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