On isothermal steady flows of incompressible, pressure-thickening - PowerPoint PPT Presentation

On isothermal steady flows of incompressible, pressure-thickening and shear-thinning fluids and their Galerkin approximation. orfer 1 , 5 M. Lanzend¨ presenting what he understood from alek 1 , 2 and M. Bul´ cek 1 , 2 , J. M´ ıˇ and collaborated on with A. Hirn 3 and J. Stebel 2 , 4 . 1 Faculty of Mathematics and Physics, Charles University in Prague 2 Jindˇ rich Neˇ cas Center for Mathematical Modeling 3 University of Heidelberg, 4 Mathematical Institute, AS CR 5 Institute of Computer Science, AS CR M. Lanzend¨ orfer et al. (ICS AS CR) Incompressible piezoviscous fluids January 13, 2011 1 / 14

➊ Motivated by several practical applications, we consider ➋ incompressible fluids with viscosity depending on shear rate and on pressure. The latter dependence, in particular, leads to difficulties in both theory and numerical simulations. We will focus on isothermal steady flows of a subclass of such fluids; ➌ briefly discuss the known results on existence of weak solutions; ➍ show the connection of the viscosity–pressure relation with the inf–sup inequality and the stable Galerkin discretization; ➎ mention the relation of inf–sup inequality to the pressure boundary conditions. ➏ We will advert to open problems and disclose some troubles occurring in numerical experiments (motivated by the lubrication problems). M. Lanzend¨ orfer et al. (ICS AS CR) Incompressible piezoviscous fluids January 13, 2011 2 / 14

Incompressible fluids with viscosity depending on pressure and shear rate Mathematical formulation inside (0 , T ) × Ω: div v v v = 0 ∂ τ v v v + div( v v v ⊗ v v v ) − div S S S = −∇ π + f f f , v ) | 2 ) D S S S = 2 ν ( π, | D D D ( v v D D ( v v v ) M. Lanzend¨ orfer et al. (ICS AS CR) Incompressible piezoviscous fluids January 13, 2011 4 / 14

Applications: lubrication problems, journal bearing M. Lanzend¨ orfer et al. (ICS AS CR) Incompressible piezoviscous fluids January 13, 2011 5 / 14

Applications: lubrication problems, journal bearing M. Lanzend¨ orfer et al. (ICS AS CR) Incompressible piezoviscous fluids January 13, 2011 5 / 14

Applications: lubrication problems, journal bearing M. Lanzend¨ orfer et al. (ICS AS CR) Incompressible piezoviscous fluids January 13, 2011 5 / 14

Viscosity and volume variation with pressure for squalane (representing a low viscosity paraffinic mineral oil, see S. Bair, Tribology Letters , 2006). 8 10 7 10 1 6 10 0.8 viscosity [mPa s] 5 10 V / V 0 0.6 4 10 0.4 3 10 0.2 2 10 1 10 0 0 200 400 600 800 1000 1200 0 100 200 300 400 p [MPa] p [MPa] M. Lanzend¨ orfer et al. (ICS AS CR) Incompressible piezoviscous fluids January 13, 2011 6 / 14

Viscosity for SAE 10W/40 reference oil RL 88/1 , (partly) by Hutton, Jones, Bates, SAE , 1983 M. Lanzend¨ orfer et al. (ICS AS CR) Incompressible piezoviscous fluids January 13, 2011 8 / 14

Incompressible fluids with viscosity depending on pressure and shear rate Mathematical formulation inside (0 , T ) × Ω: div v v v = 0 ∂ τ v v v + div( v v v ⊗ v v v ) − div S S S = −∇ π + f f f , v ) | 2 ) D S S S = 2 ν ( π, | D D D ( v v D D ( v v v ) Viscosity formulas used in applications � ∼ exp( απ ) , v ) | 2 ) = ν = ν ( π, | D D D ( v v p − 2 v ) | 2 ) 2 , ∼ (1 + | D D ( v D v 1 < p < 2 M. Lanzend¨ orfer et al. (ICS AS CR) Incompressible piezoviscous fluids January 13, 2011 9 / 14

Incompressible fluids with viscosity depending on pressure and shear rate Mathematical formulation inside (0 , T ) × Ω: div v v v = 0 ∂ τ v v + div( v v v ⊗ v v v v ) − div S S S = −∇ π + f f f , v ) | 2 ) D S S S = 2 ν ( π, | D D D ( v v D D ( v v ) v Problem well-posedness—first observations ν = ν ( π ) ◮ M. Renardy, Comm. Part. Diff. Eq. , 1986. ◮ F. Gazzola, Z. Angew. Math. Phys. , 1997. ◮ F. Gazzola, P. Secchi, Navier–Stokes eq.: th. and num. meth. 1998. M. Lanzend¨ orfer et al. (ICS AS CR) Incompressible piezoviscous fluids January 13, 2011 9 / 14

Incompressible fluids with viscosity depending on pressure and shear rate Mathematical formulation inside (0 , T ) × Ω: div v v v = 0 ∂ τ v v v + div( v v v ⊗ v v v ) − div S S S = −∇ π + f f f , v ) | 2 ) D S S S = 2 ν ( π, | D D D ( v v D D ( v v v ) Problem well-posedness—first positive results ∂ S S S � ∂ S S S � p − 2 p − 2 D | 2 ) D | 2 ) � � D ∼ (1 + | D D � ≤ γ 0 (1 + | D D 1 < p < 2 2 4 � � ∂ D D ∂π � ◮ M´ alek, Neˇ cas, Rajagopal, Arch. Rational Mech. Anal. , 2002. ◮ Hron, M´ alek, Neˇ cas, Rajagopal, Math. Comput. Simulation , 2003. ◮ M´ alek, Rajagopal, Handbook of mathematical fluid dynamics , 2007. M. Lanzend¨ orfer et al. (ICS AS CR) Incompressible piezoviscous fluids January 13, 2011 9 / 14

