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Impact of Microdrops on Solids James Sprittles & Yulii Shikhmurzaev Failure of conventional models All existing models are based on the contact angle being a function of the contact line speed and material properties: The experimental investigations of both Bayer & Megaridis 06 and Sikalo et al 02 using millimetre sized drops have shown this assumption incorrect – see figure 2. Additionally, conventional models predict an infinite pressure at the contact line and/or the incorrect kinematics there, see Shikhmurzaev 2007. U 5. Qualitative agreement between a simulation using the conventional model with experiments of Dong 06. Water drops of radius 25 microns at impact speed 12.2m/s with equilibrium contact angle of 88 degrees. Typical microdrop simulation (blue) compared to experiment (black). Recent experiments show that all current models of drop impact and spreading are fundamentally flawed. The error will be considerable for the micron scale drops encountered in ink-jet printing. We are developing a universal computational platform, implementing a new model, to describe such experiments. U, cm/s U Contact angle dependence on the flow field Experiments of Blake & Shikhmurzaev 02 and Clarke & Stattersfield 06 demonstrated that in curtain coating the contact angle is dependent on the flow field and, in particular, the flow rate – see figure 4. The only model to predict this effect is the Interface Formation Model (IFM), see Shikhmurzaev 2007, which we are applying to drop impact and spreading. In the bulk: On liquid-solid interfaces: On free surfaces: At contact lines: θdθd e2e2 e1e1 n n f (r, t )=0 References Bayer & Megaridis, J. Fluid Mech., 558, 2006.Shikhmurzaev, Capillary Flows with Forming Interfaces, 2007. Blake & Shikhmurzaev, J. Coll. Int. Sci., 253, 2002.Sikalo et al, Exper. Therm. Fluid Sci., 25, 2002. Clarke & Stattersfield, Phy. Fluids, 18, 2006.Sprittles & Shikhmurzaev, Phy. Rev. E., 76, 2007. Dong, PhD, 2006. Extensions The development mode provides a conceptual framework for additional physical/chemical effects including thermal effects and contact angle hysteresis. Substrates of Variable Wettability Such chemically altered solids are naturally incorporated into the IFM and can have a considerable impact on the flow field (Sprittles & Shikhmurzaev 07). Porous Substrates One of the many possible generalisations of our work is the extension of the solid from impermeable to porous. Interface formation - qualitatively Fluid particles are advected through the contact line from the liquid-gas to the liquid-solid interface. Near the contact line the interface is out of equilibrium and, notably, the surface tension takes finite time/distance to relax to its new equilibrium value – see figure 6. To model this process surface variables are introduced, the surface density and the surface velocity. Surface equation of state: Incompressible Navier-Stokes Surface mass continuity Normal and tangential stress Generalised Navier Young equation Normal velocity Mass balance Problem formulation: `Darcy type’ eqn. Computational We are developing a multi-purpose finite element code extending the spine method devised by Ruschak then improved by Scriven and co-workers. This is capable of simulating flows where the standard approach fails, such as pinch-off of liquid drops and coalescence of drops. Results from a drop impact and spreading simulation are shown in figure 5. A snapshot of the finite element mesh during a simulation can be seen in figure 7. 2. Contact angle-speed plot for 2 mm water droplets impacting at different Weber number based on impact speed (Bayer & Megaridis 06). 4. Dynamic contact angle as a function of coating speed for different flow rates (Blake & Shikhmurzaev 02). 1. Sketch of a spreading droplet. 3. Curtain coating geometry in a frame moving with the contact line. 6. Sketch of flow in the contact line region showing how surface tension relaxes over a finite distance. Governing equations 7. Computational mesh of nodes on triangular elements. Solid Gas Liquid Kinematic U, m/s

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