In the case of marginal stability, there corresponds to each value of k a critical value of the control parameter, say the temperature difference ∆T in B´enard’sproblem, a characteristic
Trang 1138 6 Instabilities and Pattern Formation
generally a complex and k -dependent quantity The k dependence of the
growth rate describes the spatial symmetry of the system; in rotationally
invariant systems, the σ n’s will only depend on the modulus|k|, whereas in
anisotropic systems like nematic liquid crystals there is an angle between the
direction of anisotropy and k
The requirement that the field equations have non-trivial solutions leads
to an eigenvalue problem for the σ n’s The stability problem is completely
determined by the sign of the real part of the σ n’s:
• If one single Re σ n > 0, the system is unstable.
• If Re σ n < 0 for all the values of n, the system is stable.
• If Re σ n = 0, the stability is marginal or neutral.
In the case of marginal stability, there corresponds to each value of k a critical
value of the control parameter, say the temperature difference ∆T in B´enard’sproblem, a characteristic velocity in flows through a pipe or the angular
velocity in Taylor’s problem, for which Re σ n = 0 All these critical values
define a curve of marginal stability, say ∆T vs k , whose minimum (∆Tc, kc)determines the critical threshold of instability In stability problems, it isconvenient to work with non-dimensional control parameters like the Rayleigh
number Ra in B´ enard’s instability, the Reynolds number Re for the transition from laminar to turbulent flows, or the Taylor number Ta in presence of
rotation; therefore, the marginal curves and the corresponding critical valuesare generally expressed in terms of these non-dimensional quantities
In several problems, it is postulated that Re σ = 0 implies Im σ = 0, this conjecture is called the principle of exchange of stability, which has been
demonstrated to be satisfied in the case of self-adjoint problems; in this case
a stationary state is attained after the onset of the instability If Re σ = 0 but Im σ = 0, the onset of instability is initiated by oscillatory perturbations and one speaks of overstability or Hopf bifurcation This kind of instability
is observed, for instance, in rotating fluids or fluid layers with a deformable
interface The condition Re σ n > 0 for at least one value of n is a sufficient
condition of instability; on the contrary, even when all the eigenvalues are such that Re σ n < 0, one cannot conclude in favour of stability as one can-
not exclude the possibility that the system is unstable with respect to finiteamplitude disturbances It is therefore worth to stress that a linear stability
analysis predicts only sufficient conditions of instability.
6.2 Non-Linear Approaches
As soon as the amplitude of the disturbance is finite, the linear approach isnot appropriate and must be replaced by non-linear theories Among themone may distinguish the “local” and the “global” ones In the latter, the de-tails of the motion and the geometry of the flow are omitted, instead attention
Trang 2is focused on the behaviour of global quantities, generally chosen as a positivedefinite functional A typical example is Lyapounov’s function; according to
Lyapounov’s theory, the system is stable if there exists a functional Z ing Z > 0 and dZ/dt ≤ 0 In classical mechanics, an example of Lyapounov’s
satisfy-function is the Hamiltonian of conservative systems Glansdorff and Prigogine
(1971) showed that the second variation of entropy δ2S provides an
exam-ple of Lyapounov’s functional in non-equilibrium thermodynamics The mainproblems with Lyapounov’s theory are:
1 The difficulty to assign a physical meaning to the Lyapounov’s functional
2 The fact that a given situation can be described by different functionals
3 That in practice, it yields only sufficient conditions of stability
We do no longer discuss this approach and invite the interested reader toconsult specialized works (e.g Movchan 1959; Pritchard 1968; Glansdorffand Prigogine 1971) Here we prefer to concentrate on the more standard
“local” methods where it is assumed that the perturbation acts at any point
in space and at each instant of time We have seen that the solution of the
linearized problem takes the form exp(σ n t) and that instability occurs when
the growth rate becomes positive, or equivalently stated, when the
dimension-less control parameter R exceeds its critical value Rc For values of R > Rc,the hypothesis of small amplitudes is no longer valid as non-linear terms be-come important and will modify the exponential growth of the disturbances.Another reason for taking non-linear terms into account is that the linear ap-proach predicts that a whole spectrum of horizontal wave numbers becomeunstable This is in contradiction with experimental observations, which show
a tendency towards simple cellular patterns indicating that only one singlewave number, or a small band of wave numbers, is unstable
Non-linear methods are therefore justified to interpret the mechanisms curring above the critical threshold The problem that is set up is a non-lineareigenvalue problem Unfortunately, no general method for solving non-lineardifferential equations in closed form has been presented and this has moti-vated the development of perturbation techniques A widely used approach
oc-is the so-called amplitude method initiated by Landau (1965) and developed
by Segel (1966), Stuart (1958), Swift and Hohenberg (1977), and many ers It is essentially assumed that the non-linear disturbances have the sameform as the solution of the linear problem with an unknown time-dependentamplitude Explicitly, the solutions will be expressed in terms of the eigen-
oth-vectors W (z) of the linear problem in the form
a (x , z, t) = A(t) exp(ik · x)W (z), (6.7)
where A(t) denotes an unknown amplitude, generally a complex quantity.
