1. Burnett regime continuum flow equations: application of the Korteweg models School of Engineering and Physical Sciences Institute of Mechanical Process and Energy Engineering (IMPEE) Heriot-Watt University, Edinburgh Workshop "Hilbert’s Sixth Problem” University of Leicester, May 02-04, 2016, by Prof. Alexander Gorban

2. Outline Motivations: some active engineering flow issues Burnett regime flow equations: review of existing problems Korteweg models of continuum flow equations: brief history Derivation of volume diffusion continuum model Applications Conclusions

3. … because the Navier-Stokes equations — despite being exceptionally useful for modeling the weather, ocean currents, pipes, cars, airplane wings and other hydrodynamic systems, and despite the million-dollar prize offered for their exact solutions — are incomplete. July 21, 2015

4. Counterintuitive mass, momentum and heat transfer In the Crookes radiometer, invented in 1873 by Sir William Crookes, exposure to light creates a heat and pressure gradient inside the partial vacuum chamber, turning the vanes. Motivations

6. Energy harvesting and storage Micro/Nano machining Illustration by Alexandra Rabbitte Formation of structures that could be used to form sensors and actuators, cooling system etc. Shale and tight gas reservoirs consist of porous structures with pore diameter in range such that pore diameters become comparable to gas mean free path. An array of holes, each measuring approximately 125 nm in length, drilled with a computer-controlled focused ion beam.

7. A single particle distribution function: Particle density distribution function: - Lead to the : Boltzmann equation Does the equation only apply to N particles interacting in a fixed volume V ? Fluid macroscopic mass-density is then always assumed to be first moment of the distribution function in the velocity space. Original Burnett flow equations: the old problem

8. Second order approximation (Burnett regime) becomes problematic  Frame indifference ?  Second law of thermodynamics ?  Angular momentum conservation ? Original Burnett flow equations: the old problem

9. Second order approximation (Burnett regime) becomes problematic

10. Korteweg model: history  Extension of the compressible Navier–Stokes equations (Korteweg 1901): diffuse interface model for liquid–vapour flows which allows for phase transitions  They are generally incompatible with the usual continuum theory of thermodynamics  Back in the news with : - Brenner: Bi-velocity hydrodynamics - Slemrod, Karlin, Gorban … : (Boltzmann equation and rarefied gases: summation of Chapman-Enskog expansion) M. Heida & J. Málek (2010), On compressible Korteweg fluid-like materials, International Journal of Engineering Science: 48, 1313–1324

11. With thermodynamics relationships Non-equilibrium physical quantities: Additional contribution: Helmholtz free-energy contribution D.M. Anderson, G.B. McFadden, A. A. Wheeler, Annual Review of Fluid Mechanics, 1998, 30:139-65 Shear Stress = T (Korteweg) + t (NSF) constitutive equations: Korteweg model: derivation Energetic heat flux ≠ Entropic heat flux

13. Volume diffusion kinetic and continuum approach Schematic representation of target molecules surrounded by others S.K. Dadzie, J.M. Reese, C.R. McInnes, Physica A, 2008, 387:6079-6094

14. Non-equilibrium physical quantities: Mass-density Mass-based velocity Peculiar velocity A configuration density A configuration velocity A configuration peculiar velocity The difference leads to an additional flux: concentration / volume flux in non-equilibrium configurations Volume diffusion kinetic and continuum approach

15. Volume vs Mass velocity Entropy heat flux Energetic Heat flux Non-equilibrium physical quantities: constitutive equations Non-local equilibrium thermodynamic relation Entropy equation: Stress tensor Volume diffusion kinetic and continuum approach Configuration contribution

16. Mass Volume transport equation Momentum equation Entropy equation Energy equation Volume diffusion kinetic and continuum approach

17. The new volume diffusion continuum approach is perfectly stable in both space and time always Volume diffusion kinetic and continuum approach

18. Some Applications

19. Numerical Implementation in OpenFOAM Conservation equations: Continuity Momentum Energy equation Constitutive equations are of Korteweg type:

20. Nano lid-driven cavity flow subject to natural convection Boundary Conditions: Temperature Velocity in NSF Velocity in Volume Diffusion

21. (1) Phenomena of Heat transfer from cold-to-hot only occur in energetic heat flux: these are observed in the slip and transition regimes Nano lid-driven cavity flow subject to natural convection Temperature distribution and heat fluxes at Kn = 10

22. Nano lid-driven cavity flow subject to natural convection Temperature distribution and heat fluxes at Kn = 10 (1) Phenomena of Heat transfer from cold-to-hot only occur in energetic heat flux: these are observed in the slip and transition regimes (2) Entropic heat flux only flows from higher temperature to low temperature: so always consistent with the Second Law

23. Nano lid-driven cavity flow subject to natural convection Temperature distribution comparison: NSF vs Volume Diffusion: Clear differences occur in non-equilibrium regions from Kn > 0.01

24. Hagen–Poiseuille flow in slip and transition regimes: analytical solution 2D Poiseuille flow in a rectangular channel (based on the volume diffusion model) +h/2 -h/2 x y

25. Hagen–Poiseuille flow in slip and transition regimes: analytical solution S.K. Dadzie, H. Brenner, 2012, Physical Review E, 86:036318

26. Slip effects Volume diffusion Comparison with Cooper, Cruden et. al 2003 experiments: Prediction of permeability in shale rocks Hagen–Poiseuille flow in slip and transition regimes: analytical solution W Tubby, SK Dadzie, C Christou, 2 nd European Conference on Non-equilibrium Gas Flows (NEGF2015), Eindhoven, the Netherlands, 2015

27. Hagen–Poiseuille flow in slip and transition regimes: analytical solution

28. The classical shock profile problem: Mach 2.85 Shock Profile C.J. Greenshields, J.M. Reese, 2007, Journal of Fluid Mechanics, 580:407-429 Inverse shock thickness vs Mach number

29. o Future works: --- Investigate the role of each of the Korteweg stress term s and their importance and contributions in various engineering flow configurations --- Construction of mixture / multicomponent flow equations in volume diffusion theory Conclusions: o Standard methods to overcome Burnett regime continuum flow equation problems are outdated o Korteweg type of continuum flow equations may be the solutions to the description of rarefied gas flows in the Burnett regime o An additional transport equation may be required for the development of a fully consistent beyond Navier-Stokes flow equation (the volume transport equation ?!) o Entropic heat flux (equation) needs clear distinction from energetic heat flux (equation) in the Burnett regime

30. With thanks to … Phd Students and RAs: - Christou Chariton - Corey Downie - William G. Tubby and funders and supporters:

31. Hagen–Poiseuille flow in slip and transition regimes: numerical cases in OpenFOAM