Chapter 11 Boundary layer theory

14 CHAPTER 11. BOUNDARY LAYER THEORY
The velocity gradients in the x momentum equation can now be expressed
in terms of f(η),
∂u x
∂x
∂u x
∂y
∂ 2 u x
∂ 2 y
= dU(x) df
dx dη − U(x)y dδ(x) d 2 f
(11.73)
δ(x) 2 dx dη 2
= U(x)
δ(x)
d 2 f
dη 2 (11.74)
= U(x)
δ(x) 2 d 3 f
dη 3 (11.75)
These are inserted into the x momentum equation 11.66, and the result is
divided throughout by (νU(x)/δ(x) 2 ), to obtain,
d 3 ( ) 2
f
dη − δ2 dU(x) df
+ U(x)δ(x) η dδ(x) df d 2 f
3 ν dx dη ν dx dη dη 2
(
+ δ2 df d(δ(x)U(x))
f(η) + U(x)y )
dδ(x) df
ν dη dx δ(x) 2 dx dη
= 0 (11.76)
Problems:
1. Consider the uniform flow of a fluid past a flat plate of infinite extent
in the x 1 −x 3 plane, with the edge of the plate at x 1 = 0. The Reynolds
number based on the fluid velocity and the length of the plate is large.
At a large distance from the plate, the fluid has a uniform velocity U 1
in the x 1 direction and U 3 in the x 3 direction. All velocities are independent
of the x 3 direction, and the velocities U 1 and U 3 are uniform
and independent of position in the outer flow,
(a) Write down the equations of motion in the x 1 , x 2 and x 3 directions.
If the thickness of the boundary layer δ is small compared to
the length of the plate L, find the leading order terms in the
conservation equations.
(b) What is the similarity form of the equation for the momentum
equations for the velocity u 1 Make use of the similarity forms for
boundary layer flows derived in class. Do not try to solve the
equation.
(c) Find the solution for the velocity u 3 in terms of the solution for
u 1 .