the onsager principle and hydrodynamic boundary conditions ping sheng department of physics and...
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The Onsager Principle and Hydrodynamic Boundary Conditions
Ping ShengDepartment 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 Fluids20 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
• Fluid-solid boundary condition– Non-slip boundary condition implies no relative
motion at the fluid-solid interface
vv v fv
ept
• 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
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.
Ft
2
2
1B
B
k TP P FP
t k T
~ exp /eq BP F k T
…Fokker-Planck Equation
is the stationary solution
White Noise
2t t kT t t
•
-
Three points to be noted:
2 , ; ,P t t t
2
2
2 2
FA F F t
t
F
t
2
2F
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
exp
2 B
F F
k T
2exp
44 BBk T tk T t
F
•
•
•
•
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
24 nv dV v
0v
0v
v
2 0p v
2v p v NS equation
n
v
solid
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)
slipv
21
2 slips v dS v
R v
v
slip nv v
0slipv
= a length (slip length);
Non-slip boundary condition
Two Phase Immiscible Flows
• Need a free energy to stabilize the interface
2r
2
Kr d f
2 1 2 1/
fsF dS
rfsdS d dS L
2/ /K f
/n fsL K
-
-
• Total free energy
-
2 4
2 4
r uf ; (Cahn-Hilliard)
Fluid 2Fluid 11fs
2fs
is locally conserved:
Interfacial is not conserved, because nJn0 in general
Jt
22r
4 2slip
i j j id dS
rF d dS Lt t
/ t J in bulk
r JF d dS L
•
•
, but
-
-
-
- Minimize J, , w.r.t. F
2Jr
2d
M
2
2dS
222
2
Jr
4 2 2
r J2
slipi j j iF d dS dr
M
dS d dS L
- Minimize w.r.t. , J,
•
- Subsidiary incompressibility condition: 0
J M
2= + = Jt
M
Minimize w.r.t. :
Minimize w.r.t. : J
= + =t
L
-
•
•
slipn n L
2v =0p
Minimize w.r.t. :
- In the bulk
- On the boundary
•
uncompensatedYoung stress
Young equation 0L
Uncompensated Young Stress
- xfs also a peaked function
43~ cosh / 2
4 2CLf x x x
cosx d x fsL f x
acrossinterface
cos 1 1 cos cosx d fs fs d sdx L •
-
•
0 cos 1 1 0s fs fsL
• 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
2v Mt
vv v fv
ept
v Lt
slipn n L
0, J 0n n nv at boundary
•
-
-
-
symmetricCoutteV=0.25H=13.6
asymmetricCoutte V=0.20 H=13.6
profiles at different z levels
)(xvx
symmetricCoutte V=0.25 H=10.2
symmetricCoutte 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 NoBoundary Conditions
4 fE
0H
xH
Ec t
4
xE
H Ec 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