Hydrodynamic Slip Boundary Condition for the Moving Contact Line

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Hydrodynamic Slip Boundary Condition for the
Moving Contact Line

• in collaboration with
• Xiao-Ping Wang (Mathematics Dept, HKUST)
• Ping Sheng (Physics Dept, HKUST)

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No-Slip Boundary Condition
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from Navier Boundary Conditionto No-Slip
Boundary Condition
shear rate at solid surface

• slip length, from nano- to micrometer
• Practically, no slip in macroscopic flows

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No-Slip Boundary Condition ?

• Apparent Violation seen from
• the moving/slipping contact line
• Infinite Energy Dissipation
• (unphysical singularity)

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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 ?

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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

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Molecular dynamics simulationsfor 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

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Modified Lennard-Jones Potentials
for like molecules
for molecules of different species
for wetting property of the fluid
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fluid-2
fluid-1
fluid-1
dynamic configuration
f-1
f-2
f-1
f-1
f-2
f-1
symmetric
asymmetric
static configurations
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boundary layer
tangential momentum transport
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The Generalized Navier B. C.
when the BL thickness shrinks down to 0
viscous part
non-viscous part Origin?
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uncompensated Young stress
nonviscous part
viscous part
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Uncompensated Young Stressmissed in Navier B. C.

• Net force due to hydrodynamic deviation from
  static force balance (Youngs equation)
• NBC NOT capable of describing the motion of
  contact line
• Away from the CL, the GNBC implies NBC for single
  phase flows.

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Continuum Hydrodynamic ModelingComponents

• 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

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Diffuse Fluid-Fluid Interface Cahn-Hilliard free
energy (1958)
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capillary force density
is the chemical potential.
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tangential viscous stress uncompensated
Young stress
Youngs equation recovered in the static case by
integration along x
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for boundary relaxation dynamics

first-order generalization from
in equilibrium, together with


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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.

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molecular positions projected onto the xz plane
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near-total slip at moving CL
Symmetric Coutte V0.25 H13.6
no slip
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profiles at different z levels

symmetric Coutte V0.25 H13.6
asymmetricCoutte V0.20 H13.6
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symmetricCoutte V0.25 H10.2
symmetricCoutte V0.275 H13.6
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asymmetric Poiseuille gext0.05 H13.6
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The boundary conditions and the parameter values
are bothlocal properties, applicable to flows
with different macroscopic/external conditions
(wall speed, system size, flow type).
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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.

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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 multi-scale
  (interfacial thickness, slip length, and
  external confinement length scale).