Antoine Remond-Tiedrez - Department of Mathematicsremondtiedre/talks/usc-summer... · 2020. 9....

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

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



GoalAsymptotic stability of equilibrium (i.e. global well-posedness forsmall data, and decay to equilibrium).



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



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



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]



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



Surfactants



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]



Elastic sheets



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).



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



Thank you for your attention!


Antoine Remond-Tiedrez - Department of Mathematicsremondtiedre/talks/usc-summer... · 2020. 9....

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