Lagrangian and Eulerian specification of the flow field

In the Eulerian specification of a field, the field is represented as a function of position x and time t. For example, the flow velocity is represented by a function

$${\displaystyle \mathbf {u} \left(\mathbf {x} ,t\right).}$$

On the other hand, in the Lagrangian specification, individual fluid parcels are followed through time. The fluid parcels are labelled by some (time-independent) vector field x₀. (Often, x₀ is chosen to be the position of the center of mass of the parcels at some initial time t₀. It is chosen in this particular manner to account for the possible changes of the shape over time. Therefore the center of mass is a good parameterization of the flow velocity u of the parcel.)[1] In the Lagrangian description, the flow is described by a function

$${\displaystyle \mathbf {X} \left(\mathbf {x} _{0},t\right),}$$

giving the position of the particle labeled x₀ at time t.

The two specifications are related as follows:[2]

$${\displaystyle \mathbf {u} \left(\mathbf {X} (\mathbf {x} _{0},t),t\right)={\frac {\partial \mathbf {X} }{\partial t}}\left(\mathbf {x} _{0},t\right),}$$

because both sides describe the velocity of the particle labeled x₀ at time t.

Within a chosen coordinate system, x₀ and x are referred to as the Lagrangian coordinates and Eulerian coordinates of the flow respectively.

The Lagrangian and Eulerian specifications of the kinematics and dynamics of the flow field are related by the material derivative (also called the Lagrangian derivative, convective derivative, substantial derivative, or particle derivative).[1]

Suppose we have a flow field u, and we are also given a generic field with Eulerian specification F(x, t). Now one might ask about the total rate of change of F experienced by a specific flow parcel. This can be computed as

$${\displaystyle {\frac {\mathrm {D} \mathbf {F} }{\mathrm {D} t}}={\frac {\partial \mathbf {F} }{\partial t}}+\left(\mathbf {u} \cdot \nabla \right)\mathbf {F} ,}$$

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.[3]

• Brewer-Dobson Circulation
• Conservation form
• Contour advection
• Displacement field (mechanics)
• Equivalent latitude
• Generalized Lagrangian mean
• Lagrangian particle tracking
• Semi-Lagrangian scheme
• Streamlines, streaklines, and pathlines
• Trajectory (fluid mechanics)
• Stochastic Eulerian Lagrangian method
• Liouville's theorem (Hamiltonian)

• Badin, G.; Crisciani, F. (2018). Variational Formulation of Fluid and Geophysical Fluid Dynamics - Mechanics, Symmetries and Conservation Laws. Springer. p. 218. doi:10.1007/978-3-319-59695-2. ISBN 978-3-319-59694-5. S2CID 125902566.
• Landau, Lev; Lifshitz, E.M. (1987). Fluid Mechanics. Course of Theoretical Physics, Volume 6 (2nd ed.). Butterworth-Heinemann. ISBN 978-0750627672.

Objectivity in classical continuum mechanics: Motions, Eulerian and Lagrangian functions; Deformation gradient; Lie derivatives; Velocity-addition formula, Coriolis; Objectivity.