 7%. A quasi-linear unbalance module for rigid gas bearing supported rotors is presented, which considers eccentricity dependent bearing force coefficients, allowing to speed up the unbalance response analysis by four orders of magnitude. The unbalance module is compared with the full transient orbital analysis, suggesting that the quasi-linear module predicts the non-linear unbalance response with &lt; 6 % deviation for amplitudes up to ε &lt; 0.5 within the complete compressibility number range.

A new well-behaved class of compact strange astrophysical model consistent with observational data

The main focus of this paper is to discuss the solutions of Einstein’s Field Equations (EFEs) for compact spherical objects study. To supply exact solution of the EFEs, we have considered the distribution of anisotropic matter governed by a new version of Chaplygin fluid equation of state (EoS). To determine different constants, we have represented the outer space-time by the Schwarzschild metric. Using the observed values of the mass for the various strange spherical object candidates, we have expanded anisotropic emphasize at the surface to forecast accurate radius estimates. Moreover, we implement various analysis to examine the physical acceptability and stability of our suggested stellar model viz., the energy conditions, cracking method, adiabatic index, etc. Graphical survey exhibits that the obtained stellar system fulfills the physical and mathematical prerequisites of the strange astrophysical object candidates Cyg X-2, Vela X-1, 4U 1636-536, 4U 1608-52, PSR J1903+327 to examine the various physical parameters and their effects on the anisotropic stellar model. The investigation reveals that complicated geometries arise from the interior matter distribution obeys a new version of Chaplygin fluid EoS and they are physically pertinent in the investigation of discovered compact structures.

Simulation of Crude Oil Transportation with Drag Reduction Agents Using k-ϵ and k-ω Models

This work explores the possibility of using Newtonian turbulence k−ϵ and k−ω models for modelling crude oil flow in pipelines with drag reduction agents. These models have been applied to predict the friction factor, pressure drop and the drag reduction percentage. The simulation results of both models were compared with six published experimental data for crude oil flow in pipes with different types of drag reduction agents. The velocity near the wall was determined using the log law line of Newtonian fluid equation and by changing the parameter ΔB to achieve an excellent agreement with experimental data. Simulated data for k−ϵ model shows better agreement with most experimental data than the k−ω turbulence model.