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Related Concept Videos

Steady Flow of a Fluid Stream01:27

Steady Flow of a Fluid Stream

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Consider a control volume, such as a pipe with solid boundaries, through which fluid flows and changes direction due to the impulse exerted by the resulting force from the pipe walls. In steady flow, the mass of fluid entering the control volume at a given time, t, with velocity v1, is equal to the mass leaving after infinitesimal time dt, with velocity v2.
During this process, the momentum of the fluid within the control volume remains constant over the time interval dt. By applying the...
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Continuity Equation01:28

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The continuity equation asserts that the mass flow rate must remain constant for a steady flow of an incompressible fluid within a confined system. This principle applies to systems where fluid passes through varying cross-sectional areas, such as nozzles, syringes, and pipes.
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Euler's Equations of Motion01:28

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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...
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Divergence and Curl01:15

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The divergence of a vector field at a point is the net outward flow of the flux out of a small volume through a closed surface enclosing the volume, as the volume tends to zero. More practically, divergence measures how much a vector field spreads out or diverges from a given point. For an outgoing flux, conventionally, the divergence is positive. The diverging point is often called the "source" of the field. Meanwhile, the negative divergence of a vector field at a point means that the...
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Fluid flow analysis is critical in many scientific and engineering disciplines, and two principal approaches are used to describe this flow: the Eulerian and Lagrangian methods. These methods offer different perspectives on monitoring and analyzing the motion of fluids, each with distinct advantages depending on the scenario.
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Bernoulli's Equation for Flow Along a Streamline01:30

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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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Related Experiment Video

Updated: Sep 30, 2025

Experimental Investigation of Secondary Flow Structures Downstream of a Model Type IV Stent Failure in a 180° Curved Artery Test Section
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Einstein flow with matter sources: stability and convergence.

Vincent Moncrief1,2, Puskar Mondal3

  • 1Department of Mathematics, Yale University, New Haven, CT, USA.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
|March 14, 2022
PubMed
Summary

This study investigates Einstein flow with matter sources, exploring if universal homogeneity and isotropy can emerge from initial conditions that are not homogeneous or isotropic. The findings aim to reconcile complex cosmic dynamics with observed universe properties.

Keywords:
Einstein flowcosmic topologycosmological perturbation theorycosmological principle

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Related Experiment Videos

Last Updated: Sep 30, 2025

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Area of Science:

  • Cosmology
  • General Relativity
  • Mathematical Physics

Background:

  • Previous studies on vacuum Einstein flow suggested that many compact manifolds evolve towards homogeneity and isotropy.
  • These prior investigations did not incorporate matter sources, a crucial component of the physical universe.

Purpose of the Study:

  • To incorporate suitable matter sources into the Einstein flow framework.
  • To determine if a similar conclusion regarding asymptotic homogeneity and isotropy can be reached with matter present.

Main Methods:

  • Analysis of Einstein flow dynamics including matter sources.
  • Mathematical investigation of the evolution of cosmological models.

Main Results:

  • The inclusion of matter sources is investigated within the Einstein flow.
  • The study explores the conditions under which a universe can become asymptotically homogeneous and isotropic.

Conclusions:

  • The research aims to establish whether matter sources permit the emergence of large-scale homogeneity and isotropy.
  • This work contributes to understanding the evolution of the universe and its compatibility with observed properties.