Some mathematical results on mixture flows. D. B RESCH Laboratoire - PowerPoint PPT Presentation

Some multi-fluid systems Local well-posedness - A physical example of bifluid system Multi-fluid model as limit of mono-fluid system Some mathematical results on mixture flows. D. B RESCH Laboratoire de Math´ ematiques, UMR5127 CNRS Universit´ e de Savoie Mont-Blanc 73376 Le Bourget du lac France Email: didier.bresch@univ-smb.fr ANR project D YFICOLTI managed by D. L ANNES Basel, June 2017 Thanks for the invitation to A. Mazzucato, G. Alberti, G. Crippa D. Bresch Some mathematical results on mixture flows.

Some multi-fluid systems Local well-posedness - A physical example of bifluid system Multi-fluid model as limit of mono-fluid system Outline Some multi-fluid systems (miscible fluids - Mixture); 1 An example of immiscible fluid system sharing the same framework : 2 Multi-fluid models as limit of mono-fluid models. 3 Joint works with: B. D ESJARDINS , E. G RENIER and J.–M. G HIDAGLIA ; M. R ENARDY (Virginia Tech, Blacksburg) ; M. H ILLAIRET (Montpellier University), X. H UANG (Academia Sinica, Pekin) ; P. M UCHA (Warsaw university) and E. Z ATORSKA (Imperial college). D. Bresch Some mathematical results on mixture flows.

Some multi-fluid systems A model with algebraic closure Local well-posedness - A physical example of bifluid system A model with a PDE closure Multi-fluid model as limit of mono-fluid system A model with an algebraic closure (common pressure) α + + α − = 1 , ∂ t ( α + ρ + ) + div ( α + ρ + u + ) = 0 , ∂ t ( α − ρ − ) + div ( α − ρ − u − ) = 0 , ∂ t ( α + ρ + u + ) + div ( α + ρ + u + ⊗ u + ) + α + ∇ P = 0 , ∂ t ( α − ρ − u − ) + div ( α − ρ − u − ⊗ u − ) + α − ∇ P = 0 , P = P − ( ρ − ) = P + ( ρ + ) , with 0 ≤ α ± ≤ 1 . See for instance M. I SHII (1975), D.A. D REW AND S.L. P ASSMAN (1998). D. Bresch Some mathematical results on mixture flows.

Some multi-fluid systems A model with algebraic closure Local well-posedness - A physical example of bifluid system A model with a PDE closure Multi-fluid model as limit of mono-fluid system The model with algebraic closure Non-conservative, non-hyperbolic system if 0 ≤ | u + − u − | < c m with + (( α + ρ + ) 1 / 3 + ( α − ρ − ) 1 / 3 ) 3 / ( α + ρ − c 2 c 2 m = c 2 − c 2 − + α − ρ + c 2 + ) . In general, c m is large compared to u + and u − and thefore flow belongs to non-hyperbolic region. See: H.B. S TEWART , B. W ENDROFF , J. Comp. Physics , 363–409, (1984) (Appendix I). Remark: Here we consider velocity governed by Navier-Stokes type systems. It exists other PDEs governing the two velocity fields: Darcy, Brinkman etc.... D. Bresch Some mathematical results on mixture flows.

Some multi-fluid systems A model with algebraic closure Local well-posedness - A physical example of bifluid system A model with a PDE closure Multi-fluid model as limit of mono-fluid system The model with algebraic closure with extra terms α + + α − = 1 , ∂ t ( α + ρ + ) + div ( α + ρ + u + ) = 0 , ∂ t ( α − ρ − ) + div ( α − ρ − u − ) = 0 , ∂ t ( α + ρ + u + ) + div ( α + ρ + u + ⊗ u + ) + α + ∇ P + π ∇ α + = 0 , ∂ t ( α − ρ − u − ) + div ( α − ρ − u − ⊗ u − )) + α − ∇ P + π ∇ α − = 0 , P = P − ( ρ − ) = P + ( ρ + ) , with 0 ≤ α ± ≤ 1 . Remark: The terms α ± ∇ P + π ∇ α ± are usually replaced by terms such as ∇ ( α ± P ± ) + F ± int + P ± int ∇ α ± where F ± int are internal forces such as relative drag terms, P ± int modelize macroscopic terms encoding pressure at the microscopic interface between the different fluids. D. Bresch Some mathematical results on mixture flows.

Some multi-fluid systems A model with algebraic closure Local well-posedness - A physical example of bifluid system A model with a PDE closure Multi-fluid model as limit of mono-fluid system The model with algebraic closure In litterature, use Bestion term α + α − ρ + ρ − α + ρ − + α − ρ + ( u + − u − ) 2 . π = δ with δ > 1 to get hyperbolicity for small relative velocity. – See paper by M. N DJINGA , A. K UMBARO , F. D E V UYST , P. L AURENT -G ENGOUX , ISMF (2005) for geometric discussions: number of intersecting points of parabola and hyperbola in quarter plane. Extension of H.B. S TEWART , B. W ENDROFF ’s approach. – Direct study: analytical calculations. D.B., B. D ESJARDINS , J.–M. G HIDAGLIA , E. G RENIER , M. H ILLAIRET . See chapter in Handbook edited by Giga and Novotny. D. Bresch Some mathematical results on mixture flows.

