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Euler's Equations of Motion01:28

Euler's Equations of Motion

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...
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...
Hyperbolas01:30

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A hyperbola is a conic section produced when a double-napped cone is intersected by a plane at an angle steeper than the slope of the cone, such that it cuts through both nappes. This intersection yields two separate, mirror-image curves known as branches, which open away from each other along the transverse axis. The nearest points on each branch to the hyperbola’s center are termed vertices, and the distance from the center to a vertex is denoted by a. Perpendicular to the transverse axis is...
Geometry of Hyperbolas01:30

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A hyperbola consists of all points where the absolute difference of distances to two fixed points, called foci, remains constant. The standard equation isEach branch extends infinitely and approaches two asymptotes, which guide the curve’s behavior. The parameters a and b define key features: a measures the distance from the center to each vertex along the transverse axis, while b influences the slopes of the asymptotes. The asymptotes have equationsA rectangle centered at the origin with...
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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An arched gate can be effectively modeled using a hyperbolic cosine profile because this type of function is smooth and symmetric about the vertical axis. When the arch is centered at the origin, its maximum height occurs at the center point. This symmetry ensures that any height below the crown of the arch is reached at two horizontal positions that are equal in distance from the centerline but lie on opposite sides.To determine where the gate reaches a height of five meters, the height of the...

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Related Experiment Video

Updated: May 17, 2026

Investigating the Three-dimensional Flow Separation Induced by a Model Vocal Fold Polyp
09:58

Investigating the Three-dimensional Flow Separation Induced by a Model Vocal Fold Polyp

Published on: February 3, 2014

Euler and Navier-Stokes equations on the hyperbolic plane.

Boris Khesin1, Gerard Misiolek

  • 1School of Mathematics, Institute for Advanced Study, Princeton, NJ 08450, USA. khesin@math.toronto.edu

Proceedings of the National Academy of Sciences of the United States of America
|October 24, 2012
PubMed
Summary
This summary is machine-generated.

Nonuniqueness of Navier-Stokes solutions on the hyperbolic plane arises from Hodge decomposition. This issue is absent in higher dimensions (n ≥ 3), and a general Hamiltonian framework for hydrodynamics is presented.

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Last Updated: May 17, 2026

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Published on: July 19, 2016

Area of Science:

  • Fluid dynamics
  • Differential geometry
  • Mathematical physics

Background:

  • The Navier-Stokes equation governs fluid motion.
  • Leray-Hopf solutions are important in understanding fluid behavior.
  • Nonuniqueness of solutions has been observed on the 2-dimensional hyperbolic plane.

Purpose of the Study:

  • To explain the cause of nonuniqueness for Navier-Stokes solutions on the hyperbolic plane.
  • To investigate if this nonuniqueness occurs in higher dimensions.
  • To develop a general Hamiltonian framework for hydrodynamics on Riemannian manifolds.

Main Methods:

  • Utilizing Hodge decomposition to analyze the Navier-Stokes equation.
  • Comparing solution behavior on the 2-dimensional hyperbolic plane versus higher-dimensional spaces (n ≥ 3).
  • Formulating a general Hamiltonian framework for hydrodynamics.

Main Results:

  • The nonuniqueness of Leray-Hopf solutions on the hyperbolic plane is shown to be a direct consequence of the Hodge decomposition.
  • This nonuniqueness phenomenon is demonstrated not to occur on the n-dimensional hyperbolic space for n ≥ 3.
  • A general Hamiltonian framework encompassing hyperbolic hydrodynamics on complete Riemannian manifolds is described.

Conclusions:

  • Hodge decomposition is identified as the underlying reason for solution nonuniqueness in this specific setting.
  • The study establishes a dimensional threshold (n ≥ 3) where this nonuniqueness is resolved.
  • The generalized Hamiltonian framework provides a unified approach to studying hydrodynamics across different geometric settings.