1. Hydrodynamic Slip Boundary Condition for the Moving Contact Line in collaboration with Xiao-Ping Wang (Mathematics Dept, HKUST) Ping Sheng (Physics Dept, HKUST)

2. No-Slip Boundary Condition ?

3. from Navier Boundary Condition to No-Slip Boundary Condition : slip length, from nano- to micrometer Practically, no slip in macroscopic flows : shear rate at solid surface

6. No-Slip Boundary Condition ? Apparent Violation seen from the moving/slipping contact line Infinite Energy Dissipation (unphysical singularity) Are you able to drink coffee?

7. Previous Ad-hoc models: No-slip B.C. breaks down Nature of the true B.C. ? (microscopic slipping mechanism) If slip occurs within a length scale S in the vicinity of the contact line, then what is the magnitude of S ?

8. Molecular Dynamics Simulations initial state: positions and velocities interaction potentials: accelerations time integration: microscopic trajectories equilibration (if necessary) measurement: to extract various continuum, hydrodynamic properties CONTINUUM DEDUCTIONCONTINUUM DEDUCTION

9. Molecular dynamics simulations for two-phase Couette flow Fluid-fluid molecular interactions Wall-fluid molecular interactions Densities (liquid) Solid wall structure (fcc) Temperature System size Speed of the moving walls

10. Modified Lennard-Jones Potentials for like molecules for molecules of different species for wetting property of the fluid

11. fluid-1 fluid-2fluid-1 dynamic configuration static configurations symmetricasymmetric f-1f-2f-1 f-2f-1

12. tangential momentum transport boundary layer

13. The Generalized Navier B. C. when the BL thickness shrinks down to 0 viscous partnon-viscous part Origin?

14. nonviscous part viscous part uncompensated Young stress

15. Uncompensated Young Stress missed in Navier B. C. Net force due to hydrodynamic deviation from static force balance (Young’s equation ) NBC NOT capable of describing the motion of contact line Away from the CL, the GNBC implies NBC for single phase flows.

16. Continuum Hydrodynamic Modeling Components: Cahn-Hilliard free energy functional retains the integrity of the interface (Ginzburg-Landau type) Convection-diffusion equation (conserved order parameter) Navier - Stokes equation (momentum transport) Generalized Navier Boudary Condition

17. Diffuse Fluid-Fluid Interface Cahn-Hilliard free energy (1958)

18. is the chemical potential. capillary force density

19. = tangential viscous stress + uncompensated Young stress Young’s equation recovered in the static case by integration along x

20. in equilibrium, together with for boundary relaxation dynamics first-order generalization from

21. Comparison of MD and Continuum Hydrodynamics Results Most parameters determined from MD directly M and optimized in fitting the MD results for one configuration All subsequent comparisons are without adjustable parameters.

22. molecular positions projected onto the xz plane

23. Symmetric Coutte V=0.25 H=13.6 near-total slip at moving CL no slip

24. symmetric Coutte V=0.25 H=13.6 asymmetric Coutte V=0.20 H=13.6 profiles at different z levels

25. symmetric Coutte V=0.25 H=10.2 symmetric Coutte V=0.275 H=13.6

26. asymmetric Poiseuille g ext =0.05 H=13.6

27. The boundary conditions and the parameter values are both local properties, applicable to flows with different macroscopic/external conditions (wall speed, system size, flow type).

28. Summary: A need of the correct B.C. for moving CL. MD simulations for the deduction of BC. Local, continuum hydrodynamics formulated from Cahn-Hilliard free energy, GNBC, plus general considerations. “Material constants” determined (measured) from MD. Comparisons between MD and continuum results show the validity of GNBC.

29. Large-Scale Simulations MD simulations are limited by size and velocity. Continuum hydrodynamic calculations can be performed with adaptive mesh (multi-scale computation by Xiao-Ping Wang). Moving contact-line hydrodynamics is multi- scale (interfacial thickness, slip length, and external confinement length scale).