Search research articles

ABOUT JoVE

Overview Leadership Blog JoVE Help Center

AUTHORS

Publishing Process Editorial Board Scope & Policies Peer Review FAQ Submit

LIBRARIANS

Testimonials Subscriptions Access Resources Library Advisory Board FAQ

RESEARCH

JoVE Journal Methods Collections JoVE Encyclopedia of Experiments Archive

EDUCATION

JoVE Core JoVE Business JoVE Science Education JoVE Lab Manual Faculty Resource Center Faculty Site

Terms & Conditions of Use

Related Concept Videos

Euler's Equations of Motion

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...

Navier–Stokes Equations

Navier–Stokes Equations

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...

Equations of Equilibrium in Three Dimensions

Equations of Equilibrium in Three Dimensions

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...

Euler Equations of Motion

Euler Equations of Motion

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...

Newtonian Fluid: Problem Solving

Newtonian Fluid: Problem Solving

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...

Steady, Laminar Flow Between Parallel Plates

Steady, Laminar Flow Between Parallel Plates

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.

You might also read

Related Articles

Articles linked to this work by shared authors, journal, and citation graph.

Sort by

Same author

Turing complete Navier-Stokes steady states via cosymplectic geometry.

PNAS nexus·2026

Same author

Optimal metrics for the first curl eigenvalue on 3-manifolds.

Calculus of variations and partial differential equations·2025

Same author

Obstructions to Topological Relaxation for Generic Magnetic Fields.

Archive for rational mechanics and analysis·2024

Same author

Constructing Turing complete Euler flows in dimension 3.

Proceedings of the National Academy of Sciences of the United States of America·2021

Same author

Helicity is the only integral invariant of volume-preserving transformations.

Proceedings of the National Academy of Sciences of the United States of America·2016

Same author

Level density of spin chains of Haldane-Shastry type.

Physical review. E, Statistical, nonlinear, and soft matter physics·2011

Search research articles

Related Experiment Video

Updated: May 21, 2025

Magnetically Induced Rotating Rayleigh-Taylor Instability

Magnetically Induced Rotating Rayleigh-Taylor Instability

Published on: March 3, 2017

Isolated steady solutions of the 3D Euler equations.

Alberto Enciso¹, Willi Kepplinger², Daniel Peralta-Salas¹

¹Instituto de Ciencias Matemáticas, Consejo Superior de Investigaciones Cientí ficas, Madrid 28049, Spain.

Proceedings of the National Academy of Sciences of the United States of America

|March 21, 2025

Summary

Researchers found isolated smooth steady solutions for incompressible Euler equations in 3D Riemannian manifolds. These solutions exhibit chaotic dynamics and are analyzed using spectral geometry and contact topology.

Keywords:

chaotic dynamics incompressible Euler equations stationary solution

More Related Videos

Confocal Imaging of Confined Quiescent and Flowing Colloid-polymer Mixtures

Confocal Imaging of Confined Quiescent and Flowing Colloid-polymer Mixtures

Published on: May 20, 2014

Generation and Control of Electrohydrodynamic Flows in Aqueous Electrolyte Solutions

Generation and Control of Electrohydrodynamic Flows in Aqueous Electrolyte Solutions

Published on: September 7, 2018

Related Experiment Videos

Last Updated: May 21, 2025

Magnetically Induced Rotating Rayleigh-Taylor Instability

Magnetically Induced Rotating Rayleigh-Taylor Instability

Published on: March 3, 2017

Confocal Imaging of Confined Quiescent and Flowing Colloid-polymer Mixtures

Confocal Imaging of Confined Quiescent and Flowing Colloid-polymer Mixtures

Published on: May 20, 2014

Generation and Control of Electrohydrodynamic Flows in Aqueous Electrolyte Solutions

Generation and Control of Electrohydrodynamic Flows in Aqueous Electrolyte Solutions

Published on: September 7, 2018

Area of Science:

Fluid Dynamics
Differential Geometry
Dynamical Systems

Background:

The study of fluid dynamics, particularly the incompressible Euler equations, is crucial for understanding complex fluid behaviors.
Steady solutions to these equations are rare and difficult to find, especially in complex geometries.

Purpose of the Study:

To demonstrate the existence of isolated smooth steady solutions for the incompressible Euler equations in three-dimensional Riemannian manifolds.
To analyze the properties and dynamics of these unique steady states.

Main Methods:

Combining techniques from dynamical systems, spectral geometry, and contact topology.
Analyzing the Euler equations on carefully selected Riemannian manifolds.
Investigating the C1-topology of the solution space.

Main Results:

Existence of isolated smooth steady solutions for incompressible Euler equations in 3D Riemannian manifolds.
These isolated solutions possess strongly chaotic dynamics.
A related result for Euclidean space shows analytic steady solutions with restricted analytic neighbors.

Conclusions:

The interplay of dynamical systems and geometric analysis provides powerful tools for studying fluid equations.
The findings suggest a richer structure of steady solutions in fluid dynamics than previously understood.
The methods offer a framework for analyzing complex fluid behaviors in various mathematical settings.