In the linear approximation, A(t) is proportional to exp(σt) and obeys the
linear differential equation
dA(t)
Trang 3140 6 Instabilities and Pattern Formation
whereas in the non-linear regime, projection of (6.8) on the space of the W ’s
leads to a coupled system of non-linear ordinary differential equations for theamplitudes
dA(t)
where N (A, A) designates the non-linear contributions Practically, the
equa-tions are truncated at the second or third order A simple example is provided
by the following Landau relation (Drazin and Reid 1981)
= AA ∗ with A ∗ the complex conjugate of A In (6.10), one has imposed
the constraint A = −A reflecting the inversion symmetry of the field variables
like the velocity and temperature fields This invariance property is destroyed
and additional quadratic terms in A2 will be present when some materialparameters like viscosity or surface tension are temperature dependent Totake into account some spatial effects like the presence of lateral boundaries,
it may be necessary to complete the above relation (6.9) by spatial terms in
stationary solution|As| given by
which is independent of the initial value A0 This is a supercritical
sta-bility, the reference flow becomes linearly unstable at the critical point
Re σ = 0, or equivalently at R = Rc, and bifurcates on a new steady
stable branch with an amplitude tending to As When the bifurcation issupercritical, the transition between the successive solutions is continu-
ous and is called a pitchfork bifurcation as exhibited by Fig 6.1.
It is instructive to develop Re σ around the critical point in terms of the
wave number k and the dimensionless characteristic number R so that
Trang 4Fig 6.1 Supercritical pitchfork bifurcation: the solution A = 0 is linearly stable for
R < Rc but linearly unstable forR > Rc, the branching of the curve at the criticalpointR = Rcis called a bifurcation Unstable states are represented by dashed lines
and stable states are represented by solid lines
Re σ = α(R − Rc) + β(k − kc)· (k − kc) +· · · , (6.14)
where α is some positive constant When R < Rc, all perturbations
are stable with Re σ < 0; at R = Rc, the system is marginally stable
and when R increases above Rc, the system becomes linearly unstable.Combining (6.14) with (6.13) results in
As∼ (R − Rc)1/2 as R → Rc, (6.15)
indicating that the amplitude Asof the steady solution is proportional tothe square root of the distance from the critical point There is a stronganalogy with a phase transition of second order where the amplitude
Ac of the critical mode plays the role of the order parameter and theexponent 1/2 in (6.15) is the critical exponent
(2) Let us now examine the case l < 0 If Re σ > 0, both terms of
Lan-dau’s equation (6.11) are positive and|A| grows exponentially; it follows
from (6.12) that |A| is infinite after a finite time t = (2Re σ) −1ln[1− (Re σ)/(lA2)], however this situation never occurs in practice because inthis case it is necessary to include higher-order terms in |A|6
, |A|8
,
in Landau’s equation and generally no truncation is allowed A more
realistic situation corresponds to Re σ < 0; now, the two terms in the
right-hand side of (6.12) are of opposite sign Depending on whether
A0 is smaller or larger than |As| given by (6.12), we distinguish two different behaviours; for A0 < |As|, the solution given in (6.12) shows
that |A| ≈ exp[(2Re σ)t] and tends to zero as t → ∞; in contrast for A0 > |As|, the denominator of (6.12) becomes infinite after a time
t = (2Re σ) −1ln[1− Re σ/lA2] and|A| → ∞ (see Fig 6.2) In this case,
Trang 5142 6 Instabilities and Pattern Formation
Fig 6.2 Time dependence of the amplitude for two different initial values A0 in the case of a subcritical instability
the reference state is stable with respect to infinitesimally small bances but unstable for perturbations with amplitude greater than the
distur-critical value As, which appears as a threshold value This situation is
referred to as subcritical or metastable, by using the vocabulary of the
and they are represented in Fig 6.3 wherein the amplitude Asis sketched as
a function of the dimensionless number R.
