Fluid mechanics Lagrange & Euler formalism

In summary, -The Lagrange and Euler formalism represent the same physical quantities, but in different coordinate systems. -The Lagrange description uses the velocity field to represent the temperature at a point. -The Euler description uses the temperature at a point as the function of time.
  • #1
LagrangeEuler
717
20
Lagrange & Euler formalism

How we get relation

[tex](\frac{\partial T^{(L)}}{\partial t})_{r_0}=(\frac{\partial T^{(E)}}{\partial t})_{r}+\frac{\partial T^{(E)}}{\partial x}(\frac{\partial x}{\partial t})_{r_0}+\frac{\partial T^{(E)}}{\partial y}(\frac{\partial y}{\partial t})_{r_0}+\frac{\partial T^{(E)}}{\partial z}(\frac{\partial z}{\partial t})_{r_0}[/tex]

[tex]T[/tex] - temperature
[tex]t[/tex] - time
 
Physics news on Phys.org
  • #2
Lagrange representation
[tex]T=T^{L}(x_0,y_0,z_0,t)[/tex]
where coordinates [tex]x_0,y_0,z_0[/tex] are fixed.

I have a trouble with that. Can you tell me one type of this function?
 
  • #3
I think, I've explained this already some days ago in this forum.

Again, the Euler coordinates [itex]\vec{x}_0[/itex] are labels for a fixed fluid element by noting its position at a fixed (initial) time [itex]t_0[/itex]. Then you follow this fluid element along its trajectory in time. This is given by the position-vector field (as a function of the Euler coordinates) [itex]\vec{x}(t,\vec{x}_0)[/itex].

In the Lagrange description you switch to a field description, i.e., you characterize the fluid by the velocity field, where an observer at point [itex]\vec{x}[/itex] measures the velocity of the fluid element at this position at time, [itex]t[/itex]. This field we denote by [itex]\vec{v}(t,\vec{x})[/itex].

The relation to the Euler coordinates is then given by

[tex]\vec{v}_0(t,\vec{x}_0)=\frac{\partial \vec{x}(t,\vec{x}_0)}{\partial t}=\vec{v}[t,\vec{x}(t,\vec{x}_0)].[/tex]

It gives the velocity of the fluid element, which has been at position [itex]\vec{x}_0[/itex] at time [itex]t[/itex].

The same procedure holds for any other physical quantity which has a specific meaning for a given fluid element. An example is the temperature [itex]T[/itex]. In the Euler description, the temperature of the fluid element which has been at [itex]\vec{x}_0[/itex] at time [itex]t_0[/itex] at time [itex]t[/itex] is given by the function [itex]T_0(t,\vec{x}_0)[/itex].

In the Lagrange description, you consider the temperature at the position [itex]\vec{x}[/itex] of the fluid element that is there at time, [itex]t[/itex]. This function is called [itex]T(t,\vec{x})[/itex]. Again, the relation between the two points of view are given by

[tex]T_0(t,\vec{x}_0)=T[t,\vec{x}(t,\vec{x}_0)].[/tex]

Now if you want the time derivative of the temperature of the specific fluid element that has been at position [itex]\vec{x}_0[/itex] at time [itex]t_0[/itex], you have to use the chain rule of multi-variable calculus,

[tex]\partial_t T_0(t,\vec{x}_0)=\left (\frac{\partial T(t,\vec{x})}{\partial t} \right)_{\vec{x}=\vec{x}(t,\vec{x}_0)}+ \frac{\partial \vec{x}(t,\vec{x}_0)}{\partial t} \cdot [\vec{\nabla} T(t,\vec{x})]_{\vec{x}=\vec{x}(t,\vec{x}_0)}=\left (\frac{\partial T(t,\vec{x})}{\partial t} \right)_{\vec{x}=\vec{x}(t,\vec{x}_0)}+\vec{v}[t,\vec{x}(t,\vec{x}_0)] \cdot [\vec{\nabla}\vec{\nabla} T(t,\vec{x})]_{\vec{x}=\vec{x}(t,\vec{x}_0)}.[/tex]

I hope with this very detailed writing of all the arguments, the two (equivalent) descriptions of fluid mechanics and how to switch from one to the other has become more clear.
 

1. What is the difference between Lagrange and Euler formalism in fluid mechanics?

The Lagrange formalism in fluid mechanics describes the motion of individual fluid particles, while the Euler formalism describes the overall motion of the entire fluid body. In Lagrange formalism, the properties of each fluid particle are tracked over time, whereas in Euler formalism, the properties are described at specific points in space.

2. How do the Lagrange and Euler equations differ in their approach to solving fluid mechanics problems?

The Lagrange equations are based on the principle of conservation of mass, momentum, and energy for individual fluid particles, while the Euler equations use the concept of fluid flow as a continuum and solve for the overall properties of the fluid body. Lagrange equations are more suitable for problems involving small-scale fluid motion, while Euler equations are better for large-scale fluid motion.

3. What are the advantages of using the Lagrange formalism in fluid mechanics?

The Lagrange formalism allows for a more detailed analysis of the motion of individual fluid particles, making it useful for studying the behavior of fluids in complex situations such as turbulence. It also allows for the identification of specific fluid particles and tracking their properties over time.

4. What are the limitations of the Euler formalism in fluid mechanics?

The Euler formalism assumes that the fluid is continuous and does not take into account the effects of individual fluid particles. This can lead to inaccuracies in situations where the fluid is highly turbulent or has regions with varying properties. Additionally, the Euler equations are not valid for fluids with high viscosity or compressibility.

5. How are the Lagrange and Euler formalism used in practical applications of fluid mechanics?

In practical applications, both Lagrange and Euler formalism are used in conjunction with each other. The Lagrange equations are used to track individual fluid particles, while the Euler equations are used to determine overall fluid properties. This allows for a more comprehensive understanding of fluid behavior and is applied in various fields, including aerodynamics, hydrodynamics, and meteorology.

Similar threads

  • Mechanics
Replies
3
Views
878
Replies
7
Views
680
  • Mechanics
Replies
1
Views
582
Replies
14
Views
2K
Replies
3
Views
647
Replies
22
Views
311
Replies
5
Views
805
Replies
7
Views
1K
Replies
2
Views
684
  • Mechanics
Replies
24
Views
934
Back
Top