Basics of Fluid Mechanics, 2014a

234 CHAPTER 8. DIFFERENTIAL ANALYSIS
H 0
T 0
T 0 T(x)
T∞
x
Fig. -8.4. Mass flow in coating process
for example 8.2.
The temperature and mass transfer taking
place which reduces (or increases) the
thickness of the film. For this example, assume
that no mass transfer occurs or can
be neglected and the main mechanism is
heat transfer. Assume that the film temperature
is only a function of the distance
from the extraction point. Calculate the
film velocity field if the density is a function
of the temperature. The relationship between the density and the temperature is
linear as
( )
ρ − ρ ∞ T − T∞
= α
(8.II.a)
ρ 0 − ρ ∞ T 0 − T ∞
State your assumptions.
Solution
This problem is somewhat similar to Example 8.1 6 , however it can be considered as
steady state. At any point the governing equation in coordinate system that moving
with the belt is
∂ (ρU x )
∂x
+ ∂ (ρU y)
∂y
=0 (8.II.b)
At first, it can be assumed that the material moves with the belt in the x direction
in the same velocity. This assumption is consistent with the first solution (no stability
issues). If the frame of reference was moving with the belt then there is only velocity
component in the y direction 7 . Hence equation (8.II.b) can be written as
∂ρ
U x
∂x = −∂ (ρU y)
(8.II.c)
∂y
Where U x is the belt velocity.
See the resembles to equation (8.I.b). The solution is similar to the previous
Example 8.1 for a general function T = F (x).
∂ρ
∂x = α ∂F(x)
U x ∂x (ρ 0 − ρ ∞ ) (8.II.d)
Substituting this relationship in equation (8.II.d) into the governing equation results in
∂U y ρ
∂y
= α U x
∂F(x)
∂x (ρ 0 − ρ ∞ ) (8.II.e)
6 The presentation of one dimension time dependent problem to two dimensions problems can be
traced to heat and mass transfer problems. One of the early pioneers who suggest this idea is Higbie
which Higbie’s equation named after him. Higbie’s idea which was rejected by the scientific establishment.
He spend the rest of his life to proof it and ending in a suicide. On personal note, this author
Master thesis is extension Higbie’s equation.
7 In reality this assumption is correct only in a certain range. However, the discussion about this
point is beyond the scope of this section.
x