For R < RG, the basic flow is globally stable which means that all
pertur-bations, even large, decay ultimately; for RG < R < Rc, the system admits
two stable steady solutions As= 0 and the branch GD whereas CG is stable At R = Rc, the system becomes unstable for small perturbations and
un-we are faced with two possibilities: either there is a continuous transition
towards the branch CF which is called a transcritical bifurcation,
character-ized by the intersection of two bifurcation curves, or there is an abrupt jump
to the stable curve DE, the basic solution “snaps” through the bifurcation
to some flow with a larger amplitude By still increasing R, the amplitude
will continue to grow until a new bifurcation point is met If, instead, the
Trang 6Fig 6.3 Subcritical instability: the system is stable for infinitesimally small
per-turbations but unstable for perper-turbations with amplitude larger than some critical
value Solid and dot lines refer to stable and unstable solutions, respectively
amplitude is gradually decreased, one moves back along the branch EDG up
to the point G where the system falls down on the basic state A = 0 tified by the point H The cycle CDGH is called a hysteresis process and is
iden-reminiscent of phase transitions of the first order
In the present survey, it was assumed that the amplitude equation wastruncated at order 3 In presence of strong non-linearity, i.e far from thelinear threshold, such an approximation is no longer justified and the intro-duction of higher-order terms is necessary, however this would result in ratherintricate and lengthy calculations This is the reason why model equations,like the Swift–Hohenberg equations (1977) or generalizations of them (Crossand Hohenberg 1993; Bodenshatz et al 2000), have been recently proposed.Although such model equations cannot be derived directly from the usualbalance equations of mass, momentum, and energy, they capture most of theessential of the physical behaviour and have become the subject of very in-tense investigations Recent improvements in the performances of numericalanalysis have fostered the resolution of stability problems by direct integra-tion of the governing equations Although such approaches are rather heavy,costly, and mask some interesting physical features, they are useful as theymay be regarded as careful numerical control of the semi-analytical methodsand associated models
6.3 Thermal Convection
Fluid motion driven by thermal gradients, also called thermal convection, is
a familiar and important process in nature It is far from being an academicsubject Beyond its numerous technological applications, it is the basis forthe interpretation of several phenomena as the drift of the continental plates,
Trang 7144 6 Instabilities and Pattern Formationthe Sun activity, the large-scale circulations observed in the oceans, the at-mosphere, etc As a prototype of thermal convection, we shall examine thebehaviour of a thin fluid layer enclosed between two horizontal surfaces whoselateral dimensions are much larger than the width of the layer The two hori-zontal bounding planes are either rigid plates or stress-free surfaces, the lowersurface is uniformly heated so that the fluid is subject to a vertical tempera-ture gradient If the temperature gradient is sufficiently small, heat is trans-ferred by conduction alone and no motion is observed When the temperaturedifference between the two plates exceeds some critical value, the conductionstate becomes unstable and motion sets in The most influential experimentalinvestigation on thermal convection dates back to B´enard (1900) The fluidused by B´enard was molten spermaceti, a whale’s non-volatile viscous oil, andthe motion was made visible by graphite or aluminium powder In B´enard’soriginal experiment, the lower surface was a rigid plate but the upper onewas open to air, which introduces an asymmetry in the boundary conditionsbesides surface tension effects The essential result of B´enard’s experimentwas the occurrence of a stable, regular pattern of hexagonal convection cells.Further investigations showed that the flow was ascending in the centres ofthe cells and descending along the vertical walls of the hexagons Moreover,optical investigations revealed that the fluid surface was slightly depressed atthe centre of the cells.
