# Asymptotic matching

## 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 $$\sim x^{1/4}$$. 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 $$X \to \infty$$?

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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 $$x = 0$$. 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 $$X\rightarrow\infty$$ ?