1. James Sprittles ECS 2007 Viscous Flow Over a Chemically Patterned Surface J.E. Sprittles Y.D. Shikhmurzaev

2. James Sprittles ECS 2007 Overview Technologically, why are flows over patterned surfaces important? What are the issues with modelling such flows? How will a single change in wettability affect a flow? How about intermittent changes?

3. James Sprittles ECS 2007 Motivation Using Patterned Surfaces Solid 1 Solid 2 Free surface flows due to unbalanced surface tension forces. Structured film formation, virtual (capillary) walls,.. Solid 1 More Wettable

4. James Sprittles ECS 2007 Motivation: Drop Impact on Chemically Patterned Surfaces Courtesy of Darmstadt University - Spray Research Group Use surface patterning to ‘correct’ deposition. For high accuracy ink-jet printing of structures.

5. James Sprittles ECS 2007 The Problem What if there is no free surface? Do variations in the wettability affect an adjacent flow? Solid 1 Solid 2 What happens in this region? Shear flow in the far field

6. James Sprittles ECS 2007 Molecular Dynamics Simulations Courtesy of Professor N.V. Priezjev More wettable Compressed More wettable Compressed Less wettable Rarefied Less wettable Rarefied

7. James Sprittles ECS 2007 Hydrodynamic Modelling: Defining Wettability Defining wettability The Young equation: The contact line Solid 1

8. James Sprittles ECS 2007 Hydrodynamic Modelling: Which Model? No-Slip No effect Slip Models (e.g. Navier Slip) There is no theta! A Problem.. We have no tools!

9. James Sprittles ECS 2007 Qualitative Picture Bulk Fluid particles are driven into areas of differing wettability. Surface properties take a finite time to relax to their new equilibrium state. What happens when flow drives fluid particles along the interface? Mass, momentum and energy exchange between surface and bulk. The process of interface formation. Interfacial Layer: For Visualisation Only Interfacial Layer: For Visualisation Only Solid 1 Solid 2 Consider region of interest.

10. James Sprittles ECS 2007 Interface Formation Equations – Hydrodynamic of Interfaces Surface density is related to surface tension: Equilibrium surface density defines wettability: Surface possesses integral properties such as a surface tension, ; surface velocity, and surface density,. Equation of State Input of Wettability

11. James Sprittles ECS 2007 The Interface Formation Model for Constant Wettability Bulk Interfacial Layer: For Visualisation Only. In the continuum limit.. Interfacial Layer: For Visualisation Only. In the continuum limit.. If then we have Navier Slip

12. James Sprittles ECS 2007 Solid-Liquid Boundary Conditions – Interface Formation Equations Bulk Tangential velocity Surface velocity Solid facing side of interface: No-slip Layer is for VISUALISATION only. In the continuum limit…

13. James Sprittles ECS 2007 Solid-Liquid Boundary Conditions – Interface Formation Equations Bulk Continuity of surface mass Normal velocity Solid facing side of interface: Impermeability Layer is for VISUALISATION only. In the continuum limit…

14. James Sprittles ECS 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: Boundary Conditions –Shear flow in the far field, which, using gives:

15. James Sprittles ECS 2007 Results - Streamlines Consider solid 1 (x 0). Coupled, nonlinear PDEs were solved using the finite element method.

16. James Sprittles ECS 2007 Results – Different Solid Combinations Consider different solid combinations.

17. James Sprittles ECS 2007 Results – Size of The Effect 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.

18. James Sprittles ECS 2007 Results - The Generators of Slip Variations in slip are mainly caused by variations in surface tension. 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.

19. James Sprittles ECS 2007 Periodically Patterned Surface Consider Solid 1 More Wettable. Consider a=1 -> Strips Have Equal Width.

20. James Sprittles ECS 2007 Results - Streamlines Solid 2 less wettable Qualitative agreement

21. James Sprittles ECS 2007 Results – Velocity Profiles Tangential (slip) velocity varies around its equilibrium value of u=9.8. Fluxes are both in and out of the interfacial layer. Overall mass is conserved.

22. James Sprittles ECS 2007 Conclusions + Further Work IFM is able to naturally incorporate variations in wettability. This effect is qualitatively in agreement with molecular dynamics simulations and is here realised in a continuum framework. Effects cannot be captured with ‘Slip Models’. Full parametric investigation is in: Sprittles & Shikhmurzaev, Phys. Rev. E 76, 021602 (2007). More complicated flows may now be considered.

23. James Sprittles ECS 2007 Thanks!

24. James Sprittles ECS 2007 Numerical Analysis of Formula for J Shapes are numerical results. Lines represent predicted flux Shapes are numerical results. Lines represent predicted flux

25. James Sprittles ECS 2007 Deviation of The Actual Contact Angle => Non Equilibrium Surface Tensions Left:Curtain Coating Experiments (+) vs Theory (lines) Blake et al 1999 Wilson et al 2006 Right:Molecular Dynamics Koplik et al 1989

26. James Sprittles ECS 2007 Interface Formation Equations + Input of Wettability Transition in wettability centred at x=y=0. Input of wettability