Lagrangian and Eulerian specification of the flow field
In classical field theories, the Lagrangian specification of the flow field is a way of looking at fluid motion where the observer follows an individual fluid parcel as it moves through space and time.
[1][2] This can be visualized by sitting on the bank of a river and watching the water pass the fixed location.
However, in general both the Lagrangian and Eulerian specification of the flow field can be applied in any observer's frame of reference, and in any coordinate system used within the chosen frame of reference.
The Lagrangian and Eulerian specifications are named after Joseph-Louis Lagrange and Leonhard Euler, respectively.
Leonhard Euler is credited of introducing both specifications in two publications written in 1755[3] and 1759.
[4][5] Joseph-Louis Lagrange studied the equations of motion in connection to the principle of least action in 1760, later in a treaty of fluid mechanics in 1781,[6] and thirdly in his book Mécanique analytique.
On the other hand, in the Lagrangian specification, individual fluid parcels are followed through time.
(Often, x0 is chosen to be the position of the center of mass of the parcels at some initial time t0.
Therefore, the center of mass is a good parameterization of the flow velocity u of the parcel.
giving the position of the particle labeled x0 at time t. The two specifications are related as follows:[2]
Now one might ask about the total rate of change of F experienced by a specific flow parcel.
where ∇ denotes the nabla operator with respect to x, and the operator u⋅∇ is to be applied to each component of F. This tells us that the total rate of change of the function F as the fluid parcels moves through a flow field described by its Eulerian specification u is equal to the sum of the local rate of change and the convective rate of change of F. This is a consequence of the chain rule since we are differentiating the function F(X(x0, t), t) with respect to t. Conservation laws for a unit mass have a Lagrangian form, which together with mass conservation produce Eulerian conservation; on the contrary, when fluid particles can exchange a quantity (like energy or momentum), only Eulerian conservation laws exist.
[7] [1] Objectivity in classical continuum mechanics: Motions, Eulerian and Lagrangian functions; Deformation gradient; Lie derivatives; Velocity-addition formula, Coriolis; Objectivity.
