Extra Credit Question

If you successfully solve this problem, and if I can successfully follow your logic,

I'll give you 5 extra credit points on exam 1. However, you're not allowed to

discuss the solution with anyone but Dr. Tilton.

Suppose that the Reynolds number is not small. Instead, suppose it is of order unity, i.e.

Re~ 1. If € is still small, € < 1, we can still find a simple solution to equations (6)-(8) using

what is called a "perturbation expansion." It goes like this. Assume the velocity and pressure

fields in equations (6)-(8) can be written as

u= u₁(x, y) + €u₁(x, y) + €²³u₂(x, y) + ...

(11)

(12)

v = vo(x, y) + €v₁(x, y) + €²v₂(x, y) + ...

p = po(x, y) + ep₁(x, y) + €²p₂(x, y) + ...

(13)

The idea here is that in the limit € 0, you recover the leading order solution you got in

problem 3. Now we can solve for Un, Un, and Pn, n = 1,2,... by substituting (11)-(13) into

equations (6)-(8) and gathering the terms in each equation so that they are written like

€º .….…...) + €²(....) + €²(.....) + .... = 0

(14)

For example, equation (6) can be written up to second-order in € as:

duo Əvo

+

𐐀х Əy

𐐀х

To satisfy this equation, you then argue that

+E¹

ди

0,

(31)+

ду

+

+€²

3

=

(16)

(2+x)*(2-)(-x)。

Repeating this procedure with all three equations (6)-(8), the problem will decouple in a

manner that is relatively straightforward to solve.

(Əu₂ Əv¹₂)

+

𐐀х dy

0,

= 0.

(15)

0.

(a) Use the procedure above to derive the partial differential equations and boundary conditions

that govern the zero-order problem (uo. Vo: Po), first-order problem (₁, ₁, P₁), and second-

order problem (u2, 02, P2). Describe the order in which you would go about solving these

problems. Also describe how nonlinear effects appear at the different orders.

(b) Solve the first-order problem for u₁ and v₁.