FLD-18

Navier–Stokes inertial–viscous snapshot

Re = ρ U L / μ compares ρ(u·∇)u with μ ∇²u. The local inertial-to-viscous ratio.

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FluidsNavier–Stokes

Governing equation

Re=\dfrac{\rho U L}{\mu}\sim\dfrac{|\rho(\mathbf u\cdot\nabla)\mathbf u|}{|\mu\nabla^2\mathbf u|}

$U \in \left[0.01, 40\right]$

where

\rho: Density (kg/m³)
U: Scale speed (m/s)
L: Scale length (m)
\mu: Viscosity (Pa·s)
Re: Inertia / viscous (—)
F_{in}/V: Inertial density (N/m³)
F_{vis}/V: Viscous density (N/m³)

The figure, part by part

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①

Governing law — red

Re=\dfrac{\rho U L}{\mu}\sim\dfrac{|\rho(\mathbf u\cdot\nabla)\mathbf u|}{|\mu\nabla^2\mathbf u|}

$U \in \left[0.01, 40\right]$

$U = 0.01$ $Re = 1000$

Along the curve

②

Domain of the variable — blue

U_{\min} = 0.01,\quad U_{\max} = 40

$U \in \left[0.01, 40\right]$

$U = 0.01$ $U = 0.01$

Along the curve

③

Response curve — green

Re=\dfrac{\rho U L}{\mu}\sim\dfrac{|\rho(\mathbf u\cdot\nabla)\mathbf u|}{|\mu\nabla^2\mathbf u|}

$U \in \left[0.01, 40\right]$

$U = 0.01$ $Re = 1000$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}Re}{\mathrm{d}U}

$U \in \left[0.01, 40\right]$

$U = 0.262$ $Re^{\prime} = 1\times 10^{5}$

Along the curve

Lecture brief

Historical brief

Bernoulli, Navier–Stokes, Reynolds (1883) and Stokes drag made continuum flow a dimensionless craft. The sheets compute head, drag, Re and a pedagogical NS snapshot. This sheet (FLD-18 — Navier–Stokes inertial–viscous snapshot) is the form associated with Navier–Stokes. Working symbols: $\rho$ , $U$ , $L$ , $\mu$ $\rightarrow$ $Re$ , $F_{in}/V$ , $F_{vis}/V$ . The incompressible NS equation is ∂u/∂t + (u·∇)u = −∇p/ρ + ν∇²u. This sheet is one scale estimate.

Purpose

Purpose: compute

Re

F_{in}/V

F_{vis}/V

from

\rho

U

L

\mu

in Fluid physics via

Re=\dfrac{\rho U L}{\mu}\sim\dfrac{|\rho(\mathbf u\cdot\nabla)\mathbf u|}{|\mu\nabla^2\mathbf u|}

Re = ρ U L / μ compares ρ(u·∇)u with μ ∇²u. The local inertial-to-viscous ratio. Use it when a real fluid physics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

\rho = 1000.000\,\mathrm{kg/m^{3}}

U = 1.000\,\mathrm{m/s}

L = 0.100\,\mathrm{m}

\mu = 0.001\,\mathrm{Pa·s}

, the governing relation

Re=\dfrac{\rho U L}{\mu}\sim\dfrac{|\rho(\mathbf u\cdot\nabla)\mathbf u|}{|\mu\nabla^2\mathbf u|}

yields

Re = 100000.00\,\mathrm{—}

F_{in}/V = 10000.00\,\mathrm{N/m^{3}}

F_{vis}/V = 0.10\,\mathrm{N/m^{3}}

. A blob of fluid, two force arrows, a Re bar. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Density \rho1000 kg/m³Scale speed U1 m/sScale length L0.1 mViscosity \mu0.001 Pa·s

Outputs

Inertia / viscous Re100000.00 —
Inertial density F_{in}/V10000.00 N/m³
Viscous density F_{vis}/V0.10 N/m³

Reading speed

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Narration of this film

A blob of fluid, two force arrows, a Re bar.

The incompressible NS equation is ∂u/∂t + (u·∇)u = −∇p/ρ + ν∇²u. This sheet is one scale estimate.

Reading speed

Watch on YouTube