1. The Onsager Principle and Hydrodynamic Boundary Conditions Ping Sheng Department of Physics and William Mong Institute of Nano Science and Technology The Hong Kong University of Science and Technology Workshop on Nanoscale Interfacial Phenomena in Complex Fluids 20 May 2008

2. in collaboration with:  Xiao-Ping Wang (Dept. of Mathematics, HKUST)  Tiezheng Qian (Dept. of Mathematics, HKUST)

3. Two Pillars of Hydrodynamics Navier Stokes equation Fluid-solid boundary condition –Non-slip boundary condition implies no relative motion at the fluid-solid interface

4. Non-slip boundary condition is compatible with almost all macroscopic fluid-dynamic problems –But can not distinguish between non-slip and small amount of partial slip –No support from first principles However, there is one exception  the moving contact line problem Non-Slip Boundary Condition

6. No-Slip Boundary Condition Appears to be violated by the moving/slipping contact line Causes infinite energy dissipation (unphysical singularity) Dussan and Davis, 1974

9. Two Possibilities Continuum hydrodynamics breaks down –“Fracture of the interface” between fluid and solid wall –A nonlinear phenomenon –Breakdown of the continuum? Continuum hydrodynamics still holds –What is the boundary condition?

10. Implications and Solution There can be no accurate continuum modelling of nano- or micro-scale hydrodynamics –Most nano-scale fluid systems are beyond the MD simulation capability We show that the boundary condition(s) and the equations of motion can be derived from a unified statistical mechanic principle –Consistent with linear response phenomena in dissipative systems –Enables accurate continuum modelling of nano-scale hydrodynamics

11. The Principle of Minimum Energy Dissipation Onsager formulation: used only in the local neighborhood of equilibrium, for small displacements away from the equilibrium –The underlying physics is the same as linear response Is not meant to be used for predicting global configuration that minimizes dissipation

12. Single Variable Version of the MEDP Let  be the displacement from equilibrium, and its rate. …Fokker-Planck Equation is the stationary solution White Noise -

13. Three points to be noted: is to be minimized w.r.t. (2) MEDP implies balance of dissipative force with force derived from free energy (1) (3) MEDP gives the most probable course of a dissipative process

14. Derivation of Equation of Motion from Onsager Principle Viscous dissipation of fluid flow is given by together with incompressibility condition - In the presence of inertial effect, momentum balance means By minimizing  with respect to, with the condition of (treated by using a Lagrange multiplier p), one obtains the Stokes equation  NS equation solid

15. Extension of the Onsager Principle for Deriving Fluid-solid Boundary Condition(s) If one supposes that there can be a fluid velocity relative to the solid boundary, then similar to for fluid, there should be a - Yields, together with, the boundary condition - But over the past century or more, it is the general belief that  Navier boundary condition (1823) = a length (slip length) ; Non-slip boundary condition

16. Two Phase Immiscible Flows Need a free energy to stabilize the interface - - Total free energy - ; (Cahn-Hilliard) Fluid 2Fluid 1

17.  is locally conserved: Interfacial  is not conserved, because  n J n  0 in general in bulk, but - - - - Minimizew.r.t.

18. - Minimize w.r.t. - Subsidiary incompressibility condition:

19. Minimize w.r.t. : -

20. Minimize w.r.t. : - In the bulk - On the boundary uncompensated Young stress  Young equation

21. Uncompensated Young Stress -  x  fs also a peaked function - The L(  )  x  term at the surface must accompany the capillary force density term  in the bulk - It is the manifestation of fluid-fluid interfacial tension at the solid boundary The linear friction law at the liquid solid interface and the Allen-Cahn relaxation condition form a consistent pair

22. Continuum Hydrodynamic Formulation at boundary - - -

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. Power-Law Decay of Partial Slip

28. Molecular Dynamic Confirmations

29. Implications Hydrodynamic boundary condition should be treated within the framework of linear response –Onsager’s principle provides a general framework for deriving boundary conditions as well as the equations of motion in dissipative systems Even small partial slipping is important –Makes b.c. part of statistical physics –Slip coefficient  is just like viscosity coefficient –Important for nanoparticle colloids’ dynamics Boundary conditions for complex fluids –Example: liquid crystals have orientational order, implies the cross-coupling between slip and molecular rotation to be possible

30. Maxwell Equations Require No Boundary Conditions

31. Publications A Variational Approach to Moving Contact Line Hydrodynamics, T. Qian, X.-P. Wang and P. Sheng, Journal of Fluid Mechanics 564, 333-360 (2006). Moving Contact Line over Undulating Surfaces, X. Luo, X.-P. Wang, T. Qian and P. Sheng, Solid State Communications 139, 623-629 (2006). Hydrodynamic Slip Boundary Condition at Chemically Patterned Surfaces: A Continuum Deduction from Molecular Dynamics, T. Qian, X. P. Wang and P. Sheng, Physical Review E72, 022501 (2005). Power-Law Slip Profile of the Moving Contact Line in Two-Phase Immiscible Flows, T. Qian, X. P. Wang and P. Sheng, Physical Review Letters 93, 094501-094504 (2004). Molecular Scale Contact Line Hydrodynamics of Immiscible Flows, T. Qian, X. P. Wang and P. Sheng, Physical Review E68, 016306 (2003).

32. Nano Droplet Dynamics over High Contrast Surface

33. Contact Line Breaking with High Wetability Contrast

34. Thank you