SLIDE 1 Jens Eggers
Geometric singularities
Eggers Fontelos
Singularities: Formation, Structure, and Propagation
Many key phenomena in physics and engineering are described as singularities in the solutions to the differential equations describing
- them. Examples covered thoroughly in this book include the formation
- f drops and bubbles, the propagation of a crack, and the formation
- f a shock in a gas.
Aimed at a broad audience, this book provides the mathematical tools for understanding singularities and explains the many common features in their mathematical structure. Part I introduces the main concepts and techniques, using the most elementary mathematics possible so that it can be followed by readers with only a general background in differential equations. Parts II and III require more 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
- n 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.
CAMBRIDGE TEXTS IN APPLIED MATHEMATICS
Singularities: Formation, Structure, and Propagation
- J. EGGERS AND
- M. A. FONTELOS
9781107485495 EGGERS AND FONTELOS – SINGULARITIES: FORMATION, STRUCTURE AND PROPAGATION COVER C M Y K
Cover illustration: courtesy of Nick Laan and Daniel Bonn.
SLIDE 2
Cusp universality
3/2
width r :
cusp surface of a viscous fluid caustics in a cup
SLIDE 3
Hele-Shaw cell
bubble with sink in center Polubarinova-Kochina, 1945 Galin, 1945
SLIDE 4
Cusp structure: viscous flow
coarse grain
42ψ = 0 ψ = rαf(φ)
similarity solution:
α = 1, 3/2, 2, . . .
SLIDE 5 Vx y =
1/2
6 ,
x
u Ay = -
y
u V = -
1/2
6
x y
u h A y y u V ¶ = = ¶
3/2
h y :
Local analysis
Renardy et al., JFM `91
3/2 3
4 sin 2 Ar f
SLIDE 6 2 Ca
Ce R
p
fluid
viscosity h
U
U Ca h g =
J.-T. Jeong, H.K. Moffatt, JFM `92
Jeong and Moffatt solution
holomorphic
z(θ) smooth z0(0) = 0
singularity:
SLIDE 7 Cusp geometry
1
x x aj = +
2
y y a j = +
2 / 2
y j =
2 3
/ 2 /3 x b a ej j j = + +
1
x aj =
y =
1
critical point: a =
2 / 2
y j =
3 /3
x a ej j = +
universal unfolding:
X = σ2 2 , Y = σ3 3 ± σ x = ✏X, y = ✏3/2Y
SLIDE 8
Singularity theory
g = ψ f φ−1
left-right equivalent:
F(x, u), F(x, u = 0) = f(x)
unfolding:
rk0(f) < min(n, p)
singular germ: germ unufolding Eggers, Suramlishvili, Eur. J. Mech. B, 2017 plane curves: (ϕm, ϕn),
hcf(m, n) = 1
Example:(ϕ2, ϕ5 + µ1ϕ + µ3ϕ3)
SLIDE 9 A small bubble
−1 1 −2.7 −2
s λ 3
3/2−cusp 5/2−cusp 3/2−cusp λ 3= −6/51/2 λ 3= −2
X = σ2 2 Y = σ ✓σ4 5 + λ3 3 σ2 + s ◆
(X, Y ) = ✓σ2 2 , σ 5 ⇣ σ2 − √ 5 ⌘2◆
ε 5/2
ε
