The mathematical formulae and rules that are applied to macroparticles may not be applicable while studying the behavior of microparticles. Different mathematical approaches have been designed for solving such problems, and both Eulerian and Lagrangian approaches are used while analyzing and solving such mathematical problems of micro-scale particles.

**Eulerian vs Lagrangian**

The main difference between Eulerian and Lagrangian is that in the Eulerian method, more attention is given to the flow properties of a control volume, in terms of functions of space and time. In the Lagrangian method, the flow volume is assumed to be made of a large number of particles and the individual particles are given more focus.

The eulerian mathematical approach is used to solve mathematical problems involving fluid flow or flow of a volume of particles. The flow is treated as a function of both space and time and the different properties of the flow, such as temperature is recorded and studied. In this approach, more focus is given to the actual flow.

The Lagrangian approach considers the fluid flow to be made of a large number of particles. In this approach, the fluid flow is studied by studying the individual particles, by defining the properties of flow such as the direction of motion and speed to the particles. Thus the particles are tracked as it moves through the flow volume.

**Comparison Table Between Eulerian and Lagrangian**

Parameters of Comparison | Eulerian | Lagrangian |

Definition | The mathematical approach to study the flow of particles and was proposed by Leonhard Euler | The mathematical approach used to study the flow of particles and was proposed by Louis Lagrange |

Concentration | Focus is given to the flow properties at a fixed point | Focus is given to an individual particle by defining its properties |

Approach | The point of observation is fixed and only changes in fluid flow are noted | The point of observation changes as the property values changes at different places |

Method | The flow is described in function of space and time with different properties | The flow is described in terms of individual particles with characteristic properties |

Usage | The eulerian approach is used very commonly | The Lagrangian approach is not commonly used |

**What is Eulerian?**

The mathematical approach to study the flow of particles suspended in volume that was proposed by Leonhard Euler is known as the Eulerian approach.

This approach focuses more on the actual flow of the volume than the individual particles. This is achieved by defining the flow in terms of the function of space and time and also establishing the parameters such as temperature which is related to the flow.

Thus the concentration of the approach is the flow of the particles. The observation of the flow is made by selecting a point of observation in the flow volume and fixing the point.

The parameters of the flow are recorded through the fixed point of observation and the change in these parametric values is noted down.

The observations made are extrapolated along the entire flow volume to determine the characteristics of the flow. This approach is thus mostly used to determine the flow characteristics of gaseous flow particles or microparticles suspended in constant flow environments.

This method is more commonly used than the other mathematical formulations for the study of unsteady dispersion of microparticles. As the flow patterns change constantly, hundreds of iterations are required to create a mathematical model using this method.

**What is Lagrangian?**

The Lagrangian approach is a mathematical formulation used to study the flow characteristic of a volume. The formulation was made by Louis Lagrange.

The Lagrangian method considers the flow volume to be made of a large number of particles. Thus the characteristics of the fluid flow are calculated by understanding the flow parameters of individual particles.

The approach is performed by selecting a single particle in the flow volume and fixing it on the particle. The characteristics of the flow such as the direction of motion and speed are assigned to the particle.

The motion of the particle is recorded and the changes to the parametric quantities are noted down. As the parameters of the flow change in different locations, the observations of the particle are made at different points throughout the flow volume.

Thus different observations are recorded at different points in the flow volume and the characteristics change in the flow of the particle is calculated. These changes are extrapolated throughout the entire flow volume to determine the nature of the fluid flow.

This method is not as widely used as the Eulerian method, due to the difficulty in setting up required for the observation. This method is also more prone to errors as such minute observations are difficult to be physically made.

**Main Differences Between Eulerian and Lagrangian**

- Eulerian method is mathematical formulation made by Leonhard Euler. Lagrangian method is a mathematical model made by Louis Lagrange.
- In Eulerian method more focus is given to the flow of the particles. In Lagrangian method more focus is given to the actual particles.
- The point of observation in Eulerian method is fixed. The point of observation in Lagrangian method changes with the particle.
- Eulerian method considers flow as a function of space and time. Lagrangian method considers volume flow in terms of characteristics of individual particles.
- Eulerian mathematical approach is more commonly used to determine fluid flow in a liquid or gaseous environment than Lagrangian mathematical approach.

**Conclusion**

The mathematical formulations required for solving problems that involve micro-sized particles are different from the normal mathematical methods.

Both Eulerian and Lagrangian mathematical methods analyze and find out the solutions to such conditions. They are used to solve problems of fluid flow consisting of small particles, such as gaseous flow systems.

The eulerian mathematical model is more commonly used than the Lagrangian model, as it is not as prone to errors and is also easy to perform in a controlled environment.

The Lagrangian method is more complex and requires a lot of precision for making observations and calculations of the observations made. Thus the method is more prone to errors.

**References**

https://www.sciencedirect.com/science/article/pii/004578259290042I

https://www.sciencedirect.com/science/article/pii/0021999174900515