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
in collaboration with:
 Xiao-Ping Wang (Dept. of Mathematics, HKUST)
 Tiezheng Qian (Dept. of Mathematics, HKUST)
Two Pillars of Hydrodynamics
• Navier Stokes equation
 v

  v   v   p     v  f e
 t


• Fluid-solid boundary condition
– Non-slip boundary condition implies no relative
motion at the fluid-solid interface
Non-Slip Boundary Condition
• 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
No-Slip Boundary Condition
• Appears to be violated by the
moving/slipping contact line
• Causes infinite energy dissipation
(unphysical singularity)
Dussan and Davis, 1974
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?
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
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
Single Variable Version of the MEDP
• Let  be the displacement from equilibrium, and  its rate.
  
•
F  

White Noise
  t 
  t    t    2 kT   t  t  
P kBT   2 P
1   F  


P   …Fokker-Planck
 2

t
  
kBT     
Equation
- Peq   ~ exp   F   / k BT  is the stationary solution
•
•
P2  , t  t; , t  
A
     
2
2t
     
2







 F     F   



exp  
 exp  

4
k
T

t
2
k
T
4 kBT t


B
B



F   
  F     F       2 
  t

2

F  
•
 
•
Three points to be noted:
(1)
 2
t


F

 F is to be minimized w.r.t. 
2
(2) MEDP implies balance of dissipative force with force derived from free energy
(3) MEDP gives the most probable course of a dissipative process
Derivation of Equation of Motion from
Onsager Principle
• Viscous dissipation of fluid flow is given by
n̂

2
  v   dV    nv  
4


together with incompressibility condition
v
ˆ
solid
v  0
• By minimizing  with respect to v , with the condition of   v  0 (treated
by using a Lagrange multiplier p), one obtains the Stokes equation
p  2v  0
- In the presence of inertial effect, momentum balance means
 v  p  2v
 NS equation
Extension of the Onsager Principle for Deriving
Fluid-solid Boundary Condition(s)
vslip
• If one supposes that there can be a fluid velocity relative to the solid
boundary, then similar to R  v  for fluid, there should be a
s  v  
1
2

2


dS   vslip  ;
= a length (slip length)



 
- Yields, together with   v  , the boundary condition
 vslip   nv  Navier boundary condition (1823)
- But over the past century or more, it is the general belief that vslip  0
Non-slip boundary condition
Two Phase Immiscible Flows
• Need a free energy to stabilize the interface
-
2
K

   r     dr      f   
2

-
   2  1  /  2  1  ;
• Total free energy
r
u
f      2   4 (Cahn-Hilliard)
2
4
F       dS  fs   
-      dS  fs     dr      dS  L 
   /   K2  f   / 
L  K  n   fs   / 

Fluid 1
 1fs
Fluid 2
 2fs
•  is locally conserved:


      J
t
- Interfacial  is not conserved, because
nJn0 in general
•
2
2 
 J2 

  slip 2 
   dr    i j   ji     dS       dr 
   dS  
2
M
4
2






 2 
•
  
  
F   dr      dS  L 
 t 
 t 
-
 / t     
-
F   dr   J       dS  L      


- Minimize
, but     J in bulk

  F w.r.t.
J, ,

2
 J2 

  slip 2 
  dr 
•   F   dr    i j   ji     dS  


4
2
2
M








2 
  dS     dr   J       dS  L      


 2 


- Minimize w.r.t.  , J, 

- Subsidiary incompressibility condition:   0

• Minimize w.r.t. J:
J   M 
- =

+  =    J  M  2 
t
• Minimize w.r.t.
:

 = +   =  L  
t
• Minimize w.r.t.  :
- In the bulk
p  2 v   =0
- On the boundary
  slip    n  n   L    
uncompensated
Young stress
L    0  Young equation
Uncompensated Young Stress
• 
across
interface
•
dx  L    x    cos  d   fs   1   fs   1    cos d  cos s 
L    0   coss   fs   1   fs   1  0
L    x   cosd f  x    x fs
- f  x ~
3
cosh 4  x  xCL  / 2 
4 2
- 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
Continuum Hydrodynamic Formulation
•

 v   M  2 
t
 v

  v   v   p     v    f e
 t


-

 v    L  
t
-   slip    n  n   L    
-
vn  0, J n   n   0 at boundary
v x (x) profiles at different z levels
symmetric
Coutte
V=0.25
H=13.6
asymmetric
Coutte
V=0.20
H=13.6
symmetric
Coutte
V=0.25
H=10.2
symmetric
Coutte
V=0.275
H=13.6
asymmetric
Poiseuille
g_ext=0.05
H=13.6
Power-Law Decay of Partial Slip
Molecular Dynamic Confirmations
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
Maxwell Equations Require No
Boundary Conditions
   E  4 f
xE  
 H
c t
  H  0
xH 
4
 E
E
c
c t
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).
Nano Droplet Dynamics over High Contrast Surface
Contact Line Breaking with High Wetability Contrast
Thank you