Heaving buoys in axisymmetric shallow water and the return to equilibrium problem Edoardo BOCCHI Supervisors: D. LANNES and C. PRANGE Institut de Math´ ematiques de Bordeaux 6 August, 2019 CEMRACS 2019, CIRM, Luminy Edoardo Bocchi (IMB, Bordeaux) CEMRACS 2019 Luminy, 6 August, 2019 1 / 16
z ζ e ζ ( t, X ) ζ i h ( t, X ) Ω( t ) ζ w ( t, X ) y I E Γ x Assumptions on the solid: Vertical side-walls Only vertical motion The contact line Γ does not depend on time ñ One free boundary problem: surface elevation ζ p t, X q Edoardo Bocchi (IMB, Bordeaux) CEMRACS 2019 Luminy, 6 August, 2019 2 / 16
Equations in the fluid domain Ω p t q for U : B t U ` U ¨ ∇ X,z U “ ´ 1 ρ ∇ X,z P ´ g e z in Ω p t q (1) div U “ 0 (2) curl U “ 0 (3) Boundary conditions at the surface and the bottom: ˜ ¸ ´ ∇ ζ z “ ζ, B t ζ ´ U ¨ N “ 0 with N “ (4) 1 z “ ´ h 0 , U ¨ e z “ 0 (5) Pressure in E : P e “ P atm (6) Constraint in I : ζ i p t, X q “ ζ w p t, X q (7) Jump at Γ : ζ e p t, ¨q ‰ ζ i p t, ¨q (8) P i p t, ¨q “ P atm ` ρg p ζ e ´ ζ i q ` P NH (9) Continuity of the normal velocity at the vertical walls: V ¨ ν “ V C ¨ ν (10) Edoardo Bocchi (IMB, Bordeaux) CEMRACS 2019 Luminy, 6 August, 2019 3 / 16
Shallow water approximation Regime: the wavelength L is larger than the depth h 0 , i.e. µ “ h 02 L 2 ! 1 Nonlinear shallow water equations ş ζ At precision O p µ q , h and Q “ ´ h 0 V dz solve $ B t h e ` ∇ ¨ Q e “ 0 ’ & B t Q e ` ∇ ¨ p 1 h e Q e b Q e q ` gh e ∇ h e “ ´ h e ρ ∇ P e “ 0 in E ’ P e “ P atm % $ B t h i ` ∇ ¨ Q i “ 0 ’ & B t Q i ` ∇ ¨ p 1 h i Q i b Q i q ` gh i ∇ h i “ ´ h i in ρ ∇ P i I ’ h i “ h w % B.C. at Γ : P i | Γ “ P atm ` ρg p ζ e ´ ζ i q | Γ ` P cor , Q e ¨ ν | Γ “ Q i ¨ ν | Γ . Edoardo Bocchi (IMB, Bordeaux) CEMRACS 2019 Luminy, 6 August, 2019 4 / 16
Axisymmetric case Cylindrical coordinates: U “ U p t, r, θ, z q , U “ p u r , u θ , u z q ù ñ Q “ Q p t, r, θ q , Q “ p q r , q θ q z ζ ( t, r ) ζ e ζ i h ( t, r ) ζ w ( t, r ) Ω ( t ) z = − h 0 r > R r < R r > R R R We assume that the flow is axisymmetric without swirl, which means that the flow has no dependence on the angular variable θ and u θ “ 0 . Edoardo Bocchi (IMB, Bordeaux) CEMRACS 2019 Luminy, 6 August, 2019 5 / 16
Axisymmetric case Cylindrical coordinates: U “ U p t, r, θ, z q , U “ p u r , u θ , u z q ù ñ Q “ Q p t, r, θ q , Q “ p q r , q θ q z ζ ( t, r ) ζ e ζ i h ( t, r ) ζ w ( t, r ) Ω ( t ) z = − h 0 r > R r < R r > R R R We assume that the flow is axisymmetric without swirl, which means that the flow has no dependence on the angular variable θ and u θ “ 0 . ù ñ Q p t, r q “ p q r , 0 q Edoardo Bocchi (IMB, Bordeaux) CEMRACS 2019 Luminy, 6 August, 2019 5 / 16
