Untangling Complex Systems, page 66
the density of the fluid significantly. In particular, the density declines as the temperature rises. Now,
let us consider a small parcel of the fluid at the bottom, having volume V and density d . Due to the 0
random fluctuations, the parcel of the fluid may move infinitesimally upwards. If this event occurs,
AC
C
AC
C
AC
C
Heating
FIGURE 10.10 Scheme of two adjacent roll-shaped cells in the convection of a fluid heated from the
below (on the left). On the right, frontal view of an array of adjacent cells rotating clock-wise (C) and anti-
clockwise (AC).
3 Feigenbaum demonstrated the universality of δ by using the concept of renormalization borrowed from statistical physics.
If you are interested to his proof, read Feigenbaum (1979).
4 This paragraph presents Natural Convection. Natural Convection is different from Forced Convection. In Forced Convection, the fluid of motion is promoted by an external force generated, for instance, by a pump, a fan or other devices.
5 The model system studied by Lord Rayleigh did not coincide tightly with the features of the system investigated by Bénard.
Read Box 1 of this chapter to know more about the theory explaining the experimental results obtained by Bénard.
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Untangling Complex Systems
the parcel is now surrounded by a cooler and denser ( d) portion of the fluid (with d > d ). As a result, 0
the parcel feels the buoyancy force ( F ) due to the gravity force (being g its acceleration):
b
F
(
0 )
b = − d − d
gV [10.18]
This buoyancy force pushes the parcel farther upwards. In contrast to Fb, there is the drag force ( FV) dependent on the fluid shear viscosity ( η):
FV = −η vL [10.19]
In [10.19], v represents the velocity of the parcel and L is its radius. When the drag force is larger or equal to the buoyancy force, there can be no motion. On the other hand, when the buoyancy force is
stronger than the drag one, the ordered convective motions can start. However, there exists a second
phenomenon that opposes the appearance of convection. And this is the heat conduction. The mol-
ecules in the warm parcel, in contact with a cooler surrounding, dissipate their high kinetic energy
by impact with the slower encompassing molecules. Heat flows spontaneously from the displaced
warm parcel. The time needed for the parcel to reach thermal equilibrium is inversely proportional
to the thermal diffusivity of the fluid. If this time is comparable to the time required for the parcel
to move a distance such as its diameter, the convective flow cannot be formed.
Of course, as there is the chance that a warm parcel of fluid moves upwards by fluctuations and
enters a colder region, so there exists the chance that a cool parcel moves downwards and finds a
less dense portion of the fluid. The buoyancy force will push the cool parcel farther downwards.
But the drag force will contrast its movement. Moreover, heat will flow by conduction from the sur-
rounding environment into the cooler parcel, and it will tend to annihilate the thermal gradient (see
Figure 10.11). The fluid self-organizes, and the convective rolls emerge when the buoyancy forces overcome dissipative processes due to the viscosity and heat conduction.
There exists a dimensionless parameter that allows predicting if an extended layer of fluid, hav-
ing height h and under a vertical thermal gradient (Δ T), self-organizes producing convective rolls.
This parameter is the Rayleigh number ( Ra), which is a ratio between two counteracting forces. The
first is the buoyancy force [10.18] that, expressed as a function of the isobaric thermal expansion
coefficient (β = (1/ V )(∂ V/∂ T ) = −(1/ d)(∂ d /∂ T )) and per unit of volume, becomes F
b = β gd ∆
0 T [10.20]
V
Tc
Th
FIGURE 10.11 Forces acting on two parcels of fluid. One warm parcel moves upwards, and one cool parcel
moves downwards. The formation of convective rolls depends on the action of the buoyancy force (continuous
black arrow) against the drag force (dotted grey arrows) and the heat conduction (dashed black arrows). The
dimensions of the two parcels are exaggerated with respect to the linear dimensions of the fluid.
The Emergence of Chaos in Time
329
The second is the dissipative force ( F ) estimated per unit of volume and generated by the joined
d
action of the drag force and heat conduction:
Fd
αµ d
=
0 [10.21]
V
h 3
In [10.21], α = ( k / CPd 0 ) is the thermal diffusivity that depends on the thermal conductivity ( k), the specific heat capacity at constant pressure ( C ), and the average density of the fluid ( d ); µ = η
( / d 0)
P
0
is the kinematic viscosity. It derives that
β g T
∆ h 3
Ra =
[10.22]6
µα
The onset of convection is for Ra c∼ 1708, independent of the fluid under consideration, and for
two flat rigid boundaries at the top and bottom layer surfaces (Chandrasekhar 1961). As long as
Ra is slightly above Ra , the cylindrical rolls of the convective pattern remain straight, and their
c
motion is periodic. At any fixed point in space, the temperature is constant. If we increase the
thermal gradient, another instability sets in when Ra reaches a second critical value. A wave
starts to propagate back and forth along the longest axis of the cylindrical rolls, causing the
temperature to oscillate at each point. Further heating determines the emergence of other insta-
bilities. In fact, waves with other frequencies start to propagate back and forth along each roll,
and temperature oscillates with more than one frequency in each point of the fluid. When large
Ra values are reached, the local temperatures start to change chaotically, with an infinite number
of frequencies. Finally, for even larger Ra values, the motion of the fluid turns from a laminar to
turbulent one (see Figure 10.12).
