Lagrangian and Eulerian specification of the flow field

This article is about fluid mechanics. For the use of generalized coordinates in classical mechanics, see generalized coordinates, Lagrangian mechanics and Hamiltonian mechanics

In fluid dynamics and finite-deformation plasticity 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] Plotting the position of an individual parcel through time gives the pathline of the parcel. This can be visualized as sitting in a boat and drifting down a river.

The Eulerian 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 the Lagrangian and Eulerian frame of reference. 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.

Description

In the Eulerian specification of the flow field, the flow quantities are depicted as a function of fixed position x and time t. Specifically, the flow velocity is described as u(x,t). On the other hand, in the Lagrangian specification, all fluid parcels are labelled by some vector field a, with a time-independent for each fluid parcel. Often, a is chosen to be 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 parametrization of the velocity v of the parcel. ^[1] In the Lagrangian description, the flow velocity v(a,t) is related to the position X(a,t) of the fluid parcels by:^[2]

$\mathbf{v} = \frac{\partial \mathbf{X}}{\partial t}.$

Consequently, u and v are related through

$\mathbf{u}\left(\mathbf{X}(\mathbf{a},t),t \right) = \mathbf{v}\left(\mathbf{a},t \right).$

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

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

$\frac{\mathrm{D}\mathbf{F}}{\mathrm{D}t} = \frac{\partial \mathbf{F}}{\partial t} + (\mathbf{u}\cdot \nabla)\mathbf{F}.$

This tells us that the total rate of change of some vector 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 convective rate of change of F.

See also

• Streamlines, streaklines, and pathlines
• Fluid dynamics
• Frame of reference
• Coordinate system
• Equivalent latitude

Notes

1. ^ ^{a b c d} Batchelor (1973) pp. 71–73.
2. ^ ^{a b c} Lamb (1994) §3–§7 and §13–§16.

References

• Batchelor, G.K. (1973), An introduction to fluid dynamics, Cambridge University Press, ISBN 0-521-09817-3 
• Lamb, H. (1994) [1932], Hydrodynamics (6th ed.), Cambridge University Press, ISBN 9780521458689