Nonlinear shallow water equations $ B t h e ` ∇ ¨ Q e “ 0 ’ & B t Q e ` ∇ ¨ p 1 h e Q e b Q e q ` gh e ∇ h e “ ´ h e in ρ ∇ P e “ 0 E ’ P e “ P atm % $ B t h i ` ∇ ¨ Q i “ 0 ’ & B t Q i ` ∇ ¨ p 1 h i Q i b Q i q ` gh i ∇ h i “ ´ h i in ρ ∇ P i I ’ h i “ h w % # P i | Γ “ P atm ` ρg p ζ e ´ ζ i q | Γ ` P cor , B.C. at Γ Q e ¨ ν | Γ “ Q i ¨ ν | Γ . ù Pressure eq: ´ ∇ ¨ p h w ρ ∇ P i q “ ´B 2 t h w ` ... Edoardo Bocchi (IMB, Bordeaux) CEMRACS 2019 Luminy, 6 August, 2019 6 / 16
Axisymmetric nonlinear shallow water equations B r q e ` q e $ B t h e ` r “ 0 , ’ ’ ’ ˆ q 2 ` q 2 & ˙ e e in p R, `8q B t q e ` B r ` gh e B r h e “ 0 , h e rh e ’ ’ ’ P e “ P atm % B r q i ` q i $ B t h i ` r “ 0 , ’ ’ ’ ˆ q 2 ` q 2 ˙ ` gh i B r h i “ ´ h i & in p 0 , R q i i B t q i ` B r ρ B r P i , h i rh i ’ ’ ’ h i “ h w % # P i “ P atm ` ρg p ζ e ´ ζ i q ` P cor , r “ R B.C. at q e “ q i . ´ ¯ ù for P cor „ q i 2 1 e ´ 1 conservation of the fluid-solid energy! | r “ R h 2 h 2 | r “ R i Edoardo Bocchi (IMB, Bordeaux) CEMRACS 2019 Luminy, 6 August, 2019 6 / 16
Solid motion Solid: G p t q “ p 0 , 0 , z G p t qq , U G p t q “ p 0 , 0 , 9 z G p t qq , ω “ 0 Define the displacement δ G p t q : “ z G p t q ´ z G,eq From the assumptions on the solid: h w p t, r q “ h w,eq p r q ` δ G p t q By the interior constraint h w “ h i we have also q i p t, r q “ ´ r 9 δ G p t q 2 Newton’s law for the conservation of the linear momentum ż R m : δ G “ ´ mg ` p P i ´ P atm q 0 Edoardo Bocchi (IMB, Bordeaux) CEMRACS 2019 Luminy, 6 August, 2019 7 / 16
Solid motion Solid: G p t q “ p 0 , 0 , z G p t qq , U G p t q “ p 0 , 0 , 9 z G p t qq , ω “ 0 Define the displacement δ G p t q : “ z G p t q ´ z G,eq From the assumptions on the solid: h w p t, r q “ h w,eq p r q ` δ G p t q By the interior constraint h w “ h i we have also q i p t, r q “ ´ r 9 δ G p t q 2 Newton’s law for the conservation of the linear momentum ż R m : δ G “ ´ mg ` p P i ´ P atm q 0 Using the elliptic equation on P i ˆ ˙ b p m ` m a p δ G qq : 9 δ 2 δ G p t q “ ´ c δ G p t q ` c ζ e p t, R q ` e p t, R q ` β p δ G q G p t q h 2 Edoardo Bocchi (IMB, Bordeaux) CEMRACS 2019 Luminy, 6 August, 2019 7 / 16
Writing u “ p ζ e , q e q : Fluid part (Hyperbolic IBVP) $ B t u ` A p u qB r u ` B p u, r q u “ 0 , r P p R, `8q ’ ’ e 2 ¨ u | r “ R “ ´ R & 9 (11) δ G p t q , 2 ’ ’ u p t “ 0 q “ u 0 . % Solid part (Nonlinear ODE) $ p m ` m a p δ G qq : δ G “ ´ c δ G ` c p e 1 ¨ u | r “ R ´ h 0 q ` p b p u q ` β p δ G qq 9 δ 2 G , ’ & δ G p 0 q “ δ 0 , 9 ’ δ G p 0 q “ δ 1 . % (12) Edoardo Bocchi (IMB, Bordeaux) CEMRACS 2019 Luminy, 6 August, 2019 8 / 16
