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Chapter 3
Laminar flows
3.1
Assumptions
Flow equations discussed in Chapter 2 provide analytical solutions only in some
special cases. In this chapter we shall consider the equations for incompressible
flow: (2.4), (2.22) and (2.74), assuming that all the coefficients are constant:
u
i
i
0
(3.1)
˙
u
i
u
k
u
i
k
ν
u
i
kk
ρ
1
p
i
g
i
(3.2)
ρ
c
p
˙
T
u
i
T
i
kT
ii
u
j
i
τ
ij
(3.3)
Definition 3.1.1
Laminar flow
Let’s make an assumption of
laminar flow
which states that the time scale
of changes in the flow can not be lower than the timescale of the motion of the
boundary or any external sources. In other words, if there is any repeatability in
the motion of the boundary or in the external forces then the frequencies associ
ated with either factors can not be lower than the frequencies of the flow motion.
It means that neither the boundaries not external forces can induce any addi
tional frequencies in the flow. In the limit case of nonmoving boundaries and
nonchanging forces the flow should not depend on time, which means that all
the dependent variables should become functions of spatial coordinates only.
The conditions of laminar flow defined by (3.1.1) are realized when the con
tribution of the nonlinear term in the momentum equation (3.2) is small, or when
51
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CHAPTER3.LAMINARFLOWS
Figure 3.1: Flow between parallel plates: the lower plate is at rest, the upper plate is
moving with velocity
U
.
the contribution of the viscous term
ν
u
i
kk
dominates.
This is usually the case
when nondimensional
Reynolds number
(2.109):
(3.4)
R
e
LU
ν
is small. In practical situations the ”smallness” of
R
e
corresponds to
R
e
100
.
3.2
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 Spring '11
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