Question about a scientific paper - Fluid Mechanics - Perturbation Theory

In summary, the conversation pertains to the validity of Gabriel Godin's assertion regarding the analytical solution for obtaining terms higher than first-order in the Lagrangian velocity. The question is whether Godin's statement in his article, "Such a development...additional terms" and the last sentence in the attachment, is correct or not. Some professors believe that an analytical solution is indeed possible, while the person asking the question has been unsuccessful in obtaining it through Taylor series expansions. They have been able to obtain a numerical solution, but are hoping for confirmation or an alternative solution. The article in question is "Magnitude of Stokes' drift in coastal waters" by Gabriel Godin, published in Ocean Dynamics in 1995.
  • #1
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My question pertains to the following article: http://tinyurl.com/4uw9h2a
I have attached the relevant section to this post.

My question is whether Godin's assertion is correct or not - namely the sentence "Such a development ... additional terms" and the last sentence in the attachment. That is, that the only way to obtain terms higher than first-order in the Lagrangian velocity [tex] v(r_p,t) [\tex] is through numerical integration of eq. (1). He argues that an analytical solution via Taylor series expansions is not possible due to the strong non-linear nature of the problem due to the displacement [tex] r_p-r_0 [\tex] being unknown.

After speaking to several professors in my department they are surprised that this paper even passed the review process and claim that the analytical solution is indeed possible to obtain. However, I have attempted to obtain the analytical solution via a Taylor development (perturbation theory) and have been unsuccessful thus far. I have been successful in obtaining the numerical solution for the higher order terms, but since I am publishing a paper it would be nice to have the analytical solution as well or at least a confirmation that the numerical solution is the only way to go in this case.

I am not a mathematician and this subject is rather unfamiliar to me, I hope someone can let me know whether the author is correct in his assertion or not.

Thank you.
 

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  • #2
Your link doesn't work. Could you just give the reference?
 
  • #3
cristo said:
Your link doesn't work. Could you just give the reference?

Sorry, here it is:

Gabriel Godin - Magnitude of Stokes' drift in coastal waters - Ocean Dynamics, 1995 - Springer
 

1. What is the purpose of perturbation theory in fluid mechanics?

Perturbation theory is a mathematical tool used in fluid mechanics to simplify complex problems by breaking them down into smaller, more manageable components. It allows scientists to study the behavior of a fluid system under small disturbances or changes, which can then be used to make predictions about the system as a whole.

2. How does perturbation theory differ from other mathematical methods used in fluid mechanics?

Perturbation theory differs from other methods, such as numerical simulations or analytical solutions, in that it is particularly useful for studying systems with small changes or disturbances. It is also often used to analyze nonlinear systems, where other methods may not be as effective.

3. What are some practical applications of perturbation theory in fluid mechanics?

Perturbation theory has a wide range of applications in fluid mechanics, including studying the stability of fluid flows, predicting the behavior of fluids in turbulent conditions, and analyzing the effects of small perturbations on large-scale systems. It is also commonly used in the design and optimization of various engineering systems, such as aircraft wings and hydraulic systems.

4. What are the limitations of perturbation theory in fluid mechanics?

While perturbation theory is a powerful tool, it does have its limitations. It is most effective for small perturbations, and as the magnitude of the disturbances increases, the accuracy of the predictions made by perturbation theory decreases. Additionally, perturbation theory may not be suitable for highly complex or chaotic systems.

5. How can perturbation theory be used to improve our understanding of fluid mechanics?

Perturbation theory allows scientists to break down complex fluid systems into simpler components, making it easier to study and understand their behavior. By using this method, scientists can make predictions and gain insights into the underlying physical mechanisms at play in various fluid systems. This can lead to improved designs and optimization of engineering systems, as well as a deeper understanding of fundamental principles in fluid mechanics.

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