# 4.1.7. Euler and Lagrange¶

## 4.1.7.1. What is the difference between the Euler and Lagrange approach?¶

- Lagrange:
Moving frame of reference

Elements are matter

Results in pathlines

- Euler:
Stationary frame of reference

Elements are spatial

Results in streamlines

## 4.1.7.2. What are the advantages and disadvantages between the Euler and Lagrange approach?¶

- Lagrange:
Advantage: More suitable than Euler for the dynamics of discrete particles in a fluid e.g. flow visualisation.

Disadvantage: Computationally expensive to keep track of large numbers of particles in a flow field

- Euler:
Advantage: More suitable than Lagrange for the dynamics of a fluid flow field, e.g. CFD

Disadvantage: Time step limited by grid due to stability or accuracy

## 4.1.7.3. What is the derivation of the link between the Euler and Lagrange mass conservation?¶

Eulerian mass conservation:

Lagrangian derivative (or Total/Material/Substantive derivative):

Lagrangian derivative applies to incompressible flow \(\longrightarrow \nabla \cdot \vec{u} = 0\)

Substituting Lagrangian derivative into expanded Eulerian mass conservation: