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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...

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.

Typical Model Studies

Typical Model Studies

Fluid mechanics model studies often utilize scaled-down systems to predict fluid behavior in full-scale environments, such as river flows, dam spillways, and structures interacting with open surfaces. Maintaining Froude number similarity in river models is crucial, as it replicates surface flow features like wave patterns and velocities.

Bernoulli's Equation for Flow Along a Streamline

Bernoulli's Equation for Flow Along a Streamline

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:

Bernoulli's Equation for Flow Normal to a Streamline

Bernoulli's Equation for Flow Normal to a Streamline

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.
The pressure difference depends on the fluid's velocity and radius of curvature. The pressure variation is minimal in flows with nearly straight streamlines.

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...

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Related Experiment Video

Updated: Sep 3, 2025

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

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

Published on: February 3, 2014

A Model of Interacting Navier-Stokes Singularities.

Hugues Faller¹, Lucas Fery^1,2, Damien Geneste¹

¹Service de Physique de l'État Condensé, CNRS UMR 3680, CEA, Université Paris-Saclay, 91190 Gif-sur-Yvette, France.

Entropy (Basel, Switzerland)

|July 27, 2022

Summary

We introduce pinçons, interacting singularities of Navier-Stokes equations, to model fluid dynamics. These pinçons exhibit complex behaviors like repulsion, collapse, and orientation under stochastic forcing, offering insights into turbulence.

Keywords:

non-equilibrium dynamics singularity turbulence

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Last Updated: Sep 3, 2025

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

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Published on: February 3, 2014

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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Published on: July 19, 2016

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Area of Science:

Fluid Dynamics
Mathematical Physics
Turbulence Theory

Background:

Navier-Stokes equations describe fluid motion but are complex to solve.
Existing models like Novikov's vorton model simplify Euler equations.
Understanding singularities is crucial for turbulence research.

Purpose of the Study:

Introduce a novel model of interacting singularities for Navier-Stokes equations, named pinçons.
Generalize existing models to include dissipation and non-equilibrium dynamics.
Investigate the behavior of interacting pinçons under various conditions.

Main Methods:

Developed a model of interacting singularities (pinçons) obeying local Navier-Stokes equations.
Studied pinçon dynamics, including pairs, dipoles, and interactions with regular fields.
Analyzed behavior under stochastic forcing and arbitrary intensity/orientation.

Main Results:

Pinçons exhibit non-equilibrium dynamics and generalize the vorton model.
A pinçon dipole shows initial repulsion, followed by dissipation.
Observed collapse, dipolar anti-aligned runaway, and anisotropic aligned runaway dynamics.
Pinçon collapse mirrors vortex ring reconnection characteristics and Leray scaling.

Conclusions:

The pinçon model provides a framework for studying interacting singularities in Navier-Stokes flows.
Pinçon dynamics reveal insights into turbulence, dissipation, and non-equilibrium states.
The model captures key features of vortex dynamics and reconnection.