4.1.7. Euler and Lagrange¶

What is the difference between the Euler and Lagrange approach?
What are the advantages and disadvantages between the Euler and Lagrange approach?
What is the derivation of the link between the Euler and Lagrange mass conservation?

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:

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