james sprittles bamc 2007 viscous flow over a chemically patterned surface j.e sprittles y.d....

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