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相关概念视频

Euler's Equations of Motion01:28

Euler's Equations of Motion

577
In fluid mechanics, shear stresses arise from viscosity, which represents a fluid's internal resistance to deformation. For low-viscosity fluids, like water, these stresses are minimal, simplifying flow analysis by allowing the fluid to be treated as inviscid, or frictionless. In an inviscid fluid, shear stresses are absent, leaving only normal stresses, which act perpendicularly to fluid elements. Notably, pressure — defined as the negative of the normal stress — remains...
577
Euler Equations of Motion01:19

Euler Equations of Motion

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Imagine a rigid body that is rotating at an angular velocity of ω within an inertial frame of reference. Along with this, picture a second rotating frame that is attached to the body itself. This frame moves along with the body and possesses an angular velocity of Ω. The total moment about the center of mass is calculated by adding the rate of change of angular momentum about the center of mass in relation to the rotating frame and the cross-product of the body's angular velocity...
336
Navier–Stokes Equations01:28

Navier–Stokes Equations

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For incompressible Newtonian fluids, where density remains constant, stresses show a linear relationship with the deformation rate, defined by normal and shear stresses. Normal stresses depend on the pressure exerted on the fluid and the rate of deformation in specific directions, which determines how fluid flows under varying pressures. Shear stresses, on the other hand, act tangentially across fluid layers. They explain how adjacent fluid layers slide relative to one another, connecting...
766
Deflection of a Beam01:19

Deflection of a Beam

383
Accurately determining beam deflection and slope under various loading conditions in structural engineering is crucial for ensuring safety and structural integrity. Singularity functions offer a streamlined approach to analyzing beams, especially when multiple loading functions complicate the bending moment equation.
Singularity functions, described in an earlier lesson, are powerful mathematical tools that represent discontinuities within a function commonly encountered in structural loading...
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Equations of Equilibrium in Three Dimensions01:30

Equations of Equilibrium in Three Dimensions

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When analyzing structures or systems at rest, it is necessary to ensure they are in equilibrium. This is where the vector and scalar equations of equilibrium come into play. These equations are crucial in ensuring a structure is stable and will not collapse or fall apart. The vector and scalar equations of equilibrium provide a framework for analyzing the forces acting on a body.
According to the vector equations of equilibrium, the vector sum of all the external forces acting on a body must...
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Bernoulli's Equation for Flow Along a Streamline01:30

Bernoulli's Equation for Flow Along a Streamline

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Bernoulli's equation relates the energy conservation in a fluid moving along a streamline. The equation applies to incompressible and inviscid fluids under steady flow. For such a flow, Newton's second law is applied to a small fluid element, which experiences forces due to pressure differences, gravity, and velocity variations. The force balance leads to the following form of Bernoulli's equation:
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相关实验视频

Updated: Sep 17, 2025

Investigating the Three-dimensional Flow Separation Induced by a Model Vocal Fold Polyp
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在3D欧勒方程中形成奇点,具有平滑的初始数据和边界数据.

Jiajie Chen1, Thomas Y Hou2

  • 1Courant Institute of Mathematical Sciences, New York University, New York, NY 10012.

Proceedings of the National Academy of Sciences of the United States of America
|June 27, 2025
PubMed
概括
此摘要是机器生成的。

研究人员审查了一项计算机辅助的证明,显示了2D Boussinesq和3D Euler方程中的有限时间奇点膨胀. 这解决了数学流体动力学和非线性部分微分方程的一个基本问题.

关键词:
欧勒方程 欧勒方程是什么?通过计算机辅助的证明.流体动力学的流体动力学奇点的形成 奇点的形成

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Experimental Measurement of Settling Velocity of Spherical Particles in Unconfined and Confined Surfactant-based Shear Thinning Viscoelastic Fluids
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相关实验视频

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Visually Based Characterization of the Incipient Particle Motion in Regular Substrates: From Laminar to Turbulent Conditions
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科学领域:

  • 数学流体动力学数学流体动力学
  • 非线性局部微分方程非线性局部微分方程
  • 计算数学是指计算数学.

背景情况:

  • 1757年由莱昂哈德·欧勒介绍的3D不可压缩的欧勒方程,是流体动力学和流的基础.
  • 一个关键的未解决的问题是,顺的解决方案是否可以产生有限时间奇点.
  • 这些方程与纳维埃-斯托克斯方程密切相关.

研究的目的:

  • 审查最近关于有限时间奇点形成的计算机辅助证明.
  • 为了研究2D Boussinesq和3D轴对称欧勒方程中的奇点发展.
  • 分析近似自相似的膨胀形状的非线性稳定性.

主要方法:

  • 计算机辅助的证明技术.
  • 动态重新缩放配方用于分析膨胀.
  • 大致自相似配置文件的数值构造.
  • 非线性稳定性的分析.

主要成果:

  • 对于特定的欧勒和布西内斯方程来说,有限时间的,几乎与自身相似的膨胀的演示.
  • 建立一个分析 (几乎) 自相似膨胀的框架.
  • 证实了数量构造的近似自相似型号的非线性稳定性.
  • 对于3D欧勒方程的一般奇点形成问题仍然是开放的.

结论:

  • 这项研究在流体动力学中奇点形成的长期问题上取得了重大进展.
  • 计算机辅助的证明提供了一个新的框架,并证明了近似自相似膨胀的非线性稳定性.
  • 虽然对3D欧勒方程没有提供完整的解决方案,但研究结果为潜在的奇点机制提供了洞察力.