# 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:

\[{{\partial \rho} \over {\partial t}} + \nabla \cdot (\rho \vec{u}) =
{{\partial \rho} \over {\partial t}} + (\vec{u} \cdot \nabla) \rho + (\nabla \cdot \vec{u})\rho
= 0\]

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

\[{{D \rho} \over {D t}} = {{\partial \rho} \over {\partial t}} + \nabla \cdot (\rho \vec{u}) =
{{\partial \rho} \over {\partial t}} + (\vec{u} \cdot \nabla) \rho + (\nabla \cdot \vec{u})\rho\]

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

\[{{D \rho} \over {D t}} = {{\partial \rho} \over {\partial t}} + (\vec{u} \cdot \nabla) \rho\]

- Substituting Lagrangian derivative into expanded Eulerian mass conservation:

\[{{D \rho} \over {D t}} + (\vec{u} \cdot \nabla) \rho = 0\]