Setup Goal Difficulties Key “tool” Variants
Viscous surface waves and their stability
Antoine Remond-Tiedrez
Carnegie Mellon University
USC Summer School on Mathematical Fluids, May 2017
Antoine Remond-Tiedrez (CMU) Viscous surface waves and their stability May 2017
Setup Goal Difficulties Key “tool” Variants
Setup
η(t)
d
Σ(t)
Σb
Ω(t) ⊆ R3
ρ(∂tu + (u · ∇) u
)+∇ · S = 0 in Ω(t)
∇ · u = 0 in Ω(t)
Sν = −σHν + ρgην on Σ(t)
∂tη = (u · ν)
√1 + |∇η|2 on Σ(t)
u = 0 on Σb
• S = pI − µDu is the stress tensor
• H is the mean curvature of the free surface
• ρ, σ, µ > 0 are constants
Antoine Remond-Tiedrez (CMU) Viscous surface waves and their stability May 2017
Setup Goal Difficulties Key “tool” Variants
GoalAsymptotic stability of equilibrium (i.e. global well-posedness forsmall data, and decay to equilibrium).
Antoine Remond-Tiedrez (CMU) Viscous surface waves and their stability May 2017
Setup Goal Difficulties Key “tool” Variants
Difficulties
• Semilinear problem in a time-dependent domain←→ Quasilinear problem in a fixed domain
• System of PDEs of mixed type
ρ(∂At u + (u · ∇A) u
)− µ∆Au +∇Ap = 0 in Ω ∼ parabolic in u
∇A · u = 0 in Ω
(pI − µDAu) ν = −σ ∇ ·
∇η√1 + |∇η|2
ν + ρgην on Ω ∼ elliptic in h
∂tη + (u1, u2) · ∇η = u3 on Σ ∼ hyperbolic in h
u = 0 on Σb
Antoine Remond-Tiedrez (CMU) Viscous surface waves and their stability May 2017
Setup Goal Difficulties Key “tool” Variants
Fixing the domain
Σ(t)
Σb
Ω(t)Φ−1
Σ
Σb
Ω
Φ = I + χηe3, where χ =
1 on Σ
0 on Σb
→ ∇Φ ∼ ∇η
∇A is the manifestation of ∇ in the new coordinates, i.e. ∇A is“Φ-conjugate” to ∇ (and similarly for ∂At )
∇Au := ∇(u Φ−1
) Φ ∼ (∇u) (∇η)−1
Antoine Remond-Tiedrez (CMU) Viscous surface waves and their stability May 2017
Setup Goal Difficulties Key “tool” Variants
Key “tool”: energy-dissipation relation
d
dt
(∫Ω(t)
ρ
2|u|2 +
∫R2
σ
√1 + |∇η|2 +
∫R2
g
2|η|
2)
︸ ︷︷ ︸Energy
+
∫Ω(t)
µ
2|Du|2︸ ︷︷ ︸
Dissipation
= 0
If moreover D is coercive over E , thenE +D = 0
CE ≤ D
→ E + CE ≤ 0
→ E(t) ≤ E(0)e−Ct
i.e. exponential decay to equilibrium (of small data) [Guo & Tice 2013]
Antoine Remond-Tiedrez (CMU) Viscous surface waves and their stability May 2017
Setup Goal Difficulties Key “tool” Variants
Variants
• Surfactants
• Elastic sheets
Incorporating various additional physics effects boils down totinkering with the surface energy.Recall that the surface energy associated with surface tension is∫
R2
σ
√1 + |∇η|2 =
∫Σ(t)
σ
which appears in the dynamic boundary condition via
Sν = −σHν︸ ︷︷ ︸δ(∫
Σ(t) σ)+ ρgην
Antoine Remond-Tiedrez (CMU) Viscous surface waves and their stability May 2017
Setup Goal Difficulties Key “tool” Variants
Surfactants
Antoine Remond-Tiedrez (CMU) Viscous surface waves and their stability May 2017
Setup Goal Difficulties Key “tool” Variants
SurfactantsSurfactants introduce Marangoni forces, due to surface tensiongradients. In other words, the surface energy changes∫
Σ(t)σ 7→
∫Σ(t)
σ(c)
and therefore the dynamic boundary condition changes
i.e. Sν = −σHν +ρgην on Σ(t)
becomes Sν = −σ(c)Hν +∇Σ(t)σ(c) +ρgην on Σ(t)
We must also specify the dynamics of the surfactants
∂tc +∇Σ(t) · (cu) = ∆Σ(t)c
Once again, exponential decay to equilibrium for small data can beshown [Kim & Tice 2016]
Antoine Remond-Tiedrez (CMU) Viscous surface waves and their stability May 2017
Setup Goal Difficulties Key “tool” Variants
Elastic sheets
Antoine Remond-Tiedrez (CMU) Viscous surface waves and their stability May 2017
Setup Goal Difficulties Key “tool” Variants
Elastic sheetsThe elastic sheet covering the free surface is accounted for byadding an elastic surface energy term.∫
Σ(t)σ 7→
∫Σ(t)
σ +
∫Σ(t)
1
2H2
and therefore an additional term appears in the dynamic boundarycondition
Sν = −σHν +
[∆Σ(t)H−H
(H2 − 4K
)]ν + ρgην on Σ(t)
Once again, small data decays to equilibrium exponentially fast(coming soon-ish to a journal near you).
Antoine Remond-Tiedrez (CMU) Viscous surface waves and their stability May 2017
Setup Goal Difficulties Key “tool” Variants
Elastic sheetsRecall that the total surface energy is
σ
∫Σ(t)
1 +
∫Σ(t)
1
2H2
where∫Σ(t)
1 ∼∫R2
1
2|∇η|2 since
(1 + |∇η|2
)1/2∼ 1 +
1
2|∇η|2∫
Σ(t)
1
2H2 ∼
∫R2
1
2|∇2η|2 since |H|2 ∼ |∆η|2
and
∫R2
|∆η|2 =
∫R2
|∇2η|2
Antoine Remond-Tiedrez (CMU) Viscous surface waves and their stability May 2017
Setup Goal Difficulties Key “tool” Variants
Thank you for your attention!
Antoine Remond-Tiedrez (CMU) Viscous surface waves and their stability May 2017