SLIDE 10 Elastic cusp with S. Karpitschka,
( )
x =κ 3 s +as3
( )
y =κ 2s2
x
SLIDE 11
A shock wave
a jump in density occurs at some finite time t0 ! W.C. Griffith, W. Bleakney
SLIDE 12
Burgers’ equation
u u u t x ¶ ¶ + = ¶ ¶
t,u x
u(x(ξ,t),t) = u0(ξ)
characteristic curves:
x(ξ,t) = u0(ξ)t +ξ
singularity in finite time
t0 = Minξ −1 u0′(ξ)
{ }
u0(ξ) = −s'+ as'3+…
SLIDE 13 Similarity solution
u u u t x ¶ ¶ + = ¶ ¶
[ ]
1 2
t U U t UU
a a b
a bx
¢ ¢ ¢
+ =
x u t U t
a b
æ ö ¢ = ç ÷ ¢ è ø x t b x ¢ =
t t t ¢ =
1
x u t U x t t
a a a
x
+ +
æ ö ¢ ¢ = = ç ÷ ¢ è ø
(1 ) U U UU a a x ¢ ¢
+ + =
x =
regular at
ξ = −U − CU 1+1/α , α i = 1 2i + 2 ,i = 0,1,2,… −U, α=0 ⎧ ⎨ ⎪ ⎩ ⎪
SLIDE 14 Similarity solution
t t t ¢ =
x ub ua
( )
1/2 3/2
( , ) / u x t t U x t ¢ ¢ =
ξ − Ua + CU 3
a = 0
ξ + Ub + CU 3
b = 0
ξ
ξ Ua Ub
SLIDE 15
2D structure of shock waves
courtesy of Patrice Legal
tc(y) − t0 = ay2 + O(y3), a > 0, y ∼ t'1/2
SLIDE 16 Compressible Eulerwith T. Grava
∂ρ ∂t + r · (ρv) = 0 ∂v ∂t + (v · r)v = 1 ρrp p = A γ ργ
∂φ ∂t + 1 2 |rφ|2 = A γ 1 ⇣ ργ−1 ργ−1 ⌘
v = rφ |rρ|ρ=ρ0 ! 1
∂ρ/∂y = 0, ∂x/∂ρ = 0 ∂2x/∂ρ2 = 0, ∂3x ∂ρ3 = finite!
c2
0 = ∂p
∂ρ = Aργ−1
SLIDE 17 Similarity solution
φ = |t0|2Φ(ξ, η), ξ = x + c0t0 |t0|3/2 , η = y |t0|1/2
ρ = ρ0[1 + |t0|1/2R(ξ, η) + |t0|Q(ξ, η)]
In(C ):
c0Rξ|t0|1 + |t0|1/2[⌥R 2 ± 3ξ 2 Rξ ± η 2Rη c0Qξ]+ ρ0|t0|1Φξξ + |t0|1/2[ΦξRξ + ΦξξR] = O(t00)
c0|t0|1/2Φξ + |t0|[⌥2Φ ± 3ξ 2 Φξ ± η 2Φη] + |t0| 2 Φ2
ξ =
γ 1 ⇢ (γ 1)|t0|1/2R + |t0|[(γ 1)Q + 1 2
In(B ):
U − 3ξUξ − ηUη = ±(γ + 1)UUξ
SLIDE 18 Similarity solution
∂3ξ ∂η3
!
= const
regularity condition:
ξUU − 3ξ + ηξη = ±(γ + 1)U ξ = ⌥γ + 1 2 U U 3F ⇣ η U ⌘
ξ = γ + 1 2 U − A0U 3 − A1U 2η − A2Uη2 − A3η3
ξ(U, η) :
SLIDE 19 Shock position
ξ
η
γ + 1 2 = 3A0U 2 + 2A1Uη + A2η2
U(ξ, η) vertical:
¯ ξ = ξ − ξs(η), ¯ U = U − Us(η)
¯ ξ = −A0 ¯ U ¯ U 2 − ∆2(η)
2 U − A0U 3 − A1U 2η − A2Uη2 − A3η3
SLIDE 20 Numerical simulation
with Basilisk
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 0.1 0.2 0.3 0.4 t 1/| |max 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 0.002 0.004 0.006 0.008 0.01 t Smax Level 12 Level 15 Level 18 S=0.7(t−0.511)3/2 Level 12 Level 15 Level 18 high order code 1/| |max =3.85*(t−0.511) (a) (b)
M.A. Herrada,
SLIDE 21
Parameters
before singularity
A0 A2
ξ = γ + 1 2 U − A0U 3 − A1U 2η − A2Uη2 − A3η3
SLIDE 22 Predictions
ξ = (x'− c t ' ) t '
3/2
after singularity
U = u t '
1/2
η = y' t '
1/2
SLIDE 23 Counterexample: drop coalescence
2
width r :
Aarts et al., PRL `05
