Solving Asymptotic Matching for LaTeX Boundary Layer

[rsq_a]

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 \sim x^{1/4}. Suppose that the boundary layer was of thickness x = O(\epsilon).

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

 

[dev00790]

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 x = 0. Suppose the first term of the outer solution, valid away from x = 0 was \sim x^{1/4} . Suppose that the boundary layer was of thickness x = O(\epsilon) .

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