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 . Consider the main equation:
In this case, the gravity term has been absorbed into the pressure term: , where is the gravitational scalar potential, i.e. .
Let us introduce the dimensionless variables , , and , where and are respectively a characteristic velocity and length of the system at hand:
In this non-dimenionsalization, the differential operators are scaled as follows:
Putting everything into the main Navier-Stokes equation then yields:
After dividing out , we arrive at the form of the original equation again:
The constant factor of the last term leads to the definition of the Reynolds number :
If we choose and 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 is large, viscosity only has a minor role, but when 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 :
In other words, describes the relative strength of intertial and viscous forces. Returning to the dimensionless Navier-Stokes equation:
For large , we can neglect the latter term, such that redimensionalizing yields:
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 . In this limit, the Navier-Stokes equations can be linearized: since is the advective-to-viscous ratio, implies that we can ignore the advective term, leaving:
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 :
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.