The first elegant experiments demonstrating the transition of natural convection from periodic
to chaotic states were performed by the French physicist Albert Libchaber in the 1970s and 1980s
6 The Rayleigh number can also be presented as the product of two other dimensionless parameters: the Prandtl number ( Pr), and Grashof number ( Gr).
Ra = ( Pr )( Gr )
The Prandtl number is the ratio between the time scale for the diffusion of heat ( h 2/ α) and the time scale for the diffusion of momentum ( h 2/ μ):
µ
Pr =
α
When Pr is small, it means that heat diffuses more quickly than momentum. When Pr is large, the opposite is true. The Grashof number ( Gr) is the ratio between the buoyancy force [10.18] and the drag force [10.19]:
( d − d 0 ) gV
Gr =
η vL
If we equate the drag and momentum forces η vL = d 2 2, we obtain that v = (η / Ld . Introducing this definition of 0 )
0 v L
velocity v and the definition of thermal expansion coefficient β in the equation of Grashof number, we obtain gβ T
∆ h 3
Gr =
2
η
Of course, when Gr is small, the drag force dominates over the buoyancy force, and the motion of the fluid is laminar.
On the other hand, when Gr is large, it is the buoyancy force to overwhelm the drag force, and the motion of the fluid is turbulent. In the case of Forced Convection, the Reynold number (see Box 4 of Chapter 9) is analogous to the Grashof number.
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Untangling Complex Systems
(a)
(b)
FIGURE 10.12 Examples of laminar (a) and turbulent (b) convection. The arrows represent the velocity
vectors.
when he was working at the École Normale Supérieure (EMS) of Paris. At first, he used liquid
helium at 3 K placed in a very tiny cell, having dimensions of the order of mm, surrounded by
a chamber at high vacuum (Libchaber and Maurer 1978; Maurer and Libchaber 1979). The low
temperature is guaranteed by thermal determinations of very high resolution and high accuracy. All
the thermal measures were carried out by exploiting microbolometers engraved in the cell.
TRY EXERCISE 10.8
Then, Libchaber used liquid mercury that assured the better stability of the convective rolls in the
presence of a stationary magnetic field (Libchaber et al. 1982). In both cases, by a constant increase
of the thermal gradient, Libchaber and his coworkers measured the value of Ra at the period-
doubling bifurcations. At Ra , the temperature at each point was constant over time. At Ra
,
c
0 = 2 Rac
the temperature started to oscillate. For a farther increase of the thermal gradient, there was another
bifurcation, and T began to oscillate with two frequencies. For larger Δ T, the frequencies became
four, then, eight, sixteen, … and finally infinite. The trend became chaotic. By using the critical Ra
values estimated at each bifurcation, Libchaber and coworkers could confirm the universality of the
Feigenbaum number [10.17]. Although convection is not directly related with the unimodal maps, it
shows the same feature of them as far as the evolution to chaotic conditions is concerned.
TRY EXERCISE 10.9
BOX 10.1 THE MARANGONI-BÉNARD CONVECTION
The system investigated experimentally by Bénard had one feature different from the model
system studied by Rayleigh. The difference was that the layer of the fluid was not confined
between two rigid boundaries but instead was open to the air at its upper surface. This
change was relevant for the shape of the pattern of the convective flow. In the thin layer
of the fluid heated from below and maintained open to the air above, Bénard observed
the formation of a polygonal tessellation of the fluid surface. When the pattern was fully
developed, it appeared as an almost perfect array of hexagons, as it were a honeycomb.
