On three-dimensional flows of activated fluids Josef Mlek Neas - PowerPoint PPT Presentation

On three-dimensional flows of activated fluids Josef Málek Nečas Center for Mathematical Modeling and Mathematical institute Charles University, Faculty of Mathematics and Physics - Anna Abbatiello, Tomáš Los and Ondřej Souček Jan Blechta and K.R. Rajagopal Miroslav Bulíček September 3, 2018

Section 1 Foreword

Soil liquefaction occurs when a saturated or partially saturated soil substantially loses strength and stiffness in response to an applied stress such as shaking during an earthquake or other sudden change in stress condition, in which material that is ordinarily a solid behaves like a liquid. Source: Wikipedia. Photo: Niigata earthquake 1964.

Geometry and structure of material Typical problem geometry and zoom into the structure of the granular material composed of solid grain matrix filled with an interstitial fluid. Mixture or single continuum model ?

Formulation of the problem PROBLEM div v = 0 � ∂ t v + div( v ⊗ v ) − div S = −∇ p in Q T ∂ t p f + v · ∇ p f − ∆ p f = 0 | S | ≤ τ ( p f ) ⇐ ⇒ D = O , � in Q T S = τ ( p f ) D | S | > τ ( p f ) ⇐ ⇒ | D | + 2 ν ∗ D . v · n = 0 g ( s , v τ ) = 0 ∇ p f · n = 0 on Σ T v (0 , · ) = v 0 p f = p 0 in Ω DATA ◮ Ω ⊂ R 3 , T > 0 , v 0 , p 0 , p s τ ( p f ) := ( p s − p f ) + and L. Chupin and J. Mathé, Existence theorem for homogeneous incompressible Navier-Stokes equation with variable theology, European Journal of Mechanics. B. Fluids 61 (2017) 135-143.

Two characterizations of the Bingham fluids (I) Dichotomy | S | ≤ τ ∗ ⇐ ⇒ D = O , D | S | > τ ∗ ⇐ ⇒ S = τ ∗ | D | + 2 ν ∗ D . (II) Implicit constitutive tensorial relation 2 ν ∗ D = ( | S | − τ ∗ ) + S | S | (III) Two scalar constraints | Z | ≤ τ ∗ and Z : D ≥ τ ∗ | D | Z := S − 2 ν ∗ D Earlier works & Tools: Duvaut & Lions (1976), Aubin & Frankowska (1995), Fuchs & Seregin (2000), Shelukhin (2002) - variational inequalities, calculus of multivalued functions, calculus of variations, regularity theory

Two characterizations of the Bingham fluids (I) Dichotomy | S | ≤ τ ∗ ⇐ ⇒ D = O , D | S | > τ ∗ ⇐ ⇒ S = τ ∗ | D | + 2 ν ∗ D . (II) Implicit constitutive tensorial relation 2 ν ∗ D = ( | S | − τ ∗ ) + S | S | (III) Two scalar constraints | Z | ≤ τ ∗ and Z : D ≥ τ ∗ | D | Z := S − 2 ν ∗ D Earlier works & Tools: Duvaut & Lions (1976), Aubin & Frankowska (1995), Fuchs & Seregin (2000), Shelukhin (2002) - variational inequalities, calculus of multivalued functions, calculus of variations, regularity theory

Questions Chupin and Mathé • considered Z : D = τ ( p f ) | D | , but there is one problem with their convergence argument that can be removed if Z : D ≥ τ ( p f ) | D | is used instead of Z : D = τ ( p f ) | D | • established the existence result in two dimensions. Energy equality available. Critical problem as 2d NS but with an additional nonlinearity. Questions • Is the model suitable to describe the “liquefaction"? • Tensorial response of Bingham fluid characterized by two scalar constraints (one inequality, one equality). How one can exploit it? • Is it possible to develop mathematical theory in three dimensions? .... to be continued later ....

