---
title: "Reynolds number"
date: 2021-05-04
categories:
- Physics
- Fluid mechanics
- Fluid dynamics
layout: "concept"
---

The [Navier-Stokes equations](/know/concept/navier-stokes-equations/)
are infamously tricky to solve,
so we would like a way to qualitatively predict
the behaviour of a fluid without needing the flow $\va{v}$.
Consider the main equation:

$$\begin{aligned}
    \pdv{\va{v}}{t} + (\va{v} \cdot \nabla) \va{v}
    = - \frac{\nabla p}{\rho} + \nu \nabla^2 \va{v}
\end{aligned}$$

In this case, the gravity term $\va{g}$
has been absorbed into the pressure term:
$p \to p\!+\!\rho \Phi$,
where $\Phi$ is the gravitational scalar potential,
i.e. $\va{g} = - \nabla \Phi$.

Let us introduce the dimensionless variables $\va{v}'$, $\va{r}'$, $t'$ and $p'$,
where $U$ and $L$ are respectively a characteristic velocity and length
of the system at hand:

$$\begin{aligned}
    \va{v} = U \va{v}'
    \qquad
    \va{r} = L \va{r}'
    \qquad
    t = \frac{L}{U} t'
    \qquad
    p = \rho U^2 p'
\end{aligned}$$

In this non-dimenionsalization, the differential operators are scaled as follows:

$$\begin{aligned}
    \pdv{}{t}
    = \frac{U}{L} \pdv{}{t'}
    \qquad \quad
    \nabla
    = \frac{1}{L} \nabla'
\end{aligned}$$

Putting everything into the main Navier-Stokes equation then yields:

$$\begin{aligned}
    \frac{U^2}{L} \pdv{\va{v}}{t'} + \frac{U^2}{L} (\va{v}' \cdot \nabla') \va{v}'
    = - \frac{U^2}{L} \nabla' p' + \frac{U \nu}{L^2} \nabla'^2 \va{v}'
\end{aligned}$$

After dividing out $U^2/L$,
we arrive at the form of the original equation again:

$$\begin{aligned}
    \pdv{\va{v}}{t'} + (\va{v}' \cdot \nabla') \va{v}'
    = - \nabla' p' + \frac{\nu}{U L} \nabla'^2 \va{v}'
\end{aligned}$$

The constant factor of the last term
leads to the definition of the **Reynolds number** $\mathrm{Re}$:

$$\begin{aligned}
    \boxed{
        \mathrm{Re}
        \equiv \frac{U L}{\nu}
    }
\end{aligned}$$

If we choose $U$ and $L$ appropriately for a given system,
the Reynolds number allows us to predict the general trends.
It can be regarded as the inverse of an "effective viscosity":
when $\mathrm{Re}$ is large, viscosity only has a minor role,
but when $\mathrm{Re}$ is small, it dominates the dynamics.

Another way is thus to see the Reynolds number
as the characteristic ratio between the advective term
(see [material derivative](/know/concept/material-derivative/))
to the [viscosity](/know/concept/viscosity/) term,
since $\va{v} \sim U$:

$$\begin{aligned}
     \mathrm{Re}
     \approx \frac{\big| (\va{v} \cdot \nabla) \va{v} \big|}{\big| \nu \nabla^2 \va{v} \big|}
     \approx \frac{U^2 / L}{\nu U / L^2}
     = \frac{U L}{\nu}
\end{aligned}$$

In other words, $\mathrm{Re}$
describes the relative strength of intertial and viscous forces.
Returning to the dimensionless Navier-Stokes equation:

$$\begin{aligned}
    \pdv{\va{v}}{t'} + (\va{v}' \cdot \nabla') \va{v}'
    = - \nabla' p' + \frac{1}{\mathrm{Re}} \nabla'^2 \va{v}'
\end{aligned}$$

For large $\mathrm{Re} \gg 1$,
we can neglect the latter term,
such that redimensionalizing yields:

$$\begin{aligned}
    \pdv{\va{v}}{t} + (\va{v} \cdot \nabla) \va{v}
    = - \frac{\nabla p}{\rho}
\end{aligned}$$

Which is simply the main [Euler equation](/know/concept/euler-equations/)
for an ideal fluid, i.e. a fluid without viscosity.



## Stokes flow

A notable case is so-called **Stokes flow** or **creeping flow**,
meaning flow at $\mathrm{Re} \ll 1$.
In this limit, the Navier-Stokes equations can be linearized:
since $\mathrm{Re}$ is the advective-to-viscous ratio,
$\mathrm{Re} \ll 1$ implies that we can ignore the advective term, leaving:

$$\begin{aligned}
    \boxed{
        \pdv{\va{v}}{t}
        = - \frac{\nabla p}{\rho} + \nu \nabla^2 \va{v}
    }
\end{aligned}$$

This equation is called the **unsteady Stokes equation**.
Usually, however, such flows are assumed to be steady
(i.e. time-invariant), leading to the **steady Stokes equation**,
with $\eta = \rho \nu$:

$$\begin{aligned}
    \boxed{
        \nabla p
        = \eta \nabla^2 \va{v}
    }
\end{aligned}$$

This equation is much easier to solve than the full Navier-Stokes equation
thanks to being linear,
and has some interesting properties, such as time-reversibility.



## References
1.  B. Lautrup,
    *Physics of continuous matter: exotic and everyday phenomena in the macroscopic world*, 2nd edition,
    CRC Press.
2.  R. Fitzpatrick,
    [Dimensionless numbers in incompressible flow](https://farside.ph.utexas.edu/teaching/336L/Fluid/node17.html),
    University of Texas.