A first theoretical interpretation of thermal convection was provided byRayleigh (1920), whose analysis was inspired by the experimental observa-tions of B´enard Rayleigh assumed that the fluid was confined between twofree perfectly heat conducting surfaces, and that the fluid properties wereconstant except for the mass density In Rayleigh’s view, buoyancy is thesingle responsible for the onset of instability By assuming small infinitesimaldisturbances, he was able to derive the critical temperature gradient for theonset of convection together with the wave number for the marginal mode.However, it is presently recognized that Rayleigh’s theory is not adequate toexplain the convective mechanism investigated by B´enard Indeed in B´enard’sset up, the upper surface is in contact with air, and surface tractions originat-ing from surface tension gradients may have a determinant influence on theonset of the flow By using stress-free boundary conditions, Rayleigh com-pletely disregarded this effect It should also be realized that surface tension
is not a constant but that it may depend on the temperature or (and) thepresence of surface contaminants This dependence is called the capillary orthe Marangoni effect after the name of the nineteenth-century Italian investi-gator The importance of this effect was only established more than 40 yearslater after Rayleigh’s paper by Block (1956) from the experimental point ofview Pearson (1958) made the first theoretical study about the influence ofthe variation of surface tension with temperature on thermal convection Thepredominance of the Marangoni effect in B´enard’s original experiment is nowadmitted beyond doubt and confirmed by experiments conducted recently inspace-flight missions where gravity is negligible When only buoyancy effects
Trang 8are accounted for, the problem is generally referred to as Rayleigh–B´enard’sinstability while B´enard–Marangoni is the name used to designate surfacetension-driven instability When both buoyancy and surface tension effectsare present, one speaks about the Rayleigh–B´enard–Marangoni’s instability.
6.3.1 The Rayleigh–B´ enard’s Instability:
A Linear Theory
We are going to study the instabilities occurring in a viscous fluid layer of
thickness d (between a few millimetres and a few centimetres) and infinite horizontal extent limited by two horizontal non-deformable free surfaces, the
z-axis is pointing in the opposite direction of the gravity acceleration g The
fluid is heated from below with Thand Tc, the temperatures of the lower and
upper surfaces, respectively (see Fig 6.4) The mass density ρ is assumed to
decrease linearly with the temperature according to the law
where T0 is an arbitrary reference temperature, say the temperature of the
laboratory, and α the coefficient of thermal expansion, generally a positive
quantity except for water around 4◦ C For ordinary liquids, α is of the order
of 10−3–10−4K−1.
