Lagrangian and Eulerian specification of the flow field

Leonhard Euler is credited with introducing both specifications in two publications written in 1755{{Cite journal |last=Euler |first=Leonhard |date=1757-01-01 |title=Principes généraux du mouvement des fluides |url=https://scholarlycommons.pacific.edu/euler-works/226/ |journal=Mémoires de l'académie des sciences de Berlin |pages=274–315}} and 1759.{{Cite journal |last=Euler |first=Leonhard |date=1761-01-01 |title=Principia motus fluidorum |url=https://scholarlycommons.pacific.edu/euler-works/258/ |journal=Novi Commentarii Academiae Scientiarum Petropolitanae |pages=271–311}}{{Cite book |last=Lamb |first=Sir Horace |url=https://books.google.com/books?id=237xDg7T0RkC&q=;+Lamb,+Hydrodynamics,+6th+ed.,+(Cambridge+University.+Press,+1937)+c |title=Hydrodynamics |date=1945-01-01 |publisher=Courier Corporation |isbn=978-0-486-60256-1 |language=en}} 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,{{Cite web |title=Joseph Louis de Lagrange: Mémoire sur la théorie du mouvement des fluides |url=http://sites.mathdoc.fr/cgi-bin/oeitem?id=OE_LAGRANGE__4_695_0 |access-date=2024-10-17 |website=sites.mathdoc.fr}} and thirdly in his book Mécanique analytique. In this book Lagrange starts with the Lagrangian specification but later converts them into the Eulerian specification.

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

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.) In the Lagrangian description, the flow is described by a function

$$\backslash mathbf\{X\}\backslash left(\backslash mathbf\{x\}\_0,t\backslash right),$$

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

The two specifications are related as follows:

$$\backslash mathbf\{u\}\backslash left(\backslash mathbf\{X\}(\backslash mathbf\{x\}\_0,t),\; t\; \backslash right)\; =\; \backslash frac\{\backslash partial\; \backslash mathbf\{X\}\}\{\backslash partial\; t\}\; \backslash left(\backslash mathbf\{x\}\_0,t\; \backslash right),$$

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.

{{Main|Material derivative}}

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).

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

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

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.{{cite book | last = Falkovich | first = Gregory | year = 2011 | title = Fluid Mechanics (A short course for physicists) | publisher = Cambridge University Press | isbn = 978-1-107-00575-4 }}

• 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

{{reflist}}

• Bennett, A. (2006). Lagrangian Fluid Dynamics. Cambridge, U.K.: Cambridge University Press.
• {{cite book| isbn=978-3-319-59694-5 | last1=Badin | first1=G. | last2=Crisciani | first2=F. | title=Variational Formulation of Fluid and Geophysical Fluid Dynamics - Mechanics, Symmetries and Conservation Laws | publisher=Springer | year=2018 | page=218 | doi= 10.1007/978-3-319-59695-2| bibcode=2018vffg.book.....B | s2cid=125902566 }}
• {{cite book

| first1 = Lev | last1 = Landau | author-link = Lev Landau

| first2 = E.M. | last2 = Lifshitz | author2-link = Evgeny Lifshitz

| title = Fluid Mechanics

| series = Course of Theoretical Physics, Volume 6

| edition = 2nd

| publisher = Butterworth-Heinemann

| year = 1987

| isbn = 978-0750627672

}}

[http://www.isima.fr/~leborgne/IsimathMeca/Lageul.pdf] Objectivity in classical continuum mechanics: Motions, Eulerian and Lagrangian functions; Deformation gradient; Lie derivatives; Velocity-addition formula, Coriolis; Objectivity.

Category:Fluid dynamics

Category:Aerodynamics

Category:Computational fluid dynamics