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two-dimensional cylinders that rotate about their horizontal axes. The hot mate-
rial rises along one side of the cylinder, and the cold material sinks along the
other side (Fig. 8.13(a)). As heating proceeds, these two-dimensional cylinders
become unstable, and a second set of cylindrical cells develops perpendicular
to the first set (Fig. 8.13(b)). This rectangular planform is called
bimodal
.As
the heating continues, this bimodal pattern changes into a hexagonal and then a
spoke pattern. Figure 8.13(c) shows hexagonal convection cells in plan view, with
hot material rising in the centres and cold material descending around the edges.
With heating, the fluid convects more and more vigorously, with the upgoing and
downgoing limbs of a cell confined increasingly to the centre and edges of the
cell, respectively. Finally, with extreme heating, the regular cell pattern breaks
up, and hot material rises at random; the flow is then irregular.
8.2.2 Equations governing thermal convection
The derivation and discussion of the full differential equations governing the
flow of a heated viscous fluid are beyond the scope of this topic, but it is of
value to look at the differential equations governing the simplified case of two-
dimensional thermal convection in an incompressible Newtonian viscous fluid.
The
Boussinesq approximation
to the most general convection equations is often
used to simplify numerical calculations. In that approximation the fluid is incom-
pressible (Eq. (8.26)) and the only result of a change in density considered is
buoyancy (Eq. (8.30)).
The general equation of conservation of fluid (i.e., there are no sources or
sinks of fluid, its volume is constant) is
∂
u
x
∂
x
+
∂
u
z
∂
z
=
0
(8.26)
where
u
(
u
x
,
u
z
)isthe velocity at which the fluid is flowing.
The two-dimensional heat equation in a moving medium (Eq. (7.19) with no
internal heat generation,
A
=
=
0) is
∂
∂
z
2
∂
T
∂
t
=
k
ρ
c
P
2
T
∂
x
2
+
∂
2
T
−
u
x
∂
T
∂
x
−
u
z
∂
T
(8.27)
∂
z
The horizontal equation of motion is
∂
∂
P
∂
x
=
η
2
u
x
∂
x
2
+
∂
2
u
x
∂
z
2
(8.28)
and the corresponding vertical equation of motion is
∂
P
∂
z
=
η
∂
∂
x
2
2
u
z
∂
x
2
2
u
z
+
∂
−
g
ρ
(8.29)
where
P
is the pressure generated by the fluid flow,
η
is the dynamic viscosity,
g
is
ρ
is the density disturbance. The convective
flow of the fluid is maintained by the buoyancy forces resulting from differences
the acceleration due to gravity and
