Hydrodynamic Slip Boundary Condition
for the Moving Contact Line
in collaboration with
Xiao-Ping Wang (Mathematics Dept, HKUST)
Ping Sheng (Physics Dept, HKUST)
v
slip
0 ?
No-Slip Boundary Condition
v
slip
0
from Navier Boundary Condition
to No-Slip Boundary Condition
v
slip
 ls  
 : shear rate at solid surface
ls :
slip length, from nano- to micrometer
Practically, no slip in macroscopic flows
slip
s
v
/U  l / R  0
 cos s   2   1
No-Slip Boundary Condition ?
Apparent Violation seen from
the moving/slipping contact line
Infinite Energy Dissipation
(unphysical singularity)
Are you able to drink coffee?
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 ?
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 DEDUCTION
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
Modified Lennard-Jones Potentials
U ff  4 [( / r )   ff ( / r ) ]
12
6
U wf  4 wf [( wf / r )   wf ( wf / r ) ]
12
6
 ff  1 for like molecules
 ff  1 for molecules of different species
 wf for wetting property of the fluid
fluid-1
fluid-2
fluid-1
dynamic configuration
f-1
f-2
f-1
symmetric
f-1
f-2
asymmetric
static configurations
f-1

boundary
layer
tangential momentum transport
The Generalized Navier B. C.
~f
slip
Gx  vx
~f ~
Gx   zx (0)
when the BL thickness
shrinks down to 0
Y
~
~
 zx (0)  [ z vx ](0)   zx (0)
viscous part
non-viscous part
Origin?
uncompensated Young stress
Y
Y
0
~
 (0)   (0)   (0)
zx
zx
nonviscous
part
zx
 s ,d   dx
int
0 ,Y
zx



viscous
part
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.
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
Diffuse Fluid-Fluid Interface
Cahn-Hilliard free energy (1958)
FCH
 1
2
  dr [ K ( )  f ( )]
2
  (  2  1 ) /(  2  1 )
1 2 1 4
f ( )   r  u
2
4

2
 / t  v    M 



 m [v / t  (v  )v ]

 p         m g ext
v
  FCH / 
capillary
force density
is the chemical potential.

v
slip
x
~
  zx (0)
 [ z vx ](0) 
[(  K z   wf /  ) x ](0)
= tangential viscous stress +
uncompensated Young stress
Young’s equation recovered
in the static case by integration along x


[ / t  v   ](0)
   [ K z   wf ( ) /  ](0)
for boundary relaxation dynamics
first-order generalization from
 K z   wf ( ) /   0
in equilibrium, together with
 / t  v    0

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.
molecular positions projected onto the xz plane
near-total slip

at moving CL
no slip
vx / V  1

Symmetric
Coutte
V=0.25
H=13.6
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
gext=0.05
H=13.6
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).
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.
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 multiscale (interfacial thickness, slip length, and
external confinement length scale).