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Related Concept Videos

Euler's Equations of Motion01:28

Euler's Equations of Motion

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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 uniform across...
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Navier–Stokes Equations01:28

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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...
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Basic Equation for Pressure Field01:13

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The basic equation for a pressure field in fluid mechanics captures the balance of forces within any segment of fluid, providing a foundational understanding of how pressure changes within fluids under various forces. Generally, two main types of forces act on any part of a fluid: surface forces and body forces. Surface forces arise from pressure differences across points within the fluid, which result in net forces that can vary depending on the local pressure gradient. Body forces, on the...
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Irrotational Flow01:28

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Irrotational flow is characterized by fluid motion where particles do not rotate around their axes, resulting in zero vorticity. For a flow to be irrotational, the curl of the velocity field must be zero. This imposes specific conditions on velocity gradients. For instance, to maintain zero rotation about the z-axis, the gradient condition:
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Velocity Potential01:20

Velocity Potential

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In steady, incompressible flow through a long, straight pipe with a uniform cross-section, the flow in the central region (far from the pipe walls) is irrotational. This irrotational nature means that fluid particles do not rotate around their axes, and a scalar function called the velocity potential, represented by ϕ, can be used to describe their movement. In irrotational flows, the velocity field V is defined as the gradient of the velocity potential:
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Bernoulli's Equation for Flow Normal to a Streamline01:16

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Bernoulli's equation for flow normal to a streamline explains how pressure varies across curved streamlines due to the outward centrifugal forces induced by the fluid's curvature. The pressure is higher on the inner side of the curve, near the center of curvature, and decreases outward to balance these centrifugal forces.
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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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Initial boundary-value problem for the spherically symmetric Einstein equations with fluids with tangential pressure.

Irene Brito1, Filipe C Mena1

  • 1Centro de Matemática, Universidade do Minho, 4710-057 Braga, Portugal.

Proceedings. Mathematical, Physical, and Engineering Sciences
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Researchers proved a unique solution exists for Einstein

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Area of Science:

  • General Relativity
  • Mathematical Physics

Background:

  • The Einstein equations govern spacetime dynamics.
  • Solving these equations for realistic matter distributions is complex.

Purpose of the Study:

  • To establish the existence and uniqueness of local solutions to the Einstein equations.
  • To analyze solutions for spherically symmetric fluid distributions with tangential pressure.

Main Methods:

  • Utilizing techniques for solving partial differential equations.
  • Applying initial-boundary value problem methods to Einstein's field equations.

Main Results:

  • Proved the existence of a unique, local-in-time solution.
  • Demonstrated this for spherically symmetric fluid distributions with tangential pressure near a time-like boundary.

Conclusions:

  • The mathematical framework supports unique solutions for specific astrophysical scenarios.
  • This has implications for modeling compact objects and gravitational phenomena.