where ty B is the boundary layer solution. The scale width of the western boundary layer is

Introducing a stretched coordinate n = x/&m (4.59)

the vorticity equation is reduced to fĂ,nnnn - tyB,n = 0 (4.60)

This equation is subject to the following boundary conditions. First, the boundary layer solution should be finite and it should match the interior solution at "infinity" (the outer edge of the boundary layer)

tyB ^ tyi, n ^^ (4.61) Second, the western boundary is a streamline, tyB = 0, at n = 0 (4.62) In addition, two types of boundary condition may apply:

The general solution for Eqn. (4.60) is

tyB = C1 + C2e + C3e 2 cos I â n I + C4e 2 sin I â n I

a) Applying the condition at infinity: ci = p, C2 = 0.

b) Applying the boundary condition of p = 0 at the wall: C3 = -pj.

c) If the no-slip boundary condition applies, pB,n = 0, n = 0; thus, C4 = C3/V3.

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