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I'm trying to understand boundary layer theory in fluids. All I've found are dimensional arguments, order of magnitude arguments, etc... What I'm looking for is more mathematically sound arguments. Not rigorous as in keeping track of all epsilons and deltas, but more rigorous than an heuristic argument. Hope you understand what I mean. Some free resources available on the web would be preferred, but if you can suggest book titles that's also helpful. Thanks.

Edit: The applications I have in mind are the calculation of damping in surface waves in basins of various shapes (circular, rectangular, etc).

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3 Answers 3

I will probably disappoint you now, but fluid dynamics must be approximative. You start from a model which is (to some extent) rigorous and intuitive and consists of partial differential equations - Navier-Stokes equations, continuity equation and Fourier-Kirchhoff (if heat transfer is involved) - with proper boundary conditions. However, for most of the shapes, there are no analytical solutions. Therefore, we first throw out unimportant information and make the parameters somehow independent. The standard dimensionless numbers (Re, Sc, Eu ...) are often used (because we are used to them and history proved they are most useful). The "boundary values" or "magnitude arguments" (e.g. Re>10000) are usually stated in books very approximative but model-independent. If you want more exact results, you need to numerically solve the partial differential equations (there are very user-friendly packages for this, e.g. COMSOL Multiphysics...). For some models, you can very precisely calculate e.g. heat transfer coefficient. At very high Re/Pr... you will observe those approximate model-independent trends.

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Boundary layer theory come in as a method to "simplify" the mathematics in fluid mechanics, so that it is solvable analytically. It separates the fluid as two regions:

  1. where the viscosity effect is important, i.e. boundary layer
  2. where viscosity is not important

This approach has been proven useful for large range of applications, that's why aerodynamics (for example) has flourished and matured a lot in the past century. In some cases where Navier-Stokes has an exact solution, researcher had shown that there exist a boundary layer where viscosity is important, and outer region where it is negligible. These examples are abundant in Fluid Mechanics book, such as Kundu-Cohen's, or Landau-Lifshitz's. By the nature of this theory, it would be interesting if there are rigorous expositions mathematically as you may probably want to see. However, if Landau's treatment did not satisfy you, you may want to check a book by Oleinik and Samokhin titled Mathematical Models in Boundary Layer Theory.

A probably important thing to note, boundary layer approximation may cause erroneous result due to the assumption of the thin boundary layer is not satisfied. Such an example (quite recent) can be seen in an article by Doinikov and Bouakaz in JASA: http://asadl.org/jasa/resource/1/jasman/v127/i2/p703_s1?isAuthorized=no

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I learnt boundary layer theory from Bender-Orszag, and found it fairly simple, though it gets a little more mathematically involved than I'd hope. Here's the book on google books.

I also wrote some notes about them in a course on numerical methods I took a year ago, which you could find useful if you speak Italian.

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