PatternFormationinRayleigh-BénardConvection 1

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Back Forward Benasque PHYSBIO 2003: Pattern Formation in Rayleigh-Bénard Convection - Lecture 1 1

Pattern Formation in Rayleigh-Bénard Convection

Lecture 1

Benasque PHYSBIO 2003: Pattern Formation in Rayleigh-Bénard Convection - Lecture 1 2

Pattern formation is a common feature of diverse systems driven far from equilibrium.

Rayleigh-Bénard convection has served as a useful canonical example of pattern formation.

In this lecture I plan to give a basic introduction to pattern

formation by addressing the question: “What happens to a

macroscopic system as we drive it further away from

equilibrium?”

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Back Forward

Today’s concepts:

• patterns

• equilibrium v. nonequilibrium

• linear instability

• nonlinear saturation

• stability balloons

• pattern competition

Stationary Ideal Patterns

Stripes Hexagons Squares

cf. Bajaj et al. Ahlers website Ahlers website

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Transients and disordered patterns

Onset of chaos in small systems

R = 2804 R = 6949

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Back Forward

Spatiotemporal chaos

Spiral Defect Chaos Domain Chaos

Equilibrium, Nonequlibrium, and Far From Equilibrium

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Closed (Equilibrium) Open (Far from Equilibrium)

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Back Forward

Open systems characterized by:

• Energy input, interconversion, and output

• Transport of matter and energy

Pattern formation is a common feature of systems far from equilibrium.

Equilibrium

• macroscopically uniform

• no fluxes of matter or energy

• dissipative processes return weakly perturbed system towards steady state (Onsager theory)

– Thermal conductivity → Uniform temperature – Viscosity → zero velocity

– Diffusion → uniform species concentration Typically get exponential decay to equilibrium

u( x , t) = X

n

u

n

e

ⁱ^qⁿ^·^x

e

^−κqⁿ²^t

Far from Equilibrium

• Exponential growth of disturbances

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Equilibrium Perturbations

decay Perturbations

grow

R

_c

D ri ving

Equilibrium Perturbations

decay Perturbations

grow

R

_c

D ri ving

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Back Forward

Pattern formation occurs when the

growing perturbation about the spatially uniform state has spatial structure (a mode with nonzero wave vector).

d

L A

B

x

y

z

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Dynamical Equations

I shall confine my discussion to systems far from equilibrium that are macroscopic and continuous

These are defined by dynamical equations that

• Reflect the laws of thermodynamics and the return to (local) equilibrium

• Are the familiar phenomenological equations

Leads us to the study of nonlinear, determinisitic, PDEs

Equations for Convection (Boussinesq)

σ ⁻ ¹ (∂ t + v · ∇) v = −∇ p + RT z ˆ + ∇ ² v (∂ t + v · ∇) T = ∇ ² T

∇ · v = 0 Boundary conditions

v = 0 at z = 0 , 1

T =

 



1 at z = 0

0 at z = 1

Conducting solution: v = 0, T = 1 − z

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Back Forward

A first approach to patterns: linear stability analysis

1. Find equations of motion of the physical variables u (x, y, z, t) 2. Find the uniform base solution u

_b

(z) independent of x, y, t 3. Focus on deviation from u

_b

u ( x , t) = u

_b

(z) + δ u ( x , t)

4. Linearize equations about u

_b

, i.e. substitute into equations of part (1) and keep all terms with just one power of δ u. This will give an equation of the form

∂

_t

δ u = ˆ L δ u

where L may involve u ˆ

_b

and include spatial derivatives acting on δ u 5. Since L is independent of ˆ x, y, t we can find solutions

δ u

q

( x

_⊥

, z, t) = u

q

(z) e

ⁱ^q^·^x^⊥

e

^σ^q^t

δ u

q

( x

_⊥

, z, t) = u

q

(z) e

ⁱ^q^·^x^⊥

e

^σ^q^t

Re σ

q

gives exponential growth or decay

Im σ

q

= −ω

q

gives oscillations, waves e

ⁱ⁽^q^·^x^⊥^−ω^q^t)

Im σ

q

= 0 H⇒ Stationary instability

Im σ

q

6= 0 H⇒ Oscillatory instability

For this lecture I will look at the case of stationary instability

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x

u

Exponential growth: exp[σ _q t]

λ=2π/q

Rayleigh’s Calculation

T +∆T T

δT

q

(x, z) =

q

²

+ π

²

2

cos (πz) cos (qx), δw

_q

(x, z) = q

²

cos (πz) cos (qx),

δu

_q

(x, z) = −iπq sin (πz) sin (qx).

