Chapter 11 Boundary layer theory

11.2. FALKNER-SKAN SOLUTIONS 13
location p(x, y) = p ∞ (x), and the pressure gradient in the x direction is,
∂p
∂x = ∂p ∞ dU
= −ρU (11.65)
∂x dx
When this is inserted into the momentum conservation equation 11.19, we
obtain, (
)
∂u x
ρ u x
∂x + u ∂u x
y = ρU ∂U
∂y ∂x + u x
µ∂2 (11.66)
∂y 2
The boundary conditions at the surface of are given by 11.3 and 11.4,
u x = 0 at y = 0 (11.67)
u y = 0 at y = 0 (11.68)
while the boundary condition in the limit y → ∞ is given by equation 11.5.
u x = U(x) for y → ∞ (11.69)
The fluid velocity and pressure fields are governed by equations 11.9,
11.20 and 11.66. We assume that the solution can be expressed in terms of a
similarity variable η = (y/δ(x)), where the boundary layer thickness δ(x) is a
function of x. Note that δ(x) ≪ L in the high Reynolds number limit, where
L is the characteristic length scale of the flow. The form for the velocity u x
is chosen to be,
u x = U(x) df
(11.70)
dη
Note that this is of the same form as that for the flow past a flat plate in
equation 11.27, and for the stagnation point flow in equation 11.49 where
U(x) = ˙γx. The stream function can be obtained by integrating equation
11.70 with respect to y,
ψ =
∫ y
The velocity u y in the y direction is,
u y
= − ∂ψ
∂x
0
dy u x
∫ η
= δ(x)U(x) dη ′ df
0 dη
= δ(x)U(x)f(η) (11.71)
= − d(δ(x)U(x)) f(η) + U(x)y dδ(x)
dx δ(x) 2 dx
df
dη
(11.72)