james sprittles bamc 2007 viscous flow over a chemically patterned surface j.e sprittles y.d....
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James Sprittles BAMC 2007
Viscous Flow Over a Chemically Patterned SurfaceViscous Flow Over a Chemically Patterned Surface
J.E Sprittles
Y.D. Shikhmurzaev
James Sprittles BAMC 2007
Wettability
1 2
2
More Wettable (Hydrophilic)More Wettable (Hydrophilic)
Less Wettable (Hydrophobic)Less Wettable (Hydrophobic)
Solid 1 Solid 2
James Sprittles BAMC 2007
The Problem
• How do variations in the wettability of a substrate affect the flow of an adjacent liquid?
• No slip – No effect.
1 2
2
Solid 1 Solid 2
What happens in this region?
Shear flow in the far fieldShear flow in the far field
James Sprittles BAMC 2007
Molecular Dynamics Simulations
Courtesy of Professor N.V. Priezjev
More wettable Dense => Surface tension -’veMore wettable Dense => Surface tension -’veLess wettable Rarefied => Surface tension +’veLess wettable Rarefied => Surface tension +’ve
James Sprittles BAMC 2007
Equilibrium Contact Angle and Equilibrium Surface Tension
• Require a mathematical definition of wettability.
ee 1lg1 cos
e1
lg
e1
• The Young equation:
a force balance at the contact line.
The contact lineThe contact line
James Sprittles BAMC 2007
Interface Formation
2
2
Solid 1 Solid 2
e1 e2
• Flow drives the interface out of equilibrium.
• Thermodynamics fights to return the interface to its equilibrium state.
• In the continuum approximation the microscopic layer is a surface of zero thickness.
• Surface possesses intrinsic properties such as a surface
tension, ; surface velocity, and surface density, . sv s
• Each solid-liquid interface has a different equilibrium surface tension.
Gradients in surface tension.
Microscopic interfacial layer in equilibrium.
James Sprittles BAMC 2007
Problem Formulation
• 2D, steady flow of an incompressible, viscous, Newtonian fluid over a stationary flat solid surface (y=0), driven by a shear in the far field.
• Bulk– Navier Stokes equations:
)vu, (u
• Boundary Conditions– Shear flow in the far field, which, using
gives:
.uuu,0u 2 p
.as0, 22
yxvSy
u
James Sprittles BAMC 2007
Solid-Liquid Boundary Conditions – Interface Formation Equations
,)0(ss
se2
se1
se
x
.2,1;cos
,tanh2
1
2
1
lg0
1221
i
lx
iessie
se
se
se
se
se
Equation of stateEquation of state
Transition in wettability at x=y=0.Transition in wettability at x=y=0.
Input of wettability
Input of wettability
James Sprittles BAMC 2007
Solid-Liquid Boundary Conditions – Interface Formation Equations
.2
tu2
1tv
,tu2
1uun
s
ses
e
s
ses
e
s
se
se
s
BulkBulk
Solid facing side of interface: No-slip
Solid facing side of interface: No-slip
tutv sLayer is for
VISUALISATION only.
Layer is for VISUALISATION only.
Tangential velocityTangential velocity
Surface
velocity
Surface
velocity
nt
James Sprittles BAMC 2007
Solid-Liquid Boundary Conditions – Interface Formation Equations
.v
,nu
s
se
ss
se
s
se
BulkBulk
Solid facing side of interface: Impermeability
Solid facing side of interface: Impermeability
nu
tv s ses sv
Continuity of surface mass
Continuity of surface mass
Normal velocityNormal velocity
Layer is for VISUALISATION only.
Layer is for VISUALISATION only.
James Sprittles BAMC 2007
Results• Consider solid 1 (x<0) more wettable than solid 2 (x>0).
• Coupled, nonlinear PDEs were solved using the finite element method.
James Sprittles BAMC 2007
Results
eeJ 21 coscos
• Consider the normal flux out of the interface, per unit time, J.
• We find:
• The constant of proportionality is dependent on the fluid and the magnitude of the shear applied.
James Sprittles BAMC 2007
Results - The Generators of Slip
• Results show that variations in slip are mainly caused by variations in surface tension as opposed to shear stress variations.
1) Deviation of shear stress on the interface from equilibrium.
2) Surface tension gradients.
1) Deviation of shear stress on the interface from equilibrium.
2) Surface tension gradients.
James Sprittles BAMC 2007
Conclusions + Further Work
• IFM is able to naturally incorporate variations in wettability.
• Surface interacts with the bulk in order to attain its new equilibrium state.
• Relate size of the effect back to the equilibrium contact angle.
• This effect is qualitatively in agreement with molecular dynamics simulations and is here realised in a continuum framework.
• More complicated situations may now be considered– Intermittent patterning– Drop impact on chemically patterned surfaces
James Sprittles BAMC 2007
Drop Impact on a Chemically Patterned Surface
• One is able to control droplet deposition by patterning a substrate
Courtesy of Darmstadt University - Spray Research Group
James Sprittles BAMC 2007
Thanks!
James Sprittles BAMC 2007
Numerical Analysis of Formula for J
Shapes are numerical results.
Lines represent predicted flux
Shapes are numerical results.
Lines represent predicted flux