I am currently working on a problem involving solving PDEs and using boundary layers. In this problem, $f(x,y)$ is the main dependent variable, $f(x,y)=f_R (X_R ,y)$ and $f(x,y)=f_B(x,Y_B)$ where $X_R$ and $Y_B$ are inner variables near the right-hand side and bottom side of a rectangular domain respectively, such that $x=a-\epsilon X_R$ and $y=\epsilon^{1/2}Y_B$, since there are boundary layers near these sides. Here, $\epsilon\ll 1$.
Among the boundary conditions (to be solved with corresponding rescaled PDEs and matching conditions) are Robin boundary conditions at each wall (with all quantities being dimensionless),
\begin{align*}\underset{(1{\rm a})}{\underbrace{\dfrac{\partial f_{R}}{\partial X_{R}}}}-\underset{(2{\rm a})}{\underbrace{\dfrac{\epsilon\Sigma}{\tau}}}f_{R} & =\underset{(3{\rm a})}{\underbrace{g_{1}(y)}}\text{ at }X_{R}=0\text{ (i.e. }x=a)\\ \underset{(1{\rm b})}{\underbrace{\epsilon^{1/2}\dfrac{\partial f_{B}}{\partial Y_{B}}}}-\underset{(2{\rm b})}{\underbrace{\dfrac{\epsilon\Sigma}{\varsigma}f_{B}}} & =\underset{(3{\rm b})}{\underbrace{g_{2}(x)}}\text{ at }Y_{B}=0\text{ (i.e. }y=0) \end{align*}
where $g_1(y)$ and $g_2 (x)$ are both known functions which are ${\cal O}(1)$, and $\tau,\varsigma\ll 1$ and $\Sigma$ are constants. We note that $\tau, \sigma$ and $\epsilon$ are all small parameters, and we want to use these as a basis for asymptotic analysis.
I have been assured that the correct distinguished limit in this problem is
$$ \boxed{\dfrac{\tau \Sigma}{\varsigma}\sim\epsilon^{1/2}}$$ but I am having trouble seeing where this comes from (either that, or I am overcomplicating things).
My approach for spotting distinguished limits would be by looking for sensible balances between the terms in the equations, which I do below.
- $\underline{\rm{(1a)\sim~(2a)}}:$ this gives $\epsilon\Sigma/\tau\sim 1$, which balances all three terms in the equation. In this instance the second condition would become, letting $\epsilon\Sigma/\tau = \lambda = {\cal O}(1)$, $$\underset{(1{\rm b})}{\underbrace{\epsilon^{1/2}\dfrac{\partial f_{B}}{\partial Y_{B}}}}-\underset{(2{\rm b})}{\underbrace{\dfrac{\lambda\tau}{\varsigma}f_{B}}}=\underset{(3{\rm b})}{\underbrace{g_{2}(x)}}\text{ at }Y_{B}=0\text{ (i.e. }y=0).$$ The balance $\rm{(1b) \sim (2b)}$ cannot work since then $\rm{(3b)}$ would dominate in isolation, and $g_2(x)\neq 0$. The only other balance is then $\rm{(2b) \sim (3b)}$, which suggests that $\tau = \varsigma \sigma$ where $\sigma = {\cal O}(1)$. Then, to summarise, both conditions become \begin{align*} \dfrac{\partial f_{R}}{\partial X_{R}}-\lambda f_{R} & =g_{1}(y)\text{ at }X_{R}=0,\\ \epsilon^{1/2}\dfrac{\partial f_{B}}{\partial Y_{B}}-\lambda\sigma f_{B} & =g_{2}(x)\text{ at }Y_{B}=0. \end{align*}
- $\underline{{\rm (2b)\sim(3b)}}:$ we have ${\epsilon\Sigma}/{\varsigma}\sim1$, so let ${\epsilon\Sigma}/{\varsigma}=\mu={\cal O}(1)$. Then the first condition becomes $$ \underset{(1{\rm a})}{\underbrace{\dfrac{\partial f_{R}}{\partial X_{R}}}}-\dfrac{\mu\varsigma}{\tau}\underset{(2{\rm a})}{\underbrace{f_{R}}}=\underset{{\rm (3a)}}{\underbrace{g_{1}(y)}}\text{ at }X_{R}=0. $$ If ${\rm (1a)\sim(2a)}$ then ${\varsigma}/{\tau}\sim1$, so ${\varsigma}/{\tau}=\upsilon={\cal O}(1)$. Then, all three terms balance, and both boundary conditions become \begin{align*} \dfrac{\partial f_{R}}{\partial X_{R}}-\mu\upsilon f_{R} & =g_{1}(y)\text{ at }X_{R}=0,\\ \epsilon^{1/2}\dfrac{\partial f_{B}}{\partial Y_{B}}-\mu f_{B} & =g_{2}(x)\text{ at }Y_{B}=0. \end{align*}
The above is how I would usually approach finding distinguished limits, but I cannot figure out how to get to the distinguished limit that has been provided. This leads me to wonder whether there is a problem with the procedure I have used, or whether something else has gone wrong (e.g. incomplete/incorrect understanding of what distinguished limits are). This leads me to a couple of questions:
- I am aware that in problems involving boundary layers, boundary layer widths are spotted by rescaling one of the spatial variables and finding what rescaling gives a balance where no fewer than two terms are dominant. Does the same idea work for dealing with boundary conditions such as these?
- Is there an obvious trick that I have missed?