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Back Forward

1 2 3 4 5

q

σ_q

5

-5

-10 0

R = 1 . 5 R_c

R = 0 . 5 R_c R = R_c

(σ

⁻¹

σ

q

+ π

²

+ q

²

)(σ

q

+ π

²

+ q

²

) − Rq

²

/(π

²

+ q

²

) = 0

Re σ

_q

q

R < R

_c

R = R

_c

R > R

_c

q _c

For R near R

_c

and q near q

_c

Re σ

q

= τ

₀⁻¹

[ ε − ξ

₀²

(q − q

_c

)

²

] with ε = R − R

_c

R

_c

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Re σ

_q

q

R < R

_c

R = R

_c

R > R

_c

q _c

For R near R

c

and q near q

c

Re σ

q

= τ

₀⁻¹

[ ε − ξ

0²

(q − q

c

)

²

] with ε = R − R

_c

R

_c

q

_c

q R

R_c

Re σ

_q

> 0

Re σ

_q

< 0

Setting Re σ

q

= 0 defines the neutral stability curve R = R

_c

(q) R

_c

(q) = (q

²

+ π

²

)

³

q

²

⇒ R

_c

= 27 π

⁴

4 , q

_c

= π

√ 2

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Back Forward

Linear stability theory is often a useful first step in understanding pattern formation:

• Often is quite easy to do either analytically or numerically

• Displays the important physical processes

• Gives the length scale of the pattern formation 1 /q

_c

But:

• Leaves us with unphysical exponentially growing solutions

• Not all pattern formation phenomena can be connected back to the linear onset

Nonlinearity

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q R

Re σ

_q

> 0

Re σ

_q

< 0

q

_c

q R

R

_c

Re σ

_q

> 0

Re σ

_q

< 0

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Back Forward

q

_c

q R

R

_c

Re σ

_q

> 0

Re σ

_q

< 0 R

_c

(q) or

q

_N

(R)

q

_c

q R

R

_c

band of growing solutions

Re σ

_q

> 0

Re σ

_q

< 0

q

_N-

q

_N+

R

_c

(q) or

q

_N

(R)

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q

_c

q R

q = n 2 π/l

R

_c

R δ u

Forward Bifurcation

q

_c

q R

q = n 2 π/l

R

_c

R δ u

Backward Bifurcation

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Back Forward

q

_c

q R

nonlinear states

R

_c

q

_c

q R

R

_c

Patterns exist.

Are they stable?

No patterns

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q

_c

q R

R

_c

E E E=Eckhaus

stable

unstable unstable

q

_c

q R

R

_c

ZZ Z=ZigZag

stable

unstable

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Back Forward

q

_c

q R

R

_c

ZZ E

E

stable

q

_c

q R

R

_c

E O

E=Eckhaus Z=ZigZag

SV=Skew Varicose O=Oscillatory SV

ZZ E

stable

b a n d

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q

_c

q R

R

_c

E O

E=Eckhaus Z=ZigZag

SV=Skew Varicose O=Oscillatory SV

E ZZ

stable b a n d

q

_S+

q

_S-

q

_N+

q

_N-

q R

R

_c

E ZZ E

stable

band

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Back Forward

Summary so far

• Uniform state unstable to growth of perturbation with wave vector q for R > R

c

( q )

• Neutral stability curve q

_N_±

(R) defined by R

_c

( q

_N_±

) = R

• Saturated, nonlinear states exist for q

_N−

(R) < q < q

_N₊

(R)

• Nonlinear states are stable in a restricted band of wave vectors q

_S−

(R) < q < q

_S+

(R) (the stability balloon)

• Near onset the instabilities take on a universal form (Eckhaus and ZigZag)

q

E+

− q

c

= q

c

− q

E−

= 1

√ 3 (q

N+

− q

c

) ∝ ε

¹^/²

q

_Z

− q

_c

= 0 × ε

¹^/²

+ ?? × ε

• Away from onset the specifics of the system will be important

Role of Symmetry

Full rotational symmetry in plane

q q

R

x

q

_y

No patterns Patterns

N

(a) (b)

q

c c

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At the linear level any superposition of modes with | q | ' q

_c

gives a (time dependent) solution

Nonlinearity will:

• Saturate growth

• Lead to mode competition How to proceed?

• There is in general no way to list all possible solutions of a nonlinear PDE

• Use symmetry to suggest possible structures

• Test stability

q

_x

q

_y

0 10 20 30 40 50

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Back Forward

q

_x

q

_y

0 10 20 30 40 50

q

_x

q

_y

0 10 20 30 40 50

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q

_x

q

_y

0 10 20 30 40 50

q

_x

q

_y

0 10 20 30 40 50 60

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Back Forward

q

_x

q

_y

0 10 20 30 40 50

It should be noted that many of the same questions come up in equilibrium physics in discussing crystal structure. Similarities:

1. Propose structure, and test for stability 2. Similar structures

3. Group theory is a useful tool

Differences in nonequilibrium systems:

1. No free energy to compare states, and choose between multistable states

2. Transitions often continuous (second order) or weakly first order, so that analysis in terms of wave vectors at q

c

makes sense

3. Often interested in two dimensional structures

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Anisotropy direction (only x → − x symmetry)

q

_x

q

_y

q

_x

q

_y

(a) (b)

θ

Conclusions

Introduced a class of pattern formation in systems that are far from equilibrium globally, but are near equilibrium locally, so that they are described by familiar continuum equations.

Study of behavior of nonlinear PDEs.

Asking what happens as we drive a system away from the thermodynamic equilibrium state leads to an understanding of pattern formation in terms of linear instability from a uniform state.

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