跨学科:促进学生数学核心素养发展——“直柱体的体积”教学课例的实践研究

以苏教版小学数学六年级上册第一单元中长方体和正方体体积的计算为基础,设计“直柱体的体积”课程内容,从整合内容、统整学科、技术赋能、回归生活四个角度分析了跨学科教学的实施策略,对该课程教学过程进行了回顾反思,旨在引导学生综合运用数学学科知识、跨学科知识和方法去发现问题、提出问题、分析问题和解决实际问题,促进学生数学核心素养的发展。

伪黎曼乘积空间 Npm(c)× ℝ中的λ-双调和超曲面

本文主要研究伪黎曼乘积空间Npm(c)× ℝ中的λ-双调和超曲面,给出超曲面是λ-双调和的等价方程,证得Npm(c)× ℝ中具有常平均曲率且形状算子可对角化的λ-双调和(εm+1λ≥0)超曲面要么是极小的,要么是一个直柱体。利用该结论,在角度函数为常数的假设下,对Npm(c)× ℝ中的Einstein 型 λ-双调和超曲面进行分类。特别地,我们讨论了(ℍm(c) × ℝ,gN- dt2)中至多具有两个不同主曲率的λ-双调和类空超曲面(Mm,g),在角度函数是常数且双曲角a≠0的假设下证得超曲面Mm要么是极小的,要么是一个直柱体。In this paper, we study the λ-biharmonic hypersurfaces in the pseudo-Riemannian product space Npm(c)× ℝ, and derive λ-biharmonic equation. It is shown that the λ-biharmonic hypersurfaces (εm+1λ≥0) with constant mean curvature and shape op-erator can be diagonalizable are either minimal or a vertical cylinder. Utilizing this result, the paper classifies Einstein-type λ-biharmonic hypersurfaces in Npm(c)× ℝ under the assumption of a constant angle function. In particular, it classifies at most two distinct principal curvatures of λ-biharmonic space-like hypersurfaces (Mm,g) in (ℍm(c) × ℝ,gN- dt2) , and proves that under the assumption of a constant angle and hyperbolic angle a≠0, the hypersurface Mm is either minimal or a vertical cylinder.

Local Flow Regime Transition Criteria of Gas-Liquid Two-phase Flow in Vertical Upward Tube with a Horizontal Rod

The upward multiphase cross flow and heat transfer in the vertical tube may occur in oil production and chemical facilities. In this study, the local flow patterns of an upward gas-water two phase cross flow in a vertical tube with a horizontal rod have been investigated with an optical probe and the digital high speed video system. The local flow patterns are defined as the bubble, slug, churn and annular flow patterns. Optical probe signals are ana- lyzed in terms of probability density function, and it is proved that the local flow patterns can be recognized by this method. The transition mechanisms between the different flow patterns have been analyzed and the corresponding transitional models are proposed. Finally, local flow pattern maps of the upward gas-water two-phase flow in the vertical tube with a horizontal rod are constructed.

ADirect Method for the Problem of the Solid Cylinder in Elasticity

The traditional semi inverse solution method of the Saint Venant problem and the Saint Venant principle,which were described in the Euclidian space under the Lagrange system formulation,are updated to be solved in the symplectic space under the conservative Hamiltonian system. Thus,the Saint Venant problem and the Saint Venant principle have been unified by the direct method. It is proved in the present paper that all the Saint Venant solutions can be obtained directly via the zero eigenvalue solutions and all their Jordan normal form of the corresponding Hamiltonian operator matrix and the Saint Venant principle corresponds to neglect the non zero eigenvalue solutions,where the non zero eigenvalues give the decay rates.

Analytical Study on Wave Diffraction from a Vertical Circular Cylinder in Front of Orthogonal Vertical Walls

In this paper, the principle of mirror image is used to transform the problem of wave diffraction from a circular cylinder in front of orthogonal vertical walls into the problem of diffraction of four symmetric incident waves from four symmetrically arranged circular cylinders, and then the eigenfunction expansion of velocity potential and Grafs addition theorem are used to give the analytical solution to the wave diffraction problem. The relation of the total wave force on cylinder to the distance between the cylinder and orthogonal vertical walls and the incidence angle of wave is also studied by numerical computation.

Transient analysis of Casson fluid thermo-convection from a vertical cylinder embedded in a porous medium: Entropy generation and thermal energy transfer visualization

Thermal transport in porous media has stimulated substantial interest in engineering sciences due to increasing applications in filtration systems,porous bearings,porous layer insulation,biomechanics,geomechanics etc.Motivated by such applications,in this article,a numerical study of entropy generation impacts on the heat and momentum transfer in time-dependent laminar incompressible boundary layer flow of a Casson viscoplastic fluid over a uniformly heated vertical cylinder embedded in a porous medium is presented.Darcy’s law is used to simulate bulk drag effects at low Reynolds number for an isotropic,homogenous porous medium.Heat line visualization is also included.The mathematical model is derived and normalized using appropriate transformation variables.The resulting non-linear time-dependent coupled governing equations with associated boundary conditions are solved via an implicit finite difference method which is efficient and unconditionally stable.The outcomes show that entropy generation and Bejan number are both elevated with increasing values of Darcy number,Casson fluid parameter,group parameter and Grashof number.To analyze the heat transfer process in a two-dimensional domain,plotting heat lines provides an excellent approach in addition to streamlines and isotherms.It is remarked that as the Darcy number increases,the deviations of heat lines from the hot wall are reduced.

巧用液体内部压强公式在规则固体压强

公式p=ρgh是用来计算静止液体内部压强.但是对于形状规则、质量分布均匀的直柱体,求其所产生的压强时,利用上式求解会非常简捷、方便.

An experimental investigation of dual-resonant and non-resonant responses for vortex-induced vibration of a long slender cylinder

Experimental results of the dual-resonant and non-resonant responses are presented for vortex-induced vibrations(VIV)of a long slender cylinder.The cylinder has a diameter of 10mm and a length of 3.31 m,giving an aspect ratio of 331.The cylinder was towed by a carriage with the velocity up to 1.5 m/s,with the Reynolds number varying from 2500 to 38000.Three different weights were used to provide the initial tension.Dual resonance means that resonance occurs simultaneously in both the cross-flow(CF)and in-line(IL)directions.The experiments were conducted in two stages.At the first stage,dual-resonant dynamic features of the cylinder subjected to vortex-induced excitation were investigated.The features of CF and IL vibration amplitude,motion orbits,phase angle differences,dominant frequencies and mode order numbers are presented.At the second stage of the experiments,particular emphasis was placed on non-resonant dynamic features.The variation of multi-mode modal displacement amplitudes was investigated in detail.