Part II Solution Techniques
7.3 Constructing Boundary Layer Solutions
on the entire domain 0≤x≤1. This is comparable to the exact solution (7.2), except for the absence of the exponentially smalle−2/εterm (which cannot be captured in regular expansions such as (7.6)).
154 7 Boundary Layer Theory prevent “double-counting” will produce the leading order solution on the entire domain (7.16). So, in an example with boundary layers at both the left and right boundaries, we write
ycomp(x)∼y0(x)+YBLCL +YBLCR . (7.22) This description covers broad classes of problems, but as will be seen, in some cases, steps (2, 3, 4) may become a bit intertwined. We also note:
• When boundary layers are necessary, which boundary conditions apply to the outer solution may not be immediately apparent. Hence a general form for the outer solution will be needed initially.
• The boundary conditions as well as the ODE play a role in determining the dom- inant balances.
• The location of the boundary layer and which boundary conditions apply to the inner solution might not be determined until matching is applied.
• If the inner/outer solutions are not matchable (either limit does not exist, or equation (7.13) cannot be satisfied) then the assumed choice of boundary layer positionx∗ or dominant balance may be not be right.
• While the term “boundary layer” stems from the fact that the inner domain often occurs at a boundary, in some cases, they can also occur within the domain of a problem, in which case they are sometimes calledinterior layers.
Consequently, the reader should consider steps (1)–(5) as “guidelines” that may need to be adjusted depending on the given problem; this is one of the challenging (and interesting) points of matched asymptotic expansions.
We use an example to illustrate the aspects of the above procedure. Consider the boundary value problem
εd2y d x2 +d y
d x =cosx (7.23a)
in the limitε→0 on the domain 0≤x≤π, subject to the boundary conditions y(0)=2, y(π)= −1. (7.23b)
7.3.1 The Outer Solution
Assuming that y(x)and its derivatives are bounded asε → 0, we write the outer solution as y ∼ y0(x)+εy1(x)+ε2y2(x)· · ·. Substituting into (7.23a) gives the sequence of equations
O(ε0): y0=cosx, O(ε1): y1+y0 =0, O(ε2): y2+y1 =0,
and so on for higher order equations. TheO(1)problem yieldsy0=sinx+Aand substituting this into theO(ε)equation givesy1(x)= −cosx+B. We can proceed in this way to determine as many terms as desired in the expansion of the general outer solution
yout=(sinx+A)+ε(−cosx+B)+O(ε2). (7.24) At each order, there is only a single constant of integration, A,B, . . .. Imposing the condition atx =0 from (7.23b) selects A =2, while the condition at x =π picksA= −1; the outer solution cannot satisfy both at once, and hence a boundary layer will be required. In summary, at this point, we do not know which boundary conditions will apply to the outer and which to the inner solutions.
7.3.2 The Distinguished Limits
To determine the relevant scaling of the singular solution, we writey(x)=εβY(X) andX =(x−x∗)/εα and assume thatY(X)= O(1)for X = O(1). Substituting into (7.23a) yields
ε1−2α+βY
(1)
+ε −α+βY
(2)
=cos (x∗+εαX)
(3)
, (7.25)
where 0 ≤ x∗ ≤ π. We also note that both boundary conditions (7.23b) take the formεβY =O(1), and hence any solution local to a boundary must haveβ=0. It remains to determineαfrom the possible dominant balances:
(a)Terms (2, 3):ε−α =ε0 ⇒ α=0, (b)Terms (1, 3):ε1−2α =ε0 ⇒ α=1/2, (c)Terms (1, 2):ε1−2α =ε−α ⇒ α=1.
Option (a) is the regular distinguished limit that corresponds to the outer solution.
Option (b) is not a valid balance since the neglected term (2) is not sub-dominant, O(ε−1/2) O(1). Consequently, the boundary layer must take the form given by (c) where the neglected term (3) is sub-dominant to the leading balance with O(1)O(ε−1). We note that in some problems, the dominant balances can change for different assumed positions of the boundary layer, x∗ (most notably for non- autonomous equations), but here term (3) uniformly satisfies|cos(x)| ≤1=O(ε0).
