The Algebra of Projective Spheres on Plane, Sphere and Hemisphere

Numerous authors studied polarities in incidence structures or algebrization of projective geometry [1] [2]. The purpose of the present work is to establish an algebraic system based on elementary concepts of spherical geometry, extended to hyperbolic and plane geometry. The guiding principle is: “The point and the straight line are one and the same”. Points and straight lines are not treated as dual elements in two separate sets, but identical elements within a single set endowed with a binary operation and appropriate axioms. It consists of three sections. In Section 1 I build an algebraic system based on spherical constructions with two axioms: ab = ba and (ab)(ac) = a, providing finite and infinite models and proving classical theorems that are adapted to the new system. In Section Two I arrange hyperbolic points and straight lines into a model of a projective sphere, show the connection between the spherical Napier pentagram and the hyperbolic Napier pentagon, and describe new synthetic and trigonometric findings between spherical and hyperbolic geometry. In Section Three I create another model of a projective sphere in the Cartesian coordinate system of the plane, and give methods and techniques for using the model in the theory of functions.

Particle Scale Numerical Simulation on Momentum and Heat Transfer of Two Tandem Spheroids:an IB-LBM Study

The cold fluid flowing over two hot spheroids placed in a tandem arrangement was numerically studied via a GPU-based immersed boundary-lattice Boltzmann method(IB-LBM)model.The drag coefficient and average Nusselt number of both the two spheroids were obtained with the main influencing factors investigated.To validate the IB-LBM model,several numerical case studies containing one and two spheres were firstly conducted to reach the good agreement with the previously reported data.Then,a number of simulations were further carried out which were designed by changing the particle aspect ratio(1.0≤Ar≤4.0)and inter particle distance(1.5≤ι≤7.0,whereι=L/D,D stands for the volume-equivalent sphere diameter)as well as the Reynolds number(10≤Re≤200).Their influence on the momentum and heat transfer characteristics between the solid and fluid phases was fully discussed.Numerical results show that,for all the considered Reynolds numbers and aspect ratios,the individual and total drag coefficients and average Nusselt number increase with the inter particle distance.The inter particle distance has greater influence on the drag coefficient and average Nusselt number of the trailing particle than the leading one.The drag coefficient and average Nusselt number of the trailing particle are far less than the leading one under the same working conditions.The prediction correlations for the drag coefficient and average Nusselt number of both the two spheroids were established with low deviations.At last,the influence of the relative incidence angles between the two tandem spheroids on the momentum and heat transfer was studied.It is shown that the relative incidence angles play significant roles due to the change of the frontal area of the leading spheroid with these angles.