What does the Reynolds number tell you?

It is the ratio of inertial to viscous forces in a flow; small Reynolds numbers indicate smooth laminar flow dominated by viscosity, while large values indicate inertia-dominated flow prone to turbulence.

Why are the Navier-Stokes equations so hard to solve?

They are nonlinear partial differential equations, and the nonlinear inertial term couples scales of motion, producing turbulence; the existence and smoothness of general solutions remains an open mathematical problem.

Viscous Flow and Navier-Stokes

Viscous flow accounts for internal friction in fluids; its governing equations are the Navier-Stokes equations, whose balance of inertia and viscosity is captured by the Reynolds number.

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Definition

Viscous flow is the motion of a fluid with internal friction, governed by the Navier-Stokes equations that add viscous stresses to the inviscid Euler equation, with the flow regime characterized by the Reynolds number.

Scope

This topic covers the Navier-Stokes equations adding viscous stress to Euler's equation, the no-slip boundary condition, the Reynolds number as the ratio of inertial to viscous forces, exact solutions such as Poiseuille and Couette flow, boundary-layer theory, and the onset of turbulence. It is the realistic description of fluids with internal friction.

Core questions

How do viscous stresses modify the equations of fluid motion?
What does the Reynolds number measure, and why does it govern flow behavior?
How does laminar flow give way to turbulence as the Reynolds number rises?

Key concepts

Viscosity and viscous stress
Navier-Stokes equations
No-slip boundary condition
Reynolds number
Laminar and turbulent flow
Boundary layer

Key theories

Navier-Stokes equations: Adding a viscous stress proportional to the rate of strain to Euler's equation gives the Navier-Stokes equations, the fundamental equations governing the motion of real viscous fluids.
Reynolds number and flow regimes: The dimensionless Reynolds number compares inertial to viscous forces; low values give orderly laminar flow and high values lead through instability to turbulence.

Clinical relevance

The Navier-Stokes equations are the working model of aerodynamics, hydraulics, pipe and channel flow, lubrication, and weather and ocean circulation, while the laminar-turbulent transition and boundary-layer behavior are decisive for drag, mixing, and heat transfer in engineering and geophysics.

History

Navier introduced viscous terms into the fluid equations in 1822, and Stokes gave them their rigorous continuum derivation in the 1840s. Osborne Reynolds's 1883 pipe experiments identified the dimensionless number governing the laminar-to-turbulent transition, and Prandtl's 1904 boundary-layer concept reconciled viscous and ideal flow, founding modern fluid dynamics.

Key figures

Claude-Louis Navier
George Gabriel Stokes
Osborne Reynolds
Ludwig Prandtl

Seminal works

landaufluid1987
batchelor2000

Frequently asked questions

What does the Reynolds number tell you?: It is the ratio of inertial to viscous forces in a flow; small Reynolds numbers indicate smooth laminar flow dominated by viscosity, while large values indicate inertia-dominated flow prone to turbulence.
Why are the Navier-Stokes equations so hard to solve?: They are nonlinear partial differential equations, and the nonlinear inertial term couples scales of motion, producing turbulence; the existence and smoothness of general solutions remains an open mathematical problem.