Eulerian velocities to Lagrangian velocities

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In summary, the conversation was about transforming Eulerian velocity to Lagrangian velocity. The attempt at a solution involved rearranging terms, integrating both sides, and using the initial condition to solve for a constant. The final equation for Lagrangian position was z_{1}=\frac{x_{1}}{1+tx_{1}}.
  • #1
paccali
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Homework Statement



Eulerian velocity: [tex]V_{1}=-z_{1}^{2}[/tex]

[tex]V_{1}=\frac{dz_{1}}{dt}[/tex]

[tex]z_{1}(t=0)=x_{1}[/tex]

This is supposed to become the Lagrangian velocity of:

[tex]z_{1}=\frac{x_{1}}{1+tx_{1}}[/tex]

I don't understand how to take the Eulerian velocity and transform it to Lagrangian.

Homework Equations

The Attempt at a Solution



[tex]\frac{dz_{1}}{dt}+z_{1}^{2}=0[/tex]

After this, I don't know how to take this and move forward.

I've been working the problem for a day, and I still can't get any closer. I can take Lagrangian and transform it to Eulerian, but I don't know how to do the reverse. TJ Chung's General Continuum Mechanics book is poorly developed for examples and proofs.
 
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  • #2
paccali said:

The Attempt at a Solution



[tex]\frac{dz_{1}}{dt}+z_{1}^{2}=0[/tex]

After this, I don't know how to take this and move forward.

This equation is separable, since it can be rewritten as

[tex]-\frac{dz_{1}}{z_{1}^{2}}=dt.[/tex]

You can integrate both sides and fix the integration constant with

[tex]z_{1}(t=0)=x_{1}.[/tex]
 
  • #3
Nevermind, I figured out how to solve the problem using my old differential equations textbook. In case someone is curious, here's what I did:

I rearranged the terms so that like terms were on the same side:

[tex]\frac{dz_{1}}{dt}=-z_{1}^{2}[/tex]
[tex]\frac{dz_{1}}{-z_{1}^{2}}=dt[/tex]

I then integrated each side:

[tex]\int -\frac{1}{z_{1}^{2}}dz=\int dt[/tex]
[tex]\frac{1}{z_{1}}+C_{1}=t+C_{2}[/tex]

Since both sides had constants, I dropped one, and I then used the initial condition of [tex]z_{1}(t=0)=x_{1}[/tex] to solve for C

[tex]\frac{1}{z_{1}}+C=t[/tex]
[tex]\frac{1}{z_{1}}=t-C[/tex]
[tex]z_{1}=\frac{1}{t-C}[/tex]
[tex]z_{1}(0)=x_{1}=\frac{1}{0-C}[/tex]
[tex]C=-\frac{1}{x_{1}}[/tex]

I plugged in C above and got this equation for Lagrangian position:

[tex]z_{1}=\frac{1}{t+\frac{1}{x_{1}}}=\frac{x_{1}}{1+tx_{1}}[/tex]
 
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  • #4
I just found what I did wrong, but thanks for the help nonetheless.
 

Related to Eulerian velocities to Lagrangian velocities

What is the difference between Eulerian velocities and Lagrangian velocities?

Eulerian velocities refer to the velocity of a fluid at a fixed point in space, whereas Lagrangian velocities refer to the velocity of a fluid particle as it moves through space.

Why is it important to convert between Eulerian velocities and Lagrangian velocities?

Converting between these two types of velocities is important in fluid dynamics and environmental studies where the motion of particles in a fluid is being analyzed. It allows for a better understanding of the behavior of the fluid and its effects on the surrounding environment.

What are the equations used to convert between Eulerian velocities and Lagrangian velocities?

The equation used to convert from Eulerian to Lagrangian velocities is v = u + V, where v is the Lagrangian velocity, u is the Eulerian velocity, and V is the velocity of the fluid at a fixed point. The equation used to convert from Lagrangian to Eulerian velocities is u = v - V.

How do Eulerian and Lagrangian velocities differ in terms of their use in different fields of study?

Eulerian velocities are commonly used in fluid mechanics and engineering, while Lagrangian velocities are used in meteorology and oceanography. This is because Eulerian velocities are more useful for studying the overall motion of a fluid, while Lagrangian velocities are better for tracking the movement of individual particles.

What are some real-world applications of converting between Eulerian velocities and Lagrangian velocities?

One example of a real-world application is in weather forecasting, where Lagrangian velocities are used to track the movement of air particles to predict the path of a storm. Another application is in oceanography, where Eulerian velocities are used to study ocean currents and their effects on marine life and human activities.

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