Questions Chupin and Mathé • considered Z : D = τ ( p f ) | D | , but there is one problem with their convergence argument that can be removed if Z : D ≥ τ ( p f ) | D | is used instead of Z : D = τ ( p f ) | D | • established the existence result in two dimensions. Energy equality available. Critical problem as 2d NS but with an additional nonlinearity. Questions • Is the model suitable to describe the “liquefaction"? • Tensorial response of Bingham fluid characterized by two scalar constraints (one inequality, one equality). How one can exploit it? • Is it possible to develop mathematical theory in three dimensions? .... to be continued later ....

Section 2 Viscous fluids and visco-elastic fluids

Unsteady flows of incompressible fluids Ω ⊂ R 3 Governing equations div v = 0 � ∂ v ∂t + div( v ⊗ v ) = −∇ p + div S in (0 , T ) × Ω S = S T v · n = 0 } on (0 , T ) × ∂ Ω v (0 , · ) = v 0 } in Ω A : B := � 3 Energy balance i,j =1 A ij B ij ∂ | v | 2 � | v | 2 � 1 + div 2 v + p v − S v + S : ∇ v = 0 2 ∂t d ˆ ˆ ˆ | v | 2 + 2 ( | v | 2 + 2 p )( v · n ) − 2 S : ( v ⊗ n ) = 0 S : ∇ v + dt Ω Ω ∂ Ω

Unsteady flows of incompressible fluids Ω ⊂ R 3 Governing equations div v = 0 � ∂ v ∂t + div( v ⊗ v ) = −∇ p + div S in (0 , T ) × Ω S = S T v · n = 0 } on (0 , T ) × ∂ Ω v (0 , · ) = v 0 } in Ω A : B := � 3 Energy balance i,j =1 A ij B ij ∂ | v | 2 � | v | 2 � 1 + div 2 v + p v − S v + S : ∇ v = 0 2 ∂t d ˆ ˆ ˆ | v | 2 + 2 ( | v | 2 + 2 p )( v · n ) − 2 S : ( v ⊗ n ) = 0 S : ∇ v + dt Ω Ω ∂ Ω

Unsteady flows of incompressible fluids Ω ⊂ R 3 Governing equations div v = 0 � ∂ v ∂t + div( v ⊗ v ) = −∇ p + div S in (0 , T ) × Ω S = S T v · n = 0 } on (0 , T ) × ∂ Ω v (0 , · ) = v 0 } in Ω A : B := � 3 Energy balance i,j =1 A ij B ij ∂ | v | 2 � | v | 2 � 1 + div 2 v + p v − S v + S : ∇ v = 0 2 ∂t d ˆ ˆ ˆ | v | 2 + 2 ( | v | 2 + 2 p )( v · n ) − 2 S : ( v ⊗ n ) = 0 S : ∇ v + dt Ω Ω ∂ Ω

Internal flows ˆ ˆ ˆ � � ( − S ) : ( v ⊗ n ) = ( − S ) n · v = ( − S ) v τ · v τ ∂ Ω ∂ Ω ∂ Ω S n Boundary conditions ( S n ) τ • v · n = 0 on ∂ Ω n • constitutive equation involving v τ and/or ( − S n ) τ s Ω s := ( − S n ) τ z τ := z − ( z · n ) n ∂ Ω ˆ ˆ ˆ � � ( − S ) : ( v ⊗ n ) = ( − S ) n · v = ( − S n τ · v τ ∂ Ω ∂ Ω ∂ Ω v τ = 0 no slip boundary condition s = γ ∗ v τ with γ ∗ > 0 Navier’s slip boundary condition s = 0 (perfect) slip boundary condition

