Fluid Mechanics

At an initial instant \(t_0\) , the fluid is cut into elementary particles fluids centered on a set of common point \(M_0\) and one follows the movement of the particles over time in the frame of reference (R).

At time \(t\), we can define the velocity vector \(\vec v(M_0,t)\) of the particle which was in M₀ at time \(t_0\).

It naturally defines the trajectory of a fluid particle, the place of successive positions of the particle over time.

In this case, an observer is connected to each fluid particle.

This type of description is useful when one wants to study the advection phenomena of contaminants such as scattered radioactive elements in flow and following the trajectories of fluid particles.

It is this description that is traditionally chosen in mechanics of material point or of solid for example.

To illustrate, a police officer who wants to verify that a particular car does not commit an excess of speed chooses to follow that car and measure the speed with a radar.

If the police officer is more concerned about the respect of the speed limit at a dangerous crossroads, he lays his radar and controls the speed of all the cars that will pass this place : it is the Eulerian approach of a flow.

More precisely, the fluid is described in each instant by the set of fluid particle velocities that make it up ; their initial position is noted \(\vec R_0\) (at \(t=0\)). The velocity of a particle is then :

\(\vec V(t) = \frac{{d\vec R(t)}}{{dt}} = \vec V(\vec R_0 ,t)\)

Where \(\vec R(t) = \overrightarrow {OM}\) is the radius vector of the particle at time t.

These quantities only and explicitly depend on time for a considered particle (\(\vec R(t)\) is a well known function of time).

Thus, in Lagrangian formalism, the speed of each particle depends only on the time \(t\) (the path being known) and initial coordinates of the particle.

Rather than describing the speed of a fluid particle, which provides flow characteristics as a function of time but never at the same location (the position of the particle continues to vary), the Eulerian description is to study the movement of the fluid at fixed locations.

There are many experimental techniques for measuring the velocity of a fluid at a given position (like the "hot wire" which allows to obtain the velocity of the fluid by measuring the cooling of a heated wire, or the anemometer consisting of an helice whose rotational speed provides the wind speed, ...).

Thus, one can follow the time evolution of the speed at a fixed point \(M\) : one obtains the velocity of the fluid particle which is in \(M\) at the instant \(t\) of measurement. Therefore it means each time a different particle.

The set of speed vectors of particles forms a vector field :

\(\vec v(\vec r,t) = \vec v(M,t) = \vec v(x,y,z,t)\)

depending both space and time : the variables of space \((x,y,z)\) and time \(t\) are independent variables.

The Eulerian approach describes the state of a moving fluid by associating fields : velocity field, pressure field, temperature field, ...

The Eulerian point of view is the perspective implicitly adopted in electromagnetism and thermodynamics on the diffusion phenomena.

Eulerian description is particularly suitable in the case of steady flows.

A steady flow (or flow under permanent regime) is a flow for which the velocity at any point \(M\) is independent of time :

\(\frac{{\partial \vec v}}{{\partial t}}(M) = \vec 0\)

In a steady flow, the speed is constant at any fixed point but the velocity of fluid particles varies, few exceptions, with time.

Thus a Eulerian steady flow is not a Lagrangian steady flow.

The stationary nature of flow depends on the chosen reference.

Take the case of the wake of a duck on the surface of a river : the wake, which gives the impression of following the duck is stationary in the reference related to duck but is not stationary in the reference related to the bank .