When the temperature difference ∆T = Th− Tc (typically not more than
a few ◦C) between the two bounding surfaces is lower than some critical
value, no motion is observed and heat propagates only by conduction inside
the fluid However by further increasing ∆T , the basic heat conductive state becomes unstable at a critical value (∆T )crit and matter begins to performbulk motions which, in rectangular containers, take the form of regular rollsaligned parallel to the short side as visualized in Fig 6.5, this structure is
referred to as a roll pattern Note that the direction of rotation of the cells is
unpredictable and uncontrollable, and that two adjacent rolls are rotating inopposite directions
Fig 6.4 Horizontal fluid layer submitted to a temperature gradient opposed to the
acceleration of gravity g
Trang 9146 6 Instabilities and Pattern Formation
Fig 6.5 Convective rolls in Rayleigh–B´enard’s instability
A qualitative interpretation of the onset of motion is the following Bysubmitting the fluid layer to a temperature difference, one generates a tem-perature and a density gradient A fluid droplet close to the hot lower platehas a lower density than everywhere in the layer, as density is generally adecreasing function of temperature As long as it remains in place, the fluidparcel is surrounded by particles of the same density, and all the forces acting
on it are balanced Assume now that, due to a local fluctuation, the droplet isslightly displaced upward Being surrounded by cooler and denser fluid, it willexperience a net upward Archimede’s buoyant force proportional to its vol-ume and the temperature difference whose effect is to amplify the ascendingmotion Similarly a small droplet initially close to the upper cold plate andmoving downward will enter a region of lower density and becomes heavierthan the surrounding particles It will therefore continue to sink, amplifyingthe initial descent What is observed in the experiments is thus the result
of these upward and downward motions
However, experience tells us that convection does not appear whateverthe temperature gradient as could be inferred from the above argument Thereason is that stabilizing effects oppose the destabilizing role of the buoyancyforce; one of them is viscosity, which generates a friction force directed oppo-site to the motion, the second one is heat diffusion, which tends to spread outthe heat contained in the droplet towards its environment reducing the tem-perature difference between the droplet and its surroundings This explainswhy a critical temperature difference is necessary to generate a convectiveflow: motion will start as soon as buoyancy overcomes the dissipative effects
of viscous friction and heat diffusion These effects are best quantified by theintroduction of the thermal diffusion time and the viscous relaxation time
Trang 10equilibrium Another relevant timescale is the buoyant time, i.e the time that
a droplet, differing from its environment by a density defect δρ = ρ0α∆T , needs to travel across a layer of thickness d,
This result is readily derived from Newton’s law of motion ρ0d2z/dt2= gδρ for a small volume element; a large value of ∆T means that the buoyant
time is short To give an order of magnitude of these various timescales,
let us consider a shallow layer of silicone oil characterized by d = 10 −3m,
ν = 10 −4m2s−1 , χ = 10 −7m2s−1 , it is then found that τ ν = 10−2s and
τ χ= 10 s.
The relative importance of the buoyant and dissipative forces is obtained
by considering the ratios τ ν /τBand τ χ /τBor, since they occur simultaneously,
through the so-called dimensionless Rayleigh number,
Ra = τ ν τ χ
τ2 B
desta-exceeds some critical value (Ra)c For Ra < (Ra)c, the fluid remains at rest
and heat is only transferred by conduction, for Ra > (Ra)c, there is a suddentransition to a complex behaviour characterized by the emergence of order inthe system The ratio between the dissipative processes is measured by thedimensionless Prandtl number defined as
for gases P r ∼ 1, for water P r = 7, for silicone oils Pr is of the order of 103,
and for the Earth’s mantle P r ∼ 1023
In a linear stability approach, the main problem is the determination of
the marginal stability curve, i.e the curve of Ra vs the wave number k at
σ = 0 The one corresponding to Rayleigh–B´enard’s instability is derived inthe Box 6.1
Box 6.1 Marginal Stability Curve
The mathematical analysis is based on the equations of fluid mechanics ten within the Boussinesq approximation This means first that the density
writ-is considered to be constant except in the buoyancy term; second that all thematerial properties as viscosity, thermal diffusivity, and thermal expansioncoefficient are temperature independent; and third that mechanical dissi-pated energy is negligible The governing equations of mass, momentum,and energy balance are then given by
∇ · v = 0,
Trang 11148 6 Instabilities and Pattern Formation
where use has been made of the equation of state (6.18) and where
v (v x , v y , v z ) and p designate the velocity and pressure fields, respectively.
The set (6.1.1) represents five scalar partial differential equations for the
five unknowns p, T and v x , v y , v z.