Hence our scaled equation for the inner solution is given by d2Y
d X2 +dY
d X =εcos(x∗+εX), (7.26)
wherex∗has not yet been determined.
156 7 Boundary Layer Theory
7.3.3 The Inner Solution
Having the inner problem in regular perturbation form, we expandY(X)asY ∼ Y0+εY1+ε2Y2+ · · · and substitute into (7.26) to give the system of equations
O(ε0): Y0 +Y0=0, O(ε1): Y1 +Y1=cos(x∗), O(ε2): Y2 +Y2= −sin(x∗)X, . . . .
(7.27)
TheO(1)equation yields the leading order inner solution,
Y0(X)=D+Ce−X, (7.28)
with the higher order problems producing smaller corrections to this result. The constants of integration C,D must be determined by boundary conditions or by matching with the outer solution, but this, in turn, depends on the location ofx∗.
Consider the forms of the inner domain in terms ofX =(x−x∗)/εfor different possible values ofx∗
(i) Left boundary (x∗=0):x≥0 ⇒ 0≤X <o(1/ε) (ii) Interior : 0<x∗< π ⇒ −o(1/ε) <X <o(1/ε) (iii)Right boundary (x∗=π):x≤π ⇒ −o(1/ε) <X ≤0.
These options correspond to three possible forms of the composite solution (see Fig.7.3): (a) a left boundary layer satisfying y(0) = 2 matching with an outer solution which hasA=1 in order to satisfy the right boundary condition, (b) a nar- row interior transition region connecting two outer solutions, with AL = 2 and AR = −1, and (c) a right boundary layer satisfying y(π) = −1, with an outer solution satisfyingy(0)=2.
The location of the boundary layer will be determined by the structure of (7.28) and its limiting behaviour. The exponential terme−Xin (7.28) diverges ifXis allowed to become large and negative. Suchexponentially divergingtermscannotsatisfy the
(a) (b) (c)
Fig. 7.3 Three hypothetical sketches of the conjectured inner/outer solutions for (7.23a,7.23b) with a boundary layeraat the (Left),bin the (Interior),cat the (Right)edgeof the domain
asymptotic matching condition (7.13) (withX → −∞being the appropriate form of the ‘outer limit’ process) and areun-matchable.
Consequently options (ii) and (iii) are not feasible, and we conclude that the boundary layer must be at x∗ = 0 with the left boundary condition from (7.23b) being relevant, namelyY(0)=2. Applying this condition reduces (7.28) to
Y0(X)=2+C(e−X−1), (7.29) whereCremains to be determined.
7.3.4 Asymptotic Matching
Having identified the position of the boundary layer asx∗=0, we have the leading order inner solution (7.29), valid on 0 ≤ x < O(ε), and the outer solution (7.24), valid on 0<x≤π. Since the right boundary lies in the outer domain, that boundary condition determines A= −1 in (7.24), leavingCfrom (7.29) as the last remaining unknown.
Applying the matching condition (7.13) forY0(X→ ∞)andy0(x→0)yields
Xlim→∞2+C(e−X−1)=2−C= lim
x→0sin(x)−1= −1 (7.30) and henceC =3.
7.3.5 The Composite Solution
The overlap shared in common by the leading order inner and outer solutions above is−1. Therefore we can form the boundary layer correction as
YBLC=Y0−(−1)=3e−X.
Finally, adding this correction to the outer solution yields the leading order composite solution on 0≤x ≤π(see Fig.7.4),
ycomp= −1+sin(x)+3e−x/ε. (7.31) This is in agreement with the exact solution (valid for allε >0),
y= −1+sin(x)−ε[1+cos(x)]
1+ε2 +
3+ 2ε 1+ε2
e−x/ε+e.s.t.
158 7 Boundary Layer Theory
Fig. 7.4 A plot ofycomp(x) (7.31) for a sequence of ε→0. The boundary layer becomes narrower asε decreases