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# Lagrangian and Eulerian coordinates

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### Lagrangian and Eulerian coordinates

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 position x and time t. Specifically, the flow is described by a function

$\mathbf\left\{v\right\}\left\left(\mathbf\left\{x\right\},t\right\right)$

giving the flow velocity at position x at time t.

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 a. (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 is described by a function

$\mathbf\left\{X\right\}\left\left(\mathbf\left\{a\right\},t\right\right)$

giving the position of the parcel labeled a at time t.

The two specifications are related as follows:[2]

$\mathbf\left\{v\right\}\left\left(\mathbf\left\{X\right\}\left(\mathbf\left\{a\right\},t\right),t \right\right) = \frac\left\{\partial \mathbf\left\{X\right\}\right\}\left\{\partial t\right\}\left\left(\mathbf\left\{a\right\},t \right\right)$

because both sides describe the velocity of the parcel labeled a at time t.

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

## Substantial derivative

Main article: Material derivative

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]

Suppose we have a flow field with Eulerian specification v, and we are also given some function F(x,t) defined for every position x and every time t. (For instance, F could be an external force field, or temperature.) Now one might ask about the total rate of change of F experienced by a specific flow parcel. This can be computed as

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

(where ∇ denotes the gradient with respect to x, and the operator v⋅∇ 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 v 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(a,t),t) with respect to t.

## References

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