In Pipe Flow the Reynolds number can be written in simplified form:

(1)	$\begin{array}{l}\displaystyle {\rm Re} = \frac{j_m \cdot d}{\mu(T,p)} = \frac{4 \, \dot m}{\pi \, d} \cdot \frac{1}{\mu(T,p)}\end{array}$

where

$\begin{array}{l}\displaystyle j_m = \frac{ \dot m }{ A}\end{array}$	mass flux
$\begin{array}{l}\displaystyle \dot m = \frac{dm }{ dt}\end{array}$	mass flowrate
$\begin{array}{l}\displaystyle d = \sqrt{ \frac{4 A}{\pi}}\end{array}$	characteristic linear dimension of the pipe (or exactly a pipe diameter in case of a circular pipe)
$\begin{array}{l}\mu(T,p)\end{array}$	dynamic viscosity as function of fluid temperature $\begin{array}{l}T\end{array}$ and pressure $\begin{array}{l}p\end{array}$

Derivation

(2)	$\begin{array}{l}\displaystyle {\rm Re} = \frac{u \cdot d}{\nu} = \frac{\rho \cdot u \cdot d}{\mu} = \frac{j_m \cdot d}{\mu(T,p)}\end{array}$

where

$\begin{array}{l}u\end{array}$	average cross-sectional flow velocity
$\begin{array}{l}\rho\end{array}$	fluid density
$\begin{array}{l}\nu\end{array}$	kinematic viscosity

The mass flowrate is constant along the pipe: $\begin{array}{l}\dot m = \rm const\end{array}$ .

In many engineering application the pipeline is built from inter-connected pipes or ducts with constant cross-sectional area $\begin{array}{l}A = \rm const\end{array}$ which means that mass flux is also constant along pipes: $\begin{array}{l}j_m = \rm const\end{array}$ .

Equation (1) shows that in this case a variation of Reynolds number along the pipe will be a function of fluid viscosity only: $\begin{array}{l}{\rm Re} = {\rm Re}(\mu)\end{array}$ which in turn is a function of fluid temperature $\begin{array}{l}T(l)\end{array}$ and pressure $\begin{array}{l}p(l)\end{array}$ along the pipe.

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Reynolds number in Pipe Flow

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