Some multi-fluid systems A model with algebraic closure Local well-posedness - A physical example of bifluid system A model with a PDE closure Multi-fluid model as limit of mono-fluid system A low mach number model α − + α + = 1 , ∂ t ( α + ) + div ( α + u + ) = 0 , ∂ t ( α − ) + div ( α − u − ) = 0 , ρ + ( ∂ t ( α + u + ) + div ( α + u + ⊗ u + )) + α + ∇ P + π ∇ α + = 0 , ρ − ( ∂ t ( α − u − ) + div ( α − u − ⊗ u − )) + α − ∇ P + π ∇ α − = 0 , with ρ − and ρ + constants and P the Lagrangian multiplier associated to the constraint α + + α − = 1. D. Bresch Some mathematical results on mixture flows.

Some multi-fluid systems A model with algebraic closure Local well-posedness - A physical example of bifluid system A model with a PDE closure Multi-fluid model as limit of mono-fluid system A model with an algebraic closure Hyperbolic with Bestion closure namely: π = δ α + α − ρ + ρ − α + ρ − + α − ρ + ( u + − u − ) 2 with δ > 1. Rq: We will see a model which shares the same form: The two-layers shallow-water system between rigid lids: See slide 14. In this model, π = 0 and a term cst ∇ α + appears in the + momentum component. D. Bresch Some mathematical results on mixture flows.

Some multi-fluid systems A model with algebraic closure Local well-posedness - A physical example of bifluid system A model with a PDE closure Multi-fluid model as limit of mono-fluid system A model with an algebraic closure Local well-posedness on an associated low mach number limit system, see [1] Global weak solutions if degenerate viscosities and capillarity terms, see [2] Invariant regions, see [2] Global weak solutions in one space dimension if degenerate viscosities, see [3] [1] D. B., M. R ENARDY . Well-Posedness of Two-Layer Shallow-Water Flow Between Two Horizontal Rigid Plates. Nonlinearity , 24, 1081–1088, (2011). [2] D. B., B. D ESJARDINS , J.–M. G HIDAGLIA , E. G RENIER . On Global Weak Solutions to a Generic Two-Fluid Model. Arch. Rational Mech. Anal. Volume 196, Number 2, 599-629, (2009). [3] D.B., X. H UANG , J. L I . A Global Weak Solution to a One-Dimensional Non-Conservative Viscous Compressible Two-Phase System. Comm. Math. Phys. , Volume 309, Issue 3, 737–755, (2012). D. Bresch Some mathematical results on mixture flows.

Some multi-fluid systems A model with algebraic closure Local well-posedness - A physical example of bifluid system A model with a PDE closure Multi-fluid model as limit of mono-fluid system A model with a PDE closure (equation on fraction) α + + α − = 1 , 1 ∂ t α + + u int · ∇ α + λ P ( P + − P − ) , = ∂ t ( α + ρ + ) + div ( α + ρ + u + ) = 0 , ∂ t ( α − ρ − ) + div ( α − ρ − u − ) = 0 , 1 ∂ t ( α + ρ + u + ) + div ( α + ρ + u + ⊗ u + ) + α + ∇ P + + P + λ u ( u + − u − ) , int ∇ α + = 1 ∂ t ( α − ρ − u − ) + div ( α − ρ − u − ⊗ u − ) + α − ∇ P − + P − λ u ( u − − u + ) , int ∇ α − = with u int and P ± int respectively interface velocity and interface pressures explicitely given in terms of the unknowns. D. Bresch Some mathematical results on mixture flows.

Some multi-fluid systems A model with algebraic closure Local well-posedness - A physical example of bifluid system A model with a PDE closure Multi-fluid model as limit of mono-fluid system A model with a PDE closure (equation on fraction) If λ u → 0, One-velocity field. See works by F. D IAS , D. D UTYKH and J.–M. G HIDAGLIA (2010) on a two-fluid model for violent aerated flows. See for instance: R. A BGRALL , C. B ERTHON , F. C OQUEL , S. D ELLACHERIE , D.A. D REW and S.L. P ASSMAN , Th. G ALLOU ¨ ET , M. I SHII , Ph. L E F LOCH , R. S AUREL and others for modeling and numerics. D. Bresch Some mathematical results on mixture flows.

Some multi-fluid systems A model with algebraic closure Local well-posedness - A physical example of bifluid system A model with a PDE closure Multi-fluid model as limit of mono-fluid system Letting λ u and λ p go formmaly to zero, we get a mono-fluid system with two species. If we consider the viscous version, we have α + + α − = 1 , P + ( ρ + ) = P − ( ρ − ) , ∂ t ( α + ρ + ) + div ( α + ρ + u ) = 0 , ∂ t ρ + div ( ρ u ) = 0 , ∂ t ( ρ u ) + div ( ρ u ⊗ u ) + ∇ P + ( ρ + ) − µ ∆ u + ∇ div u = 0 . – Global weak solution for such system? Yes using the framework recently developed by D.B., P.–E. Jabin. Quantitative regularity estimates on appropriate unknowns needed. Work in progress with P. Mucha and E. Zatorska Note that the pressure is not monotone in a conservative variable. D. Bresch Some mathematical results on mixture flows.