Incompressible fluids with viscosity depending on pressure and shear rate Mathematical formulation inside (0 , T ) × Ω: div v v v = 0 ∂ τ v v v + div( v v v ⊗ v v v ) − div S S S = −∇ π + f f f , v ) | 2 ) D S S S = 2 ν ( π, | D D D ( v v D D ( v v v ) on the boundary (0 , T ) × ∂ Ω = Γ D ∪ Γ N ∪ Γ P : v v v · n n n = 0 and − T T Tn n = σ v n v v on Γ N v v v = v v v D v v v v D v D v on Γ D if Γ P = ∅ , − T T Tn n n = b b b ( v v v ) on Γ P ´ Ω 0 π d x x x = 0 then ◮ Bul´ ıˇ cek, M´ alek, Rajagopal, Indiana Univ. Math. J. , 2007 ◮ Bul´ ıˇ cek, M´ alek, Rajagopal, SIAM J. Math. Anal. , 2009 M. Lanzend¨ orfer et al. (ICS AS CR) Incompressible piezoviscous fluids January 13, 2011 9 / 14

Incompressible fluids with viscosity depending on pressure and shear rate Mathematical formulation inside Ω: div v v v = 0 div( v v ⊗ v v v v ) − div S S S = −∇ π + f f f , v ) | 2 ) D S S S = 2 ν ( π, | D D D ( v v D D ( v v v ) on the boundary ∂ Ω = Γ D ∪ Γ N ∪ Γ P : v v v · n n n = 0 and − T T Tn n n = σ v v v on Γ N v v v = v v v v v D v v D v v D on Γ D if Γ P = ∅ , − T T Tn n n = b b b ( v v v ) on Γ P ´ Ω 0 π d x x = 0 x then ◮ Franta, M´ alek, Rajagopal, Proc. Royal Soc. A , 2005 ◮ M. L., Nonlin. Anal.: Real World App. , 2009 M. Lanzend¨ orfer et al. (ICS AS CR) Incompressible piezoviscous fluids January 13, 2011 9 / 14

Incompressible fluids with viscosity depending on pressure and shear rate Mathematical formulation inside Ω: div v v v = 0 div( v v v ⊗ v v v ) − div S S S = −∇ π + f f f , v ) | 2 ) D S S S = 2 ν ( π, | D D D ( v v D D ( v v v ) on the boundary ∂ Ω = Γ D ∪ Γ N ∪ Γ P : v v v · n n n = 0 and − T T Tn n = σ v n v v on Γ N v v v = v v v v v v D v D v v D on Γ D if Γ P = ∅ , − T T Tn n n = b b b ( v v v ) on Γ P ´ Ω 0 π d x x = 0 x then M. Lanzend¨ orfer et al. (ICS AS CR) Incompressible piezoviscous fluids January 13, 2011 9 / 14

Incompressible fluids with viscosity depending on pressure and shear rate Mathematical formulation inside Ω: div v v v = 0 div( v v v ⊗ v v v ) − div S S S = −∇ π + f f f , v ) | 2 ) D S S S = 2 ν ( π, | D D D ( v v D D ( v v v ) on the boundary ∂ Ω = Γ D ∪ Γ N ∪ Γ P : v v v · n n n = 0 and − T Tn T n = σ v n v v on Γ N v v v = v v v v D v v v D v v D on Γ D if Γ P = ∅ , − T T Tn n = b n b b ( v v v ) on Γ P ´ Ω 0 π d x x x = 0 then ◮ Stebel & M. L., Appl. Mat.–Czech. , in print; 2009 preprint NCMM ◮ Stebel & M. L., Math. Comput. Simulat. , submitted 2009 preprint NCMM M. Lanzend¨ orfer et al. (ICS AS CR) Incompressible piezoviscous fluids January 13, 2011 9 / 14

Basic a priori estimates Weak formulation ( q , div w w w ) Ω = 0 v v w S D v D w w f w b v w ([ ∇ v v ] v v , w w ) Ω + ( S S ( π, D D ( v v )) , D D ( w w )) Ω − ( π, div w w ) Ω = ( f f , w w ) Ω − ( b b ( v v ) , w w ) Γ P M. Lanzend¨ orfer et al. (ICS AS CR) Incompressible piezoviscous fluids January 13, 2011 10 / 14

Basic a priori estimates Weak formulation ( q , div w w w ) Ω = 0 S D v D w w f w b w ( S S ( π, D D ( v v )) , D D ( w w )) Ω − ( π, div w w ) Ω = ( f f , w w ) Ω − ( b b , w w ) Γ P M. Lanzend¨ orfer et al. (ICS AS CR) Incompressible piezoviscous fluids January 13, 2011 10 / 14

Basic a priori estimates Weak formulation ( q , div w w w ) Ω = 0 ( S S S ( π, D D D ( v v v )) , D D D ( w w w )) Ω − ( π, div w w w ) Ω = ( f f f , w w w ) Ω − ( b b b , w w w ) Γ P Test eq. by solution v ) | p ± 1 ( S S S ( π, D D D ( v v v )) , D D D ( v v v )) Ω ∼ | D D D ( v v = ⇒ D v v S � p ′ ≤ K S � D D ( v v ) � p ≤ K = ⇒ � v v � 1 , p + � S M. Lanzend¨ orfer et al. (ICS AS CR) Incompressible piezoviscous fluids January 13, 2011 10 / 14