The **Reynolds number** is a dimensionless parameter that represents the contraposition between the inertia force linked to the increase in the mass of the fluid within the boundary layer and the viscous force that tends to control the motion. Fluid flow is divided into two main types: laminar flow and turbulent flow. The typology of motion is described quantitatively (and not phenomenologically) by the Reynolds number.

The concept was introduced by George Stokes in 1851, but the Reynolds number was named by Arnold Sommerfeld in 1908 after Osborne Reynolds (1842–1912), who introduced it in 1883 by carrying out systematic experiments for the first time on the flow of fluid inside transparent tubes with a circular section with a straight axis, in which a constant flow circulated and employing a needle, a dye was injected in order to highlight the flow rate.

The transition from laminar flow to turbulent flow occurs in a transition region in which the viscous force, typical of the fluid which tends to control the motion, due to a continuous increase in the mass involved in the disordered motion of the fluid particles (i.e., as a result, the increase of the inertial force proper to the mass, which is no longer able to dampen the oscillations of the particles of fluid that wander with disordered motion), occurs the transition to turbulent flow.

So the Reynolds number is the ratio between the inertia force and a viscous force. The viscosity, especially the kinematic one, is responsible for the type of motion: whether laminar or turbulent; as it represents an indication of how much the motion or non-motion is transmitted inside the fluid. The inertial force is related to the mass of the fluid in transit at a certain speed. The viscous force, on the other hand, is linked to the product of dynamic viscosity, the speed gradient normal to the surface and to the surface itself.

The juxtaposition of these two forces shows that the relevant quantities for the determination of the Reynolds number are: the numerator mass and acceleration, and the denominator is the viscosity and a spatial gradient of velocity normal to the sliding surface as the speed perturbation, which is it is propagating inside the fluid, it has a velocity component (from bottom to top) normal to the direction of propagation. Therefore:

\[Re=\dfrac{ma}{\mu\dfrac{u}{L}L^2}=\dfrac{\rho L u}{\mu}=\dfrac{Lu}{\nu}\]

where \(u\) is the flow speed of the fluid, \(\mu\) the dynamic viscosity, \(\nu\) the kinematic viscosity, \(t\) is the time and \(L\) the characteristic linear dimension (for example the section of a circular tube).

## Characteristic properties of the Reynolds number

The study of the relationship that expresses the Reynolds number shows that it is directly proportional to the density of the fluid, its velocity and the characteristic linear dimension, while it is inversely proportional to the viscosity. The values of the Reynolds number are to be considered “low“ or “high“ in relation to a specific system, in which are fixed:

- the geometry of the body hit by the flow;
- the nature of the fluid;
- the operating conditions (temperature and pressure) under which the experience takes place.

In particular, the value of the Reynolds number that divides laminar flow from turbulent flow depends on the geometric shape of the body (or set of bodies) in correspondence of which the fluid passes and the orientation of the body relative to the flow (with the exception of spherical bodies, being symmetrical in all directions). So if you consider for example the case of a fluid moving on the external surface of a sphere, a tube, a cube, or a set of tubes, you will have for each case a different Reynolds number at which the laminar/turbulent transition takes place. The transition between laminar flow and turbulent flow can be foreseen by using the Moody diagram, with which the viscous friction coefficient can be calculated starting from the values of the Reynolds number and the relative roughness.

As regard to the dependence of the Reynolds number on the flow speed of the fluid, having fixed the geometry of the system, the composition of the fluid, the temperature and pressure, it is possible to say that fluid in laminar flow has a lower velocity compared to the same fluid in turbulent flow, or in other words, the transition between laminar flow and turbulent flow occurs by increasing the velocity of the fluid.

## Magnetic Reynolds number

The **magnetic Reynolds number** (R_{m}) is the magnetic analog of the Reynolds number, a fundamental dimensionless group that occurs in magneto-hydro-dynamics. It gives an estimate of the relative effects of advection or induction of a magnetic field by the motion of a conducting medium, often a fluid, to magnetic diffusion. It is typically defined by:

\[\mathrm{R}_{\mathrm{m}}=\dfrac{UL}{\eta}\sim \dfrac{\mathrm{induction}}{\mathrm{diffusion}}\]

where: \(U\) is a typical velocity scale of the flow; \(L\) is a typical length scale of the flow; \(\eta\) is the magnetic diffusivity. The mechanism by which the motion of a conducting fluid generates a magnetic field is the subject of dynamo theory. When the magnetic Reynolds number is very large, however, diffusion and the dynamo are less of a concern, and in this case focus instead often rests on the influence of the magnetic field on the flow.