Energy estimates and constitutive equations Ω ⊂ R 3 • Governing equations div v = 0 � in (0 , T ) × Ω ∂ v S = S T ∂t + div( v ⊗ v ) = −∇ p + div S , v · n = 0 } on (0 , T ) × ∂ Ω v (0 , · ) = v 0 } in Ω D := 1 � ∇ v + ( ∇ v ) T � • Energy equality valid for t ∈ (0 , T ] 2 ˆ t ˆ t ˆ ˆ � v ( t ) � 2 s · v τ = � v 0 � 2 2 + 2 S : D + 2 2 0 Ω 0 ∂ Ω • To close the system we add a material dependent relation involving S and D we add a material dependent relation involving s and v τ Constitutive equations

Energy estimates and constitutive equations Ω ⊂ R 3 • Governing equations div v = 0 � in (0 , T ) × Ω ∂ v S = S T ∂t + div( v ⊗ v ) = −∇ p + div S , v · n = 0 } on (0 , T ) × ∂ Ω v (0 , · ) = v 0 } in Ω D := 1 � ∇ v + ( ∇ v ) T � • Energy equality valid for t ∈ (0 , T ] 2 ˆ t ˆ t ˆ ˆ � v ( t ) � 2 s · v τ = � v 0 � 2 2 + 2 S : D + 2 2 0 Ω 0 ∂ Ω • To close the system we add a material dependent relation involving S and D we add a material dependent relation involving s and v τ Constitutive equations

Energy estimates and constitutive equations Ω ⊂ R 3 • Governing equations div v = 0 � in (0 , T ) × Ω ∂ v S = S T ∂t + div( v ⊗ v ) = −∇ p + div S , v · n = 0 } on (0 , T ) × ∂ Ω v (0 , · ) = v 0 } in Ω D := 1 � ∇ v + ( ∇ v ) T � • Energy equality valid for t ∈ (0 , T ] 2 ˆ t ˆ t ˆ ˆ � v ( t ) � 2 s · v τ = � v 0 � 2 2 + 2 S : D + 2 2 0 Ω 0 ∂ Ω • To close the system we add a material dependent relation involving S and D we add a material dependent relation involving s and v τ Constitutive equations

Classes of constitutive equations div v = 0 ∂ v S = S T ∂t + div( v ⊗ v ) = −∇ p + div S , ( 1 ) G ( S , D ) = O implicit algebraic equations ∗ ∗ ∗ ( 2 ) G ( S , S , D , D ) = O A an objective time derivative rate type viscoelastic fluids ∗ ∗ ( 3 ) G ( S , S , D , D ) − ∆ S = O rate type viscoelastic fluids with stress diffusion ∗∗ ∗ ∗∗ ∗ ( 4 ) G ( S , S , S , D , D , D ) = O rate type viscoelastic fluids of higher order

Classes of constitutive equations div v = 0 ∂ v S = S T ∂t + div( v ⊗ v ) = −∇ p + div S , ( 1 ) G ( S , D ) = O implicit algebraic equations ∗ ∗ ∗ ( 2 ) G ( S , S , D , D ) = O A an objective time derivative rate type viscoelastic fluids ∗ ∗ ( 3 ) G ( S , S , D , D ) − ∆ S = O rate type viscoelastic fluids with stress diffusion ∗∗ ∗ ∗∗ ∗ ( 4 ) G ( S , S , S , D , D , D ) = O rate type viscoelastic fluids of higher order

Section 3 Implicit constitutive equations and implicitly stated boundary conditions

G ( S , D ) = O KR Rajagopal (2003) S = 2 ν D Navier-Stokes 2 ν ( | S | 2 , | D | 2 ) D = 2 α ( | S | 2 , | D | 2 ) S generalized viscosity 2 ν D = ( | S | − σ ∗ ) + S Bingham | S | 2 ν S = ( | D | − d ∗ ) + 1 D Euler/Navier-Stokes | D | K. R. Rajagopal: On implicit constitutive theories. Appl. Math. , 48 (2003) 279—319. J. Málek, V. Průša : Derivation of equations of continuum mechanics and thermodynamics of fluids, Handbook of Mathematical Analysis in Mechanics of Viscous Fluids , (eds.Y. Giga, A. Novotný), Springer available online (2017)