Geometric singularities Cusp universality surface of a viscous - PowerPoint PPT Presentation

9781107485495 EGGERS AND FONTELOS – SINGULARITIES: FORMATION, STRUCTURE AND PROPAGATION COVER C M Y K CAMBRIDGE TEXTS Jens Eggers Fontelos Eggers Many key phenomena in physics and engineering are described as IN APPLIED singularities in the solutions to the differential equations describing MATHEMATICS them. Examples covered thoroughly in this book include the formation of drops and bubbles, the propagation of a crack, and the formation of a shock in a gas. Aimed at a broad audience, this book provides the mathematical Singularities: tools for understanding singularities and explains the many common features in their mathematical structure. Part I introduces the main Structure, and Propagation Formation, Structure, concepts and techniques, using the most elementary mathematics Singularities: Formation, possible so that it can be followed by readers with only a general background in differential equations. Parts II and III require more and Propagation specialized methods of partial differential equations, complex analysis, and asymptotic techniques. The book may be used for advanced fluid mechanics courses and as a complement to a general course on applied partial differential equations. CAMBRIDGE TEXTS IN APPLIED MATHEMATICS EDITORIAL BOARD Professor M. J. Ablowitz, Department of Applied Mathematics, University of Colorado Boulder, USA Professor S. Davis, School of Engineering Sciences and Applied Mathematics, Northwestern University, USA Professor E. J. Hinch, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, UK Professor A. Iserles, Department of Applied Mathematics and Theoretical Physics, University of Cambridge, UK Dr J. Ockendon, Mathematical Institute, University of Oxford, UK Professor P. J. Olver, School of Mathematics, University of Minnesota, USA The aim of this series is to provide a focus for publishing textbooks in applied mathematics at the advanced undergraduate and beginning graduate level. The books are devoted to covering certain mathematical techniques and theories and exploring their applications. J. EGGERS AND Cover illustration: courtesy of Nick Laan M. A. FONTELOS and Daniel Bonn. Geometric singularities

Cusp universality surface of a viscous fluid cusp : 3/2 width r caustics in a cup

Hele-Shaw cell bubble with sink in center Polubarinova-Kochina, 1945 Galin, 1945

Cusp structure: viscous flow 4 2 ψ = 0 coarse grain similarity solution: ψ = r α f ( φ ) α = 1 , 3 / 2 , 2 , . . .

Local analysis = - = - 1/2 u V u 6 Ay , y x ¶ = h u 6 A y = 1/2 x ¶ y u V y 3/2 : h y f y = Vx - 3/2 3 4 Ar sin 2 Renardy et al., JFM `91

Jeong and Moffatt solution J.-T. Jeong, H.K. Moffatt, JFM `92 holomorphic h fluid U = U Ca g viscosity h - p = 2 Ca R Ce z ( θ ) smooth singularity: z 0 (0) = 0

Cusp geometry a = critical point: 0 = + a j 1 x x = a j x y = ✏ 3 / 2 Y 0 1 x = ✏ X, 1 y = = + a j 0 y y X = σ 2 Y = σ 3 0 2 2 , 3 ± σ 2 / 2 = j 3 /3 y = ej + j x a universal 2 / 2 = ej = + j j + j 2 3 x b / 2 a /3 y unfolding:

Singularity theory g = ψ � f � φ − 1 left-right equivalent: singular germ: rk 0 ( f ) < min( n, p ) unfolding: F ( x , u ) , F ( x , u = 0) = f ( x ) plane curves: ( ϕ m , ϕ n ) , hcf(m , n) = 1 Example: ( ϕ 2 , ϕ 5 + µ 1 ϕ + µ 3 ϕ 3 ) germ unufolding Eggers, Suramlishvili, Eur. J. Mech. B, 2017

A small bubble 3/2 − cusp λ 3 X = σ 2 0 2 5/2 − cusp λ 3 = − 2 − 2 ✓ σ 4 ◆ 5 + λ 3 3 σ 2 + s Y = σ 3/2 − cusp λ 3 = − 6/5 1/2 − 2.7 ✓ σ 2 ⌘ 2 ◆ ⇣ √ 2 , σ σ 2 − ( X, Y ) = 5 5 − 1 0 1 s ε 5/2 ε

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A shock wave a jump in density occurs at some finite time t 0 ! W.C. Griffith, W. Bleakney

Burgers’ equation ¶ ¶ characteristic curves: u u + = u 0 ¶ ¶ x ( ξ , t ) = u 0 ( ξ ) t + ξ t x singularity in finite time t,u u ( x ( ξ , t ), t ) = u 0 ( ξ ) { } t 0 = Min ξ − 1 u 0 ′ ( ξ ) x u 0 ( ξ ) = − s ' + as ' 3 + …

