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Related Concept Videos

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...
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Newtonian Fluid: Problem Solving01:18

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Newtonian fluids exhibit a constant viscosity, meaning their shear stress and shear strain rate are directly proportional. This property ensures a predictable and stable response to applied forces, maintaining a linear relationship between force and flow. Examples include water, air, and light oils, consistently demonstrating this proportional behavior regardless of external conditions.
A velocity gradient forms within the fluid when a Newtonian fluid is placed between two parallel plates, with...
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Divergence and Stokes' Theorems01:06

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The divergence and Stokes' theorems are a variation of Green's theorem in a higher dimension. They are also a generalization of the fundamental theorem of calculus. The divergence theorem and Stokes' theorem are in a way similar to each other; The divergence theorem relates to the dot product of a vector, while Stokes' theorem relates to the curl of a vector. Many applications in physics and engineering make use of the divergence and Stokes' theorems, enabling us to write...
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Bernoulli's Equation for Flow Normal to a Streamline01:16

Bernoulli's Equation for Flow Normal to a Streamline

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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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Turbulent Flow: Problem Solving01:09

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Carbonation is a process used to dissolve carbon dioxide gas in a liquid, commonly used in the production of carbonated beverages. Achieving efficient carbonation requires careful control of temperature, pressure, and flow conditions. By adjusting these parameters, carbonation efficiency can be maximized, producing a higher concentration of CO2 in the liquid.
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Steady, Laminar Flow Between Parallel Plates01:17

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Understanding steady, laminar flow between parallel plates is essential for analyzing and designing flow in narrow rectangular channels, commonly found in various water conveyance and drainage systems. The Navier-Stokes equations govern fluid motion and are generally challenging to solve due to their nonlinearity. However, simplifications are possible in certain cases, like the steady laminar flow between parallel plates. For this scenario, we assume steady, incompressible, laminar flow.
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Magnetically Induced Rotating Rayleigh-Taylor Instability
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The Navier-Stokes regularity problem.

James C Robinson1

  • 1Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, UK.

Philosophical Transactions. Series A, Mathematical, Physical, and Engineering Sciences
|June 9, 2020
PubMed
Summary
This summary is machine-generated.

The existence and uniqueness of solutions for the 3D Navier-Stokes equations remain unproven. This review highlights rigorous results and the connection between solution regularity and uniqueness.

Keywords:
Clay Millennium ProblemNavier–Stokes equationsregularity and uniqueness

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Area of Science:

  • Fluid dynamics
  • Mathematical physics
  • Partial differential equations

Background:

  • The Millennium Prize Problems include the existence and smoothness of solutions to the Navier-Stokes equations.
  • Understanding these solutions is crucial for fluid flow modeling.

Purpose of the Study:

  • To review rigorous mathematical results on the existence and uniqueness of solutions for the 3D Navier-Stokes equations.
  • To emphasize the relationship between solution regularity and uniqueness.

Main Methods:

  • Review of existing literature on the mathematical theory of Navier-Stokes equations.
  • Analysis of rigorous results concerning solution properties.

Main Results:

  • No definitive proof guarantees unique, globally existing solutions for smooth initial conditions.
  • Key theorems and inequalities related to solution regularity are presented.
  • The link between the regularity of solutions and their uniqueness is a central theme.

Conclusions:

  • The mathematical understanding of Navier-Stokes solutions is an active and challenging research area.
  • Further research is needed to establish the existence and uniqueness of solutions.
  • The regularity of solutions is intrinsically tied to their uniqueness properties.