Problem 5.10
Problem
5.10
[Difficulty: 2]
5.10
Given:
Approximate profile for a laminar boundary layer:
U y
u
δ  c x (c is constant)
δ
Find:
(a) Show that the simplest form of v is
v
u y

4 x
(b) Evaluate maximum value of v/u where δ = 5 mm and x = 0.5 m
Solution:
We will check this flow field using the continuity equation
Governing
Equations:

u    v    w    0 (Continuity equation)
x
y
z
t
Assumptions:
(1) Incompressible flow (ρ is constant)
(2) Two dimensional flow (velocity is not a function of z)
u v

0
x y
Based on the two assumptions listed above, the continuity equation reduces to:
u u d
Uy 1 
Uy
v
Uy
u

  2  cx 2  


3 Therefore from continuity:
3
x  dx
2

y
x
2cx 2
2cx 2
1
The partial of u with respect to x is:
Integrating this expression will yield the y-component of velocity:

v




U y
3
2  c x
2
U y
2
Now due to the no-slip condition at the wall (y = 0) we get f(x) = 0. Thus: v 
v
δ
The maximum value of v/U is where y = δ: v ratmax 

u
4 x
v ratmax 
dy  f ( x ) 
3
4  c x
U y
3
4  c x
U y


y
1 4 x
2
5  10
2
c x

 f ( x)
2
u y
4 x
(Q.E.D.)
v
u y

4 x
2
3
m
4  0.5 m
v ratmax  0.0025