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## Viscous Flow Over a Chemically Patterned Surface

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**Viscous Flow Over a Chemically Patterned Surface**J.E. Sprittles Y.D. Shikhmurzaev 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? James Sprittles ECS 2007**MotivationUsing Patterned Surfaces**• Free surface flows due to unbalanced surface tension forces. • Structured film formation, virtual (capillary) walls,.. Solid 1 More Wettable Solid 2 Solid 1 James Sprittles ECS 2007**Motivation:Drop Impact on Chemically Patterned Surfaces**• Use surface patterning to ‘correct’ deposition. • For high accuracy ink-jet printing of structures. James Sprittles ECS 2007 Courtesy of Darmstadt University - Spray Research Group**What if there is no free surface?**Do variations in the wettability affect an adjacent flow? The Problem What happens in this region? Shear flow in the far field Solid 2 Solid 1 James Sprittles ECS 2007**Molecular Dynamics Simulations**More wettable Compressed Less wettable Rarefied James Sprittles ECS 2007 Courtesy of Professor N.V. Priezjev**Hydrodynamic Modelling:Defining Wettability**• Defining wettability The contact line Solid 1 • The Young equation: 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! James Sprittles ECS 2007**Qualitative Picture**• Consider region of interest. • What happens when flow drives fluid particles along the interface? • Mass, momentum and energy exchange between surface and bulk. • The process of interface formation. • Fluid particles are driven into areas of differing wettability. • Surface properties take a finite time to relax to their new equilibrium state. Bulk Interfacial Layer: For Visualisation Only Solid 2 Solid 1 James Sprittles ECS 2007**Interface Formation Equations – Hydrodynamic of Interfaces**• Surface possesses integral properties such as a surface tension, ; surface velocity, and surface density, . • Surface density is related to surface tension: Equation of State • Equilibrium surface density defines wettability: Input of Wettability James Sprittles ECS 2007**The Interface Formation Modelfor Constant Wettability**• If then we have Navier Slip Bulk Interfacial Layer: For Visualisation Only. In the continuum limit.. James Sprittles ECS 2007**Solid-Liquid Boundary Conditions – Interface Formation**Equations Tangential velocity Surface velocity Bulk Layer is for VISUALISATION only. In the continuum limit… Solid facing side of interface: No-slip James Sprittles ECS 2007**Solid-Liquid Boundary Conditions – Interface Formation**Equations Normal velocity Continuity of surface mass Bulk Layer is for VISUALISATION only. In the continuum limit… James Sprittles ECS 2007 Solid facing side of interface: Impermeability**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: James Sprittles ECS 2007**Results - Streamlines**• Consider solid 1 (x<0) more wettable than solid 2 (x>0). • Coupled, nonlinear PDEs were solved using the finite element method. James Sprittles ECS 2007**Results – Different Solid Combinations**• Consider different solid combinations. 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. 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. James Sprittles ECS 2007**Periodically Patterned Surface**• Consider Solid 1 More Wettable. • Consider a=1 -> Strips Have Equal Width. James Sprittles ECS 2007**Results - Streamlines**Solid 2 less wettable Qualitative agreement 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. 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. James Sprittles ECS 2007**Thanks!**James Sprittles ECS 2007**Numerical Analysis of Formula for J**Shapes are numerical results. Lines represent predicted flux 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 James Sprittles ECS 2007**Interface Formation Equations + Input of Wettability**Transition in wettability centred at x=y=0. Input of wettability James Sprittles ECS 2007