In the basic unperturbed state, the fluid is at rest and temperature is
conveyed by conduction, so that the solutions of (6.1.1) are simply
is the positive temperature difference between the lower and upper
bound-aries Designating by v = v −0, T = T −Tr, p = p −prthe infinitesimallysmall perturbations of the basic state, we can linearize (6.1.1) and obtainthe following set for the perturbed fields
where e z is the unit vector pointing opposite to g We now determine the
corresponding boundary conditions If we assume that the thermal tivity at the limiting surfaces is much higher than in the fluid itself, anythermal disturbance advected by the fluid will be instantaneously smoothed
conduc-out so that T will vanish at the bounding surfaces Since the horizontal
boundaries are assumed to be free surfaces, the shearing stress is zero atthe surface; when use is made of the equation of continuity (6.1.3), this
condition is identical to setting ∂2v
z /∂z2= 0 together with v z= 0, as the
surfaces are non-deformable Summarizing, the boundary conditions are
x = v y = v z = 0, which combined with
the continuity condition yields
v
z = ∂v z /∂z = 0, T = 0 at perfectly heat conducting rigid walls.
(6.1.7)
Trang 12At adiabatically isolated walls, the condition T = 0 will be replaced by
∂T /∂z = 0, expressing that the surface is impermeable to heat flow When
the boundary surface is neither perfectly heat conducting nor adiabaticallyisolated, heat transfer is governed by Newton’s cooling law −λ∂T/∂z = h(T −T ∞ ) where λ is the heat conductivity of the fluid, h the so-called heat transfer coefficient, and T ∞the temperature of the outside world.
A further simplification of the set (6.1.3)–(6.1.5) is obtained by applying
twice the operator rot ( ≡ ∇×) on the momentum equation and using the
continuity equation, we are then left with the two following relations in the
where ∇2 ≡ ∂2/∂x2 + ∂2/∂y2 denotes the horizontal Laplacian whereas
β = ∆T /d We now make the variables dimensionless by introducing the
where W (Z) and Θ(Z) are the amplitudes of the perturbations, k x and k y
are the dimensionless wave numbers in the directions x and y, respectively, and σ is the dimensionless growth rate Since the present problem is self- adjoint, it can be proved (Chandrasekhar 1961) that σ is a real quantity.
There exist several ways to make the variables dimensionless, the choicemade here seems to be one of the most preferred Sometimes, it is alsopreferable to describe the horizontal periodicity of the flow by means of the
wavelength λ = 2π/k rather than the wave number as it provides a direct
measure of the dimensions of the cells
Substitution of solutions (6.1.11) and (6.1.12) in (6.1.8) and (6.1.9)leads to the following amplitude differential equations
(D2− k2)(D2− k2− σ)W = Ra k2Θ, (6.1.13)
(D2− k2− σ P r)Θ = −W, (6.1.14)
where D stands for d/dZ and k2= k2
x + k2y , Ra and Pr denote the
dimen-sionless Rayleigh and Prandtl numbers defined by (6.21) and (6.22) Theboundary conditions corresponding to two perfectly heat conducting freesurfaces are
Trang 13150 6 Instabilities and Pattern Formation
W = D2W = 0, Θ = 0 at Z = 0 and Z = 1. (6.1.15)
The latter are satisfied for solutions of the form W = A sin πZ, Θ =
B sin πZ, which substituted in (6.1.13) and (6.1.14) lead to the following algebraic equations for the two unknowns A and B:
A + (π2+ k2+ σ)B = 0, (6.1.16)
(π2+ k2)(π2+ k2+ σP r −1 )A − k2Ra B = 0.
