Categories: Fluid dynamics, Fluid mechanics, Physics.

The 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) to the 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 for an ideal fluid, i.e. a fluid without viscosity.

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.

- B. Lautrup,
*Physics of continuous matter: exotic and everyday phenomena in the macroscopic world*, 2nd edition, CRC Press. - R. Fitzpatrick, Dimensionless numbers in incompressible flow, University of Texas.

© Marcus R.A. Newman, a.k.a. "Prefetch".
Available under CC BY-SA 4.0.