A hyperbolic model for viscous fluids First numerical examples ILYA PESHKOV CHLOE, University of Pau, France EVGENIY ROMENSKI Sobolev Institute of Mathematics, Novosibirsk, Russia

Part 1 Theory PESHKOV, ROMENSKI, IWNET, 5-10/07/2015

What are we going to present? A unified VISCOSITY-coefficient-free model for Solids & Fluids First order PDEs, Hyperbolic, Causal No explicit flow division on Equilibrium and non-Equilibrium, Newtonian and non-Newtonian Fully consistent with Thermodynamics Free of empirical relations (closed to what called “from first principles”) PESHKOV, ROMENSKI, IWNET, 5-10/07/2015

Two ways to derive a continuum model, Navier-Stokes example Empirical approach to describe particle interactions by means of Newton’s law of viscosity Acausal theory (parabolic) Postulates of Continuum Mechanics Second order PDE Issues with computing, discontinuities, mesh quality With phenomenological viscosity coefficient Only equilibrium flows OR Material particle or Fluid parcel, Fluid element Try to develop an “Ab initio” approach free of the above shortcomings … PESHKOV, ROMENSKI, IWNET, 5-10/07/2015

“Ab initio” approach, an objective alternative to the viscosity In 1930th, Yakov Frenkel developed a microscopic theory of liquids, and proposed a fundamental characteristic “particle settled life time” gases liquids solids time Debye frequency Liquids are solids if observation time less then Ref: Brazhkin et al 2012 “Two liquid states of matter…”, Phys. Rev. E

“Ab initio” approach, an objective alternative to the viscosity In 1930th, Yakov Frenkel developed a microscopic theory of liquids, and proposed a fundamental characteristic “particle settled life time” We apply Frenkel’s idea to continuum modeling. Now, molecules = material particles or fluid parcels Simple Consequences: Macro-flow is the particle rearrangement process No free volume particles have to deform During the time all connections are conserved The longer the bigger resistance to shearing defines the smallest length scale (particle scale)

“Ab initio” approach, an objective alternative to the viscosity In 1930th, Yakov Frenkel developed a microscopic theory of liquids, and proposed a fundamental characteristic “particle settled life time” We apply Frenkel’s idea to continuum modeling. Now, molecules = material particles or fluid parcels Deformed particle Undeformed Distortion (non-symmetric)

Particle rearrangements and Strain dissipation Deformed particle Undeformed Shear fluctuations cannot propagate across the slip plane Particles in Equilibrium Flow slip They dissipate here Distortions and are incompatible (local field)

But how to derive the governing equations? Godunov form of conservation laws: Convex potential Conservation laws An extra cons. law Symmetric hyperbolic form Ref: Godunov, “An interesting class of quasi-linear systems”, 1961 Only conservation laws in Lagrangian frame admit the Godunov structure… PESHKOV, ROMENSKI, IWNET, 5-10/07/2015

But how to derive the governing equations? Lagrangian frame Two requirements : Conservative form (or divergent form) An extra conservation law First and Second laws : Examples: Elastoplasticity Viscous fluids Multiphase flow Electromagnetism Flows in porous media Arbitrary Number of ODEs Lagrangian equations can be rigorously derived from the variational principle Ref: Godunov, Mikhailova, Romenski “Systems of thermodynamically coordinated laws ….”, 1996

But how to derive the governing equations? Eulerian frame After the Lagrange-to-Euler transformation: the momentum equation is: The same structure we have in GENERIC approach Ref: Godunov, Mikhailova, Romenski “Systems of thermodynamically coordinated laws ….”, 1996 Ref: Peshkov, Grmela, Romenski “Irreversible mechanics and thermodynamics of ….”, 2014

Governing equations Momentum Equation for the distortion Mass conservation Entropy equation Ref: Godunov, “Elements of continuum mechanics”, 1978 (in Russian) Ref: Godunov, Romenski, “Elements of continuum mechanics….”, 2003 PESHKOV, ROMENSKI, IWNET, 5-10/07/2015

Equation of State, Closure Total energy conservation Equation of State equilibrium viscous kinetic The simplest example of the Viscous energy micro meso macro transversal sound velocity at the equilibrium PESHKOV, ROMENSKI, IWNET, 5-10/07/2015

Equation of State, Closure Total energy conservation Equation of State equilibrium viscous kinetic The simplest example of the Viscous energy micro meso macro For example Classical (equilibrium) pressure Non-equilibrium pressure

slip Strain dissipation equilibrium viscous kinetic micro meso macro Shear fluctuations dissipate here PESHKOV, ROMENSKI, IWNET, 5-10/07/2015

slip Strain dissipation Equation for the distortion here The dissipative dynamics is also generated by the Energy potential: slip Shear fluctuations dissipate here Energy potential generates both Non-dissipative and Dissipative dynamics PESHKOV, ROMENSKI, IWNET, 5-10/07/2015

Energy transformation Total energy conservation equilibrium viscous kinetic Equations of State meso micro macro Dissipative Non Dissipative Entropy growths here Entropy is constant here External energy supplied to fluid PESHKOV, ROMENSKI, IWNET, 5-10/07/2015

Part 2 Numerical Examples PESHKOV, ROMENSKI, IWNET, 5-10/07/2015

Newton’s law of viscosity, simple shear Air viscosity Newton’s law Ref: Peshkov, Romenski, Continuum Mech. and Therm., 2014

Newton’s law of viscosity, arbitrary shear Dotted lines Ref: Peshkov, Romenski, Continuum Mech. and Therm., 2014

Dispersion relations for viscous gas, longitudinal sound EOS Ideal gas Equilib. long. sound Phase velocity, [m/s] Air-like gas frequency, [s-1] zoomed plot Phase velocity, [m/s] Quasilinear form, near equilibrium frequency, [s-1] Ref: Muracchini, Ruggeri, Seccia, Wave motion, 1992

Dispersion relations for viscous gas, transversal sound EOS NS Ideal gas Phase velocity, [m/s] Equilib. long. sound Air-like gas frequency, [s-1] zoomed plot Phase velocity, [m/s] Quasilinear form, near equilibrium NS frequency, [s-1] Ref: Muracchini, Ruggeri, Seccia, Wave motion, 1992

Rayleigh–Taylor instability, Viscous gas with a big viscosity No slip No slip Solver No Navier-Stokes solver Godunov-type 2nd order Finite-Volume method No slip No slip No slip EOS gravity Ideal gas No slip No slip Ref: Shi Jin, “Runge-Kutta methods for hyperbolic cons. laws with stiff relax. terms”, JCP, 1995

Rayleigh–Taylor instability, Viscous gas with a big viscosity No slip No slip Copper bar impact at 190 m/s No slip No slip No slip gravity A unified model for fluids and solids (only different EoS and ) No slip No slip Ref: Barton, Romenski, Commun. Comput. Phys. , 2012

Thank you for your attention! Conclusion A unified model for fluids (Newtonian/non-Newtonian) and solids has been proposed no viscosity and other phenomenological concepts no empirical relations The user should provide EoS (total energy potential) and particle settled life time First numerical examples were presented Thank you for your attention! PESHKOV, ROMENSKI, IWNET, 5-10/07/2015