Cours : Dynamique Non-Lin eaire Laurette TUCKERMAN - PowerPoint PPT Presentation

Cours : Dynamique Non-Lin´ eaire Laurette TUCKERMAN laurette@pmmh.espci.fr Rolls, stripes and their instabilities Sources: R. Hoyle, Pattern Formation: An Introduction to Methods, Cambridge 2006 M. Cross, http://www.its.caltech.edu/ ∼ mcc

Pattern formation No horizontal BCs (horizontally homogeneous domain) = ⇒ eigenvectors are e i q · x with x ≡ ( x, y ) , q ≡ ( q x , q y ) Linear instability depends on magnitude q ≡ | q | , not on orientation of q q u q e i q · x Near threshold, final solution = � Nonlinear terms = ⇒ relative magnitudes of different u q = ⇒ pattern Mathematics not yet ready to choose between all possible patterns = ⇒ Pose problem on fixed lattice and seek specified pattern, i.e. stripes, rect- angles, squares, hexagons

Swift-Hohenberg equation Instability of trivial state u = 0 to eigenvectors e i q · x with growth rate σ ( q ) σ depends on q 2  ⇒ σ ( q ) = a 0 + a 2 q 2 − q 4  σ > 0 for q ∼ q c  = = µ − ( q 2 c − q 2 ) 2 σ < 0 for | q | ≫ 1

Swift-Hohenberg equation − ( q 2 c − q 2 ) 2 σ = µ ↓ ↓ ∂ t ∆ Add saturating cubic term to halt exponential growth near q c � 2 u − u 3 q 2 � ∂ t u = µu − c + ∆ Add quadratic term to obtain hexagons Include q c and q ′ c to obtain quasipatterns Derived by J. Swift and P.C. Hohenberg (Phys. Rev. A 15, 319 (1977)) to describe pattern formation in convection

Patterns produced by Swift-Hohenberg equation Stripes Hexagons Zigzag instability Quasicrystals

Stripes or Rolls � 2 u − u 3 q 2 � ∂ t u = µu − c + ∆ q c = 1 Domain of length L = 24 c − q 2 ) 2 Amplitude u = Steady bifs at µ q = ( q 2 � µ − ( q 2 c − q 2 ) 2

⇒ q = n 2 π Periodic boundary conditions in x with wavelength L = L Allowed q closest to q c = 1 = ⇒ smallest critical µ λ n = L q n = 2 nπ µ n = ( q 2 c − q 2 n ) 2 n n L 4 6.0 1.05 0.01 3 8.0 0.79 0.15 n λ n q n µ n 5 4.8 1.31 0.51 (1 2 − 1 2 ) 2 = 0 2 π 2 12.0 0.52 0.53 1 1 (1 2 − 2 2 ) 2 = 9 π 1 24.0 0.26 0.87 2 2 (1 2 − 3 2 ) 2 = 64 3 2 π/ 3 3 6 4.0 1.57 2.15 7 3.4 1.83 5.56 L = λ c = 2 π 8 3.0 2.09 11.47 9 2.7 2.36 20.72 L = 24 ≈ 4 λ c = 8 π length scale for x q c λ c √ √ RB convection free-slip BCs depth π/ 2 2 RB convection rigid BCs depth π 2 Swift-Hohenberg equation 2 π 1 2 π

Periodic BCs = ⇒ circle pitchforks Neumann BCs ( ∂ x u | 0 ,π = 0 ) = ⇒ ordinary pitchforks L ≈ λ c = ⇒ thresholds separated = ⇒ only first bifurcation important L ≫ 1 = ⇒ thresholds close = ⇒ discretization sometimes neglected

Square patterns ⇒ evecs e ± ix 1 and e ± ix 2 Periodicity L x = L y = λ c = 2 π = u ( x 1 , x 2 , t ) = z 1 ( t ) e ix 1 + ¯ z 1 ( t ) e − ix 1 + z 2 ( t ) e ix 2 + ¯ z 2 ( t ) e − ix 2 Symmetry group of square lattice generated by: D 4 : rotation S π/ 2 and reflection κ in x 1 two-torus T 2 : translations by p = ( p 1 , p 2 ) in x 1 and x 2 u ( x 2 , − x 1 , t ) = z 1 ( t ) e ix 2 + ¯ z 1 ( t ) e − ix 2 + z 2 ( t ) e − ix 1 + ¯ z 2 ( t ) e ix 1 S π/ 2 u ( x 1 , x 2 , t ) ≡ u ( − x 1 , x 2 , t ) = z 1 ( t ) e − ix 1 + ¯ z 1 ( t ) e ix 1 + z 2 ( t ) e ix 2 + ¯ z 2 ( t ) e − ix 2 κu ( x 1 , x 2 , t ) ≡ P p 1 ,p 2 u ( x 1 , x 2 , t ) ≡ u ( x 1 + p 1 , x 2 + p 2 , t ) z 1 ( t ) e i ( x 1 + p 1 ) + ¯ z 1 ( t ) e − i ( x 1 + p 1 ) + z 2 ( t ) e i ( x 2 + p 2 ) + ¯ z 2 ( t ) e − i ( x 2 + p 2 ) =   S π/ 2 ( z 1 , z 2 ) ≡ (¯ z 2 , z 1 )   = ⇒ κ ( z 1 , z 2 ) ≡ (¯ z 1 , z 2 ) P p 1 ,p 2 ( z 1 , z 2 ) ≡ ( e ip 1 z 1 , e ip 2 z 2 )  

