Transcript Document
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
l s
: shear rate at solid surface
l s : slip length, from nano- to micrometer
Practically, no slip in macroscopic flows
v
slip
/ U ls / 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
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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 )
12
U wf 4 wf [( wf / r )
12
( / r ) ]
6
ff
wf ( wf / r ) ]
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
G x vx
~f
~
G x zx ( 0 )
when the BL thickness
shrinks down to 0
Y
~
~
zx ( 0 ) [ z v x ]( 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)
1
2
FCH d r [ K ( ) f ( )]
2
( 2 1 ) /( 2 1 )
f ( )
1
2
r
2
1
4
u
4
2
/ t v M
m [v / t (v )v ]
p
v
FCH /
m g ext
capillary
force density
is the chemical potential.
v
slip
x
~
zx ( 0 )
[ z v x ]( 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).