Inclassical field theories, theLagrangian specification of the flow field is a way of looking at fluid motion where the observer follows an individualfluid parcel as it moves through space and time.^[1][2] Plotting the position of an individual parcel through time gives thepathline of the parcel. This can be visualized as sitting in a boat and drifting down a river.

TheEulerian specification of the flow field is a way of looking at fluid motion that focuses on specific locations in the space through which the fluid flows as time passes.^[1][2] This can be visualized by sitting on the bank of a river and watching the water pass the fixed location.

The Lagrangian and Eulerian specifications of the flow field are sometimes loosely denoted as theLagrangian and Eulerian frame of reference. However, in general both the Lagrangian and Eulerian specification of the flow field can be applied in any observer'sframe of reference, and in anycoordinate system used within the chosen frame of reference. The Lagrangian and Eulerian specifications are named afterJoseph-Louis Lagrange andLeonhard Euler, respectively.

These specifications are reflected incomputational fluid dynamics, where "Eulerian" simulations employ a fixedmesh while "Lagrangian" ones (such asmeshfree simulations) feature simulation nodes that may move following thevelocity field.

Leonhard Euler is credited with 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 theprinciple of least action in 1760, later in a treaty of fluid mechanics in 1781,^[6] and thirdly in his bookMécanique analytique.^[5] In this book Lagrange starts with the Lagrangian specification but later converts them into the Eulerian specification.^[5]

In theEulerian specification of afield, the field is represented as a function of positionx and timet. For example, theflow velocity is represented by a function$${\displaystyle \mathbf{u}\left(\mathbf{x},t\right).}$$

On the other hand, in theLagrangian specification, individual fluid parcels are followed through time. The fluid parcels are labelled by some (time-independent) vector fieldx₀. (Often,x₀ is chosen to be the position of the center of mass of the parcels at some initial timet₀. 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 velocityu 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 labeledx₀ at timet.

The two specifications are related as follows:^[2]$${\displaystyle \mathbf{u}\left(\mathbf{X}({\mathbf{x}}_0,t),t\right)=\frac{\mathrm{\partial }\mathbf{X}}{\mathrm{\partial }t}\left({\mathbf{x}}_0,t\right),}$$because both sides describe the velocity of the particle labeledx₀ at timet.

Within a chosen coordinate system,x₀ andx are referred to as theLagrangian coordinates andEulerian coordinates of the flow respectively.

The Lagrangian and Eulerian specifications of thekinematics anddynamics of the flow field are related by thematerial derivative (also called the Lagrangian derivative, convective derivative, substantial derivative, or particle derivative).^[1]

Suppose we have a flow fieldu, and we are also given a generic field with Eulerian specificationF(x, t). Now one might ask about the total rate of change ofF experienced by a specific flow parcel. This can be computed as$${\displaystyle \frac{\mathrm{D}\mathbf{F}}{\mathrm{D}t}=\frac{\mathrm{\partial }\mathbf{F}}{\mathrm{\partial }t}+\left(\mathbf{u}\cdot \mathrm{\nabla }\right)\mathbf{F},}$$where ∇ denotes thenabla operator with respect tox, and the operatoru⋅∇ is to be applied to each component ofF. This tells us that the total rate of change of the functionF as the fluid parcels moves through a flow field described by its Eulerian specificationu is equal to the sum of the local rate of change and the convective rate of change ofF. This is a consequence of thechain rule since we are differentiating the functionF(X(x₀, t), t) with respect tot.

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]

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

• Bennett, A. (2006).Lagrangian Fluid Dynamics. Cambridge, U.K.: Cambridge University Press.
• Badin, G.; Crisciani, F. (2018).Variational Formulation of Fluid and Geophysical Fluid Dynamics - Mechanics, Symmetries and Conservation Laws. Springer. p. 218.Bibcode:2018vffg.book.....B.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.

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

• Fluid dynamics
• Aerodynamics
• Computational fluid dynamics

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