Seek terms (up to cubic) which commute with P p 1 ,p 2 ( z 1 , z 2 ) z 1 → e ip 1 z 1 z 2 → e ip 2 z 2 z 1 → e − ip 1 ¯ z 2 → e − ip 2 ¯ ¯ z 1 ¯ z 2 z 1 → e ip 1 z 1 e − ip 1 ¯ z 1 → e − ip 1 ¯ z 1 e − ip 1 ¯ z 1 z 1 → e ip 1 z 1 e ip 1 z 1 z 1 ¯ z 1 z 1 ¯ ¯ z 1 z 2 → e ip 2 z 2 e − ip 2 ¯ z 2 → e − ip 2 ¯ z 2 e − ip 2 ¯ z 2 z 2 → e ip 2 z 2 e ip 2 z 2 z 2 ¯ z 2 z 2 ¯ ¯ z 2 z 2 → e − ip 1 ¯ z 1 e − ip 2 ¯ z 1 z 2 → e ip 1 z 1 e ip 2 z 2 z 1 ¯ ¯ z 2 z 2 → e ip 1 z 1 e − ip 2 ¯ z 1 z 2 → e − ip 1 ¯ z 1 e ip 2 z 2 z 1 ¯ z 2 ¯ z 1 z 1 → e ip 1 z 1 ¯ z 2 z 1 → e ip 1 z 2 ¯ z 1 ¯ z 1 z 1 z 2 ¯ z 2 z 1 z 1 z 2 → e ip 2 z 1 ¯ z 2 z 2 → e ip 2 z 2 ¯ z 1 ¯ z 1 z 2 z 2 ¯ z 2 z 2 Commute with κ = ⇒ coefficients real Commute with S π/ 2 = ⇒ equality of coefficients z 1 = µz 1 − ( a 1 | z 1 | 2 + a 2 | z 2 | 2 ) z 1 ˙ z 2 = µz 2 − ( a 2 | z 1 | 2 + a 1 | z 2 | 2 ) z 2 ˙ Same as equation for Hopf bifurcation with O (2) symmetry! rolls in x 1 , x 2 direction → left, right travelling waves → r 1 , r 2

Solutions: • Rolls in x 1 direction ( r 1 � = 0 , r 2 = 0 ) • Rolls in x 2 direction ( r 1 = 0 , r 2 � = 0 ) • Squares with r 1 = r 2

Square patterns u ( x 1 , x 2 , t ) = z 1 ( t ) e ix 1 + ¯ z 1 ( t ) e − ix 1 + z 2 ( t ) e ix 2 + ¯ z 2 ( t ) e − ix 2 = r ( t ) e ix 1 + r ( t ) e − ix 1 + r ( t ) e ix 2 + r ( t ) e − ix 2 = 2 r ( t )(cos( x 1 ) + cos( x 2 )) � x 1 + x 2 � x 1 − x 2 � � = 4 r ( t ) cos cos 2 2 Nodal lines u = 0 are diagonals with slopes ± 1 : x 1 + x 2 = π + 2 nπ, x 1 − x 2 = π + 2 nπ

Container with square symmetry Eigenvector ψ 1 with two rolls along x 1 = ⇒ Rotated eigenvector ψ 2 with two rolls along x 2 = ⇒ Four equivalent branches resembling ± ψ 1 , ± ψ 2 generated at bifurcation ψ 1 + ψ 2 is also an eigenvector, oriented along diagonal Four other branches resembling ± ( ψ 1 + ψ 2 ) , ± ( ψ 1 − ψ 2 ) Equivalent to each other but not to branches oriented along x 1 or x 2 Different stability, secondary bifurcations Two sets of four branches generated at pitchfork Marangoni convection: results from temperature dependence of surface tension at free surface Fluid dragged along surface = ⇒ vertical motion (conservation of mass) Eigenvector with full D 4 symmetry = ⇒ transcritical bifurcation

Eigenvectors of Marangoni convection in container with square horizontal cross section From Bergeon, Henry, Knobloch., Phys. Fluids 13, 92 (2001)

