Variational methods in fluid-structure interactions: Dynamics, dissipation, constraints, and Darcy’s law in moving media Vakhtang Putkaradze Mathematical and Statistical Sciences, University of Alberta, Canada July 7, 2017; DarrylFest70: ICMAT, Madrid Joint work with Francois Gay-Balmaz, (CNRS and ENS, Paris) Akif Ibragimov (Texas Tech) and Dmitry Zenkov (NCSU) 1 Supported by NSERC Discovery grant and University of Alberta 2 FGB & VP, Comptes Rendus M´ ecanique 342 , 79-84 (2014) J. Nonlinear Science , March 18 (2015); ecanique , Comptes Rendus M´ (2016) dx.doi.org/10.1016/j.crme.2016.08.004 Vakhtang Putkaradze Variational methods in fluid-structure interactions
Outline 1 Problem formulation: tubes conveying fluid 2 Variational derivation of tube-fluid equations 3 Discretization of tube-fluid equations with examples 4 Compressible gas in stretchable tube 5 Introduction of dissipation 6 A simple problem: tube pendulum with a droplet 7 Poromechanics and Darcy’s law as dynamical limit 8 Conclusions and open questions Vakhtang Putkaradze Variational methods in fluid-structure interactions
Variational treatment of a tube conveying fluid Figure: Image of a garden hose and its mathematical description No friction in the system for now, incompressible fluid, Reynolds numbers ∼ 10 4 (much higher in some applications), general 3D motions Hose can stretch and bend arbitrarily (inextensible also possible) Cross-section of the hose changes dynamically with deformations: collapsible tube Vakhtang Putkaradze Variational methods in fluid-structure interactions
Previous work Constant fluid velocity in the tube , 2D dynamics: English: Benjamin (1961); Gregory, Pa¨ ıdoussis (1966); Pa¨ ıdoussis (1998); Doare, De Langre (2002); Flores, Cros (2009), . . . Russian: Bolotin (?) (1956), Svetlitskii (monographs 1982, 1987), Danilin (2005), Zhermolenko (2008), Akulenko et al. (2015) . . . Hard to generalize to general 3D motions Not possible to consistently incorporate the cross-sectional dynamics Elastic rod with directional (tangent) momentum source at the end – the follower-force method, see Bou-Rabee, Romero, Salinger (2002), critiqued by Elishakoff (2005). Shell models: Paidoussis & Denise (1972), Matsuzaki & Fung (1977), Heil (1996), Heil & Pedley (1996) , . . . : Complex, computationally intensive, difficult (impossible) to perform analytic work for non-straight tubes. 3D dynamics from Cosserat’s model (Beauregard, Goriely & Tabor 2010): Force balance, not variational, cannot accommodate dynamical change of the cross-section. Variational derivation: FGB & VP (2014,2015). Vakhtang Putkaradze Variational methods in fluid-structure interactions
Variational treatment of changing cross-sections dynamics Mathematical preliminaries: Rod dynamics is described by SE (3)-valued functions (rotations and 1 translations in space) π ( s , t ) = (Λ , r )( s , t ). Fluid dynamics inside the rod is described by 1D diffeomorphisms 2 s = ϕ ( a , t ), where a is the Lagrangian label. Conservation of 1-form volume element (fluid incompressibility) 3 defined through a holonomic constraint: � � d r � � � � Q 0 ◦ ϕ − 1 ( s , t ) ∂ s ϕ − 1 ( s , t ) Q := A � = (1) � � � ds where area A depends on the deformations of the tube. Alternatively, evolution equation for Q is ∂ t Q + ∂ s ( Qu ) = 0. 4 Note that commonly used Au =const does not conserve volume for 5 time-dependent flow. See e.g. [Kudryashov et al , Nonlinear dynamics (2008)] for correct derivation in 1D. Vakhtang Putkaradze Variational methods in fluid-structure interactions
Mathematical preliminaries: Geometric rod theory for elastic rods I Purely elastic Lagrangian r , r ′ , Λ , ˙ Λ , Λ ′ ) L = L ( r , ˙ Use SE (3) symmetry reduction [Simo, Marsden, Krishnaprasad 1988] (SMK) to reduce the Lagrangian to ℓ ( ω , γ , Ω , Γ ) of the following coordinate-invariant variables (prime= ∂ s , dot= ∂ t ): Γ = Λ − 1 r ′ , Ω = Λ − 1 Λ ′ , (2) r , ω = Λ − 1 ˙ γ = Λ − 1 ˙ Λ . (3) Note that symmetry reduction for elastic rods is left-invariant (reduces to body variables). Notation: small letters (e.g. ω , γ ) denote time derivatives; capital letters (e.g. Ω , Γ ) denote the s -derivatives. Vakhtang Putkaradze Variational methods in fluid-structure interactions
