Boundary Layer Theory

[gerbil]

Hello All,

 

I have a question related to the fascinating topic of "Boundary Layer Theory".

Take for example the simple equation

 

mx'' + kx' + cx = 0 where k and c are finite constants and (m -> 0), (The derivative sign (') is by t)

x(0) = 0

x'(0) = A > 0.

 

In order to achieve a comprehension on what's going on near the point t = 0 when (m -> 0), it is suggested to take the change of variables t* = t/m.

By that the former equation becomes,

 

x'' + kx' + mcx = 0 (Now, the derivative sign (') is by t*). Now, moving into the limit or alternatively noticing that mcx is negligible, we can drop it and we get,

 

x'' + kx' = 0 and now the solution is fairly easy.

 

Books call this solution: "The inner solution" of the equation, since this solution describes the solution inside a boundary layer.

 

My question is as follows:

 

The idea of taking the change of variables t* = t/m, is just a mathematical trick or does it have a meaning (e.g. streching of the time scale, etc.)?

If it has a meaning, i'd like to know its intuition.

 

Best,

Miki