Basics of Fluid Mechanics, 2014a

9.3. NUSSELT’S TECHNIQUE 301
boundary conditions are
U x (y =0)
U 0x
= f(x)
μ ∂U x
∂x (y = h) =τ 0 g(x)
Now it is very convenient to define several new variables:
(9.22)
U =
where :
U x(x)
U 0x
(9.23)
x =
x h
y
= y h
The length h is chosen as the characteristic length since no other length is provided.
It can be noticed that because the units consistency, the characteristic length can be
used for “normalization” (see Example 9.11). Using these definitions the boundary and
initial conditions becomes
U x(y=0)
U 0x
= f ′ (x)
hμ
U 0x
∂U x
∂x (y =1)=τ 0 g ′ (x)
(9.24)
It commonly suggested to arrange the second part of equation (9.24) as
∂U x
∂x (y =1)=τ 0 U 0x
hμ g′ (x) (9.25)
Where new dimensionless parameter, the shear stress number is defined as
τ 0 = τ 0 U 0x
hμ
(9.26)
With the new definition equation (9.25) transformed into
∂U x
∂x (y =1)=τ 0 g ′ (x) (9.27)
Example 9.11:
Non–dimensionalize the following boundary condition. What are the units of the coefficient
in front of the variables, x. What are relationship of the typical velocity, U 0 to
U max ?
(
U x (y = h) =U 0 ax 2 + b exp(x) )
(9.XI.a)