Mathematical preliminaries: Geometric rod theory for elastic rods II Euler Poincar´ e theory: [Holm, Marsden, Ratiu 1998]. For elastic rods: compute variations as in [Ellis, Holm, Gay-Balmaz, VP and Ratiu, Arch. Rat.Mech. Anal. , (2010)]: consider Σ = Λ − 1 δ Λ ∈ so (3) and Ψ = Λ − 1 δ r ∈ R 3 , and ( Σ , Ψ ) ∈ se (3). δ ω = ∂ Σ δ γ = ∂ ψ ∂ t + ω × Σ , ∂ t + γ × Σ + ω × ψ (4) δ Ω = ∂ Σ δ Γ = ∂ ψ ∂ s + Ω × Σ , ∂ s + Γ × Σ + Ω × ψ , (5) Compatibility conditions (cross-derivatives in s and t are equal) Ω t − ω s = Ω × ω , Γ t + ω × Γ = γ s + Ω × γ . � Critical action principle δ ℓ d t d s = 0+ (4,5) give SMK equations. � � δℓ � � δℓ � � � 0 = δ ℓ d t d s = δ ω , δ ω + δ Ω , δ Ω + . . . � = � linear momentum eq , Ψ � + � angular momentum eq , Σ � d t d s Vakhtang Putkaradze Variational methods in fluid-structure interactions
Mathematics preliminaries: incompressible fluid motion Following Arnold (1966), describe a 3D incompressible fluid motion by Diff Vol group r = ϕ ( a , t ). Eulerian fluid velocity is u = ϕ t ◦ ϕ − 1 ; symmetry-reduced � u 2 d r . Lagrangian is ℓ = 1 / 2 Variations of velocity are computed as η = δϕ ◦ ϕ − 1 ( s , t ) , δ u = η t + u ∇ η − η ∇ u . (6) Incompressibility condition � � ∂ r � � J = � = 1 ⇒ Lagrange multiplier p . � � ∂ a � � Euler equations: δ ℓ d V d t = 0 with (6) and ( ?? ) ∂ u ∂ t + u · ∇ u = −∇ p , div u = 0 Further considerations: α -model, Complex fluids etc. : D. D. Holm & many others Vakhtang Putkaradze Variational methods in fluid-structure interactions
Garden hoses: Lagrangian and symmetry reductions Symmetry group of the system (ignoring gravity for now) 1 G = SE (3) × Diff A ( R ) = SO (3) � R × Diff A ( R ) . (7) Position of elastic tube and fluid: 2 �� � � � � � � , π · r f ◦ ϕ − 1 ( s , t ) ( π, ϕ ) · Λ 0 , r t , 0 , r f = π · Λ 0 , r t , 0 . � �� � � �� � right invariant left invariant Velocities: 3 � � = d � � r ( s , t ) , r ◦ ϕ − 1 ( s , t ) v r , v f dt � � r ◦ ϕ − 1 ( s , t ) + r ′ ( s , t ) u ( s , t ) = ˙ r ( s , t ) , ˙ . (8) Change in cross-section A = A ( Ω , Γ ) 4 Incompressibility condition J = A ( s , t ) ∂ a ∂ s | Γ | = 1 with Lagrange 5 multiplier µ (pressure) ∂ Q ∂ t + ∂ ∂ s ( Qu ) = 0 , with Q = A | Γ | . (9) Vakhtang Putkaradze Variational methods in fluid-structure interactions
Equations of motion � δℓ � δℓ � � ( ∂ t + ω × ) δℓ δ ω + γ × δℓ δ Ω − ∂ Q δ Γ − ∂ Q δ γ + ( ∂ s + Ω × ) ∂ Ω µ + Γ × ∂ Γ µ = 0 � δℓ � ( ∂ t + ω × ) δℓ δ Γ − ∂ Q δ γ + ( ∂ s + Ω × ) ∂ Γ µ = 0 m := 1 δℓ m t + ∂ s ( mu − µ ) = 0 , Q δ u ∂ t Q + ∂ s ( Qu ) = 0 , Q = A | Γ | Compatibility condition: Λ st = Λ ts , r st = r ts ∂ t Ω = ω × Ω + ∂ s ω , ∂ t Γ + ω × Γ = ∂ s γ + Ω × γ Assume A = A ( Ω , Γ ) , symmetric tube with axis E 1 for Lagrangian ℓ ( ω , γ , Ω , Γ , u ) � � = 1 − λ | Γ − E 1 | 2 � α | γ | 2 + � � + ρ A ( Ω , Γ ) | γ + Γ u | 2 − � � I ω , ω J Ω , Ω | Γ | d s . 2 See FGB & VP for linear stability analysis, nonlinear solutions etc. Vakhtang Putkaradze Variational methods in fluid-structure interactions
Non-conservation of energy Define the energy function � δℓ � L � δ ω · ω + δℓ δ γ · γ + δℓ e ( ω , γ , Ω , Γ , u ) = d s − ℓ ( ω , γ , Ω , Γ , u ) δ u u 0 and boundary forces at the exit (free boundary) � � � F u := δℓ F Γ := δℓ δ Γ − µ∂ Q F Ω := δℓ δ Ω − µ∂ Q � � � δ u u − µ Q s = L , s = L , s = L . � � � ∂ Γ ∂ Ω Then, the energy changes according to � T � d s =0 � dt e ( ω , γ , Ω , Γ , u ) = ( F Ω · Ω + F Γ · Γ + F u u ) s = L d t . � 0 The system is not closed and the energy is not conserved. Similar statement is true for variational discretization. Vakhtang Putkaradze Variational methods in fluid-structure interactions
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