The center of each hexagon is a region of warm upwelling fluid that reaches the surface;
it is dragged to the perimeter of the polygon, where it sinks because cooled down. There
was also a slight depression of the upper free surface of the fluid at each cell center. When
the fluid is a thin layer and is maintained open to the air, the surface tension of the fluid
( Continued)
The Emergence of Chaos in Time
331
BOX 10.1 (Continued) THE MARANGONI-BÉNARD CONVECTION
is the primary factor responsible for the convection pattern and not the buoyancy. Surface
tension ( γ) is the force per unit of length whose effect is to minimize the surface area of
fluid. It is responsible for the spherical shapes of liquid drops. Surface tension plays like the
propulsive force in the Bénard convection because it varies with temperature: in particular,
it decreases when the fluid is warmed. Therefore, when the temperature of the surface of
the liquid is not uniform, local gradients of surface tension are generated. The gradients
give rise to flows on the surface. These flows are communicated to the bulk of the fluid as a
result of its shear viscosity (see Figure in this B10.1).
Surface of the fluid
Local cold spot
Local hot spot
Warm plate
FIGURE B10.1 Convection induced by the local surface tension gradients. The surface tension in
local cold spots is larger than in local hot spots.
The mechanism for the onset of convection is similar to that proposed by Rayleigh when the
buoyancy is the driving force. However, in this case, the driving force is the surface tension.
The surface tension force ( F ) per unit volume of fluid is
st
Fst
dγ T
∆
T
∆
=
γ
[B1.1]
V
=
2
2
dT h
T h
being h the thickness of the layer, and Δ T the thermal gradient between the bottom and the
top of the fluid. Friction and thermal diffusion oppose the action of the surface tension.
Both effects combine to yield a dissipative force ( F ) per unit of volume that is equal to
d
F
3
(
)
(
)
d V = αµ d 0 h (remember equation [10.21]). The ratio between Fst V and Fd V is the Marangoni number (named after the Italian physicist lived between the nineteenth and the
twentieth centuries):
γ Th T
∆
Ma =
[B1.2]
αµ d 0
The Marangoni number is the control parameter for the convection driven by the surface ten-
sion. When Ma is above a critical value Ma , which depends on the boundary conditions, but
c
it is usually ∼80 (Bragard and Velarde 1997), the surface tension dominates and overcomes
the dissipative effects of viscous drag and thermal diffusion.
TRY EXERCISE 10.10
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Untangling Complex Systems
10.6 THE ENTROPY PRODUCTION IN THE NONLINEAR
REGIME: THE CASE OF CONVECTION
In the previous paragraph, we learned that when a fluid is heated from below but the thermal gradi-
ent Δ T is small, heat is transferred through conduction. At larger Δ T, above a critical value Δ T , conc
vection becomes the fastest mechanism of heat transfer. There exists a dimensionless parameter that
measures the ratio between the rates of heat transfer by convection and conduction. This parameter
is the Nusselt number ( Nu):
convection heat transfer rate
Nu =
[10.23]
conduction heat transfer rate
When the Rayleigh number is less than its first critical value, Ra , Nu
c
< 1, and conduction is more
effective in transferring heat. When Ra ∼ Ra , Nu
, Nu
c
∼ 1. When Ra > Rac
> 1 and convection is
more efficient.
In Chapter 3, we knew that when conduction is the most effective mechanism of heat transfer
because Δ T is small and we are not very far-from-equilibrium, the system evolves spontaneously to
a stable stationary state where the Entropy Production P* is minimized. What happens when Δ T is
large, and convection becomes the fastest mechanism of heat transfer? In other words, how does P*
change when we are very far-from-equilibrium?
In the case of heat conduction in a system of volume V and surface area A, the Entropy Production is
2
1
1
*
diS
P =
= Jq∇ dV = Lq ∇
∫
∫ dV [10.24]
dt
T
T
V
V
because we are in linear regime and J
1 )
q = Lq∇ ( T = − k∇ T .
How does the Entropy Production P* change over time? We consider the time derivative of
equation [10.24] to answer this question. Assuming L constant over the temperature range of the
q
system, we obtain:
dP*
1 ∂
1
1
= 2 Lq∇
∇
dV 2 J
dt
∫
∫
T t
∂
=
∇ ∂
dV [10.25]
T
q
t
∂ T
V
V
Equation [10.25] transforms in [10.26], after integration by parts:7
dP*
1
1
=
∂
2∫ J
q n dA −
∂
2∫ J
∇ qdV [10.26]
dt
t
∂ T
t
∂ T
A
V
In [10.26], n is the unit vector perpendicular to the surface A and pointed outwards. The first integral of equation [10.26] vanishes when the thermal gradient is maintained constant at the boundary of
7 Integration by parts is a method to solve integrals when the integrand is a product of two functions, for example, x ⋅ y.
The method derives from the formula for the derivative of a product: d ( x ⋅ y) = ydx + xdy. After integrating both terms of the equation, we obtain ∫ d ( x ⋅ y) = x ⋅ y = ∫ d
