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

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

3. 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

4. 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

5. “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

6. “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)

7. “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)

8. 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)

9. 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

10. 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

11. 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

12. 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

13. 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

14. 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

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

16. 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

17. 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

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

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

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

21. 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

22. 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

23. 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

24. 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

25. 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