¢ = - Similarity solution t t t 0 ¶ ¶ u u æ ö æ x ö x + = u 0 ¢ = ¢ a ¢ ¢ a x t b a + = x x = = 1 u t U u t U 1 x t ç ÷ ç ÷ ¶ ¶ t x ¢ ¢ a b + è è t t ø ø [ ] ¢ ¢ - a + + a x + = ¢ ¢ ¢ ¢ a - U (1 ) U a b UU - 0 - a + bx + = 1 2 t U U t UU 0 ⎧ regular at 1 ξ = − U − CU 1 + 1/ α , α i = 2 i + 2 , i = 0,1,2, … ⎪ ⎨ x = 0 ⎪ − U , α =0 ⎩

¢ = - Similarity solution t t t 0 U a U b ua ub ξ + U b + CU 3 b = 0 ξ x ξ − U a + CU 3 x ξ a = 0 ( ) ¢ ¢ = 1/2 3/2 u x t ( , ) t U x t / only stable solution!

2D structure of shock waves courtesy of Patrice Legal t c ( y ) − t 0 = ay 2 + O ( y 3 ), a > 0, y ∼ t ' 1/2

Compressible Euler with T. Grava ∂ρ p = A ∂ t + r · ( ρ v ) = 0 γ ρ γ ∂ t + ( v · r ) v = � 1 ∂ v ρ r p v = r φ ∂φ A ∂ t + 1 2 | r φ | 2 = � ⇣ ⌘ ρ γ − 1 � ρ γ − 1 0 γ � 1 ∂ρ / ∂ y = 0 , ∂ x/ ∂ρ = 0 | r ρ | ρ = ρ 0 ! 1 ∂ 3 x ∂ 2 x/ ∂ρ 2 = 0 , ∂ρ 3 = finite ! 0 = ∂ p ∂ρ = A ρ γ − 1 c 2 0

Similarity solution ξ = x + c 0 t 0 y φ = | t 0 | 2 Φ ( ξ , η ) , η = | t 0 | 3 / 2 , | t 0 | 1 / 2 ρ = ρ 0 [1 + | t 0 | 1 / 2 R ( ξ , η ) + | t 0 | Q ( ξ , η )] � c 0 | t 0 | 1 / 2 Φ ξ + | t 0 | [ ⌥ 2 Φ ± 3 ξ 2 Φ η ] + | t 0 | 2 Φ ξ ± η 2 Φ 2 ξ = In(B c 2 ⇢ � ( γ � 1) | t 0 | 1 / 2 R + | t 0 | [( γ � 1) Q + 1 γ 2 � 3 γ + 2 + O ( t 0 3 / 2 ) 0 � � R 2 ] � γ � 1 2 ): 2 ± 3 ξ � c 0 R ξ | t 0 | � 1 + | t 0 | � 1 / 2 [ ⌥ R 2 R ξ ± η 2 R η � c 0 Q ξ ]+ In(C ρ 0 | t 0 | � 1 Φ ξξ + | t 0 | � 1 / 2 [ Φ ξ R ξ + Φ ξξ R ] = O ( t 0 0 ) ): U − 3 ξ U ξ − η U η = ± ( γ + 1) UU ξ

Similarity solution ξ ( U, η ) : ξ U U − 3 ξ + ηξ η = ± ( γ + 1) U ⇣ η ξ = ⌥ γ + 1 ⌘ U � U 3 F U 2 ∂ 3 ξ � regularity condition: ! � = − F 000 ( a ) = const � ∂η 3 � 0 ξ = γ + 1 U − A 0 U 3 − A 1 U 2 η − A 2 U η 2 − A 3 η 3 2

Shock position ξ = γ + 1 U − A 0 U 3 − A 1 U 2 η − A 2 U η 2 − A 3 η 3 2 γ + 1 = 3 A 0 U 2 + 2 A 1 U η + A 2 η 2 U ( ξ , η ) vertical: 2 η ξ ¯ ¯ ξ = ξ − ξ s ( η ) , U = U − U s ( η ) � ¯ ξ = − A 0 ¯ ¯ U 2 − ∆ 2 ( η ) � U

Numerical simulation M.A. Herrada, G. Pitton with Basilisk (a) Level 12 0.4 Level 15 Level 18 0.3 high order code 1/| � � | max 1/| � � | max =3.85*(t − 0.511) 0.2 0.1 0 0.4 0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.5 0.51 0.52 0.53 0.54 0.55 t (b) 0.01 Level 12 0.008 Level 15 Level 18 0.006 S=0.7(t − 0.511) 3/2 S max 0.004 0.002 0 0.4 0.41 0.42 0.43 0.44 0.45 0.46 0.47 0.48 0.49 0.5 0.51 0.52 0.53 0.54 0.55 t

Parameters A 0 A 2 before singularity ξ = γ + 1 U − A 0 U 3 − A 1 U 2 η − A 2 U η 2 − A 3 η 3 2

Predictions after singularity ξ = ( x ' − c t ' ) t ' 3/2 U = u t ' 1/2 η = y ' t ' 1/2

Counterexample: drop coalescence Aarts et al., PRL `05 2 : width r