= Reynolds number
{c}
{title2=$\mathrm{Re}=\rho UL/\mu=UL/\nu$}
{wiki}
The <Reynolds number> $\operatorname{Re}=UL/\nu$ compares inertial and viscous effects. Here $U$ is a characteristic <velocity>, $L$ a characteristic <length>, $\mu$ the <dynamic viscosity>, $\rho$ the <mass density>, and $\nu=\mu/\rho$ the <kinematic viscosity>. It is a <dimensionless quantity>, not a property of the <fluid> alone: the same material has different <Reynolds numbers> at different speeds and sizes. Specify the chosen scales when reporting it.
The comparison comes from the <Navier-Stokes equation>. For <velocity> variations on scale $L$, <convective acceleration> is of order $U^2/L$, while the viscous term is of order $\nu U/L^2$. Their ratio is $UL/\nu$. Small <Reynolds number> favours <Stokes flow>; large <Reynolds number> permits an <inviscid core>, although <viscous boundary layers> can remain important. Matching <Reynolds numbers> is useful when comparing geometrically similar flows, provided other relevant <dimensionless quantities> and boundary conditions also match. https://uclnatsci.github.io/2021/Electromagnetism-Fluids-and-Waves/fluids/scaling.html[UCL, Scaling arguments, section 34.1.1]{ref}
In the constriction example under <incompressible flow>, use the mean axial <velocity> and pipe diameter, $L=D=2R$. Taking $\rho=1000\,\mathrm{kg\,m^{-3}}$ and a stipulated <dynamic viscosity> $\mu=0.001\,\mathrm{Pa\,s}$ gives
$$
\mathrm{Re}_1=\frac{1000(0.20)(0.020)}{0.001}=4000,\qquad
\mathrm{Re}_2=\frac{1000(0.80)(0.010)}{0.001}=8000.
$$
Halving the radius at fixed <volumetric flow rate> doubles this diameter-based <Reynolds number>, because mean <velocity> increases fourfold. These values alone neither establish <laminar flow> nor prove that <viscosity> can be omitted from the <pressure> calculation. Increasing $\mu$ to $0.10\,\mathrm{Pa\,s}$ with the same imposed flow gives $40$ and $80$, but strengthens viscous effects; small <Reynolds number> does not justify <Bernoulli's principle>.
The name recalls Osborne Reynolds's 1883 experimental investigation of water motion and resistance in pipes. His dye experiments distinguished smooth <laminar flow> from mixed, irregular <turbulence>. https://cir.nii.ac.jp/crid/1362544419765899904[Reynolds, 1883, original paper abstract and bibliographic record]{ref} https://uclnatsci.github.io/2021/Electromagnetism-Fluids-and-Waves/fluids/scaling.html[UCL, Scaling arguments, section 34.1.4]{ref} There is no universal transition value applicable to every geometry and disturbance environment. https://www.damtp.cam.ac.uk/user/hkm2/PDFs/Moffatt_1977_GordonandBreach_Slogfdatohdt_149.pdf[H. K. Moffatt, transition discussion, p. 208]{ref}
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