Non-trivial solutions demand that the determinant of the coefficients
vanishes, which results in the following dispersion relation between k, σ,
Ra, and P r:
(π2+ k2)σ2+ (π2+ k2)2(1 + P r)σ + P r[(π2+ k2)3− k2Ra] = 0 (6.1.17)
By setting σ = 0 in (6.1.17), one obtains the marginal curve (Ra)0 vs k
determining the Rayleigh number at the onset of convection (Fig 6.6); it isindependent of the Prandtl number and given by
(Ra)0= (π
2+ k2)3
Fig 6.6 Marginal stability curve for Rayleigh–B´enard’s instability in a horizontal
fluid layer limited by two stress-free perfectly heat conducting surfaces
Trang 14The solutions of (6.1.17) can be written as
unconditionally stable by heating from above
For slightly supercritical conditions, where Ra is close to (Ra)0, (6.24)reads as
σ = (π2+ k2) Ra − (Ra)0
(Ra)0(1 + P r −1), (6.25)
which shows that, for Ra larger than (Ra)0, the amplitude of the disturbancesamplifies exponentially and the basic state is unstable However, for suchvalues, the non-linear terms become important and the linear analysis ceases
to be valid
The minimum value of the marginal curve is obtained from relation (6.23)
by differentiation with respect to k; setting this result equal to zero gives the critical wave number kc at which the curve (Ra)0(k) is minimum, and the corresponding critical Rayleigh number (Ra)c For the present problem, it isfound that
kc= π/ √
2 = 2.21, (Ra)c= 27π4/4 = 657.5 (free–free boundary conditions).
(6.26)The critical Rayleigh number allows us to determine the critical temperaturedifference at which the system changes from the state of rest to the state
of cellular motion, the critical wave number provides information about the
horizontal periodicity of the patterns at the onset of convection, kcrepresentsthe most dangerous mode picked up by the fluid For other boundary condi-tions, the calculations are more complicated but the procedure remains valid,the marginal curves will have approximately the same form as in Fig 6.6 withcritical values given by
kc= 2.68, (Ra)c= 1, 100.6 (rigid–free boundaries),
kc= 3.117, (Ra)c= 1, 707.7 (rigid–rigid boundaries).
As expected, stability is reinforced (larger (Ra)c value) in presence of rigidsurfaces as the fluid motion is more strongly inhibited by the viscous forces
At the same time the dimensions of the cells (larger kcvalue) are diminished:more energy must now be dissipated to compensate for the larger release ofenergy by buoyancy, clearly narrower cells are associated with greater dis-
sipation and energy release It is also worth to note that, for Ra > (Ra)
Trang 15152 6 Instabilities and Pattern Formationand according to the linear theory, a continuous spectrum of modes becomesunstable from which follows that the observed pattern should be very intri-cate This is in contradiction with the observation that the flow prefers rathersimple cellular forms; the reason for this discrepancy must once more be at-tributed to the omission of non-linear terms Moreover, the linear theory isunable to predict the particular pattern (either rolls, squares, or hexagons)selected by the fluid; this is so because the eigenvalue problem is degenerate,
which means that to one eigenvalue Ra there corresponds an infinite number
of possible patterns with the same wave number k The reason why a
partic-ular pattern is selected can only be understood from a non-linear approach,which shows that, for the present problem, two-dimensional parallel rolls arethe preferred patterns as confirmed by experimental observations
6.3.2 The Rayleigh–B´ enard’s Instability:
A Non-Linear Theory
We now examine the behaviour of the amplitude of the disturbance beyondthe critical point Returning to the linear theory where the amplitude is
supposed to behave as A(t) ∼ exp σt, we may write that, at threshold, the
relevant differential equation is
dA
where σ is given by (6.25) For supercritical Rayleigh numbers, the amplitude
will then increase exponentially but non-linear self-interaction between modes
becomes important giving raise to higher-order terms in A n We are then led
to an amplitude equation of the form suggested by Landau, i.e
dA
where l is a positive constant to be determined from the boundary conditions whereas σ is the growth rate corresponding to the most dangerous mode kc(= π/ √
2 for free–free boundaries) In virtue of (6.25) and for large values of
Pr as in silicone oils, σ is given by
in which
measures the relative distance from the critical point There is no term in A2
in (6.28) because it is assumed that the convective pattern is such that, byreversing the fluid velocities, the same pattern is observed; this implies that
(6.28) must be invariant with respect to the symmetry A = −A Of course