Writing u “ p ζ e , q e q : Fluid part (Hyperbolic IBVP) $ B t u ` A p u qB r u ` B p u, r q u “ 0 , r P p R, `8q ’ ’ e 2 ¨ u | r “ R “ ´ R & 9 (11) δ G p t q , 2 ’ ’ u p t “ 0 q “ u 0 . % Solid part (Nonlinear ODE) $ p m ` m a p δ G qq : δ G “ ´ c δ G ` c p e 1 ¨ u | r “ R ´ h 0 q ` p b p u q ` β p δ G qq 9 δ 2 G , ’ & δ G p 0 q “ δ 0 , 9 ’ δ G p 0 q “ δ 1 . % (12) Theorem (E.B. ’18) Local well-posedness of the coupled system (11) - (12) for compatible initial data u 0 , δ 0 , δ 1 and u 0 P H k r pp R, `8qq with k ě 2 . Edoardo Bocchi (IMB, Bordeaux) CEMRACS 2019 Luminy, 6 August, 2019 8 / 16
Return to equilibrium It consists in dropping the solid with no initial velocity from a non-equilibrium position into a fluid initially at rest. Initial data Solid: δ G p 0 q “ δ 0 ‰ 0 , 9 δ G p 0 q “ 0 Fluid: h e p 0 , r q ” h 0 , ζ e p 0 , r q ” 0 , q e p 0 , r q ” 0 Edoardo Bocchi (IMB, Bordeaux) CEMRACS 2019 Luminy, 6 August, 2019 9 / 16
Return to equilibrium It consists in dropping the solid with no initial velocity from a non-equilibrium position into a fluid initially at rest. Initial data Solid: δ G p 0 q “ δ 0 ‰ 0 , 9 δ G p 0 q “ 0 Fluid: h e p 0 , r q ” h 0 , ζ e p 0 , r q ” 0 , q e p 0 , r q ” 0 ñ Compatibility conditions are NOT satisfied Different approach: linearized equations in the exterior domain nonlinear equations in the interior domain Edoardo Bocchi (IMB, Bordeaux) CEMRACS 2019 Luminy, 6 August, 2019 9 / 16
Hydrodynamical linear-nonlinear model (L-NL) r P p R, `8q B t ζ e ` B r q e ` q e $ r “ 0 & B t q e ` gh 0 B r ζ e “ 0 % r P p 0 , R q B t h i ` B r q i ` q i $ r “ 0 ’ ’ & ˆ q 2 ` q 2 ˙ ` gh i B r h i “ ´ h i i i B t q i ` B r ρ B r P i ’ ’ % h i rh i r “ R q e | r “ R “ ´ R 9 δ G p t q , P i | r “ R “ P atm ` ρg p ζ e ´ ζ i q | r “ R ` P cor 2 Edoardo Bocchi (IMB, Bordeaux) CEMRACS 2019 Luminy, 6 August, 2019 10 / 16
Focus on the solid equation ˆ ˙ b p m ` m a p δ G qq : δ 2 9 δ G p t q “ ´ c δ G p t q ` c ζ e p t, R q ` e p t, R q ` β p δ G q G p t q h 2 Edoardo Bocchi (IMB, Bordeaux) CEMRACS 2019 Luminy, 6 August, 2019 11 / 16
Focus on the solid equation ˆ ˙ b p m ` m a p δ G qq : δ 2 9 δ G p t q “ ´ c δ G p t q ` c ζ e p t, R q ` e p t, R q ` β p δ G q G p t q h 2 Exterior problem: B t ζ e ` B r q e ` q e $ r “ 0 q e | r “ R “ ´ R & 9 B.C. δ G p t q . 2 B t q e ` v 2 0 B r ζ e “ 0 , % Edoardo Bocchi (IMB, Bordeaux) CEMRACS 2019 Luminy, 6 August, 2019 11 / 16
Focus on the solid equation ˆ ˙ b p m ` m a p δ G qq : δ 2 9 δ G p t q “ ´ c δ G p t q ` c ζ e p t, R q ` e p t, R q ` β p δ G q G p t q h 2 Exterior problem: B t ζ e ` B r q e ` q e $ r “ 0 q e | r “ R “ ´ R & 9 B.C. δ G p t q . 2 B t q e ` v 2 0 B r ζ e “ 0 , % Linear wave equation B tt ζ e ´ v 2 0 ∆ r ζ e “ 0 Edoardo Bocchi (IMB, Bordeaux) CEMRACS 2019 Luminy, 6 August, 2019 11 / 16
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