Bifurcation diagram of Marangoni convection in container with square cross section

Hexagons u ( x, y, t ) = z 1 ( t ) e i k 1 · x + z 2 ( t ) e i k 2 · x + z 3 ( t ) e i k 3 · x + c.c. Groups D 6 and T 2 generated by S 2 π/ 3 ( z 1 , z 2 , z 3 ) ≡ ( z 3 , z 1 , z 2 ) κ ( z 1 , z 2 , z 3 ) ≡ (¯ z 1 , ¯ z 2 , ¯ z 3 ) P p ( z 1 , z 2 , z 3 ) ≡ ( e i k 1 · p z 1 , e i k 2 · p z 2 , e i k 3 · p z 3 ) k 1 + k 2 + k 3 = 0 = ⇒ quadratic terms allowed! z 3 → e − i k 2 · p ¯ z 2 e − i k 3 · p ¯ z 3 = e − i (k 2 +k 3 ) · p ¯ z 3 = e i k 1 · p ¯ z 2 ¯ ¯ z 2 ¯ z 2 ¯ z 3 Can include ¯ z 2 ¯ z 3 in evolution equation for z 1

µ − b | z 1 | 2 − c ( | z 2 | 2 + | z 3 | 2 ) � � z 1 = ˙ z 1 + a ¯ z 2 ¯ z 3 µ − b | z 2 | 2 − c ( | z 3 | 2 + | z 1 | 2 ) � � z 2 = ˙ z 2 + a ¯ z 3 ¯ z 1 µ − b | z 3 | 2 − c ( | z 1 | 2 + | z 2 | 2 ) � � z 3 = ˙ z 3 + a ¯ z 1 ¯ z 2 z j = r j e iφ j Hexagons: r 1 = r 2 = r 3 = r � � r + ri ˙ = ( µ − ( b + 2 c ) r 2 ) r + ar 2 e − i ( φ 1 + φ 2 + φ 3 ) ˙ φ 1 r = ( µ − ( b + 2 c ) r 2 ) r + ar 2 cos( φ 1 + φ 2 + φ 3 ) ˙ φ 1 = − ar 2 sin( φ 1 + φ 2 + φ 3 ) r ˙ Steady states: Φ ≡ φ 1 + φ 2 + φ 3 = 0 , π = ⇒ cos(Φ) = ± 1  � � a 2 + 4 µ ( b + 2 c ) 1 � − a ±  0 = µ − ( b + 2 c ) r 2 ± ar = b +2 c ⇒ r = � � a 2 + 4 µ ( b + 2 c ) 1 � + a ±  b +2 c

Hexagons Up hexagons: Φ = 0 Down hexagons: Φ = π

Bifurcation Diagram Up- and down-hexagons bifurcate transcritically from 0 at µ = 0 Saddle-node bifurcation at µ = − a 2 / (4( b + 2 c )) , where r = a Rolls created at pitchfork bifurcation Rectangles created in secondary bifurcation from roll branch

Squares and hexagons in simulation of Faraday experiment Boxes supporting the periodic patterns in the square and hexagonal cases.

Square Pattern in Faraday Simulation From Perinet, Juric, Tuckerman, J. Fluid Mech. 635, 1 (2009)

Hexagonal Patterns: interface and velocity fields

Hexagonal pattern in Faraday simulation

Squares, stripes, hexagons in a granular layer From Bizon, Shattuck, Swift, McCormick & Swinney, Patterns in 3D vertically oscillated granular layers: simulation and experiment , Phys. Rev. Lett. 80, 57 (1998).

Spherical Symmetry

Bifurcation diagram for ℓ = 6 : Matthews, Phys. Rev. E, 2003 and Nonlinearity, 2003

Instabilities of roll patterns Swift-Hohenberg equation reproduces instabilities of striped (roll) patterns: (E) Eckhaus: change in wavelength (Z) zigzag: sinusoidal in-phase oscillations along roll axes (SV) skew-varicose: sinusoidal out-of-phase oscillations along roll axes (CR) cross-roll: appearance of perpendicular rolls (OS) oscillatory: time-dependent oscillations along roll axes skew-varicose Vertically vibrated granu- lar layer. From de Bruyn, Bizon, Shattuck, Goldman, Swift, Swinney, Phys. Rev. Lett. 81, 1421 (1998) cross-roll

Busse Balloon for RB Convection (F. Busse & R. Clever, 1967–79) Rayleigh cross-roll knot zig-zag Prandtl skew-varicose oscillatory wavenumber From F.H. Busse, Transition to turbulence in Rayleigh-B´ enard convection, in Hydrodynamic In- stabilities and the Transition to Turbulence, ed. by H.L. Swinney and J.P. Gollub, Springer, 1981.