Mathematical Analysis and Computational Simulations for flows of - PowerPoint PPT Presentation

Mathematical Analysis and Computational Simulations for flows of incompressible fluids Josef M´ alek, Charles University, Prague Computational Methods with Applications , Harrachov, August 21, 2007 • Mechanics of Incompressible Fluids based • Mathematical analysis of models (existence) on work by K.R. Rajagopal Steady flows – Framework (steady internal flows) – Navier-Stokes fluid – Constitutive equations (Expl, Impl) – Shear-rate dependent (power-law-like) fluids – Hierarchy of Power-law-like Fluids – Pressure dependent fluid – Compressible vrs Incompressible fluids – Pressure (and shear-rate) dependent fluid – Maximization of the rate of dissipation – Implicit Power-law-like Fluids – Implicit constitutive theories – Concluding remarks

1 Incompressible Fluid Mechanics: Basic Framework/1 Goal: To describe flows of various fluid-like-materials exhibiting so different and so fascinating phenomena yet share one common feature: these materials are well approximated as incompressible . SubGoal: To understand theoretical foundations and mathematical properties of these models. ̺ t + div ̺ v = 0 ( ̺ v ) t + div( ̺ v ⊗ v ) = div T + ̺ f T · D − ρdψ dt = ξ with ξ ≥ 0 • ̺ density • v = ( v 1 , v 2 , v 3 ) velocity • ψ Helmholtz free energy • f = ( f 1 , f 2 , f 3 ) external body forces • T = ( T ij ) 3 • T · D stress power i,j =1 Cauchy stress • ξ rate of dissipation • D := D ( v ) := 1 / 2( ∇ v + ( ∇ v ) T ) p := 1 • Homogeneous fluids (the density is constant) = ⇒ div v = tr D = 0 3 tr T pressure

2 Incompressible Fluid Mechanics: Basic Framework/2 Steady flows in Ω ⊂ R 3 (time discretizations lead to similar problems) div v = 0 div( v ⊗ v ) − div S = −∇ p + f in Ω v = 0 on ∂ Ω T · D = ξ with ξ ≥ 0 Constitutive equations: Relation between the Cauchy stress T , of the form T = − p I + S , and the symmetric part of the velocity gradient D := D ( v ) g ( T , D ) = 0 Power-law-like Rheology (broad, accessible) • Implicit • Explicit S = µ ( . . . ) D – Navier-Stokes Fluids S = µ ∗ D – Fluids with the yield stress (Bingham, – Power-law Fluids S = µ ∗ | D | r − 2 D Herschel-Bulkley) – Power-law-like Fluids S = µ ( | D | 2 ) D – Fluids with activation criteria – Implicit Power-law-like Fluids – Fluids with shear-rate dpt viscosity – Fluids with S = µ ( p ) D – Fluids with the yield stress – Fluids with S = µ ( p, | D | 2 ) D – Fluids with activation criteria

3 Framework is sufficiently robust to model behavior of various type of fluid-like materials. Models are frequently used in many areas of engineering and natural sciences: mechanics of colloids and suspensions, biological fluid mechanics (blood, synovial fluid), elastohydrodynamics, ice mechanics and glaciology, food processing

4 Fluids with shear-rate dependent viscosities ν ( | D | 2 ) If v = ( u ( x 2 ) , 0 , 0) , then | D ( v ) | 2 = 1 / 2 | u ′ | 2 ... shear rate. • ν ( | D | 2 ) = µ ∗ | D | r − 2 • ν ( | D | 2 ) = µ ∗ 0 + µ ∗ 1 | D | r − 2 1 < r < ∞ r > 2 • power-law model • Ladyzhenskaya model (65) • ν ( | D | 2 ) ց as | D | 2 ր • (Smagorinskii turbulence model: r = 3 ) • shear-rate thinning fluid ( r < 2 )

5 Fluids with the yield stress or the activation criteria/discontinuous stresses • threshold value for the stress to start flow • Bingham fluid • Herschel-Bingham fluid • drastic changes of the properties when certain criterion is met • formation and dissolution of blood • chemical reactions/time scale

6 Discontinuous stresses described by a maximal monotone graph T12 Tmax shear rate shear rate T1 T T2

7 Implicit power-law-like fluids � � � � S � � � � � � � � ���� ���� � � � � � � � � � � � � � � � � � � � � ���������������������������� ���������������������������� � � � � K=|u’(r)| � �

8 Fluids with pressure-dependent viscosities ν ( p ) ν ( p ) = exp( γp ) Bridgman(31): ”The physics of high pressure” Cutler, McMickle, Webb and Schiessler(58) Johnson, Cameron(67), Johnson, Greenwood(77), Johnson, Tewaarwerk(80) elastohydrodynamics: Szeri(98) synovial fluids No global existence result. • Renardy(86), local, ( ν ( p ) → 0 as p → ∞ ) p • Gazzola(97), Gazzola, Secchi(98): local, severe restrictions

9 Fluids with shear- and pressure-dependent viscosities ν ( p, | D | 2 ) η 0 − η ∞ ν ( p, | D | 2 ) = ( η ∞ + 1 + δ | D | 2 − r ) exp( γ p ) Davies and Li(94), Gwynllyw, Davies and Phillips(96) p ν ( p, | D | 2 ) = c 0 r = 1 Schaeffer(87) - instabilities in granular materials | D | r − 2 ν ( p, | D | 2 ) = ( A + (1 + exp( α p )) − q + | D | 2 ) 2 0 ≤ q ≤ 1 r − 1 2 − rA (2 − r ) / 2 α > 0 , A > 0 1 ≤ r < 2 2 α Q.: Is the dependence of the viscosity on the pressure admissible in continuum mechanics and thermodynamics?

