Asymptotic matching

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Main Question or Discussion Point

There seems to be something wrong with the LaTeX

This seems like an absolutely trite question, but I can't seem to figure it out.

Suppose you had a boundary-layer problem, say at x = 0. Suppose the first term of the outer solution, valid away from x = 0 was [tex]\sim x^{1/4}[/tex]. Suppose that the boundary layer was of thickness [tex]x = O(\epsilon)[/tex].

Suppose that you have solved for the inner solution near x = 0. What would be the required matching condition?

So in this example, we would re-scale [tex]x = \epsilon X[/tex]. Then wouldn't the inner solution need to behave like, [tex]y \sim \epsilon^{1/4} X^{1/4}[/tex] as [tex]X \to \infty[/tex]? But this doesn't seem possible, since we can't allow the inner solution to blow up. What is the correct matching condition as [tex]X \to \infty[/tex]?
 
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I've amended your equations, so they are viewable.

This seems like an absolutely trite question, but I can't seem to figure it out.

Suppose you had a boundary-layer problem, say at [tex]x = 0[/tex]. Suppose the first term of the outer solution, valid away from [tex]x = 0[/tex] was [tex]\sim x^{1/4}[/tex] . Suppose that the boundary layer was of thickness [tex] x = O(\epsilon)[/tex] .

Suppose that you have solved for the inner solution near [tex]x = 0.[/tex] What would be the required matching condition?

So in this example, we would re-scale [tex]x = \epsilon X[/tex] . Then wouldn't the inner solution need to behave like, [tex] y \sim \epsilon^{1/4} X^{1/4}[/tex] as [tex] X \rightarrow \infty[/tex] ? But this doesn't seem possible, since we can't allow the inner solution to blow up. What is the correct matching condition [tex]X\rightarrow\infty[/tex] ?
 

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