� � 10 Compressible vrs Incompressible Compressible fluid T = T ( ρ, ∇ v ) = ⇒ T = T ( ρ, D ( v )) I D ) D 2 T = α 0 ( ρ, I D , I I D , I I I D ) I + α 1 ( ρ, I D , I I D , I I I D ) D + α 2 ( ρ, I D , I I D , I I I D := 1 [tr D ] 2 − tr D 2 I D := tr D , I , I I I D := det D 2 Linearized model: T = ˆ α 0 ( ρ ) I + λ ( ρ )[tr D ] I + 2 µ ( ρ ) D , Navier-Stokes fluid Usually: ˆ α 0 ( ρ ) is − p ( ρ ) thermodynamic pressure constitutive relation for p ( ρ ) needed to close the model = ⇒ λ, µ depends on pressure Incompressible fluid I D ) D 2 T = − p I + ˆ α 1 ( I I D , I I I D ) D + ˆ α 2 ( I I D , I I Drawbacks: • p in general is not the mean normal stress • Can ˆ α i depend on p ? No, as the derivation based on the principle: Constraints do no work

11 Maximization of the rate of dissipation subject to two constraints Assume ξ = 2 ν (tr T , | D | 2 ) | D | 2 and ψ = ψ ( θ, ρ ) = const Maximizing ξ with respect to D on the manifold described by the constraints (1) ξ = T · D (2) div v = tr D = 0 results to ∂ D − λ 1 I − λ 2 ( T − ∂ξ ∂ξ ∂ D ) = 0 = ⇒ p := 1 T = − p I + 2 ν ( p, | D | 2 ) D with 3 tr T Viscosity: resistance between two sliding surfaces of fluids

12 Implicit theories Derivation of the model from the implicit constitutive eqn g ( T , D ) = 0 Isotropy of the material implies α 0 I + α 1 T + α 2 D + α 3 T 2 + α 4 D 2 + α 5 ( T D + DT ) + α 6 ( T 2 D + DT 2 ) + α 7 ( T D 2 + D 2 T ) + α 8 ( T 2 D 2 + D 2 T 2 ) = 0 α i being a functions of ρ, tr T , tr D , tr T 2 , tr D 2 , tr T 3 , tr D 3 , tr( T D ) , tr( T 2 D ) , tr( D 2 T ) , tr( D 2 T 2 ) For incompressible fluids T = 1 3 tr T I + ν (tr T , tr D 2 ) D

13 Analysis of PDEs - Existence of (weak) solution - steady flows Z • Mathematical consistency of the model (well-posedness), existence for any data in a reasonable function space • Notion of the solution: balance equations for any subset of Ω ⇐ ⇒ weak solution ⇐ ⇒ FEM • Choice of the function spaces • To know what is the object we approximate • Ω ⊂ R 3 open, bounded with the Lipschitz boundary ∂ Ω div v = 0 in Ω (1) − div(2 ν ( p, | D ( v ) | 2 ) D ( v )) + div( v ⊗ v ) = −∇ p + f 1 v = 0 on ∂ Ω pdx = p 0 (2) | Ω | Ω (0) ν ( ... ) = ν 0 NSEs (1) ν ( ... ) = ν ( | D | 2 ) (2) ν ( ... ) = ν ( p, | D | 2 ) (3) discontinuous (implicit) power-law fluids

14 NSEs − µ ∗ ∆ v + div( v ⊗ v ) = −∇ p + f in Ω div v = 0 Step 1. Finite-dimensional approximations: ( v N , p N ) - Fixed-point iterations sup N �∇ v N ) � 2 2 + sup N � p N � 2 2 ≤ K Step 2. Uniform estimates: Step 3. Weak compactness: ∇ v N → v and p N → p weakly in L 2 Step 4. Nonlinearity tretaed by compact embedding: v N → v strongly in L 2 Step 5. Limit in approximations as N → ∞ µ ∗ ( ∇ v N , ∇ ϕ ) − ( v N ⊗ v N , ∇ ϕ ) = ( p N , div ϕ ) + � f , ϕ � ∀ ϕ ( v , p ) is a solution

15 µ ( | D ( v ) | 2 ) = µ ∗ | D ( v ) | r − 2 D ( v ) - Critical values for the power-law index − µ ∗ div( | D ( v ) | r − 2 D ( v )) + div( v ⊗ v ) = −∇ p + f in Ω div v = 0 r ≤ K - expect that v ∈ W 1 ,r Energy estimates: sup N �∇ v N � r 0 , div (Ω) • Equation for the pressure: p = ( − ∆) − 1 div div( v ⊗ v − µ ∗ | D ( v ) | r − 2 D ( v )) 2(3 − r ) and | D ( v ) | r − 2 D ( v ) ∈ L r ′ with r ′ := r/ ( r − 1) 3 r W 1 ,r ֒ → L 3 r/ (3 − r ) = ⇒ v ⊗ v ∈ L r 3 r ⇒ r ≥ 9 r − 1 ≤ 2(3 − r ) ⇐ 5 • If v , ϕ ∈ W 1 ,r then v ⊗ v · ∇ ϕ ∈ L 1 only for r ≥ 9 5 • For r ≥ 9 5 the energy equality holds, higher differentiability accessible (useful tools) – Compactness of the quadratic nonlinearity requires W 1 ,r ֒ → L 2 : holds for r ≥ 6 → ֒ 5 Analysis easier for r ≥ 9 / 5 and more difficult for r ∈ (6 / 5 , 9 / 5) , in both cases more difficult than NSEs